ORNL-TM-1545 by Jon Morrow

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Contract No. W-7405-eng-26

General Engineering and Construction Division

DESIGN STUDY OF A HEAT-EXCHANGE SYSTEM FOR ONE MSBR CONCEPT by the GE&C Division Design Analysis Section

i i

’

SEPTEMBER 1967

OAK RIDGE NATIONAL LABORATORY I

Oak Ridge, Tennessee operated by UNION CARBIDE CORPORATION f o t the U.S. ATOMIC ENERGY COMMISSION

iii

The individuals who participated in this study performed by the General Engineering and Construction Division Design Analysis Section are listed below. C. E. Bettis R. J. Braat z G. A. Cristy D. A. Dyslin 0, A. Kelly T. W. Pickel L, R. Shobe A, E. Spaller w. c. Stoddart

CONTENTS

...... .................... INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . DESIGN APPROACH . . . . . . . . . . . . . . . . . . . . Heat-Exchange System . . . . . . . . . . . . . . . . . . F a c t o r s Affecting Design of Heat Exchangers . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . Maintenance Philosophy . . . . . . . . . . . . . . . Feasibility . . . . . . . . . . . . . . . . . . . . Heat-Transfer and Pressure-Drop Calculations . . . . . . S t r e s s Analyses . . . . . . . . . . . . . . . . . . . . DESIGN FOR PRIMARY HEAT EXCHANGERS . . . . . . . . . . . CaseA . . . . . . . . . . . . . . . . . . . . . . . . . Case B . . . . . . . . . . . . . . . . . . . . . . . . . Design Variables . . . . . . . . . . . . . . . . . . Calculatory Procedures . . . . . . . . . . . . . . . R e s u l t s of Calculations . . . . . . . . . . . . . . DESIGN FOR BLANKET-SALT HEAT EXCHANGERS . . . . . . . . Case A . . . . . . . . . . . . . . . . . . . . . . . . . Case B . . . . . . . . . . . . . . . . . . . . . . . . . DESIGN FOR BOILER-SUPERHEATER EXCHANGERS . . . . . . . . Case A . . . . . . . . . . . . . . . . . . . . . . . . . Case B . . . . . . . . . . . . . . . . . . . . . . . . . Case C . . . . . . . . . . . . . . . . . . . . . . . . . DESIGN FOR STEAM REHEATER EXCHANGERS . . . . . . . . . . CaseA . . . . . . . . . . . . . . . . . . . . . . . . . CaseB . . . . . . . . . . . . . . . . . . . . . . . . . CaseC . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract

. 3.

. .

3 6

6 12 12 12 13

14 20 22 22

26 28 30 32 35

35 39

46 46

52 53 56 56 62 63

Page

........... .... CALCULATIONS FOR BLANKET-SALT HEAT EXCHANGER . CALCULATIONS FOR BOILER SUPERHEATER EXCHANGER . . . . . . . . . . . . . . . . . . . CALCULATIONS FOR STEAM REHEATER EXCHANGER . . . CALCULATIONS FOR REHEAT-STEAM PREHEATER . . . . NOMENCLATURE . . . . . . . . . . . . . . . . .

8. DESIGN FOR REHEAT-STEAM PREHEATERS A p p e n d i x A. CALCULATIONS FOR PRIMARY HEAT EXCHANGER A p p e n d i x B. A p p e n d i x C. A p p e n d i x D. Appendix E. A p p e n d i x F.

71 111

140 164 184 198

vii LIST OF TABLES Table Number

Title

Page Number

Values for Fuel-, Blanket-, and Coolant-Salt Properties Used in Preliminary Calculations for MSBR-Heat Transfer Equipment

Primary Heat Exchanger Design Data for Case A

Primary Heat Exchanger Design Data for Case B

Blanket-Salt Heat Exchanger Design Data for Case A

Blanket-Salt Heat Exchanger Design Data for Case B

Boiler-Superheater Design Data for Case A

Boiler-Superheater Stress Data for Case B

Boiler-Superheater Design Data for Case C

Steam Reheater Exchanger Design Data for Case A

Steam Reheater Stress Data for Case B

Steam Reheater Exchanger Design Data for Case C

Design Data for the Reheat-Steam Preheater

LIST OF FIGURES Figure Number

Page Number

Title Flow Diagram for the Case-A Heat-Exchange System

Flow Diagram for the Case-B Heat-Exchange System

Primary Fuel-Salt-to-Coolant-Salt Heat Exchanger for Case A

Primary Fuel-Salt-to-Coolant-Salt Heat Exchanger for.Case B

Blanket-Salt Heat Exchanger for Case A

Blanket-Salt Heat Exchanger for Case B

Boiler-Superheater Exchanger

Steam Reheater Exchanger

Reheat-Steam Preheater Exchanger

DESIGN STUDY OF A HEAT-EXCHANGE SYSTEM FOR ONE MSBR CONCEPT

Abstract

The heat-exchange system for one concept of a 1000-Mw(e) nuclear power plant using a molten-salt breeder reactor has been studied. The system has five types of heat exchangers to transfer the heat generated in the reactor core to the supercritical steam energy required to drive the turbine for the generation of electrical power. The two major design approaches reported here are for flow circuits in which heat is transferred from the molten core fuel and fertile blanket salts to the molten coolant salt and then to the supercritical fluid. The Case-A system involves relatively high fueland blanket-salt pressures in the reactor core. These pressures are reduced in the Case-B system by reversal of the flows of the fuel and blanket salts through the reactor core and the respective pumps and exchangers, while the operating pressures of the coolant-salt system are raised above those in the Case-A system. The criteria used, assumptions made, relationships employed, and the results obtained in the design for each of the five types of exchangers used in these cases are reported. Although the resulting design for the Case-B heat-exchange system and the exchangers appears to be the most workable one, further experimental and analytical investigations are needed before the designs for these exchangers can be finalized.

INTRODUCTION

Thermal-energy molten-salt breeder reactors (MSBR) are being studied to assess their economic and nuclear performance and to identify important design problems. One such study made at Oak Ridge National Laboratory The initial (ORNL) was of a conceptual loOO-Mw(e) MSBR power plant.' reference design for this plant employs a molten-salt breeder reactor with a two-region fluid-fuel concept that has fissile material in the

'P. R. Kasten, E. S. Bettis, and R. C. Robertson, "Design Studies of 1000-Mw(e) Molten-Salt Breeder Reactors," USAEC Report ORNL-3996, Oak Ridge National Laboratory, August 1966.

core stream and fertile material in the blanket stream.

This study

encompassed all of the major equipment required for a complete power station, including the components of the heat-exchange system.

The

design studies for the heat exchangers are presented here in greater detail to document some of the unique considerations involved in the transfer of heat between different molten-salt systems and between molten salt and water or steam. The reference 1000-Mw(e) plant'

has four heat-exchange loops, and

the design for this heat-exchange system is referred to here as Case A. Five types of heat exchangers are used in this system to accomplish the two principal transfers of heat:

(1) from the fuel and blanket salts to

the coolant salt and (2) from the coolant salt to supercritical fluid. After an evaluation of this concept was made, the possibility of improving the system by changing the operating pressures of the fuel-, blanket-, and coolant-salt systems became apparent.

The resulting design for a

reverse-flow system with lower fuel- and blanket-salt pressures in the reactor core and higher operating pressures in the coolant-salt system is presented here as Case B.

Because of the possibility of developing

a coolant salt with a lower and more favorable freezing-point temperature, another modification of the Case-A system was considered briefly. This is referred to as the Case-C design.

The design criteria, assump-

tions, calculatory procedures, and the resulting design data are discussed for each of the five types of exchangers in these heat-exchange systems

3 2.

SUMMARY

In the Case-A design of the heat-exchange system for the reference lOOO-Mw(e) MSBR power plant, five types of heat exchangers are used in each of the four heat-exchange loops. Each loop has one primary fuelsalt-to-coolant-saltexchanger, one blanket-salt exchanger, four boiler superheater exchangers, two steam reheater exchangers, and two reheatsteam preheaters to transfer the heat in the fuel and blanket salts to the coolant salt and from the coolant salt to the supercritical fluid. The fuel salt circulates through graphite tubes in the reactor core with a maximum pressure of 95 psi, leaves the core, and passes through the primary exchanger where some of its heat is transferred to the coolant salt. Upon leaving the primary exchanger, the fuel salt enters the suction side of the fuel-salt pump and is discharged back to the reactor core. The blanket salt circulates through the reactor core outside the graphite tubes with a maximum pressure of 115.8 psi. Upon leaving the core, it passes through the blanket-salt exchanger to release some of its heat to the coolant salt, enters the suction side of the blanket-salt pump, and is discharged back to the reactor. Coolant salt is circulated through the primary exchanger and the blanket-salt exchanger in series. A major portion (87%) of the coolant salt is then circulated through the boiler superheater exchangers where its heat is released to the supercritical fluid, while a smaller portion (13%) of the coolant salt is circulated through the reheaters. Reheatsteam preheaters are required in the steam system to raise the temperature of the exhaust steam from the high-pressure turbine before it enters the reheaters. A modification of the Case-A design that would eliminate the need for the reheat-steam preheaters and for direct-contact heating of the supercritical fluid before it enters the boiler superheaters was considfly. Case C involves lowering the inlet temperatures of the steam reheaters and the boiler superheaters to study the effects of a coolant salt with a lower freezing-point temperature. As the studies progressed and more understanding of the overall system was obtained, the fact that the pressure of the fuel salt is 4

8 I

4 higher than that of the coolant salt at the same point in the primary exchanger and the relatively high pressures of the molten salts in con-

tact with graphite in the reactor core caused concern. The Case-B system was developed to obtain a more desirable pressure arrangement. In the Case-B system, the operating pressures of the coolant-salt system were raised to assure that any leakage in the overall system would be from the coolant-salt system into the fuel- or blanket-salt systems. To lower the pressure on the graphite tubes in the reactor core and keep salt penetration at a minimum, the flows of the fuel and blanket salts through the reactor core and the respective pumps and exchangers are reversed from those in Case A.

The result is that the maximum pressure

of the fuel salt in the reactor core is 18 psi, and the maximum pressure

of the blanket salt is 32.5 psi. The Case-B heat-exchange system involved redesign of the primary, blanket-salt, boiler superheater, and steam reheater exchangers. The design for the Case-B primary exchanger is an improvement over that for Case A because the expansion bellows was eliminated and differential thermal expansion between the inner and outer tubes is accommodated by the use of sine-wave shaped tubes in the inner annulus. The blanketsalt exchanger was improved in Case B by the addition of a floating head to accommodate differential thermal expansion between the tubes and shell and to reduce cyclic fatigue. These improvements seem well suited for ultimate application in the power-plant heat-exchange system. The design criteria, calculatory procedures, and the results of the calculations for each type of exchanger are given in this report, and many of the detailed calculations for the Case-B exchangers are appended. While no optimization can be claimed for these exchanger designs, we feel that the concepts are sound and reasonable. However, there are two major uncertainties in the heat-transfer calculations. Accurate values of the molten-salt physical constants are not known, and the heat-transfer correlations for our application need further verification. The values for physical constants and tolerances for values of viscosity, density, and thermal capacity that we used were provided us by the designers of the MSBR during the early stages of the study. Recent developments have revealed uncertainties in viscosity and thermal conductivity that could

invalidate the existing designs. Therefore, before final designs for the heat exchangers can be presented with confidence, experimental investigations should be made to 1. 2.

accurately determine the physical properties of molten fluoride salts; investigate the heat-transfer properties of molten salts to extend the work of MacPherson and Yarosh;'

check and extend the heat-transfer and pressure-drop equations developed by Bergelin et al."'

to include disk and doughnut baffles;

4. investigate heat transfer from molten salt to supercritical fluid, in %-

particular the heat transfer during the transition from pressurized water to supercritical fluid, if a practical experiment can be devised; 5.

determine the tendency of both bent and straight tubes to vibrate under various conditions of diameter, pitch, support, length, and baffle shape and spacing; and to

determine the possible vibration effects of direct mixing of super-

critical fluid with pressurized water. Analytical studies should also be made of the

1. possible errors to arrive at the amount of contingency that should be designed into each exchanger to provide a trustworthy design,

G. e c

cyclic conditions in the system,

vibration in each exchanger to assure adequate tube life,

4. possibility of applying recently developed knowledge of tube configurations that enhance heat transfer, 5 . loads on exchangers imposed by methods of support and the piping restraints, and 6 . the maintenance considerations.

'R. E. MacPherson and M. M. Yarosh, "Development' Testing and Performance Evaluation of Liquid Metal and Molten-Salt Heat Exchangers," Internal Document, Oak Ridge National Laboratory, March 1960. *O. P. Bergelin, G . A. Brown, and A. P. Colburn, "Heat Transfer and Fluid Friction During Flow Across Banks of Tubes -V: A Study of a Cylindrical Baffled Exchanger Without Internal Leakage," Trans. ASME, 7 6 : 841-

850 (1954).

30. P. Bergelin, K. J. Bell, and M. D. Leighton, "Heat Transfer and Flui-dFriction During Flow Across Banks of Tubes -VI: The Effect of Internal Leakages Within Segmentally Baffled Exchangers," Trans. ASME, 8 0 : 53-60 ( 1 9 5 8 ) .

6 3.

The reference design'

DESIGN APPROACH

for the LOOO-Mw(e) MSBR power plant involves

a two-region two-fluid system with the fuel salt and the blanket salt in the reactor core separated by graphite tubes.

The fuel salt consists of

uranium fluoride dissolved in a carrier salt of lithium and beryllium fluorides, and the blanket salt contains thorium fluoride dissolved in a similar carrier salt.

The energy generated in the reactor fluid is trans-

ferred to a coolant-salt circuit that couples the reactor to a supercritical steam cycle.

This reference design employs one reactor with four

heat-exchange loops. Since a high plant-availability factor in the power plant is important to the maintenance of low power costs, a modular-type design for the MSBR plant was also considered. This modular plant would have four separate and identical reactors with their separate salt circuits.

The

desirability of the modular-type design and consideration of the size of the components and pipe required for the reference plant influenced the selection of the number of heat exchangers to be used in the system. Regardless of the number of reactors to be used in the 1000-Mw(e) plant, the study reported here is based on the use of four heat-exchange modules in the system. A

Heat-Exchange System

Each of the four modules or loops of the reference heat-exchange system contains one primary fuel-salt-to-coolant-salt exchanger, one blanket-salt exchanger, four boiler superheater exchangers, two steam reheater exchangers, and two reheat-steam preheater exchangers.

These

exchangers are housed in temperature-controlled shielding cells that can be heated to temperatures up to 1000째F with either gas or electric 'P. R. Kasten, E. S. Bettis, and R. C. Robertson, "Design Studies of 1000-Mw(e) Molten-Salt Breeder Reactors," USAEC Report ORNL-3996, Oak Ridge National Laboratory, August 1966.

7 heaters.

This simplifies the system considerably by eliminating the need

for auxiliary heaters on each piece of equipment and the need to insulate each individual component. The flow diagram of the heat-exchange system for the reference design,I Case A, is shown in Fig. 1. Each heat transfer stage consists of a number of identical exchangers represented in Fig. 1 as a single i

typical exchanger. The temperature values shown in Fig. 1 constitute the basis from which development of the designs for the various heat exchangers was begun. Some of these values were derived from known properties of molten-salt systems, and some were derived from simple heat and material balances. Values in the steam system were selected to make use of the existing steam-cycle design in the Tennessee Valley Authority's 900-Mw(e) Bull Run steam plant. This design was lnodified to increase the power rating to 1000 W(e).

In the resulting design for the Case-A system, the fuel salt leaving the reactor core at a pressure of 90 psi enters the primary exchanger at a temperature of 1300OF and a pressure of 85.8 psi and is circulated through the tubes in the exchanger where some of its heat is released to the coolant salt.

It then enters the suction side of the fuel-salt pump

and is discharged back to the reactor at a temperature of 1000째F and a

pressure of 95 psi. The heat from the blanket salt is transferred to the coolant salt in the blanket-salt heat exchanger. The blanket salt leaving the reactor at a pressure of 95.8 psi enters the exchanger at a temperature of 1250OF and a pressure of 90.3 psi, is circulated through the

tubes of the exchanger where some of its heat is given up to the coolant salt, enters the suction sid of the blanket-salt pump, and is discharged rature of 1150째F and a pressure of 97.8 psi. back to the reactor at a t These transfers of heat to the coolant-salt system represent one of the two direct steps involved in converting the heat generated in the reactor core to the energy required to drive the steam turbine. The coolant salt in the system enters the primary exchanger at a temperature

$6'

of 850째F and a pressure of 79.2 psi and leaves it at a temperature of llll째F and pressure of 28.5 psi. This coolant salt is then circulated through the blanket-salt exchanger, entering at a temperature of 1lllOF and pressure of 26.5 psi and leaving at a temperature of 1125'F and a

-COOLANT SALT P U M P

I I

e--.1 -

85f

GASEOUS F I S S I O N PRODUCTS DISPOSAL

1125OF

I I 1 I I

I i BOILER SUPERHEATER

i REHEAT STEAM I I

PREHEATER

L+---J

REHEATER

Flow D i a g r a m for the C a s e - A H e a t - E x c h a n g e S y s t e m .

BOOSTER PUMP

FUEL SALT BLANKET SALT COOLANT SALT STEAM WATER

CONDENSATE & MAKE-UP

8 Z F

FUEL S A L T HT. EXCHG.& PUMP

F i g . 1.

MIXING TEE

---

----------_____

pressure of 9 psi.

The second direct heat-conversion step is performed

when the major portion (87%) of the coolant salt is circulated through the boiler superheater exchangers where its heat is released to the supercritical fluid. The remaining portion (13%) of the coolant salt in the system is used to reheat the exhaust steam from the high-pressure turbine in the steam reheater exchanger. There is a possibility that the coolant salt in the steam reheater could be cooled to a temperature too close to its freezing point by the much cooler steam fed directly from the turbine

To avert possible freezing of the coolant salt, a reheat-steam preheater was included in the system wherein heat from a relatively small quantity of the high-temperature supercritical fluid raises the temperature of the reheat steam. For the same reason, the temperature of the supercritical fluid is raised before it enters the boiler superheaters by direct-contact heating. The studies of the MSBR heat-exchange system progressed to a point where a better evaluation of the temperatures and pressures in the overall system could be made. The most objectionable single feature of the Case-A system is that there are points in the primary heat exchanger where the pressure of the fuel salt is higher than that of the coolant salt, particularly where the fuel salt enters at a pressure of 85.8 psi and the coolant salt exits at a pressure of 28.5 psi. When efforts were directed toward correcting this feature, other areas where improvements in the system could be made became evident. The revised design that we refer to as the Case-B systemwas developed by the reactor designers to improve the overall operation of the heat-exchange system. The Case-B system required the redesign of the primary, blanket-salt, boiler superheater, and steam reheater exchangers, and the resulting flow diagram for the system is shown in Fig. 2 . exhaust.

The operating pressures of the coolant-salt system in the Case-B design were raised to assure that any leakage in the overall system would be from the coolant-salt system into the fuel or blanket salt system, The Case-B system has a further advantage over the Case-A system in that the pressures of the molten salts in contact with the graphite tubes in the reactor core are lower. At the top of the reactor where Y

ORNL DWG 67-6818

I I

+ I

252 PSI 1125OF

&y+

It I

-J

---BOILER SUPER HEATER

BLANKET S A L T 1000째 HT. EXCHG. & PUMP

208 1125OF

---

' I

IiI I

'*_. ,

CONDENSAT

REHEAT STEAM F W F A T F R. .P R.._.

R E HEATER

BOOSTER PUMP

MIXING TEE

-LEGENDASEOUS FlSSl RODUCTS DISPOSAL

L----------

+-d

Fig. 2.

E L SALT HT. EXCHG.& PUMP

FUEL SALT BLANKET SALT COOLANT SALT STEAM WATER

-----

-------

----

Flow Diagram for the Case-B Heat-Exchange System.

pressures are at a minimum, the pressure of the fuel salt in the graphite tubes in the Case-A system is 83.5 psi, while that of the blanket salt outside the tubes is 95.8 psi. At the bottom of the reactor where the pressures are maximum, the pressure of the fuel salt is 95 psi and the corresponding pressure of the blanket salt is 115.8 psi. In the Case-B system, where the pressures are at a minimum at the top of the reactor, the pressure of the fuel salt circulating through the graphite tubes is 6.5 psi, while the pressure of the blanket salt outside the tubes at that point is 12.5 psi. Q

At the bottom of the reac-

tor where the pressures are maximum, the pressure of the fuel salt is 18 psi and the corresponding pressure of the blanket salt is 32.5 psi.

This lowering of the pressure of the fuel and blanket salts in the reactor for the Case-B system is accomplished by reversing the flow of the fuel and blanket salts through the respective pumps and exchangers and through the reactor core. The fuel salt leaving the reactor core at a temperature of 1300째F and a pressure of 13 psi enters the suction side of the fuel-salt pump and is discharged into the primary heat exchanger at a pressure of 146 psi.

It is circulated through the tubes of the

exchanger and then reenters the reactor at a temperature of lOOO'F and a pressure of 18 psi.

The blanket salt leaving the reactor core at a

temperature of 1250'F and a pressure of 12.5 psi enters the suction side of the blanket-salt pump and is discharged into the blanket-salt exchanger at a pressure of 111 psi. It is circulated through the tubes in the exchanger and then reenters the reactor at a temperature of 1150째F and a pressure of 14.5 psi. To study the effects of having a coolant salt with a lower and more favorable freezing-point temperature in the system, another modification of the Case-A heat-exchange system was developed. Case C involves lowering the inlet temperatures of the steam reheaters and the boiler superheaters, eliminating the reheat-steam preheaters and the need for directcontact heating of the supercritical fluid before it enters the superheaters. The turbine exhaust steam is fed directly to the steam reheaters at a temperature of 552'F, and the supercritical fluid enters the boiler superheaters at a temperature of 580'F.

12 Factors Affectinn Design of Heat Exchangers

Several factors other than operating temperatures and pressures also had to be taken into account before the designs for the various components of the heat-exchange system could be developed.

Those factors

included consideration of the materials that could be used to construct the components, the maintenance philosophy to be followed, and the feasibility of the concept as a whole. 3

Materia 1s V

Metal surfaces in frequent contact with molten fluoride salts must have corrosion resistance not provided by conventional materials. I

Hastel-

loy N, originally developed as INOR-8 specifically for use with molten fluoride salts, was designated by the designers of the MSBR as the structural material for all components in the fuel, blanket, and coolant systems that are in contact with molten salts. The preheater and high-temperature steam piping will be made of Croloy, 2 1/4% chrome and 1% molybdenum.

Carbon steel can be used for

those surfaces in contact with water at temperatures below 700째F, as specified in Section I11 of the ASME Boiler and Pressure Vessel Code. 2

Maintenance Philosophy Certain features of the designs for the heat exchangers are governed by the maintenance philosophy to be applied to them.

We believe that the

reliability of the exchangers can be held at a high level through quality control in design and fabrication.

Therefore, the maintenance philosophy

we adopted predicates that defective exchangers will be replaced with new ones rather than being repaired in place.

This attitude is neces-

sary for the primary and blanket exchangers because of the high level

2ftASME Boiler and Pressure Vessel Code, Section 111, Nuclear Vessels," The American Society of Mechanical Engineers, New York, 1965.

of radioactivity that they will incur during operation at full power. When an exchanger must be removed from its temperature-controlled shielding cell, it will have

be placed in another shielded cell for an

indefinite period. . Those exchangers that contain no fuel or blanket salt are not likely to reach as high a level of radioactivity during operation as the primary and blanket exchangers, but the level reached will probably be high enough to prevent direct repair in place.

If they fail, these exchangers will also be replaced, and after the required decay time, they will be repaired. Removal and subsequent repair of the exchangers in the steam system constitute major operations, but no provisions for repairing them have yet been made.

In the final design, such provisions would be based on indus-

trial experience and practice with conventional heat exchangers that are operated in the same pressure and temperature range as those used in this application,

Feasibi1ity Some of the features of the designs for the heat exchangers that had to be investigated to demonstrate the feasibility of the concept are worth mentioning here. "

In each of the exchangers that have molten salt on the

shell side, a baffle extends across the entire cross-sectional area of the shell at a distance of 0 . 5 in. from the tube sheet. This provides a stagnant layer o f molten salt between the tube sheet and the circulating molten salt, and this insulating layer of salt serves to reduce the temperature drop across the t sheet connection is tobe

e sheet, As conceived, the tube-to-tuberolling at two places within the thick-

ness of the tube sheet and welding at the end of the tube. Trepanning the tube sheet around each tube makes a reliable tube-to-tube-sheet weld possible.

In cases where the shell-side fluid travels for considerable distance without baffling, small ring baffles are used to break the flow between the outermost tubes and the shell. In some cases, long tube lengths are unsupported by baffles. '

Although the drawings do not show it, these tubes

can be held in place by some form of wire-mesh tube support.

14 Heat-Transfer and Pressure-Drop Calculations

The values for the physical properties of the fuel, blanket, and coolant salt that were used in the preliminary calculations were provided by the designers of the MSBR.

These are given in Table 1.

Because

of the unusual situations involved in this heat-exchange system, we searched the literature to determine correlations between reported conditions and those of our particular application and to select the physical properties most nearly suited to those involved in our application. From the material searched, we developed the bases for our heat-transfer and pressure-drop calculations. For calculations involving heat transfer by forced convection through a molten-salt film inside a tube with a small diameter, we used data reported by MacPherson and Yar~sh.~From the data, the following equation was written. hidi = 0.000065 (NRe)l k where hi

*43

(N )O Pr

- heat transfer coefficient inside tube, Btu/hr* fi? *OF,

d. = inside diameter of tube, 1

k = thermal conductivity, Btu/hr*fi? *OF per ft, NRe

- Reynolds number,

NPr - Prandtl number.

Equation 1 was used when N < 10,000 and Eq. 2, the Dittus-Boelter Re equation, was used when N 2 10,000. Re hidi = 0.023(NRe) k

Q 08

=R. E, MacPherson and M. M. Yarosh, "Development Testing and Performance Evaluation of Liquid Metal and Molten-Salt Heat Exchangers," Internal Document, Oak Ridge National Laboratory, March 1960.

C')

Table 1. Values for Fuel-, Blanket-, and Coolant-Salt Properties Used in Preliminary Calculations for MSBR-Heat-Transfer Equipment Fuel Salt 0

Reference temperature, F t, approximate 0

Liquidus temperature, F

Blanket Salt

Coolant Salt

1150

1200

LiF-BeF, -UF4

LiF-ThF4-BeFa

988 NaF-NaBF,

34.3 842

102.6 1040

68.3 579

(a)

Density, lb/f$ Viscosity, lb/hr' ft Thermal conductivity Btu/hr*ft?moF per ft

127 +- 6 27 k 3

277 k 14(a) 38 k 19( a)

about 125 12

1.5(b)

1 . 5 ( ' )

l . 3 ( ' )

Heat capacity, Btu/lb*OF

0.55

0.14

0.22

0.055

about 0.41

("Internal memo MSBR-D-24 from Stanley Cantor to E. S. Bettis July 15, 1965, Subject: Physical Property Estimates of MSBR Reference-Design Fuel and Blanket Salts. (b)

A value of 4.0 Btu/hr*ft**'F per ft was used for the thermal conductivity of the fuel salt in the calculations for the Case-A primary exchanger. Subsequent to the calculatory work done for the Case-A primary exchanger, the improved estimated value of 1.5 was obtained and used. (c)W. R. Gambill, "Prediction of Thermal Conductivity of Fused Salts, 'I Internal Document, Oak Ridge National Laboratory, August 1956.

In all instances of baffled flow, we made use of the work of 0. P. Bergelin et al.4 j 5 for both the heat-transfer and the pressure calculations. Four out of five of the heat exchangers have coolant salt on the shell side, and in each of these four exchangers, the flow of the coolant salt is directed by baffles. A relation between a heat transfer factor J and the Reynolds number for such baffled flow is illustrated in graphic form in Fig. 11 of Ref. 4 . Based on the outside diameter of the tube, the Reynolds number, NRe =

(3)

where d = outside diameter of the tube, 0

G =

mass velocity of fluid, lb/hr-fe,

pb = viscosity at temperature of bulk fluid, lb/hr'ft.

The heat transfer factor for the window area,

where h W C P Gm k

= heat transfer coefficient for window area, Btu/hr*f$-째F, = specific heat,

mean mass velocity of fluid, lb/hr*fte, = thermal conductivity, Btu/hr-f$*OF per ft,

= viscosity at temperature of bulk fluid, lb/hr-ft,

pi = viscosity at temperature of tube surface, lb/hr.ft.

40. P. Bergelin, G . A. Brown, and A. P. Colburn, "Heat Transfer and Fluid Friction During Flow Across Banks of Tubes -V: A Study of a Cylindrical Baffled Exchanger Without Internal Leakage," Trans. ASME, 7 6 : 841-850

(1954)

50. P. Bergelin, K. J. Bell, and M. D. Leighton, "Heat Transfer and Fluid Friction During Flow Across Banks of Tubes -VI: The Effect of Internal Leakages Within Segmentally Baffled Exchangers," Trans. ASME,

80:

53-60 ( 1 9 5 8 ) .

"cur

The h e a t t r a n s f e r f a c t o r f o r t h e cross-flow area,

J=%(P) (z) a13

0 -14

C G

(5)

P B

'In some instances, values of J were read from t h e graph4 and t h e h e a t t r a n s f e r c o e f f i c i e n t s were determined from t h e equation.

In other

instances, p a r t i c u l a r l y where machine computation w a s used, t h e followihg approximate equations derived from t h e graph were used t o determine J. For 800

-< NRe

For 100

-< NRe

lo5, J

= 0.346

NRe-O

800, J = 0.571 NRe-'

*38a.

(7)

*466.

The value of t h e h e a t t r a n s f e r c o e f f i c i e n t f o r t h e window area, hw, and t h e value f o r t h e cross-flow area, hg, were then combined i n Eq. 8 t o determine t h e t o t a l h e a t - t r a n s f e r c o e f f i c i e n t . h a = h a ' + h a t t B B w w ' where a = t h e heat t r a n s f e r surface, f t ? / f t . These r e l a t i o n s h i p s were used t o determine values f o r t h e h e a t t r a n s f e r c o e f f i c i e n t s on t h e s h e l l s i d e of b a f f l e d exchangers.

The d a t a

reported by B e r g e l i ~ ~ , and ~ t h e r e f o r e t h e r e l a t i o n s h i p s given above, were based on work with half-moon shaped b a f f l e s with s t r a i g h t edges, whereas i n t h e study reported here, d i s k and doughnut b a f f l e s were used i n a l l t h e exchangers except t h e superheater.

The a d a p t a t i o n of B e r g e l i n ' s

d a t a t o d i s k and doughnut b a f f l e s involved c e r t a i n i n t e r p r e t a t i o n s .

The

"number of major r e s t r i c t i o n s encountered i n c r o s s flow" r e f e r r e d t o i n - B e r g e l i n ' s work was i n t e r p r e t e d f o r tubes arranged i n t r i a n g u l a r a r r a y as being t h e number of rows o f tubes, r, i n c r o s s flow. and doughnut b a f f l e s varies i n d i r e c t i o n through 360'.

Cross flow f o r d i s k For 30째 of change

i n d i r e c t i o n , t h e d i s t a n c e between tube rows w i l l vary from t h e p i t c h p t o 0 . 8 6 6 ~ . Therefore, an average of t h e s e two extremes w a s used t o c a l -

(9)

iu I

18 Where an arrangement has the tubes placed in concentric circular rings, r is simply the number of rings in the cross-flow area. In calculating the flow area,

AW,

for the fluid moving outward from

the doughnut opening of diameter Ddâ&#x20AC;&#x2122; we made use of the approximate equation for the number of tubes, n, in a circular band with a width equal t o 1 pitch, p. xDdp n=-=-.

rrD d 0.99 0.9P For the values of Dd involved, the factor 0 . 9 lies between 0 . 8 and 1; our choice of 0.9 was arbitrary. Then, A,/X

rrDd

- nd

0 â&#x20AC;&#x2122;

where

X = the baffle spacing, d

the outside diameter of the tube.

The number of tubes of given size and pitch arranged in triangular array that can fit into a cylindrical shell,

where

correction factor,

D = the diameter of the cylindrical shell, p = pitch of the tubes. The value of K decreases with the increasing ratio between D and p.

The

value K = 1.15 is sometimes used for triangular array, and the value

K = 1.12 was used in the calculations for the reheater exchanger. This compromise value has been checked by graphic methods and found reasonable for our applications. The number of tubes passing through the window in a doughnut baffle, n = A /0.866$ W

In this case, there is no space devoid of tubes near the outside of the circular window as is found in the case of the cylindrical shell.

For heat transfer calculations involving parallel flow on the shell side, we used the data reported by Short.6 The heat transfer coefficient outside of the tubes, 0.6

h0 = O . l d60 k ( z )

0.33

(9) .

This equation was used in the design calculations for the reheat-steam preheater and the parallel-flow portion of the Case-A primary exchanger. The superheater and the preheater both involve supercritical fluid, and the data reported by Swenson et

was used extensively in calcu-

lating the heat transfer coefficients for those components. The correlation recommended in this work is given below. hidi

0 .613

-ki where 0 h. = heat transfer coefficient inside tube, Btu/hr-f?- F, 1

d. = inside diameter of tube, 1 = thermal conductivity inside tube, Btu/hr*ft? *OF per ft, ki G = mass velocity of fluid, lb/hr-ft?, = viscosity of fluid at temperature inside tube, lb/hr*ft, pi Hi = enthalpy at temperature inside tube, Btu/lb,

! ,

* '

= enthalpy a t temperature of bulk f l u i d , Btu/lb,

ti = temperature

tb = temperature of b = specific volume Vb v = specific volume of fluid inside tube, fP/lb. - i

'B. E. Short, "Flow Geometry and Heat Exchanger Performance," Chem. Eng. Progr., 61(7): 63-70 (July 1965).

*.7H. S. Swenson, C. R. Kakarala, and J. R. Carver, "Heat Transfer to Supercritical Water in Smooth-Bore Tubes," Trans. ASPIE, Ser. C: J. Heat Transfer, 87(4): 477-484 (November 1965).

20 The physical p r o p e r t i e s of s u p e r c r i t i c a l f l u i d under various conditions o f p r e s s u r e and temperature were taken from d a t a reported by Keenan and Keyes'

and by Nowak and Grosh.' The p r e s s u r e drops i n t h e s h e l l s i d e of t h e exchangers were calcu-

l a t e d by using B e r g e l i n ' s equations.*

APcro ssflow

= 0 . 6 r p-

vg2

B 2gc

and

PV,a

indow = (2

+ 0.6rJ

, 2gC

where

r = number o f r e s t r i c t i o n s , V,

= mean v e l o c i t y , f t / s e c ,

vB =

cross-flow velocity, f t l s e c .

S t r e s s Analyses

Stress c a l c u l a t i o n s were made f o r each of t h e h e a t exchangers.

Where

possible, t h e shear s t r e s s theory of f a i l u r e w a s used as t h e f a i l u r e c r i t e r i o n , t h e s t r e s s e s were c l a s s i f i e d , and t h e l i m i t s of s t r e s s i n t e n s i t y were determined i n accordance with t h e methods prescribed i n Section I11 of t h e ASME Boiler and Pressure Vessel Code?

The analyses were performed by t r e a t i n g thermally induced s t r e s s e s a s secondary s t r e s s e s i n a single-cycle a n a l y s i s r a t h e r than i n a multicyclic analysis.

This procedure should a s s u r e adequate s t r e n g t h so t h a t

f u t u r e analyses based on c y c l i c c o n s i d e r a t i o n s should not r e s u l t i n severe r e v i s i o n s t o t h e geometries o f t h e exchangers.

A departure from

t h e maximum shear stress theory of f a i l u r e occurred i n t h e a n a l y s i s o f ~

~_____

'J. H. Keenan and F. G. Keyes, Thermodynamic P r o p e r t i e s of Steam, John Wiley and Sons, New York, 1936.

'E. S. Nowak and R. J. Grosh, "An I n v e s t i g a t i o n of C e r t a i n Thermodynamic and Transport P r o p e r t i e s of Water and Water Vapor i n t h e Critical Region," USAEC Report ANI,-6064, Argonne National Laboratory, October 1959.

21 the tube sheets of the exchangers.

These were analyzed by using the maximum normal stress theory of failure and the allowable stress intensity values applicable to an analysis based on the maximum shear stress theory of failure. No consideration was given to the support of these vessels or to the loads induced by piping restraints. The thicknesses of the heads, shells, tube sheets, and tubes were chosen to withstand the maximum pressure and thermal stresses at the respective temperatures in the material. The exchangers were tentatively designed, and the stresses caused by differential expansion and discontinuities were checked to be sure that the allowable stress at the maximum temperature had not been exceeded. These analyses reported here are preliminary, but their extent is considered sufficient to appraise the integrity of the conceptual designs for the heat exchangers. More rigorous analyses would be required to investigate cycling and off-design-point operations such as startup or a hazardous incident.

4. DESIGN FOR PRIMARY HEAT EXCHANGERS

The four primary fuel-salt-to-coolant-salt heat exchangers are shell-and-tube two-pass exchangers with disk and doughnut baffles.

The

basic geometry of the top-supported vertical exchangers, the material to be used to fabricate them, and the size of the tubes were established by the designers of the MSBR.

The structural material selected for the pri-

mary exchangers was Hastelloy N, and the tube chosen was one with an outside diameter of 3/8 in. and a wall thickness of 0.035 in.

Case A

The criteria governing the design for the primary heat exchangers for Case A that were fixed by the system are 1.

the temperatures and pressures of the incoming and outgoing coolant salt,

the temperatures and pressures of the incoming and outgoing fuel salt,

the flow rates of the coolant salt and the fuel salt, and

4. the total heat to be transferred. The design developed for the Case-A primary heat exchanger is shown in Fig. 3.

The exchanger is about 5.5 ft in diameter and 18.5 ft high,

including the bowl of the circulating pump.

The inlet of the fuel-salt

pump is connected to the exchanger, and the fuel salt from the reactor enters the exchanger from the 18-in.-diameter inner passage of the concentric pipes connecting the reactor and exchanger.

The fuel salt flows

downward in the exchanger through the rows of tubes in the outer annular section, and upon reaching the bottom of the shell, it reverses direction and moves upward through the tubes in the center section. The coolant salt enters the exchanger at the top and flows downward, countercurrent to the flow of the fuel salt, through the center section. Upon reaching the bottom of the shell, the coolant salt turns and flows upward around the tubes in the outer annular section and leaves the exchanger through the annular collecting ring at the top.

ORNL Dug. 65-12379

'b'

LS; During the preliminary stages of the design for the Case-A primary heat exchangers, a computer study was made to determine the effects on the area of the heat-transfer surface of varying the pitch, the baffle size and spacing, and the ratio of the number of tubes in the two sections. The computer code established the minimum baffle spacing limited by thermal stress in the tubes and then increased the spacing as necessary to remain within the pressure-drop limitations. The thermal-stress criteria for the computer code were based on the worst possible conditions in each section of the exchanger, and as a result, the annular section was designed without baffles. It was therefore necessary to make %and" calculations for an exchanger with several baffles in the bottom of the annular section. This changed the baffle spacing in the center section, the ratio of the tubes in the two sections, and the length of the exchanger. The resulting design data for the Case-A primary exchanger are given in Table 2 . Table 2. Primary Heat Exchanger Design Data for Case A

Number required

Shell-and-tube two-pass vertical exchanger with disk and doughnut baffles 4

Rate of heat transfer, each, Mw Btu/hr

528.5 1.8046 x 109

Shell-side conditions Cold fluid Entrance temperature,OF Exit temperature, OF Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, psi Mass flow rate, lb/hr Tube-side conditions Hot fluid Entrance temperature, OF Exit temperature, OF Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, psi Mass flow rate, lb/hr

Coolant salt 850

1111 79.2 28.5 50.7 1.685 x lo7

Fuel salt 1300 1000 85.8 0 85.8 1.093 x lo7

'w Table 2 (continued) Mass velocity, lb/hr*f? Center section Annular section Velocity, fps Center section Annular section

5.95 x 106 5.175 x 106 13.0 11.3

Tube material

Hastelloy N

Tube OD, in.

0.375

Tube thickness, in.

0.035

Tube length, tube sheet to tube sheet, ft Center section Annular

13.7 11.7

Shell material

Hastelloy N

Shell thickness, in.

0.5

Shell ID, in. Center section Annular section

40.25 66.5

Tube sheet material Tube sheet thickness, in. Top annular section Bottom annular section Top and bottom center section Number of tubes Center section Annular section

Hastelloy N 3.62 1.75 1 .o 3624 4167

Pitch of tubes, in.

Center section

0.625

Annular section

0.750

Total heat transfer area per exchanger, fra Center section Annular section Total Type of baffle

4875 4790 9665 Tube outside diame :er Disk and doughnut

Number of baffles Center section Annular section

5 2

Baffle spacing, i n . Center section Annular section

27.4 21

Basis for area calculation

26 Table 2 (continued) Disk OD, in. Center section Annular section

30.6 55.8

Doughnut ID, Center section Annular section

25 .O 51.0

Overall heat transfer coefficient, U, Btu/hr- f3

1106

Maximum stress intensity,a psi Tube Calculated

Pa = 413; (Pm + Q) = 12,017 Pm = Sm = 4600; (Pm + Q) =

A1 lowable

3sm = 13,800 Shell Calculated

= 6156; (Pm

+ Q)

= 21,589

Pa = Sm = 12,000; (Pm

Allowable

Q) =

3s = 36,000 m Maximum tube sheet stress, psi Calculated Allowable

10,750 10,750

a The symbols are those of Section I11 of the ASME Boiler and Pressure Vessel Code, where = primary membrane stress intensity, m Q = secondary stress intensity,

S = allowable stress intensity. m

B Case

In the reverse-flow heat-exchange system of Case B, the fuel-salt and blanket-salt flows were reversed from those of Case A, and the operating pressures of the coolant-salt system were increased.

This involved

redesign of the primary heat exchanger, as well as some of the other components of the system.

The design for the primary heat exchanger for

Case B is illustrated in Fig. 4.

27 ORNL DWG 61-6819

FUEL SALT LEVEL (DUMP)

FUEL SALT LEVEL (OPERATING)

P’ cFUE SALT

COOLANT SALT FROM *STEAM HEAT EXCHANGER

BAFFLE

-DOUGHNUT

--LONGITUDINAL

Fig. 4.

BAFFLE

COOLANT SALT TO DRAIN TANK

Primary Fuel-Salt-to-Coolant-Salt Heat Exchanger for Case B.

As may be seen i n Fig. 4, t h e flow of the f u e l s a l t i n t h e exchanger f o r Case B i s reversed from t h a t i n Case A, with t h e o u t l e t of t h e f u e l s a l t pump connected t o t h e exchanger.

Fuel s a l t e n t e r s t h e exchanger i n

t h e i n n e r annular region, flows downward through t h e tubes, and then upward through t h e tubes i n t h e o u t e r annular region before e n t e r i n g the reactor.

The coolant s a l t e n t e r s t h e exchanger through t h e annular

v o l u t e a t t h e top.

It then flows downward through the b a f f l e d o u t e r

region, r e v e r s e s t o flow upward through t h e b a f f l e d i n n e r annular region, and e x i t s through a c e n t r a l pipe. I n t h i s design, a f l o a t i n g head i s used t o reverse t h e flow of t h e f u e l s a l t i n t h e exchanger.

This w a s done t o accommodate t h e d i f f e r e n t i a l

thermal expansion between t h e tubes and t h e s h e l l and c e n t r a l pipe. expansion bellows o f t h e Case-A design was

The

eliminated, and d i f f e r e n t i a l

thermal expansion between t h e i n n e r and o u t e r tubes i s accommodated by using sine-wave type bent tubes i n t h e inner annulus and s t r a i g h t tubes i n t h e o u t e r annulus.

Doughnut-shaped b a f f l e s are used i n both annuli.

Those i n t h e inner annulus have no overlap so t h a t t h e bent tubes have

a longer unrestrained bend.

Des i m Variables Before making any d e t a i l e d c a l c u l a t i o n s f o r t h e design of t h e ,

primary heat exchanger f o r Case B, w e fixed a number of t h e design varia b l e s on t h e b a s i s of our own judgment. Tube Pitch.

As previously s t a t e d , t h e tubes i n t h e inner annulus

are bent and those i n the o u t e r annulus are s t r a i g h t .

To s i m p l i f y t h e

bend schedule f o r t h e tubes i n t h e inner annulus, t h e s e tubes are placed i n concentric circles w i t h a constant d e l t a radius and a n e a r l y constant circumferential pitch.

The bends i n t h e tubes are made i n t h e c y l i n d r i -

cal s u r f a c e t h a t l o c a t e s each r i n g of tubes.

A r a d i a l spacing of 0.600

in. w a s s e l e c t e d f o r these bent tubes i n t h e inner annulus as being t h e minimum spacing p r a c t i c a l f o r assembly.

The spacing w a s increased t o

0.673 in. i n t h e circumferential d i r e c t i o n because t h e d i s t a n c e between

t h e tubes a t t h e bends is somewhat less than t h e c i r c u m f e r e n t i a l pitch.

The tubes in the outer annulus are located on a triangular pitch. The dimension for this pitch, 0.625 in., was selected on the basis of the calculations done for the Case-A primary exchanger. This spacing resulted in an efficient use of the shell-side pressure drop. Number of Tubes. The number of tubes in each annular region is determined by the allowable pressure drop on the inside of the tubes. Since for Case B, the heat-transfer efficiency of the inner annulus needed to be lowered to minimize the temperature drop across the tube walls, the number of tubes chosen for the outer annulus was less than the number chosen for the inner annulus. This number of tubes in the outer annulus was further reduced until all of the allowable pressure drop for the fuel salt was utilized. The resulting number of tubes in the outer annulus was 3794 with 4347 tubes in the inner annulus. Lengths of Annular Regions. The length of the exchanger and the number and size of the baffles determines the flow conditions for the coolant-salt pressure drops and the heat-transfer coefficients. Preliminary calculations showed that the exchanger would be approximately 15 ft long, with a recess of about 1 ft for the pump.

A 14-to-15 ratio

was established for the lengths of the inner and outer annuli. Baffle Spacing and Size. The baffle spacing in the inner annulus is limited by the length required for the unrestrained bends in the tubes and the maximum allowable temperature drop across the tube wall. The use of smaller distances between baffles results in larger outside film coefficients and therefore smaller film drops. This increases the temperature drop across the tube wall. The number of baffles selected for use in the inner annulus was four.

In the outer annulus, an attempt was made to select the baffle size and spacing combinations that would result in efficient use of the available shell-side pressure drop. Ten baffles were used in the outer annulus.

t;-

Calculatory Procedures After selecting the just-described design variables, the pressure drops, the required length of the exihanger, and the stresses in the tubes were calculated for the selected conditions, and these conditions were adjusted where necessary. Then the thicknesses required for the shell, skirt, tube sheets, and head of the exchanger were calculated. The detailed heat-transfer, pressure-drop, and stress-analysis calculations are given in Appendix A. Pressure Drops. The pressure drops for the selected conditions were calculated, and where necessary, these conditions were adjusted to obtain the allowable pressure drops. The pressure drops inside the tubes were calculated by using Fanning's equation, =

[(y) & , +

where

P = pressure, psi, f = friction factor = 0.0014

+ (0.125/N

O ,)'3a

Re L = length of tube, ft, = inside diameter of tube, di G = mass velocity, lb/hr'ft?, p = density, lb/fi?, g = gravitational conversion constant, lb,- ft/lbf-ser? C

The pressure drops in the coolant salt outside the tubes were calculated by using Bergelin's AP

equati0ns.l

crossflow

= 0.6rp

vB" and 2gC

PVm'

= (2

Mwindow

+ 0.6rJ

2=C

IO. P. Bergelin, G . A. Brown, and A. P. Colburn, "Heat Transfer and Fluid Friction During Flow Across Banks of Tubes -V: A Study of a CylindriBaffled Exchanger Without Internal Leakage," Trans. ASME, 76: 841-850 (1954)

where r = number of restrictions, Vm = mean velocity, ft/sec,

= cross-flow velocity, ft/sec.

These equations developed by Bergelin were the result of his experimental work with straight-edged baffles.

However, we decided that the flow

conditions for doughnut baffles would be similar to those for straightedged baffles and that the same pressure-drop equations would be applicable. Length of Exchanger. t

The required length of the exchanger for the

selected conditions was calculated, and the selected conditions were adjusted until the calculated length equaled the assumed length.

Six

equations that state the conditions necessary for heat balance in the exchanger were combined, which resulted in a single equation with the length of the exchanger as a function of all the other variables.

The

development and use of this length equation is given in more detail in Appendix A. Stress in Tubes.

The stresses in the tubes were calculated, and if

the allowable stresses were exceeded, the selected conditions were changed to lower the stress.

The conceptual design for the exchanger

was analyzed to determine the stress intensities existing in the tubes of the exchanger that were caused by the action of pressure, thermal â&#x20AC;?

gradients, and restraints imposed on the tubes by other portions of the

exchanger.

The major stresses produced in the tubes are

primary membrane stresses caused by pressure,

secondary stresses caused by temperature gradients across the tube wall,

discontinuity stresses at the junction of the tube and tube sheet, and

4. secondary stresses at the mid-height of the inner tubes caused by differential expansion of the inner and outer tubes.

The analysis of the inner tubes was made by using an energy method with a simplified model of the tubes.

These calculations for the stresses

are given in more detail in Appendix A.

. 32

bâ&#x20AC;? Thicknesses.

The thicknesses required f o r t h e s h e l l , s k i r t , tube

sheets, and head were calculated.

The conceptual design f o r t h e

exchanger w a s analyzed t o determine t h e stress i n t e n s i t i e s o r maximum normal stresses e x i s t i n g i n t h e heat exchanger s h e l l s and tube s h e e t s t h a t are caused by t h e a c t i o n of pressure and t o determine t h e loads caused by t h e a c t i o n of o t h e r portions of t h e exchanger.

The major

stresses i n t h e s h e l l are 1.

primary membrane stresses caused by pressure and

d i s c o n t i n u i t y stresses a t t h e junction of t h e tube s h e e t s and s h e l l .

The i n l e t and e x i t s c r o l l s o r t o r o i d a l turn-around chamber were n o t considered i n t h i s analysis.

The shear stress theory of f a i l u r e was used

as t h e f a i l u r e c r i t e r i o n , and t h e stresses were c l a s s i f i e d and t h e l i m i t s

of stress i n t e n s i t y w e r e determined i n accordance w i t h Section If1 of t h e ASMF, Boiler and Pressure Vessel Code. The tube s h e e t s were designed by using t h e maximum normal stress theory of f a i l u r e and ligament e f f i c i e n c i e s based on Section V I 1 1 of t h e ASME B o i l e r and Pressure Vessel Code.

The tube s h e e t s were considered

t o be simply supported, and t h e pressure caused by t h e tube loads and pressures w e r e used t o determine a uniform e f f e c t i v e pressure t o be used i n t h i s analysis.

Results of Calculations

The c a l c u l a t i o n s described i n t h e preceding paragraphs are given i n d e t a i l i n Appendix A, and t h e r e s u l t i n g design d a t a f o r t h e primary heat exchanger f o r Case B are given i n Table 3.

Table 3.

Primary Heat Exchanger Design Data for Case B Shell-and-tube two-pass vertical exchanger with doughnut baffles

Number required

Four

Rate of heat transfer, each Mw Btu/hr

528.5 1.8046 x 109

Shell-side conditions Cold fluid 0 Entrance temperature, F Exit temperature, OF Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, psi Mass flow rate, lb/hr

Coolant salt 850 1111 198 164 34 1.685 x lo7

Tube-side conditions Hot fluid Entrance temperature, OF Exit temperature, OF Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, p s i Mass flow rate, lb/hr Tube material

Fuel salt 1300 1000 146 50 96 1.093 x lo7 Hastelloy N

Tube OD, in.

0.375

Tube thickness, in.

0.035

Tube length, tube sheet to tube sheet, ft Inner annulus Outer annulus

15.286 16.125

Shell material

Hastelloy N

Shell thickness, in. Tube sheet material

1 66.7 Hastelloy N

Tube sheet thickness, in. Top outer annulus Top inner annulus Floating head

2.5 3.5

Shell ID, in.

Number of tubes Inner annulus Outer annulus

1.5

4347

3794

. 34 Table 3 (continued) Pitch of tubes, in. Inner annulus Radial Circumferential Outer annulus, triangular Type of baffle

0.600 0.673 0.625 Doughnut

Number of baffles Inner annulus Outer annulus

4 10

a Maximum stress intensity, psi Tube Calculated Allowable Shell Calculated Allowable

m' = 285; (Pm + Q) = 6504 Pm = Sm = 5850; (Pm + Q) = 3s = 17,500 m Pm = 6470; (Pm + Q) = 9945 Pm = Sm = 18,750; (Pm + Q) = 3Sm = 56,250

Maximum tube sheet stress, calculated and allowable, psi Inner annulus Outer annulus Floating head a The symbols are Pressure Vessel Code, P = m Q = S

3500 17,000 10,000

those of Section I11 of the ASME Boiler and where primary membrane stress intensity, secondary stress intensity, and allowable stress intensity.

DESIGN FOR BLANKET-SALT HEAT EXCHANGERS

Heat accumulated in the blanket salt while it is circulating around the reactor core is transferred to the coolant salt by means of four shell-and-tube one-shell-pass two-tube-pass exchangers with disk and doughnut baffles. The basic geometry of the top-supported vertical exchangers, the material to be used to fabricate them, the tube size, and the approximate pressure drops were established by the designers of the MSBR.

All surfaces of the blanket-salt exchanger that contact

fluoride salts were to be made of Hastelloy N, the outside diameter of the tubes was specified as 0.375 in., and the wall thickness as 0.035 in. The blanket salt would circulate inside the tubes and the coolant salt through the shell. The pressure drop in the tubes was to be approximately 90 psi, and the pressure drop in the shell would be about 20 psi.

Physical-property data pertaining to the blanket salt and the coolant salt were also supplied by the designers of the MSBR.

Case A

The criteria governing the design for the blanket-salt exchangers for Case A that were fixed by their function in the overall system are the 1. inlet temperature of the blanket salt, 2.

outlet temperature of the blanket salt,

mass flow rate of the blanket salt,

4. inlet temperature of the coolant salt, tlet temperature of the coolant salt, s s flow rate the coolant salt, and 7. the total heat be transferrr As shown in Fig. 5 , th

tion of the blanket-salt heat exchanger

for Case A is similar to that of the primary heat exchanger for Case A; that is, two passes were used on the tube side with the tubes arranged in two concentric regions and with the blanket-salt pump located at the top of the inner annulus. The inlet of the blanket-salt pump is connected to

36 ORNL Dug. 65-12380

SALT PUMP

COOLANT SALT TOSTE&lU HEAT EXCHANGER

BAFFLE

-FROM

COOLANT SALT PRIMARY EXCHANGER

DRAIN

<TO

Fig. 5 .

BLANKET DRAIN TANKS

Blanket-Salt Heat Exchanger for Case A.

t h e exchanger, and t h e blanket s a l t from t h e r e a c t o r e n t e r s t h e exchanger and moves downward through t h e tubes i n t h e o u t e r annular region and then upward through t h e t u b e s i n t h e inner annular region t o t h e pump suction. S t r a i g h t tubes with two tube s h e e t s a r e used r a t h e r than U-tubes t o permit drainage of t h e blanket s a l t . The coolant s a l t passes through t h e primary h e a t exchangers and t h e b l a n k e t - s a l t h e a t exchangers i n s e r i e s .

However, u n l i k e t h e primary

exchanger, a s i n g l e c o o l a n t - s a l t pass on t h e s h e l l s i d e o f t h e blanketi

s a l t exchanger was judged adequate on t h e b a s i s of t h e h e a t load, temperat u r e conditions, and t h e advantage o f f e r e d by t h e s i m p l i f i c a t i o n of t h e design f o r t h e exchanger. Before any values could be generated f o r t h e design of t h e b l a n k e t - s a l t h e a t exchanger f o r Case A, i t was necessary t o t e n t a t i v e l y f i x t h e values of some a d d i t i o n a l design v a r i a b l e s .

From s e v e r a l approximations, a

t r i a n g u l a r tube p i t c h of 0.8125 i n . was chosen, and i t w a s a l s o decided t h a t t h e r e would be an equal number of tubes i n each annulus.

Disk and

doughnut b a f f l e s were s e l e c t e d t o improve t h e s h e l l - s i d e h e a t t r a n s f e r c o e f f i c i e n t and t o provide t h e necessary tube support.

B a f f l e s on t h e

s h e l l s i d e of t h e tube s h e e t s reduce t h e temperature d i f f e r e n c e a c r o s s t h e s h e e t s t o keep thermal s t r e s s e s w i t h i n t o l e r a b l e l i m i t s . area-to-shell-cross-sectional-area

An open-

r a t i o of 0.45 was s e l e c t e d f o r t h e

baffles. With t h e s e data fixed, values were c a l c u l a t e d f o r t h e 1.

number of tubes p e r ann

diameter of t h e s h e l l ,

s i z e and spacing of t h e

l e n g t h of t h e tubes,

thickness of t h

thickness o f t h

thickness and s

These design values were

pendent upon c a l c u l a t i o n s and upon an a n a l y s i s

of t h e i n d i v i d u a l h e a t - t r a n s f e r c o e f f i c i e n t s ,

s h e l l - s i d e and tube-side

p r e s s u r e drops, and t h e thermal and mechanical s t r e s s e s .

The r e s u l t i n g

design d a t a developed f o r t h e b l a n k e t - s a l t h e a t exchanger f o r Case A a r e given i n Table 4.

. 38

Table 4 .

Blanket-Salt Heat Exchanger Design Data for Case A

Shell-and-tube one-shellpass two-tube-pass exchanger, with disk and doughnut baffles Number required

Rate of heat transfer per unit, Mw Btu/hr

27.75 9.47 x 107

Shell-side conditions Cold fluid Entrance temperature, OF Exit temperature, OF Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, psi Mass flow rate, lb/hr

Coolant salt 1111 1125 26.5 9 17.5 1.685 x lo7

Tube-side conditions Hot fluid Entrance temperature, OF Exit temperature, 0 F Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, psi Mass flow rate, lb/hr Mass velocity, lb/hr-fid Velocity, fps Tube material

Hastelloy N

Tube OD, in.

0.375

Tube thickness, in.

0.035

Tube length, tube sheet to tube sheet, ft

8.25

Shell material Shell thickness, in.

Hastelloy N 0.25

Shell ID, in.

36.5

Tube sheet material

Hastelloy N

Tube sheet thickness, in.

1 .o

Number of tubes

1641 ( 4 2 0 per pass)

Pitch of tubes, in.

0.8125

Total heat transfer area, fid

1330

Basis for area calculation

Outside diameter

Blanket salt 1250 1150 90.3 0 90.3 4 . 3 x 106 10.48 x 10s 10.5

cd .

Table 4 (continued) Type of baffle

Disk and doughnut

Number of baffles

Baffle spacing, in.

24.75

Disk OD, in.

26.5

Doughnut ID, in. Overall heat transfer coefficient, U, Btu/hr*fP a Maximum stress intensity, psi Tube Calculated

Allowable

1016

Pm Pm

= =

411; (Pm + Q) = 7837 S = 6500; (Pm + Q) = m

3 sm = 19,500

She11 Calculated AL lowable

= 1663;

(Pm + Q)

Pm = Sm = 12,000; (Pm

11,140

Q) =

3Sm = 36,000

Maximum tube sheet stress, psi Calculated Allowable

2217 5900 at 1200째F

a The symbols are those of Section I11 of the ASME Boiler and Pressure Vessel Code, where Pm = primary membrane stress intensity, Q = secondary stress intensity, and Srn = allowable stress intensity.

In the reverse-flow heat exchange system of Case B, the flow of the blanket salt through the exchanger was sed from that of Case A. The redesigned blanket-salt

lustrated in Fig. 6 .

let of the blanket-salt

onnected to the exchanger and the blanket ulus, flows downward, reverses

salt enters the tubes i

The out-

direction, and flows upward through.the tubes in the outer annulus and out to the reactor. The coolant salt from the primary exchanger enters the bottom of the blanket-salt exchanger to flow upward through a central

40 ORNL DWG 67-6820

PUMP

SALT

FUEL

Fig. 6.

Blanket-Salt Heat Exchanger for Case B.

CI . I

pipe and then make a single pass downward on the shell side of the exchanger before going out to the coolant-salt pumps. A significant improvement over the Case-A design for the blanketsalt heat exchanger was the inclusion of a floating head in the Case-B design to reverse the flow of the blanket salt in the exchanger. This was done to accommodate thermal expansion between the tubes and the shell and the central pipe.

However, unlike the Case-B primary exchanger,

straight tubes are used in both annuli of the blanket-salt exchanger for Case B. Doughnut and disk baffles are also used in the design for the Case-B blanket-salt heat exchanger. i

i :

The reversal of the flow direction of the blanket salt, the increase in operating pressures, and the physical changes in the blanket-salt heat exchanger did not influence its heat-transfer or pressure-drop characteristics. Therefore, the calculatory procedures used'to determine the design variables for the Case-B blanket-salt heat exchanger were basically the same as those used for the Case-A exchanger. The heat-transfer and pressure-drop calculations and the calculations performed for the stress analysis for the Case-B design are given in Appendix B. The calculations performed to determine the number of tubes to be used in each annulus of the Case-B blanket-salt exchanger were based on the straightforward relationship between the mass flow rate, tube size, linear velocity, and density. The number of tubes per pass resulting from these Calculations was 810. Determination of the geometry of the shells and baffles followed readily once the number of tubes was estab-

lished Calculating the baffle spacing and tube length that fulfills the heat-transfer and pressure-d heat transfer coefficient fo

equirements was quite involved. The blanket salt inside the tubes was

calculated by using an equation derived from heat transfer data on son and Yarosh.' molten salt provided by In this equation, the ansfer coefficient blanket salt inside the tubes,

'R. E. MacPherson and M. M. Yarosh, "Development Testing and Performance Evaluation of Liquid Metal and Molten-Salt Heat Exchangers," Internal Document, Oak Ridge National Laboratory, March 1960.

hi = 0.000065 -(NReY k di

043

(NpJ

where k = thermal conductivity, Btu/hr.ft2e0F per ft, di = inside diameter of tube,

se= Reynolds number, NPr = Prandtl number,

The value determined for hi was 2400 Btu/hr0ft2-OF. The resistance across the tube wall was then easily evaluated by using the conductivity equation, and this was found to be 2 . 8 x

The first step in determining the heat transfer coefficient and the 4

pressure drop for the coolant salt outside the tubes was to plot a curve of the outside film resistance as a function of the baffle spacing.

Data

for the curve were generated by calculating the outside film resistance for various assumed baffle spacings.

The method used was based on an

adaptation of the work on cross-flow exchangers done by Bergelin et a1.2,3

At this point, it was possible to establish whether the baffle spacing was limited by thermal stress in the tube wall or by the allowable shellside pressure drop.

The outside film resistance, Ro, was evaluated for

the maximum temperature drop (46OF) across the tube wall, and the corresponding baffle spacing from the curve was approximately 6 in.

The

pressure drop at a baffle spacing of 6 in. exceeded the allowable of 20 psi.

Therefore, the pressure drop was limiting and the baffle spacing

was greater than 6 in. The next step in the calculatory procedures was to develop the equations given below that relate the baffle spacing, the outside film resistance, and the shell-side pressure drop.

L = 6.74

(0.85 x 104)R0

0. P. Bergelin, G. A. Brown, and A. P. Colburn, 'Tieat Transfer and Fluid Friction During Flow Across Banks of Tubes -V: A Study of a Cylindrical Baffled Exchanger Without Internal Leakage," Trans. ASME, 76: 841-850 ( 1 9 5 4 ) .

P. Bergelin, K. J. Bell, and M. D. Leighton, "Heat Transfer and Fluid Friction During Flow Across Banks of Tubes -VI: The Effect of Internal Leakages Within Segmentally Baffled Exchangers," Trans. ASME, 80:

53-60 ( 1 9 5 8 ) .

where L = length o f t h e tube, f t , and RO

= thermal f i l m r e s i s t a n c e o u t s i d e tubes.

L P = ~(0.32 x

L + (F

1) (2.40 x

lo6)[&)

where

P = s h e l l - s i d e p r e s s u r e drop, p s i , L = length of tube, f t ,

X = b a f f l e spacing, GB = cross-flow mass v e l o c i t y , l b / h r * f $ , Gm = mean mass v e l o c i t y , l b / h r * f p .

These equations and t h e curve were then used t o determine a b a f f l e spacing a t which t h e s h e l l - s i d e p r e s s u r e drop w a s w i t h i n t h e maximum allowable of 20 p s i .

Using four b a f f l e s a t a spacing of 1.65 f t , t h e

p r e s s u r e drop a c r o s s t h e s h e l l w a s 14.55 p s i .

With t h e tube l e n g t h

e s t a b l i s h e d a t 8.3 f t , t h e pressure drop through t h e tubes was determined from t h e Darcy equation and found t o be 90.65 p s i . Analysis of t h e thermal s t r e s s e s i n t h e exchanger required determfnat i o n of t h e temperature of t h e s a l t between t h e tube passes.

The tempera-

t u r e was evaluated by a t r i a l - a n d - e r r o r c a l c u l a t i o n involving a h e a t balance between t h e two tube passes, and i t s value w a s determined as 1184'F. The design data f o r t h e b l a n k e t - s a l t h e a t exchanger for Case B t h a t r e s u l t e d from t h e s e c a l c u l a t i o n s are given i n Table 5. Table 5.

Blanket-Salt Heat Exchanger Design Data f o r Case B

Type

Shell-and-tube one-shellp a s s two-tube-pass exchanger with d i s k end doughnut baffles

Number required

Rate of h e a t t r a n s f e r p e r u n i t ,

27.75

But/hr

9.471 x lo7

44 Table 5 (continued) Shell-side conditions Cold fluid Entrance temperature, 0 F Exit temperature, OF Entrance pressure, a psi Exit pressure,a psi Pressure drop across exchanger,b psi Mass flow rate, lb/hr

Coolant salt 1111 1125 138 129 15 1.685 x lo7

Tube-side conditions Hot fluid Entrance temperature, OF Exit temperature, OF Entrance pressure,a psi Exit pressure,a psi Pressure drop across exchanger,b psi Mass flow rate, lb/hr Ve locity ft/s ec

Blanket salt 1250 1150 111 20 91 4.3 x 106 10.5

Tube material

Hastelloy N

Tube OD, in.

0.375

Tube thickness, in.

0.035

Tube length, tube sheet to tube sheet, ft

8.3

Shell material

Hastelloy N

Shell thickness, in.

0.25

Shell ID, in.

40.78

Tube sheet material

Hastelloy N

Tube sheet thickness, in.

Number of tubes Inner annulus Outer annulus

810 810

Pitch of tubes, in.

0.8125

Total heat transfer area, ft?

1318

Basis for area calculation

Tube OD

Type baffle

Disk and doughnut

Number of baffles

Baffle spacing, in.

19.80

Disk OD, in.

33.65

Doughnut ID, in.

31.85

Overall heat transfer coefficient, U, Btu/hr f?t

1027

v c

Table 5 (continued) Maximum stress intensity,c psi Tube Calculated

Pm = 841; (Pm + Q) = 4833 Pm = Sm = 11,400; (Pm + Q) 3s = 34,200 m

A1lowable Shell Calculated Allowable

m' = 3020; (Pm + Q ) = 7913

Pm = Sm = 12,000; (Pm + Q) = 3 sm = 36,000

Maximum tube sheet stress, calculated and allowable, psi Top annular Lower annular

8500 6500

a Includes pressure caused by gravity head. bPressure l o s s caused by friction only. c

The symbols are those of Section I11 of the ASME Boiler and Pressure Vessel Code where

Pm = primary membrane stress intensity, Q = secondary stress intensity, and Sm = allowable stress intensity. I

jI _ fI 1

i i i i

i b i

DESIGN FOR BOILER-SUPERHEATER EXCHANGERS

Sixteen, four in each heat-exchange module, vertical U-tube U-shell superheater exchangers are used to transfer heat from the coolant salt to feedwater.

The feedwater enters the superheater at a temperature of

700째F and a pressure of 3766 psi and leaves at a temperature of lOOO*F and a pressure of 3600 psi as supercritical fluid. The four superheater exchangers in each module of the heat-exchange system are supplied by one variable-speed coolant-salt pump.

Case A

The location of the four superheater exchangers in each module of the heat-exchange system for Case A is illustrated in Fig. 1.

The

criteria governing the design for the boiler-superheater exchangers for Case A that were fixed by the system are the 1.

temperature and pressure of the incoming salt,

temperature and pressure of the outgoing salt,

3. 4.

temperature of the incoming feedwater, temperature and pressure of the outgoing supercritical fluid,

maximum drop in pressure of the supercritical fluid across the

exchanger,

flow rate of the salt,

flow rate of the feedwater, and

the total heat transferred.

The type exchanger chosen for the system is a vertical U-tube oneshell-pass and one-tube-pass unit, as illustrated in Fig. 7.

The tubes

and shell are fabricated of Hastelloy N because of its compatibility with molten salt and the supercritical fluid.

The U-shaped cylindrical shell

of the exchanger is about 18 in. in diameter, and each vertical leg is about 34 ft high, including the spherical head.

Segmental baffles were

used to improve the heat-transfer coefficient on the shell side to the extent permitted by pressure drop in the salt stream and thermal stresses

u .

47 ORNL Dwg. 65-12383

SUPER CRITICAL FLUID INLET

SUPER CRITICAL FLUD OUTLET +-

COOLANT SALT INLET

COOLANT SALT

T E ROD 8 SPACER

ler-Superheater Exchanger.

. 48

in the tube wall. A baffle on the shell side of each tube sheet provides a stagnant layer of salt that helps reduce stresses in the sheet caused by temperature gradients. The coolant salt can be completely drained from the shell, and the feedwater can be partially removed from the tubes by gas pressurization or by flushing. Complete drainability was not considered a mandatory design requirement. Design variables that had to be determined for the boiler-superheater exchanger were the

1. number of tubes, 2. tube pitch, 3.

length of tubes,

4. thickness of the wall, 5 . thickness of tube sheet, 6. baffle size and spacing, 7. diameter of shell, 8 . thickness of shell, and 9 . thickness and shape of head. Because of marked changes in the physical properties of water as its temperature is increased above the critical point at supercritical pressures, the temperature driving force and heat-transfer coefficient vary considerably along the length of the exchanger. This conditions required that the heat-transfer and pressure-drop calculations for the superheater exchanger be made on incremental lengths of the exchanger. Since the heat balances, heat transfer equations, and pressure drop equations had to be satisfied for each increment and for the entire exchanger, an iterative procedure was programmed for the CDC 1604 computer to perform the necessary calculations. This procedure varies the number of tubes, the total length of the exchanger, and the number and spacing of baffles to satisfy the heat-transfer, pressure-drop, and thermal-stress requirements. The input information, a simplified outline of the program, and the output information are given in Appendix C. The values for the physical properties of the coolant salt used in the calculations were provided by the MSBR designers and are given in Table 1. The calculations of physical properties for supercritical steam were included in the computer program as subroutines. The values

49 f o r s p e c i f i c volume and enthalpy as functions of temperature and p r e s s u r e were taken from t h e work of Keerkan and Reyes,'

and t h e values f o r thermal

conductivity and v i s c o s i t y were taken from d a t a reported by Nowak and Gro sh .2 An a d a p t a t i o n of Eqs. 3 through 11 and Eqs. 14 and 15 discussed i n Chapter 3 was used t o c a l c u l a t e t h e s h e l l - s i d e h e a t t r a n s f e r c o e f f i c i e n t s and p r e s s u r e drops.

The heat t r a n s f e r c o e f f i c i e n t on t h e i n s i d e of t h e

tubes w a s c a l c u l a t e d by using Eq. 13 given i n Chapter 3.

The p r e s s u r e

drop on t h e i n s i d e of t h e tubes was c a l c u l a t e d by using Fanning's equation, with t h e f r i c t i o n f a ~ t o r , ~ f = 0.00140

0.125

*3a

where 'i di

= v i s c o s i t y of f l u i d a t temperature i n s i d e tube, l b / h r . f t , = i n s i d e diameter of tube,

G = mass v e l o c i t y , l b / h r * f $ .

The thermal r e s i s t a n c e of t h e tube w a l l was c a l c u l a t e d f o r each increment of tube length by using t h e thermal conductivity of Hastellog N evaluated a t t h e average temperature of t h e tube w a l l of each p a r t i c u l a r increment.

The procedure used t o determine t h e allowable temperature

drop a c r o s s t h e tube w a l l was based on t h e requirements set f o r t h i n Section I11 of t h e ASME B o i l e r and Pressure Vessel Code.

The thermal

stresses were t r e a t e d as secondary membrane plus bending s t r e s s e s w i t h

t h e maximum allowable value being equal t o t h r e e t i m e s t h e aIlowable membrane stress f o r Hastelloy N minus t h e s t r e s s e s caused by pressure. lJ.

H. Keenan and F,

Keyes, Thermodynamic P r o p e r t i e s of Steam,

John Wiley and Sons, Ne

2E. S. Nawak and R rosh, "An h v e s t i g a t i o n of Certain Thermoi e s of Water and Water Vapor i n the Critical dynamic and Transport P Region," USAEC Report ANL-6064, Argonne National Laboratory, October 1959. ' 5 . H. Perry, Editor, p. 383 i n Chemical Engineer's Handbook, 3rd ed., M c G r a w - H i l l , New York, 1950.

u From an equation reported by Harvey,* the thermal stress,

where

= tangential or axial thermal stress in tube wall, psi,

(2 =

coefficient of thermal expansion, in./in:

0F,

ET = modulus of elasticity of tube, psi, to = temperature outside tube, 0F, ti = temperature inside tube, OF,

u = Poisson's ratio,

do = outside diameter of tube, di = inside diameter of tube. This equation and the allowable thermal stress as defined above were used to calculate the allowable temperature drop across the thbe wall. The computer program was run for several cases to investigate the parameters, and "hand" calculations were made to check the program. However, it should be noted that this work did not constitute a complete parameter study or optimization of the design variables.

The resulting

design data for the boiler-superheater exchanger for Case A are given in Table 6. *J. F. Harvey, P. 63 in Pressure Vessel Design, D. Van Nostrand Company, New Jersey, 1963. Table 6.

Boiler-Superheater Design Data for Case A U-tube U-shell exchanger with cross- flow baffles

Number required

Rate of heat transfer, each, Mw

Btu/hr

120.9 4.13 x 108

Table 6 (continued) ~~

Shell-side conditions Hot fh i d Entrance temperature, OF Exit temperature, OF Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, psi Mass flow rate, lb/hr

Coolant salt 1125 850 149 90.9 58.1 3.6625 x 106

Tube-side conditions Cold fluid Entrance temperature, OF Exit temperature, O F Entrance pressure, psi Exit pressure, psi Pressure drop across exchanger, psi Mass flow rate, lb/hr Mass velocity, lb/hr*ft? Tube material

Supercritical fluid 700 1000 3766.4 3600 166.4 6.3312 x lo5 2.78 x 106 Hastelloy N

Tube OD, in.

0.50

Tube thickness, in.

0.077

Tube length, tube sheet to tube sheet, ft

63.81

Shell material

Hastelloy N

Shell thickness, in.

0.375

Shell ID, in.

18.25

Tube sheet material Tube sheet thickness, in.

Hastelloy N

Number of tubes

349

Pitch of tubes, in. Total heat transfer area, f P Basis for area calculation

0.875

4.75

2915 Outside surface

Crossflow 9 Variable

Type of baffle Number of baffles Baffle spacing .

Overall heat Btu/hr'ft?

t, U,

1030

. *

Table 6 (continued) Maximum stress i n t e n s i t y , Tube Calculated

psi Pm = 6748; (P + Q) = 40,665 m Pm = Sm = 16,000; (Pm + Q) =

A 1 lowable

Shell Calculated

= 48,000

+ Q) =

8543

= Sm = 10,500; (Pm

+ Q)

Pm = 3775; (Pm

m '

Allowable

3Sm = 31,500 i

Maximum tube sheet stress, p s i Calculated A 1lowable

~16,600 16,600

%e symbols a r e those o f Section I11 of t h e ASME B o i l e r and P r e s s u r e Vessel Code, where = primary membrane s t r e s s i n t e n s i t y ,

m '

Q = secondary stress i n t e n s i t y ,

S = allowable stress i n t e n s i t y . m

Case B

A s previously discussed, t h e o p e r a t i n g p r e s s u r e s of t h e coolants a l t system were raised i n t h e Case-B design f o r t h e heat-exchange system.

The coolant salt e n t e r s t h e superheater exchanger a t t h e same >

temperature as i n Case A but a t a p r e s s u r e of 251.5 p s i , and i t leaves t h e exchanger a t t h e same temperature as i n Case A b u t a t a p r e s s u r e of 193.4 p s i .

Thus, t h e

p r e s s u r e drop a c r o s s t h e exchanger, 58.1 p s i , and

t h e h e a t t r a n s f e r are t h e same f o r Case B as they a r e f o r Case A.

How-

ever, t h e d i f f e r e n c e between t h e p r e s s u r e l e v e l i n Case B and t h a t f o r Case A r e s u l t e d i n d i f f e r e n t stress i n t e n s i t i e s f o r t h e two cases.

The

maximum p r e s s u r e drop through t h e tubes w a s s p e c i f i e d as 200 p s i , b u t t h i s value had t o be reduced t o prevent exceeding t h e l i m i t a t i o n placed on t h e maximum h e i g h t o f t h e exchanger.

53 An analysis was made of the stress intensities in the tube sheets, tubes, shells, high-pressure heads, shell-to-tube-sheet junctions, and tube-to-tube-sheet junctions. However, the discontinuity stresses were not analyzed at.the junctions of the high-pressure heads and shells or at the junctions involving the entrance and exit of coolant salt or supercritical fluid. These calculations performed for the Case-B boiler-superheater exchanger are given in Appendix C, and the resulting design data are given in Table 7. All other superheater design data for Case B are the same as for Case A. Table 7. Boiler-Superheater Stress Data for Case B a Maximum stress intensity, psi Tube Pm = 13,843; (Pm + Q) = 40,662 Calculated A1lowab1e

= Sm = 16,000; (Pm m 3sm = 48,000

Shell Calculated

+ Q)

6372; (Pm + Q) = 14,420 pm = sm = 10,500; (pm + Q) = 3sm = 31,500 Pm =

Allowable

Maximum tube sheet stress, psi Calculated Allowable

<16,600 16,600

aThe symbols are those of Section I11 of the ASME Boiler and Pressure Vessel Code, where Pm = primary membrane stress intensity, Q = secondary stress intensity, and

S = allowable stress intensity. m

Case C

m e C is ,asically a modification of the Case-A heat-exchange system that was studied briefly to determine the effect of having a coolant salt with a lower liquidus point. To study this condition, the temperature

of t h e feedwater e n t e r i n g t h e b o i l e r superheater was lowered from 700째F t o 580째F,

and t h e entrance p r e s s u r e w a s s e t a t 3694 p s i .

With t h e feed-

water leaving t h e superheater as s u p e r c r i t i c a l f l u i d a t a temperature of 1000째F and p r e s s u r e of 3600 p s i , t h e coolant s a l t e n t e r i n g a t a temperat u r e of 1125'F

and a p r e s s u r e o f 149 p s i leaves t h e exchanger a t a tem-

p e r a t u r e of 850째F and p r e s s u r e o f approximately 145 p s i . The computer program was used t o s i z e an exchanger f o r t h e s e condit i o n s , and t h e r e s u l t i n g design d a t a f o r t h e Case-C b o i l e r superheater

are given i n Table 8. Table 8.

Boiler Superheater Design Data f o r Case C

U-tube U-shell exchanger w i t h cross-flow b a f f l e s Number required

Rate of h e a t t r a n s f e r , each Mw Btu/hr

114.7 3.914 x 108

S h e l l - s i d e conditions Hot f l u i d Entrance temperature, OF Exit temperature, OF P r e s s u r e drop a c r o s s exchanger, p s i

Coolant s a l t 1125 850 3.9

Tube-side c o n d i t i o n s Cold f l u i d Entrance temperature, 0 F E x i t temperature, OF Entrance pressure, p s i E x i t pressure, p s i Pressure drop a c r o s s exchanger, p s i

Supercritical fluid 580 1000 3694 3600 9.4

Tube material

Hastelloy N

Tube OD, i n ,

0.50

Tube thickness, i n .

0.077

Tube length, tube sheet t o tube sheet, f t

62.64

S h e l l material

Hastelloy N

S h e l l thickness, i n .

0.375

S h e l l . ID, i n .

Tube s h e e t material

Hastelloy N

55 Table 8 (continued) Number o f tubes

323

Pitch o f tubes, i n .

0.875

Total heat transfer area, f?

2648

Basis for area calculation

Outside o f tubes

Type o f b a f f l e

Cross flow

Number o f b a f f l e s

Baffle spacing, i n .

Variable

Overall heat transfer c o e f f i c i e n t , U, Btu/hr* ft?

785

56 7.

DESIGN FOR STEAM REHEATER EXCHANGERS

Eight, two i n each module o f t h e heat-exchange system, v e r t i c a l shell-and-tube exchangers t r a n s f e r h e a t from t h e coolant salt t o t h e exhaust steam from t h e high-pressure t u r b i n e . t h e exchanger a t a temperature o f 1125'F

The coolant salt e n t e r s

and leaves a t a temperature of

850째F, having e l e v a t e d t h e temperature of t h e steam from 650째F t o 1000째F.

Case A

The l o c a t i o n of t h e two steam r e h e a t e r exchangers i n each module of the heat-exchange system f o r Case A i s i l l u s t r a t e d i n Fig. 1.

The c r i t e r i a

governing t h e design f o r t h e reheater exchangers f o r Case A t h a t were fixed by the system a r e t h e incoming salt,

temperature and p r e s s u r e of t h e

temperature and p r e s s u r e of t h e outgoing salt,

temperature and p r e s s u r e of t h e incoming steam,

temperature and p r e s s u r e of t h e outgoing steam,

flow rate of t h e s a l t ,

flow rate of t h e steam, and

t o t a l heat transferred.

Following a procedure o u t l i n e d by Kays and London,'

we s e l e c t e d t h e

general type of exchanger t o be used f o r t h e s e conditions.

The r e h e a t e r

exchangers a r e s t r a i g h t counterflow ones with b a f f l e s , as shown i n Fig. 8. The s t r a i g h t s h e l l occupies less c e l l volume than a U-tube U-shell design and r e q u i r e s s l i g h t l y l e s s c o o l a n t - s a l t inventory.

The steam e n t e r s t h e

bottom of t h e exchanger a t a p r e s s u r e of 580 p s i , flows upward through t h e tubes, and l e a v e s t h e top of t h e exchanger a t a p r e s s u r e of 567.1 psi.

The coolant s a l t e n t e r s t h e upper p o r t i o n of t h e exchanger a t a

p r e s s u r e o f 106 p s i , flows downward around t h e tubes, and leaves t h e lW. M. Rays and A. L. London, Compact Heat Exchangers, 2nd ed., McGraw H i l l , New York, 1964.

57 O W L Dwg. 65-12381

/STEAM

OUTLET

TUBE SHEET

SALT INLET

TIE ROD

SPbICE R

TUBULAR SHfLL

Fig. 8. Steam Reheater Exchanger.

bottom p o r t i o n of t h e exchanger a t a pressure of 89.6 p s i .

A special

Lri'

d r a i n pipe a t t h e bottom of t h e exchanger p e r m i t s drainage of t h e coolant salt. Disk and doughnut b a f f l e s w e r e s e l e c t e d f o r t h e exchanger because, i n our opinion, t h i s design r e s u l t s i n a more e f f i c i e n t exchanger than one i n which straight-edged b a f f l e s are used.

B a f f l e s on t h e s h e l l s i d e

of t h e tube s h e e t s provide a stagnant l a y e r of coolant s a l t t o reduce t h e thermal stresses i n t h e s h e e t s .

The o u t s i d e diameter of t h e tubes

s t i p u l a t e d by t h e designers of t h e MSBR is 3 / 4 in.,

and t h e tubes and

s h e l l are f a b r i c a t e d of Hastelloy N because of i t s c o m p a t i b i l i t y w i t h molten s a l t and steam. The design v a r i a b l e s t h a t had t o be determined f o r t h e steam r e h e a t e r exchanger w e r e t h e 1.

number of tubes,

tube length,

w a l l thickness of tubes,

b a f f l e s i z e and spacing,

thickness of tube sheets, and

thickness and shape of head.

To determine t h e s e v a r i a b l e s , i t w a s necessary t o c a l c u l a t e t h e h e a t t r a n s f e r c o e f f i c i e n t s , pressure drops, and t h e thermal and mechanical

stresses.

These c a l c u l a t i o n s are given i n Appendix D.

From a consideration of stresses caused by t h e temperatures and pressures t o be encountered, a tube w a l l thickness of 0.035 i n . was chosen.

From t h e s t i p u l a t e d tube s i z e ( 3 / 4 i n . OD),

t h e steam p r e s s u r e drop,

and an assumed tube length, t h e number of tubes required was c a l c u l a t e d . This c a l c u l a t e d value was c o r r e c t e d by i t e r a t i o n , and the number of tubes

w a s f i x e d a t 628.

A t r i a n g u l a r tube p i t c h of 1 i n . was s e l e c t e d , and t h e

i n s i d e diameter of t h e s h e l l w a s c a l c u l a t e d t o be 28 i n .

Then, by using

t h e Dittus-Boelter equation, Eq. 2 of Chapter 3, t h e h e a t t r a n s f e r c o e f f i c i e n t f o r t h e steam i n s i d e t h e tubes was c a l c u l a t e d as 409 Btu/hr*ft2-*F. Design c a l c u l a t i o n s f o r t h e s h e l l - s i d e flow of coolant s a l t were somewhat more involved.

Turbulent flow is d e s i r a b l e f o r good heat

t r a n s f e r on t h e s h e l l s i d e .

W e t h e r e f o r e a r b i t r a r i l y chose a Reynolds

number of 7500 f o r p a r a l l e l flow through both t h e i n t e r i o r and e x t e r i o r

59 The Reynolds number chosen i s w e l l above t h e number a t

b a f f l e windows.

t h e threshold o f turbulence, and t h e r e s u l t i n g s i z e o f t h e exchanger is not extreme.

From t h i s number and t h e mass flow r a t e fixed by t h e system,

t h e flow area i n t h e two b a f f l e windows w a s determined a s 0.764 ft?

Because we judged i t t o be good design p r a c t i c e and because i t s i m p l i f i e d c a l c u l a t i o n of t h e s h e l l - s i d e h e a t - t r a n s f e r c o e f f i c i e n t , we made t h e cross-flow a r e a equal t o t h e flow a r e a through t h e b a f f l e windows:

0.764

Knowing t h e b a f f l e s i z e s and p i t c h , we were then a b l e t o determine

ft?.

t h e b a f f l e spacing, which i s 12.4 i n .

The s h e l l - s i d e h e a t - t r a n s f e r c o e f f i c i e n t w a s c a l c u l a t e d by u s i n g an a d a p t a t i o n of Eqs. 3 through 11 discussed i n Chapter 3, and t h i s c o e f f i c i e n t i s 2240 Btu/hr*f@

Knowing t h e s h e l l - s i d e h e a t - t r a n s f e r coef-

f i c i e n t , t h e h e a t - t r a n s f e r c o e f f i c i e n t f o r t h e steam i n s i d e t h e tubes

(409 Btu/hr.fi? *OF), t h e thermal conductivity of t h e Hastelloy N, and t h e t o t a l h e a t t r a n s f e r r e d p e r hour i n t h e exchanger, t h e length o f t h e tubes w a s c a l c u l a t e d .

+LJ, +L =iy i -k T( a m* hOa UatnL

where L = l e n g t h of tube, f t , U = o v e r a l l h e a t - t r a n s f e r c o e f f i c i e n t , Btu/hr* ft? a

= t o t a l h e a t t r a n s f e r surface, f i ? / f t ,

n = number of tubes, hi = h e a t - t r a n s f e r c o e f f i c i e n t f o r steam i n s i d e tubes,

= h e a t t r a n s f e r s u r f a c e i n s i d e tubes, f ? / f t ,

k = thermal conductivity, Btu/hr.f?

-OF p e r

ft,

T = t h i c k n e s s of tube

= mean h e a t - t r a n s f e m ho = h e a t - t r a n s f e r coe aO

n t s a l t i n shell,

= shell-side heat t =

7*78 lo' (0,01373 628

= 21.70 f t .

0.00150 + 0.00227)

60 Conventional means were used t o c a l c u l a t e t h e steam-side pressure

drop f o r t h e r e h e a t e r exchanger, and t h e t o t a l w a s found t o be 12 p s i . The pressure drop on t h e s a l t s i d e w a s c a l c u l a t e d by making use of an adaptation of Eqs. 14 and 15 discussed i n Chapter 3.

The s a l t - s i d e

pressure drop i s 11.4 p s i . I n designing t h e exchanger, a length of s h e l l w a s allowed f o r the i n l e t and o u t l e t p i p e s t h a t w a s somewhat l a r g e r than t h e b a f f l e spacing. This increased t h e o v e r a l l length o f t h e tubes from 21.7 and 22.1 f t . This small increment i n t h e length w a s neglected i n s o f a r as i t s e f f e c t upon pressure, v e l o c i t y , etc.,

i s concerned.

The Case A design d a t a f o r

t h e steam r e h e a t e r exchanger are given i n Table 9. Table 9.

Steam Reheater Exchanger Design Data f o r Case A S t r a i g h t tube and s h e l l with d i s k and doughnut b a f f l e s

Number required

Rate o f heat t r a n s f e r p e r u n i t ,

Mw Btu/hr Shell-side conditions Hot f l u i d Entrance temperature, OF E x i t temperature, OF Entrance pressure, p s i E x i t pressure, p s i Pressure drop across exchanger, p s i Mass flow r a t e , lb/hr Mass v e l o c i t y , l b / h r * f 6â&#x20AC;&#x2122; Tube-side conditions Cold f l u i d 0 Entrance temperature, F E x i t temperature, OF Entrance pressure, p s i E x i t pressure, p s i Pressure drop a c r o s s exchanger, p s i Mass flow rate, lb/hr Mass v e l o c i t y , l b / h r * f $ Velocity, f p s

36.25 1.24

109

Coolant s a l t 1125 850 106 94.6 11.4 1.1 x 106 1.44 x 106 Steam 650 1000 580 568

12 6.3 x 105 3.98 x 106 145

Tube material

Hastelloy N

Tube OD, in.

0.75

61 Table 9 (continued) Tube thickness, in. Tube length, tube sheet to tube sheet,

0.035

Shell material Shell thickness, in. Shell ID, in.

Hastelloy N 0.5 28

Tube sheet material

Hastelloy N

Tube sheet thickness, in.

4.75

Number of tubes

628

Pitch of tubes, in. Total heat transfer area, ft?

1.o 2723

Basis for area calculation

Outside of tubes

Type of baffle

Disk and doughnut

Number of baffles Baffle spacing, in.

10 and 10 12.375

Disk OD, in.

24.3

Doughnut ID, in. Overall heat transfer coefficient, U, Btu/hr-f? a Maximum stress intensity, psi Tube Calculated

16.9 285

22.1

5243; (P

= S

Allowable

m' 3s

She11 Calculated

= 14,500;

15,091

(P, + Q) =

43,500

+Q) = 14,751 = 10,600; (Pm +Q) =

= 4350; (Pm

Pm =

Allowable

+ Q)

3sm = 31,800

Maximum tube sheet stress, psi Calculated Allowable

9 600 9600

%he symbols are those of Section 111 of the ASME Boiler and Pressure Vessel Code, where P = primary membrane stress intensity,

;h, j j

Q = secondary stress intensity, Sm = allowable stress intensity.

. 62

Case B

I n t h e Case-B design f o r t h e heat-exchange system, t h e o p e r a t i n g p r e s s u r e s o f t h e coolant s a l t were r a i s e d .

The coolant s a l t e n t e r s t h e

steam r e h e a t e r exchanger a t t h e same temperature as i n Case A (1125OF)

but a t a pressure of 208.5 p s i , and it leaves t h e exchanger at t h e same temperature as i n Case A (850OF) but a t a pressure o f 197.1 p s i .

Thus,

t h e pressure drop a c r o s s t h e exchanger, 11.4 p s i , and t h e h e a t t r a n s f e r a r e t h e same f o r Case B as f o r Case A, but t h e d i f f e r e n c e between t h e pressure l e v e l s of Case A and Case B r e s u l t e d i n d i f f e r e n t stress inten-

s i t i e s f o r the two cases. An a n a l y s i s w a s made of t h e stress i n t e n s i t i e s e x i s t i n g throughout

t h e r e h e a t e r f o r t h e higher p r e s s u r e conditions, but t h e e f f e c t s o f t h e design f o r t h e c o o l a n t - s a l t entrance and e x i t p i p e s and t h e steam d e l i v e r y and e x i t plenums upon t h e s h e l l stresses were not considered.

These cal-

c u l a t i o n s are given i n Appendix D, and t h e r e s u l t i n g stress d a t a a r e given i n Table 10.

A l l o t h e r steam r e h e a t e r design d a t a f o r Case B are t h e

same as f o r Case A. Table 10.

Steam Reheater S t r e s s Data f o r Case B

Maximum stress i n t e n s i t y , a p s i Tube Calculated

Pm = 4349; (Pm + Q) = 13,701 Pm = Sm = 14,500; (Pm -t Q) =

Allowable

3Sm = 43,500' !

She 11 Calculated

= 6046.5; (P + Q) = 17,165 m' m Pm = S = 10,600; (Pm + Q)= m 3sm = 31,800

A 1lowable Maximum tube sheet stress, p s i Calculated Allowable

<lo, 500 10,500

%he symbols are those o f Section I11 of t h e ASME Boiler and Pressure Vessel Code, where = primary membrane s t r e s s i n t e n s i t y , pm Q = secondary s t r e s s i n t e n s i t y , and

= allowable stress i n t e n s i t y .

63 Case C

The Case-C modification of the Case-A design for the heat-exchange system was studied to determine the effect of having a coolant salt with a lower liquidus point. To effect this, the temperature of the steam entering the reheater exchanger was lowered from 650 to 551째F, and the entrance pressure was set at 600 psi. With the steam leaving the reheater at a temperature of 1000째F and pressure of 584.8 psi, the coolant salt entering at a temperature of 1125'F

and pressure of 106 psi leaves at a temperature of 850째F and a pressure of 92.2 psi. The design data for the

Case-C reheater are given in Table 11. Table 11.

Steam Reheater Exchanger Design Data for Case C Straight tube and shell with disk and doughnut baffles

Number required

Rate of heat transfer, each Mw

Btu/hr

48.75 1.66 x 108

Shell-side conditions Hot fluid Entrance temperature, OF Exit temperature, OF Pressure drop across exchanger, psi Tube-side conditions Cold fluid Entrance temperature, OF Exit temperature, OF Entrance pressure, psi Pressure drop across exchanger, psi Tube material

Hastelloy N

Tube OD, in.

0.75

Tube thickness, in.

0.035 23.4

Tube length, tube sheet to tube sheet, ft

Coolant sa 1t 1125 850 13.8 Steam 551 1000 600 15.2

Shell material

Hastelloy N

Shell thickness, in.

0.5

Shell ID, in.

Table 11 (continued) Tube sheet material

Hastelloy N

Number o f tubes

6 20

P i t c h of tubes, i n .

T o t a l h e a t t r a n s f e r area, ft?

2885

Basis f o r area c a l c u l a t i o n

Outside o f tubes

Type o f b a f f l e s

Disk and doughnut

Number o f b a f f l e s

13 and 14

Baffle spacing, in.

Disk OD, i n .

21.3

Doughnut I D , in.

18.5

Overall h e a t t r a n s f e r c o e f f i c i e n t , U, Btu/hr f t?

289

65 8.

DESIGN FOR REHEAT-STEAM PREHEATERS

Eight reheat-steam p r e h e a t e r s , two i n each module of t h e heat-exchange system, a r e used t o h e a t t h e exhaust steam from t h e high-pressure t u r b i n e before i t e n t e r s t h e r e h e a t e r s t o a s s u r e t h a t t h e coolant s a l t w i l l not be cooled below i t s l i q u i d u s p o i n t . The p r e h e a t e r s a r e p a r t of t h e steam power system, and s i n c e they do not come i n t o c o n t a c t with molten s a l t s , they a r e f a b r i c a t e d of Croloy. Being a p a r t of t h e steam system, t h e p r e h e a t e r s are unaffected by t h e Case-B design f o r t h e heat-exchange system, and t h e i r design i s t h e same f o r both Case A and Case B. Thus, t h e i r l o c a t i o n i n a module o f t h e system may be seen i n e i t h e r Fig. 1 o r 2.

However, t h e need f o r the p r e h e a t e r s i s eliminated i n t h e Case-C

modification of t h e Case-A h e a t exchange system. T h r o t t l e steam o r s u p e r c r i t i c a l f l u i d a t a temperature of 1000째F and a p r e s s u r e of 3600 p s i i s used t o h e a t t h e exhaust steam from t h e highp r e s s u r e t u r b i n e , and t h e p r e h e a t e r s had t o be designed f o r t h i s high temperature and p r e s s u r e . h e a t e r i s shown i n Fig. 9.

A conceptual drawing of t h e reheat-steam pre-

The exchanger s e l e c t e d i s a one-tube-pass

one-shell-pass counterflow type with U-tubes and U-shell.

S e l e c t i o n of

a U-shell r a t h e r than a divided c y l i n d r i c a l s h e l l permits t h e use o f smaller diameters f o r t h e heads and reduces t h e t h i c k n e s s e s r e q u i r e d f o r t h e heads and t h e tube s h e e t s .

Each v e r t i c a l l e g of t h e U-shell i s about

21 i n . i n diameter and t h e o v e r a l l h e i g h t is about 15 f t , including t h e s p h e r i c a l heads.

The U-tubes have an o u t s i d e diameter of 318 i n . and a

w a l l t h i c k n e s s of 0.065 i n . They a r e l o c a t e d i n a t r i a n g u l a r a r r a y with a p i t c h of 3/4 i n . No b a f f l e s a r e used i n t h i s design, but t h e tubes a r e supported a t intervals. The s u p e r c r i t i c a l f l t h e tubes, back up, and

869째F and a p r e s s u r e of

i n l e t below t h e head a t

e n t e r s t h e head region, flows down through

s from t h e o p p o s i t e head a t a temperature of psi.

The t u r b i n e exhaust steam e n t e r s a s i d e

mperature of 551째F and a p r e s s u r e of 595.4 psi,

flows around t h e tubes c o u n t e r c u r r e n t to t h e flow of t h e s u p e r c r i t i c a l f l u i d , and leaves t h e exchanger through a side o u t l e t on t h e o p p o s i t e l e g below t h e head a t a temperature of 650째F and a p r e s s u r e of 590 p s i .

66 ORNL Drrg 65-12382

SUPER -CRITICAL F U i O OiJTlET

STEAM INLEl

Fig. 9.

Reheat-Steam Preheater Exchanger.

The heat-transfer and pressure-drop calculations made for the reheat-steam preheater are given in Appendix E.

The heat transfer coef-

ficient for the supercritical-fluid film inside the tubes was calculated by using the Dittus-Boelter equation, Eq. 2 discussed in Chapter 3 of this report.

The reheat steam flows outside and parallel to the tubes,

and the heat-transfer coefficient for the film outside the tubes was calculated by using both Eq. 2 and Eq. 12 of Chapter 3. Equation 12 gave the most conservative value and it was used in the design calculations. Pressure drops in the tubes and in the shell were calculated by using the Darcy equation for the friction loss; four velocity heads to account

67 f o r t h e i n l e t , e x i t , and r e v e r s a l l o s s e s ; and a c o r r e c t i o n f a c t o r f o r changes i n k i n e t i c energy between t h e i n l e t and e x i t of t h e exchanger. An a n a l y s i s of t h e s t r e s s i n t e n s i t i e s i n t h e tubes, tube sheets, s h e l l s , aad high-pressure heads and of t h e discontinuity-induced s t r e s s e s a t t h e i r j u n c t i o n s w a s made.

The d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n

of t h e high-pressure head and t h e s h e l l , t h e entrance l i n e , and t h e e x i t l i n e were not considered i n t h i s a n a l y s i s .

These s t r e s s c a l c u l a t i o n s a r e

a l s o given i n Appendix E, and t h e r e s u l t i n g design d a t a f o r t h e reheatsteam p r e h e a t e r a r e given i n Table 12. Table 12.

Design Data f o r t h e Reheat-Steam P r e h e a t e r One-tube-pass one-shell-pass U-tube U-shell exchanger with no b a f f l e s

Number required

Rate of h e a t t r a n s f e r , each Mw Btu/hr

12.33 4.21 x 107

She11-s i d e conditions Cold f l u i d 0 Entrance temperature, F 0 Exit temperature, F Entrance pressure, p s i E x i t pressure, p s i P r e s s u r e drop a c r o s s exchanger, p s i Mass flow r a t e , lb/hr Mass v e l o c i t y , l b / h r * ft?

Steam 551 650 595.4 590 .O 5.4 6.31 105 3.56 x 105

Tube-side c o n d i t i o n s Hot fluid 0 Entrance temperature, F E x i t temperature, OF Entrance pressure, p s i E x i t pressure, p s i P r e s s u r e drop a c r o s s exchanger, p s i Mass flow r a t e , Ib/hr Mass v e l o c i t y , l b / h r * ff? Velocity, f p s

S u p e r c r i t i c a l water 1000 869 3600 3535 65 3.68 x 105 1.87 x 106 93.5

Tube material

Cro l o y

Tube OD, i n .

0.375

Tube thickness, i n .

0.065

Tube length, tube sheet t o tube sheet, ft

13.2

Table 12 (continued) Shell material

Croloy

Shell thickness, in.

7116

Shell ID, in.

20.25

Tube sheet material

Croloy

Tube sheet thickness, in.

6.5

Number of tubes

603

Pitch of tubes, in.

0.75

Total heat-transfer area, ft?

781

Basis for area calculation

Tube OD

Type of baffle

None

Overall heat transfer coefficient, U, Btu/hr 'f? Maximum stress intensity, a p s i Tube Calculated

162

Pm = 10,503; (Pm + Q) = 7080

Sm = 10,500 @ 961째F;(Pm + Q) = 3Sm = 31,500

Allowable

'm-

Shell Calculated

Pm = 14,375; (Pm + Q) = 33,081 Pm =

A1 lowab le

sm =

15,000 @ 650째F;

(Pm + Q) = 3sm = 45,000 i

Maximum tube sheet stress, psi Calculated Allowable

7800 7800 @ 1000째F

%e symbols are those of Section I11 of the ASME Boiler and Pressure Vessel Code, where m'

= primary membrane stress intensity,

Q = secondary stress intensity, and

Sm = allowable stress intensity.

APPENDICES

'LL

Appendix A CALCULATIONS FOR PRIMARY HEAT EXCHANGER

Pressure-Drop and Heat-Transfer Calculations f o r Case B The Case B design f o r t h e primary f u e l - s a l t - t o - c o o l a n t - s a l t exchanger i s i l l u s t r a t e d i n Fig. 4 of Chapter 4.

heat

The v a l u e s assumed t o

determine t h e design v a r i a b l e s f o r t h i s two-pass shell-and-tube exchanger with d i s k and doughnut b a f f l e s a r e t a b u l a t e d below.

a (oa)

= length of o u t e r annulus = 16.125 f t ,

= l e n g t h of i n n e r annulus = 15.050 f t ,

N(oa) = number of b a f f l e s i n o u t e r annulus = 10, = number of b a f f l e s i n inner annulus = 4, (ia) X(oa) = b a f f l e spacing i n o u t e r annulus = 1.466 f t , X ( i a ) = b a f f l e spacing i n i n n e r annulus = 1.466 f t and 3.396 f t , N

p(oa) = tube p i t c h i n o u t e r annulus = 0.625 i n . ( t r i a n g u l a r ) , p(ia)

= tube p i t c h i n i n n e r annulus

= 0.600 i n . r a d i a l and 0.673 i n . circumferential,

n(oa) = number of tubes i n o u t e r annulus = 3794, n(ia)

= number of tubes i n i n n e r annulus =

4347.

The assumed l e n g t h f o r $(oa) of 16.125 f t r e s u l t e d i n a c a l c u l a t e d length of 16.11 f t .

Therefore, t h e 16.125-ft l e n g t h w a s used f o r t h e o u t e r

annulus, and t h e r e s u l t i n g l e n g t h for t h e i n n e r a n n u l u s was 15.050 ft. With t h e s e v a r i a b l e s e s t a b l i s h e d , t h e pressure-drop and h e a t - t r a n s f e r c a l c u l a t i o n s were made, and t

terms used . i n t h e s e c a l c u l a t i o n s a r e

defined i n Appendix F. Pressure-Drop C a l c u l a t i o n s C a l c u l a t i o n of t h e t o t a l p r e s s u r e drop i n s i d e t h e tubes involved t h e determination of t h e p r

u r e drop i n s i d e t h e tubes o f both t h e o u t e r

and i n n e r annuli, and c a l c u l a t i o n o f t h e t o t a l p r e s s u r e drop o u t s i d e t h e tubes o r on t h e s h e l l s i d e involved t h e determination of t h e s h e l l - s i d e p r e s s u r e drop i n both t h e o u t e r and i n n e r annuli.

The p r e s s u r e drop i n s i d e t h e tubes i n

P r e s s u r e Drop I n s i d e Tubes. t h e o u t e r annulus,

where f = t h e f r i c t i o n f a c t o r = 0.0014

+ 0.125/(NRe)'

*",

L = length of tubes, f t , di = i n s i d e diameter of tubes, G = mass v e l o c i t y of f l u i d i n s i d e tubes, l b / h r e f 9 , p = d e n s i t y of f l u i d i n s i d e tubes, lb/f$,

and

= g r a v i t a t i o n a l conversion constant, l b m * f t / l b f*se$

To determine t h e f r i c t i o n f a c t o r , the Reynolds number, diG NRe = 9 P

where di = 0.0254167 f t , p =

27 l b / f t * h r , and

G = flow rate of f l u i d i n s i d e tubes/flow area,

where t h e flow area, A = 3794 tubes (5.07374 x lo-* f e / t u b e ) = 1.9250 ft?,

Therefore, N

= 5345, and t h e f r i c t i o n f a c t o r , Re f = 0.0014 + 0.125(5345)-0~3" = 0.009416.

mi(oa)=[4(0*009416)(16*125) 0.02542

5.678 x 106 (127) 3*6(144)64,4 '03 = 54.69 p s i .

For t h e p r e s s u r e drop i n s i d e t h e tubes i n t h e i n n e r annulus,

L n

(%a) (i 4

= 15.286 f t ,

= 4347,

A = t h e flow a r e a i n s i d e t h e tubes = 2.2056 fid

Gi(lf3

= 4.956 x ldj l b / f @ - h r ,

NRe = 4665, and

f = 0.0014

0.125(4665)*'3a

= 0.00981.

73 mi(ia)

= 41.19 p s i

The t o t a l p r e s s u r e drop i n s i d e t h e tubes, mi( t o t a l )

= 54.69

Pressure Drop Outside Tubes.

41.19 = 95.88 p s i

The s h e l l - s i d e flow p a t t e r n of t h e cool-

a n t s a l t i n t h e primary h e a t exchanger i s i l l u s t r a t e d i n Fig. A.l.

The

h o r i z o n t a l c r o s s - s e c t i o n a l p o r t i o n of t h i s i l l u s t r a t i o n shows t h e o u t e r annulus, oa, t o be divided i n t o t h r e e types of flow regions, and t h e s e a r e numbered 1 through 3 from t h e o u t s i d e i n .

Flow region 2 o r t h e middle

region is e s t a b l i s h e d by t h e overlap of t h e a l t e r n a t e l y spaced b a f f l e s , and t h e r e i s only cross flow i n t h i s region.

Regions 1 and 3 c o n s i s t of

b a f f l e windows i n which t h e r e i s a combination of p a r a l l e l and cross flow. I

ORNL OWG 67-6566

INNER ANNULUS

OUTER ANNULUS

(ia)

Fig. A.1.

Shell-Side Flow P a t t e r n of Coolant S a l t i n Primary Exchanger.

L)â&#x20AC;&#x2DC; From Eq. 15 in Chapter 3 , the shell-side pressure drop in the window region of the outer annulus, aPo(oa)

= (2

3- 1

PV,â&#x20AC;?

+ 0.6rw

)

3-1

2gc

where rw = the number of restrictions in the window region, p = the density of the coolant salt = 125 lb/fP,

= mean velocity of the fluid, ft/sec, and = gravitational conversion constant, lbm*ft/lbf set3

The volumetric flow rate of the coolant salt, q =r

(

0 sec/hr) = 37.444 fP/sec

With the parallel-flow areas being based on a velocity of 15 ft/sec, the average flow area in the outer annulus is the average of the cross-flow and the parallel-flow areas.

where

S = the cross-sectional area of the window region W

= parallel-flow areas in regions 3 and

= 2.496 f p ,

SB = the cross-sectional area of the cross-flow region = cross-flow area in region 2 = 7.708 f?

Therefore,

= [ 2 . 4 9 6 ( 7 . 7 0 8 ) ] 0 * 6 = 4.386 fiZ

and the mean velocity of the coolant salt in the outer annulus,

vm =

37*444 = 8.537 ft/sec 4.386

The number of restrictions in the window region,

where

D = the outside diameter of the outer annulus region, 0 Di = the inside diameter of the inner annulus region, and p = the pitch of the tubes.

For o u t e r annulus region 3, r

59.11

53.05 = 5.196

- 1.866(0.625)

and f o r t h e o u t e r annulus region 1, rw

66.70 61.39 = 4.553 1.866(0.625)

Therefore, t h e number of cross-flow r e s t r i c t i o n s i n regions 1 and 3, rW

(5.196 + 4.553)0.5 = 4.875

3- 1

The s h e l l - s i d e p r e s s u r e drop i n the o u t e r annulus b a f f l e window flow regions 3 and 1,

= 4.838 p s i

From Eq. 14 i n Chapter 3, t h e s h e l l - s i d e p r e s s u r e drop i n t h e crossflow region of t h e o u t e r annulus, aPo(oa) 2

= 0.6r p

vz . 2gc

The v e l o c i t y of the coolant s a l t i n region 2,

V = 37*444 fPâ&#x20AC;&#x2122;sec 7.708 ff3

= 4.858 f t / s e c

and t h e number of cross-flow r e s t r i c t i o n s i n region 2,

61.39 59.11 = 1.955 , r B -- 1,866(0.625)

0.373 p s i

As may be seen i n Fig. A.l,

t h e h o r i z o n t a l c r o s s - s e c t i o n a l a r e a of

t h e inner annulus, i a , i s divided roughly i n t o two flow regions caused by t h e a l t e r n a t e spacing of t h e doughnut b a f f l e s with v i r t u a l l y no overlap.

These two regions, 4 and 5, c o n s i s t of b a f f l e

t h e r e i s a combination of p a r a l l e l and c r o s s flow. i n t h e i n n e r annulus is 15.856 f t ? .

windows i n which

The cross-flow a r e a

The p a r a l l e l - f l o w a r e a i n region 5

76 of t h e i n n e r annulus i s 4.529 f t 2 , and t h e p a r a l l e l - f l o w area i n region 4

of t h e inner annulus i s 4.559 ft?

The average flow areas i n region 5

and 4, sm5

m '4

= [4.529(15.856)]0*s

= 8.474 ft? and

= [4.559(15.856)]0'6

= 8.502 ft?

The mean v e l o c i t i e s of t h e coolant s a l t i n regions 5 and 4,

37*444 4.419 vm5=8.474=

-- 37*444 = 8.502

'm4

f t l s e c , and

4.404 f t / s e c

The pressure drop i n t h e window areas 5 and 4 o f t h e inner annulus, *o

(ia)

+ 0.6(10) 125(4*404)"

= 2

*o(m4

125(4.419)a 0e6(15) 2(32.2)(144) 2 (32.2) (144)

= 2.895 p s i and

= 2.091 p s i

To determine t h e t o t a l s h e l l - s i d e pressure drop i n both t h e o u t e r and i n n e r annuli, assume t h e pressure drop a t t h e entrance and t h e turnarounds t o be proportional t o AP

*en

& turn

o(oa)2

+ r

turn

/restrictions.

Thus,

(mi?1

2.689 + 3.029(0,373) 1.142

= 1.868 p s i

Assume a leakage f a c t o r of 0.52 f o r leakage between t h e tubes and b a f f l e s and between t h e b a f f l e s and t h e s h e l l .

Then t h e t o t a l s h e l l - s i d e p r e s s u r e

drop f o r both t h e o u t e r and i n n e r annuli, *o(

t o t a l ) = (ZAP05

= [2(2.895) = 34.42 p s i

2hpo4

lomo(3, 1)

2(2.091

lO(4.838)

lW02

=en

& t u r n)O .52

+ l l ( 0 . 3 7 3 ) + 2(1.868)]0.52

ci .

Heat-Transfer Calculations The heat t r a n s f e r c o e f f i c i e n t s i n s i d e t h e tubes, across t h e tube wall, and o u t s i d e t h e tubes were determined f o r both t h e o u t e r and i n n e r annuli of t h e Case-B primary heat exchanger.

The o v e r a l l heat t r a n s f e r

c o e f f i c i e n t s were then determined f o r both annuli, and these values were used with t h e mass-flow and h e a t - t r a n s f e r equations developed t o determine the required length of the tubes f o r the primary exchanger. From Eq. 1 i n Chapter 3, t h e h e a t

Heat Transfer Coefficients.

t r a n s f e r c o e f f i c i e n t i n s i d e t h e tubes i n t h e o u t e r annulus,

= 0.000065 -(N,)l hi(oa) di = 0.000065

*43

) O *4

di d 0

o.02542

= 1672 Btu/hr.ff?

*OF

The heat t r a n s f e r c o e f f i c i e n t i n s i d e t h e tubes i n t h e inner annulus, h

i( l a )

= 0 .007806(NRe)l

*43

= 0.007806 (4665)l = 1377 Btu/hr-f$ -OF

The heat t r a n s f e r c o e f f i c i e n t across the tube w a l l f o r both annuli, d k

% = f0 i , e

where T = thickness o f t h e tube w a l l and

d d = m

- di d

d. 0

hw =

0,305 (11.6) 0.375 In 0.305 0.375(0.0029167)

0.375

= 3602 B t u / h r * f p â&#x20AC;&#x2DC;OF

78 From Eq. 4 i n Chapter 3 where p i / b

= 1 f o r t h i s case, the heat

t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes, C GJ

= 0.168865

where f o r 800

NRe

5 105,

J = 0.346(NRe)d*382

f o r 100

NRe

-< 800,

J = 0.571(NRe)"

A leakage f a c t o r of 0.80 was assumed f o r leakage between t h e tubes and b a f f l e s and between the b a f f l e s and s h e l l .

Thus, t h e h e a t t r a n s f e r

c o e f f i c i e n t o u t s i d e t h e tubes, ho = 0.1350465

For t h e heat t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes i n t h e window

regions

3 and 1 of t h e o u t e r annulus, t h e mass v e l o c i t y of t h e coolant s a l t i n the shell.

1*685 lo7 lb'hr 4.386 ft?

G =

NRe

-- -Gdo - - (3.8418 p

= 3.8418 x

106 l b / f ? * h r

x 106)0.03125 =

and

J = 0.346(10005)4*382

= 0.01024

Therefore, h

o(oa),

= 0.13504(3.8418

x 106)(0.01024)

JI I

= 5312 Btu/hr*ft?*OF

I n the cross-flow region 2 of t h e o u t e r annulus, t h e mass v e l o c i t y of t h e coolant s a l t i n t h e s h e l l ,

6 = -

NRe

7.708 f 2.1860(0.03125) 12

J = 0.346(5693)-O

= 2.1860 x l@l b / f P 9hr

= 5693, and = 0.01272

u . f

. 79

-6,

The h e a t t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes i n region 2 of t h e outer annulus

, x 106) (0.01272)

= 0.13504(2.1860

ho ( 0 4 L r)

= 3755 Btu/hr*fI?*OF

To determine t h e heat t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes i n t h e inner annulus, t h e average of the mean flow a r e a s i n regions 5 and 4 of the inner annulus,

*m(ia) There fore,

G =

8.474 + 8.502 = 8.488 ft2 2

= 1.9852 x

NRe

106 l b / f ? * h r

(1.9852 x 106)0.03125 = 5170, and 12 E

0.346(5170)40.382 = 0.01322

The heat t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes i n t h e inner annulus,

= 0.13504(1.9852 x 106)(0.01322)

ho ( i a )

= 3544 Btu/hr.ft? * O F

Overall Heat Transfer C o e f f i c i e n t s .

The o v e r a l l heat t r a n s f e r

c o e f f i c i e n t i n t h e o u t e r annulus,

U (04

+ -1

1 h i + q

The average h e a t t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes i n the o u t e r annulus

n h

= Oav

(oa),

+ n

:o(oa)3 total n

h (04,

(04

3155(5312) + 639(3755) 3794

= 5050Btu/hr* f t ? * O F

o(oaI2

80 Therefore, U

(04

1 1672

1 1 +-+3602

1 = 931 Btu/hr'ft? *OF

5050

The o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t i n t h e i n n e r annulus, U

(ia) -

1 1377

1 = 778 Btu/hr-f?'OF

1 +-+3602

3544

Determination of Length o f Tubes The h e a t - t r a n s f e r and mass-flow equations developed i n t h e following m a t e r i a l were used t o c a l c u l a t e t h e required l e n g t h of t h e tubes i n t h e Case-B primary h e a t exchanger.

The heat t r a n s f e r rate i n t h e o u t e r

annulus,

where u(Oa)

= t h e o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t i n t h e o u t e r annulus,

Btu/hr= ft? *OF,

(4= tFX tCX

tF out t~ i n

h e a t flow a r e a i n o u t e r annulus, f ? ,

= temperature of t h e f u e l s a l t a t turnaround p o i n t i n exchanger, 0

= temperature of coolant s a l t a t turnaround point, = temperature of f u e l s a l t a t o u t l e t ,

= temperature o f coolant s a l t a t i n l e t ,

The h e a t t r a n s f e r r a t e i n t h e i n n e r annulus,

The t o t a l h e a t t r a n s f e r r a t e ,

F, 0

F, I

The heat t r a n s f e r r a t e i n t h e o u t e r annulus, Q(04

= (wcp)F(tpX

(A.4)

tF out) ,

and

where (WCp)F

= product of t h e flow r a t e , W, and t h e s p e c i f i c heat, CP’

(WCp)c

f o r the fuel s a l t , = product of the flow r a t e , W, and t h e s p e c i f i c h e a t , CP’ f o r t h e coolant s a l t ,

The h e a t flow a r e a of t h e o u t e r annulus, (A. 6)

A ( 0 4 = &A(,,) where Kl = a constant used f o r t h e s e equations only.

By d e f i n i t i o n ,

and

S u b s t i t u t i n g i n Eq. A . l ,

Q(oa) - U(oa)A(oa)Atm(oa)

(A. l a )

and s u b s t i t u t i n g Eq. A . 6 i n t o Eq. A . l a , (A. lb)

Similarly, (A.2a) Combining Eqs. A.lb and A.2a and e l i m i n a t i n g A(ia), Q(oa) (o a)1 ‘ At m (o a

Q(ia) ( ia f t rn (ia)

(A. 2b)

S u b s t i t u t i n g Eq. A.3 i n t o Eq. A.2b, Q(oa) (oa) hAtm(oa)

Qtotal

- Q(oa)

(ialAtm( i a )

From Eqs. A.4 and A.5, t FX - tF o u t

- ,%EL (WCp)F

and

Subtracting,

Defining a second constant used f o r t h e s e equations only,

Therefore,

- tCX

xeQ(oa)

- tC i n

tF out

-(A. 8 ) *

From previously s t a t e d d e f i n i t i o n s

S u b s t i t u t i n g Eq. A.8 i n t o Eq. A.9, Atm(ia)

Atm(oa)

- t

(% Q(oa) (kQ<oa)

+ t

F out + t t~ i n F out

C in

t~ i n

- tC out - t

t~ o u t

c in

or Atm(ial =

Qioa)

- t

% Q(oa)

Atm(oa)

c out

t~ i n +

+ t

out

(A. 10)

2tF out

S u b s t i t u t i n g Eq. A.10 i n t o Eq. A.7,

Equation A . l l w i l l be used t o determine t h e h e a t t r a n s f e r r a t e i n t h e o u t e r annulus, Q

S u b s t i t u t i n g i n t o Eq. A.6, (04

Kl =

A(oa) *(i a )

- 3793 tubes (16,125 f t ) - 4347 tubes (15.286 ft)

= o.92045

84 Substituting values i n t o the equation defining &,

Id, = (1.093 x 107)(0.55) = 2.1599 x 10-

1 (1.685 x 107)(0.41)

and i n t o

'(â&#x201A;Źa) U

Kl (04

778 = 0.90788 931(0.92045)

Substituting values i n Eq. A . l l ,

Qtotal + (1300

Q(oa) 1 + 0.90788

- 850 -

1.8046 x 109 + (1.56952 x 1O1O)

1 + 0.90788

1111)l

2.1599 x loa + 2(1000 850) Q(oa) 2.1599 x lo-*

+ 1000

+ (1.38895 x 1O1O)

By t r i a l and error, Q(oa)

= 8.9391

x 108 Btu/hr

From Eq. A.3, Q(ia)

- Q(oa) = 1.8046 x 1O9 - 8.9391 x 108

Qtotal

= 9.1069 x 108 Btu/hr

From Eq. A . 4 , t

Fx = 1149'F,

Atm( i a )

and from Eq. A.5, tCX = 979'F.

- 979) 2+ (1000 - 850)

(1149

and (1300

1111) + (1149 2

- 979)

Then,

= 160~F

= 179.5~F

The length o f the tubes i n the outer annulus,

8.9391 x 108

- 3794(931)n(0.03125) (160)

= 16.11 f t

Using t h e previously assumed r a t i o , '(ia)l'(oa)

= 14/15 and adding 0.236 f t

f o r tube bends, t h e length of t h e tubes i n t h e inner annulus,

L(ia)

] [g l? 15 ( o a )

= [+6.ll)]

+ 0.236 f

0.236 = 15.27 f t

9.1069 x 108 - 4347 (788)n(O .03125) (179.5)

15.28 f t

These c a l c u l a t e d lengths may be favarably compared with t h e l e n g t h s assumed = 15.286 f t . A f t e r studying = 16.125 f t and L f o r t h e tubes, L (ia) (04 t h e r e s u l t s of previous i t e r a t i o n s , t h e l e n g t h s e s t a b l i s h e d for t h e tubes i n t h e primary exchanger a r e L(Oa) i

L(ia)

i .

16.125 f t and

= 15.286 f t

S t r e s s Analysis f o r Case B

The f u e l s a l t e n t e r s

rimary h e a t exchanger i n the i n n e r annular 0

region a t a temperature of 1300 F and a p r e s s u r e o f 147 p s i , flows down through the 4347 bent tubes 15.05 f t , reaches t h e f l o a t i n g head a t a 0

temperature of 1149 F and a p r e s s u r e of 119 p s i , r e v e r s e s d i r e c t i o n and

flows upward through t h e 3794 s t r a i g h t tubes i n the o u t e r annulus 16.125 f t , and leaves t h e exchanger a t a temperature of 1000째F and a p r e s s u r e of

86 50 p s i .

"he coolant s a l t e n t e r s t h e exchanger through t h e annular v o l u t e

a t t h e top, e n t e r s t h e o u t e r annulus a t a temperature o f 850째F and a p r e s s u r e o f 194 p s i , flows downward t o t h e bottom tube sheet, r e v e r s e s d i r e c t i o n a t a temperature of 979'F

and a p r e s s u r e of 181 p s i , flows

upward through t h e i n n e r annulus and e x i t s i n t o t h e c e n t r a l p i p e a t a temperature o f l l l l 째 F and a p r e s s u r e of 161 p s i . of t h e o u t e r annulus of t h e exchanger is 66.7 in.,

The i n s i d e diameter t h e o u t s i d e diameter

of t h e i n n e r annulus i s 53.05 in., and t h e o u t s i d e diameter of t h e c e n t r a l pipe is 22 i n .

The stress a n a l y s i s f o r t h e Case-B primary h e a t exchanger c o n s i s t e d of a determination of t h e s t r e s s e s produced i n t h e tubes, t h e s h e l l s , and t h e tube sheets.

The shear s t r e s s theory of f a i l u r e was used as t h e

f a i l u r e c r i t e r i o n , and t h e s t r e s s e s were c l a s s i f i e d and l i m i t s of stress i n t e n s i t y were determined i n accordance with Section I11 of t h e ASME B o i l e r and Pressure Vessel Code.

The terms used i n t h e s e c a l c u l a t i o n s

a r e defined i n Appendix F.

S t r e s s e s i n Tubes The s t r e s s e s developed i n t h e tubes t h a t were i n v e s t i g a t e d a r e t h e

1. primary membrane s t r e s s e s caused by pressure, 2.

secondary s t r e s s e s caused by t h e temperature g r a d i e n t a c r o s s t h e tube w a l l ,

d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n of t h e tubes and tube sheets, and

secondary s t r e s s e s caused by t h e d i f f e r e n c e i n growth between t h e tubes and t h e s h e l l . Primary Membrane S t r e s s e s .

The primary membrane s t r e s s e s caused

by p r e s s u r e are t h e hoop stress and t h e l o n g i t u d i n a l s t r e s s .

The hoop

stress,

87 where

a = i n s i d e r a d i u s of tube = 0.1525 in., b = o u t s i d e r a d i u s o f tube = 0.1875 in., Pi = p r e s s u r e i n s i d e tubes, p s i , = p r e s s u r e o u t s i d e tubes, p s i ,

r = r a d i u s o f tube a t p o i n t under i n v e s t i g a t i o n , i n .

‘Ih

0.02326Pi

- 0.03516P

0.0119

= 1.954Pi

2.955P

To o b t a i n a maximum value, s e t ‘Ih

(Pi

- P )(0.000819) 0

O +

P (0.0119)

+ 0.0688 =

= 4.913Pi

= 0.0232.

- 5.913P0

The l o n g i t u d i n a l stress i n t h e tubes w a s assumed t o be caused by t h e p r e s s u r e drop i n t h e tubes.

The l o n g i t u d i n a l stress,

= 1.954AP

‘IL =

b2-a2 -

0.02326aP 0.0119

The tube-side and s h e l l - s i d e pressures, t h e p r e s s u r e drop i n the

tubes, and t h e c a l c u l a t e d hoop and l o n g i t u d i n a l stresses a t p e r t i n e n t l o c a t i o n s a r e t a b u l a t e d below. Tube- Side I

Shell- Side Pressure (psi)

Outer tubes Inlet Outlet Inner tub e s Inlet Outlet

P r e s s u r e Drop i n tubes (psi)

‘Ih

‘I

(psi)

119 50

181 184

69 69

-486 -842

135

147 119

161 181

28 28

-230 -486

55 55

135

Secondary S t r e s s e s Caused by Temperature Gradient.

The s t r e s s e s a t

t h e i n s i d e s u r f a c e of t h e tube t h a t a r e caused by t h e temperature g r a d i e n t a c r o s s t h e w a l l of t h e tube,l

and t h e s t r e s s e s a t t h e o u t s i d e s u r f a c e of t h e tube t h a t are caused by t h e temperature g r a d i e n t a c r o s s t h e w a l l of t h e tube, At"L

-- Atah

= kAt[l ' P - P I n ka ]

where

k =

= 2(1

o!E - 0.3)(0.207)

= 3.4m,

= t av tube o

Ri + Rw + Ro

(ti

- to)

The temperatures, r e s i s t a n c e s , and physical-property values a t t h e p e r t i n e n t l o c a t i o n s t h a t a r e required t o determine t h e secondary s t r e s s e s caused by t h e temperature g r a d i e n t are given i n Table A . l . Using t h e values given i n Table A . l ,

t h e s t r e s s e s c a l c u l a t e d f o r t h e i n s i d e and

o u t s i d e s u r f a c e s of t h e tubes a t t h e p e r t i n e n t l o c a t i o n s a r e given below.

Location Outer tubes Inlet Outlet Inner tubes Inlet Outlet

I n s i d e Surface

Outside Surface

At "h

At h'

AtaL

(psi)

At'L (psi)

(Psi)

688 682

-6660 -5851

+5721 +5027

+5721 +SO27

686 688

-6188 -5600

+5316

+5316 +4811

k -

A811

'J. F. Harvey, Pressure Vessel Design, D. Van Nostrand Company, New Jersey, 1963.

Table A . l .

Data Required t o Determine Secondary S t r e s s e s Caused by Temperature Gradient

Outer tubes - Inlet Outlet Inner tub e s Inlet Outlet

1149 1000

979 850

60 60

28 28

20 20

1033 897

44 39

7.52 7.27

26.5 27.2

1300 1149

1111 979

73 73

28 28

1173 1034

41 37

7.77 7.52

25.6 26.5

90 Discontinuity S t r e s s e s .

It w a s assumed t h a t t h e tube sheet i s very

r i g i d w i t h r e s p e c t t o t h e tube so that t h e r e i s no d e f l e c t i o n o r r o t a t i o n of t h e j o i n t a t t h e junction.

+, and

The d e f l e c t i o n produced by t h e moment,

t h e d e f l e c t i o n produced by t h e force,

d e f l e c t i o n produced by t h e p r e s s u r e load, must remain zero.

+, must

+, and

be equal t o t h e

t h e slope a t t h e junction

The d e f l e c t i o n caused by pressure,

and from pages 12, 126, and 127 i n Ref. 2, t h e d e f l e c t i o n s caused by t h e f o r c e and t h e moment,

4F=w and

where

El? = 12(1 3 )

' (0.17)'

(0 .035)'

= 16.66.

S e t t i n g t h e d e f l e c t i o n caused by p r e s s u r e equal t o t h e d e f l e c t i o n caused by M and F,

The slope produced by F must be counteracted by M;"

t h a t is,

F - ZAM=O. S u b s t i t u t i n g f o r D and solving t h e s e l a s t two equations f o r M and F,

" S . Timoshenko, Strength of Materials, P a r t 11, 3rd ed., D. Van Nostrand Company, New York, 1956.

From page 271 i n Ref. 3, t h e longi-ndinal stress,

and t h e hoop s t r e s s e s ,

2Pr = 9.714P , T = 2M h 2 r = Pr = 4.587P , T T

-- -2FT

Fh ' M h'

M'hsec

hr =

u( M(JL)

= 2.647P

The s t r e s s e s c a l c u l a t e d f o r t h e p e r t i n e n t l o c a t i o n s a r e tabulated below. The stress values l i s t e d f o r t h e l o n g i t u d i n a l s t r e s s component caused by t h e moment,

and t h e secondary hoop s t r e s s component caused by

t h e moment, Muhsec, a r e compressive on t h e inner s u r f a c e of t h e tube and t e n s i l e on t h e o u t e r surface of t h e tube.

Locat ion Outer tubes Inlet Outlet Inner tubes Inlet Outlet

Differential Pres sure (psi)

- 62

- 134

14 -62

Mh'

ML ' (psi)

M'hsec

Fh '

(psi)

547 1182

-284 -615

164 355

124 547

64 -284

- -602 1302 - 136 -602

Buckling of Tubes Caused by Pressure.

37 164

Where t h e e x t e r n a l p r e s s u r e

i s g r e a t e r than t h e i n t e r n a l pressure, t h e s t r e s s on t h e tube w a l l must, be checked t o a v e r t p o s s i b l e c o l l a p s e o f t h e tubes. -T= DO

0*035 2(0.1875)

0.0933

From Fig. UG31 i n Section V I 1 1 of t h e ASME B o i l e r and P r e s s u r e Vessel Code f o r T/Do = 0.0933,

the. minimum design s t r e s s required t o

match t h e design p r e s s u r e d i f f e r e n t i a l a t t h e p e r t i n e n t l o c a t i o n s and t h e maximum allowable design s t r e s s e s f o r Hastelloy N are t a b u l a t e d below.

3R. J. b a r k , Formulas H i l l , New York, 1954.

r S t r e s s e s and S t r a i n , 3rd. ed., McGraw

Location Inner tubes Inlet Outlet Outer tubes Inlet Outlet

avET

(psi)

Minimum Design S t r e s s

for m Hastelloy N (psi) S

14 -62

c1000 -1000

1173 1034

>14000

-62 134

-1000 -1400

1034 897

>14000

Li‘

-7000

>14000

Secondary S t r e s s e s Caused by Growth Difference.

Since t h e average temperature of t h e o u t e r tubes i s d i f f e r e n t than t h e average temperature o f t h e i n n e r tubes, t h e r e w i l l be d i f f e r e n t i a l growth stresses.

- €iLi-t

a 0 +0L0 ’

where

AL = u n r e s t r a i n e d thermal growth o f l e n g t h o f tubes,

E =

s t r a i n on tubes required by compatability Conditfon,

L = length of tubes, = i n n e r tubes, and i subo = o u t e r tubes.

sub

where

a = normal stress and

E = modulus of e l a s t i c i t y of tubes. There fore, *

uiLi = AL +E t o

“OL0

E ’

and

n iu i = n0 u0 where n = t h e number o f tubes.

n u =- i i

n0 ’

AL - - uiLi = AL +- ni‘iLfD i E t o n E

’

,-.

= tAL i

[ y ( + + 0L i )

ui n

-t

tA L0)

From t h e hot length of the tubes, the mean temperature of t h e tubes, and t h e c o e f f i c i e n t of thermal expansion f o r t h e tubes, these d a t a were obtained f o r t h e Case-B primary h e a t exchanger.

AT., = 0,1067 f t and

= 0.1189 f t .

(26 x 106)(0.1067 0.1189) 4347(16.13) 15.05 3794 +

= -9460 p s i

Since t h e sign i s negative, t h i s s t r e s s was assumed wrong and ai i s tension. !

- -4347 - 3794

(9460) = 10,838 p s i (compression).

Compressive buckling might occur r e a d i l y i f t h e b a f f l e spacing i s large, and t h e c r i t i c a l b a f f l e spacing,

n(26 x 106)(5.46 x lo*) 144( 10838) (0.0119)

X = 1.55 f t

= 2.4

This i s only s l i g h t l y l e s s than t h e b a f f l e schedule determined i n t h e h e a t - t r a n s f e r c a l c u l a t i o n s and s i n c e buckling i s of concern i n t h i s design, an a n a l y s i s of t h

tube concept was made.

f o r t h i s a n a l y s i s i s shown i n Fig. A.2.

fie diagram

94 ORNL DWG 67-7057

Fig. A . 2 . Diagram for Stress Analysis of Bent Tube Concept for Case-B Primary Heat Exchanger.

For t h i s a n a l y s i s , t h e energy caused by d i r e c t stress and shear w i l l be neglected.

Therefore,

Because of symmetry,

Pds 2EI

Mdx u = 2 [,,+2

(+d

Since Mo does no work,

For 0

<X<

For c

d <X <c +2'

d For c + 2 H3

<X<

= No = Mo

- F2 ( +~ R

= Mo

s i n p)

- PR(1 -

COS

R(l

+ d,

F - -(c 2

- R s i n 0) - P[2R(1 -

F - -(c 2

R s i n 0)

PR(1

cos cl)

cos 0

2 cos f3)

COS

â&#x201A;ŹI)]

F i s a dummy load used only t o determine t h e h o r i z o n t a l movement of p o i n t D.

Therefore, t h e terms involving F are s e t as being equal t o zero. cid

EI-=2 aMO

s % $ d x + 2 J 0

M s 0d s

- PR(l + a

COS

-2

COS

a ) ] (l)(R)(-de)

and

Therefore,

Modx + R J(Mo

+ PR cos f3)dp

+ R

po-

- PR

e +

COS

2PR

COS

a)d6 = 0

M0c +

RMoa -

a + M0RM p P a na + 2 ~ P a cos = 0

P P a + PR? s i n

- ~Psi -

2pPa c

2PPa(l cos c + 2w

2 P P a cos

+ 2m

The deflection of point A i n the direction of P,

- -aU - aP

There fore,

a u = 2 C/ g % d ~ + 2 1 c+dM g d s . Emp = E1

ap 0

0 !

Since M

has been determined, for 0

K =

for c

2PPa 2 p P a cos a! c + 2Ra

and

< X < c,

-c +2Rza 2Ra

Ma!

COS

. 3

d <X <c +2'

b =

2PPa c

- 2PPCx cos a - PR + PR + 2Ra

COS

or 2PPa

COS

+ PRc

& 3 =

COS

- PRc - 2 P P a

and

% =2Pa

COS

c + 2Ra

+ RC

COS

c + 2Ra

2R?a

-c-

COS Cy

. Y

- < X < c + d,

d and for c + 2

Thus, the expression for E16P'

(2mC O S a

(c + 2wl2

COS

2 ~ CaO S

After expansion and

pRp + 2Ra)Z

'I(e@

-k

8@c) cos 2 a

12ac

(16C?c

+ (16e +

24$c

+ -3)R3

sin

2a + (ac8c)cos2a

cos a

I)$-

10c2

€02de.

. 98 . -.

The d i f f e r e n t i a l tube growth,

â&#x20AC;&#x2DC;P = t& -o A I t4 i = 0.1189

0.1067 = 0.0122 f t o r 0.1464 i n .

Bending t h e inner tubes and solving f o r the maximum moment p r e s e n t i n these tubes,

% = Setting 0 =

2ppa! c

-+2pRza! 2Ra

â&#x20AC;&#x2DC;Os

- pR - PR cos 0

oO,

& = -2PR(1

2PR cos 01

- cos a!)(:

The maximum f l e x u r e s t r e s s , M

uflex

---z

-2PR(1

- cos a)[: &]

and

Then,

Assume t h a t t h e maximum o f f s e t i s 8 i n . 2R(1 Assume t h a t a! = 30

0) = 8

COS

= 0.5236 radians.

2R(1 R=--

When R = 30 in.,

- 0.866)

0.134

= 8

29.2 i n

t h e o f f s e t = 2(30)(0.134)

= 8.05 i n .

d = 2R s i n a! = 2(30)(0.5)

Since 2(d

c) = 72 in.,

i n t h e inner annulus,

= 30 in.

t h e value c = 5 in. was used.

-E = 26 x

For t h e tubes

1@ p s i

-I = n(rt - 'i")

n[ ( 0 ~ 8 7 5 ) ~( 0 ~ 5 2 51 )~

0.78539(0.0012359

5.458 x 10"

- 0.0005'408)

in.4

Solving f o r P,

+ 2(30)(0.5236)?

P = ( 26 x 106)(5.45 x lO4)(-0.1464)r5

a, 4 1

(30I4Cf(R,

- -3*3963 = -0.0459 - [73.97]

= (0.3675)

l b (tension)

20.708 36.416

= 0.20898 i n , - l b .

The f l e x u r e s t r e s s ,

- -Mc'flex

(0.20898) (0.1875)

- 5.458

lod

? .

+ 71.75

psi,

and t h e s t r e s s i n s i d e t h e tubes,

+58.36 p s i

The ' d i r e c t s t r e s s ,

'direct

'ii =

0.0459

rr(O.0119)

= +1.23 p s i

which i s n e g l i g i b l e . Secondary S t r e s s Caused by Pressure.

The l o n g i t u d i n a l load on t h e

tubes caused by t h e p r e s s u r e d i f f e r e n t i a l on t h e bottom f l o a t i n g head,

F = (191.- m j ( a = 72[(number of

head

- annulus a r e a + area of

(enclosed a r e a of tubes) 3

= 72[(3794 + 4347) (0.11045)'] = 64,740 l b (compressive)

There fore, AP'L

= 7 2 p s i (compressive) on

a l l t h e tubes.

tubes)

100

Stresses i n Shell The stresses produced i n t h e shells a r e 1. primary membrane s t r e s s e s caused by p r e s s u r e and 2.

d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n of t h e s h e l l and tube sheet. Primary Membrane S t r e s s e s .

t h e exchanger i s 1 i n . thick.

It was assumed t h a t t h e o u t e r s h e l l of

The hoop s t r e s s component,

and t h e l o n g i t u d i n a l s t r e s s component,

= APD = 3235 p s i

P'L

For t h e i n s i d e s h e l l s e p a r a t i n g t h e i n n e r annulus and t h e c e n t r a l tube by which t h e coolant s a l t leaves t h e exchanger, t h e hoop stress component,

and t h e l o n g i t u d i n a l stress component,

= 4T OPD = 3762 p s i ,

fL For t h e 1-in.-thick the

s h e l l s e p a r a t i n g t h e o u t e r and i n n e r annuli,

hoop s t r e s s component, Pa,

- - APD = - 2T

(-194

+ 161)(52.5) 2(1)

= -866 p s i

and t h e l o n g i t u d i n a l stress component,

= 433 p s i

For t h a t p o r t i o n of t h e s e p a r a t o r s h e l l above t h e head of t h e i n n e r annulus,

the hoop stress component,

and t h e l o n g i t u d i n a l stress component, PUL

= 2546 p s i

. .

101 Discontinuity S t r e s s e s i n ShelL-at Junction of S h e l l and Tube Sheet. It was assumed t h a t t h e tube s h e e t i s very r i g i d with r e s p e c t t o t h e

shell.

Therefore, t h e d e f l e c t i o n and r o t a t i o n of the s h e l l a t the j o i n t

a r e zero.

The r e a c t i o n s on the s h e l l from t h e tube sheet a r e t h e moment

and the force, M

and

F =

P 5;

where

Therefore, M =

194 2 (0.0484)

= 2003 i n . - l b / i n .

and F=-=

0.22

882 l b / i n .

For t h e o u t e r s h e l l of t h e exchanger, t h e secondary l o n g i t u d i n a l s t r e s s e s ,

MaL =

6M 7

6 0= 1

12024 p s i ( t e n s i l e i n s i d e , compressive o u t s i d e )

and t h e hoop s t r e s s e s , M'hsec

M h' Fh'

u ( M(IE) = 0.3(12024) = 3607 p s i ( t e n s i l e i n s i d e ,

compressive outside)

--= 2m2

- 2FXr -- T

2(2003)(000484) = 194 p s i (uniform t e n s i l e ) 1

2(882)(o*22)(66*7) = 12939 p s i (uniform compressive) 2(1)

The i n n e r s h e l l betwee ' s t r i c t l y a tube t h a t w i l l b

e i n n e r annulus and t h e c e n t r a l tube i s f e c t e d g r e a t l y by t h e method of anchorage

and t h e f i n a l design of t h e exchanger. continuity stresses f o r ' t h e i

Therefore, t h e secondary d i s -

r s h e l l o r tube were not considered i n

t h i s analysis. For t h e upper p o r t i o n of t h e s e p a r a t o r s h e l l above t h e top of t h e i n n e r annulus head, the constant,

102

M = - -P

2ha

194 = 1539.6 in.-lb/in. 2(0.0630)

and

194 0.251

772.96 l b / i n .

Jj'=-=-=

The secondary l o n g i t u d i n a l stresses,

6M MbL=F=

61O= 9237.8

p s i (compressive inside, t e n s i l e outside)

and the hoop s t r e s s e s ,

u ) = 2771.3 p s i (compressive inside,

= u(

Mbh sec flh=

t e n s i l e outside)

2(1539b6)(0*063) 1

194 p s i (uniform compressive)

Stresses i n Tube Sheets

The tube s h e e t s included i n t h i s a n a l y s i s of the Case-B primary heat exchanger are t h e top sheet f o r t h e o u t e r annulus, t h e top sheet f o r t h e inner annulus, and t h e bottom sheet across both o u t e r and inner annuli. Outer Annulus Top Tube Sheet.

The a c t i n g pressure, AP = 194 p s i on

t h e top tube sheet of t h e o u t e r annulus, and t h e a r e a of a c t i o n f o r t h i s p r e s sure,

-A = A - Ai 0

= 3500

- 2165 - 3794(0.11045)

= 915.55 in." ,The r e s u l t i n g force,

F = APA = 194(915.55)

= 177,617 l b

103 and the uniformly distributed effective pressure,

AP =

= 133 psi

17761711334.8

The ratio of the average ligament stress in the tube sheets, which are strengthened and stiffened by the rolled-in tubes, to the stress in flat unperforated plates of like dimensions equals the inverse of the ligament efficiency if the inside diameter of the tubes is used as the diameter of the penetrations (page 106 in Ref. 1); that is,

where @ = the inverse of the ligament efficiency,

= average ligament stress in tube sheet,

(J' (I

= stress in unperforated plate,

p = tube.pitch, di =.inside diameter of tubes. The inverse of the ligament efficiency,

0.625 - 0.305) ---0.370 -- 1.95 .

0.625 = (0.625

The value of 2.0 was used for @:

To determine the thickness required for

the outer annulus upper tube sheet,

From Case 76 in Ref. 3,

for DoafDia = 66.7/52.5

2 1-27, and at

a temperature of 1000 F, the allowable stress, S = 17,000 psi.

.P,.=

2(0.03)

(133) (66.7)a 17000

35,501 - 2.08 E .

The thickness used, T = 1.5 in.

17,0OOv-

Therefore

104 Inner Annulus Top Tube Sheet.

, -

The a c t i n g pressure, AP = 161

- 147

= 14 psi, and t h e area of a c t i o n f o r t h i s pressure,

-A = A, - Ai - + = 2165 - 397.6 - 4347(0.11045) 1287.3 in.'

The r e s u l t a n t force,

F = APA = 18,022 l b

and t h e uniformly d i s t r i b u t e d e f f e c t i v e pressure,

A P = 18022

1767.4

= 10.2 p s i

The inverse of t h e ligament e f f i c i e n c y 0.674 0.305)

= (0.674

1.826 (use 2.0)

= 52.5122 = 2.386, p 2 0.2438; and a t a temperature At DialDcenter tube of 1300째F, t h e allowable stress, S = 3500 p s i . Therefore, t h e required

thickness of t h e i n n e r annulus top tube sheet,

2 (0.2438) (10.2) (52.2)= = 3.92 i n . 3500

The thickness used, T = 2.5 i n . I

Bottom Tube Sheet.

AP = 181

The a c t i n g p r e s s u r e on t h e bottom tube sheet,

119 = 62 p s i , and t h e load caused by t h e tubes = 0 .

The

e f f e c t i v e pressure,

= 44 p s i

=E 66.7122 = 3.03, $ = 0.306; and a t t h e o p e r a t i n g For 'oaIDcenter tube temperature, t h e allowable stress, S = 10,000 psi.. Therefore, t h e

thickness required f o r the bottom tube sheet,

105

2(0.306)(44)(66.7)2

10,000

= 11.88 in.

The thidcness used, T = 3.5 i n . F u r t h e r a n a l y s i s of t h e design f o r t h e Case-B primary h e a t exchanger t h a t includes s t r e s s concentration, non-uniform loading, edge r e s t r a i n t s , and thermal s t r e s s e s w i l l be required when t h e design is f i n a l i z e d and cyclical considerations a r e investigated.

Summary of Calculated S t r e s s e s Calculated S t r e s s e s i n Tubes,

The values c a l c u l a t e d f o r t h e s t r e s s e s

t h a t a r e uniform a c r o s s t h e tube w a l l and t h e i r l o c a t i o n s a r e given i n Table A.2.

The c a l c u l a t e d s t r e s s e s on t h e i n s i d e and o u t s i d e s u r f a c e s

o f t h e tubes a r e given i n Table A.3, and t h e c a l c u l a t e d stress i n t e n s i t i e s f o r t h e tubes are compared with t h e allowable s t r e s s i n t e n s i t i e s i n Table A.4. .2.

Calculated S t r e s s e s Uniform Across Tube Wall M h'

P"h (psi)

(psi)

-284

55 55 55

-230 -358 -486

-72 -72 -72

-602

-284 -615

135 135

-486 -842

-72 -72

-602 -1302

(psi)

I n n e r t?bes Inlet Mid-height Outlet O u t e r tubes

-64

Inlet Outlet

P"L (psi)

Locat ;Con

0P"L

Fh ' (psi)

- 136

. 106

Table A.3.

Calculated S t r e s s e s on I n s i d e and Outside Surfaces of Tubes

Locat i o n I n n e r tubes i n l e t Inside surface Outside s u r f a c e Inner tubes mid-height Inside surface Outside s u r f a c e Inner t u b e s o u t l e t Inside surface Outside s u r f a c e Outer tubes i n l e t Inside surface Outside s u r f a c e Outer tubes o u t l e t Inside surface Out s i d e s u r f ace

MaL (psi)

124 +124

M'hsec (Psi) -37 +37

At'L (psi)

At'h (Psi)

-6188 +5316

-6188

-5894 +5064

-5894 +SO64

-5600 +4811

(Psi)

+5316

547 +547

164 +164

-547 +547

164 +164

-6660 +5721

-1182 +1182

-355 +355

-5851 +5027

-5851 +SO27

Lk'L

+58 --+72

Table A.4. Calculated S t r e s s I n t e n s i t i e s €or Tubes of Case-B Primary Exchanger Compared With Allowable S t r e s s I n t e n s i t i e s Primary t

Locat ion I n n e r tubes i n l e t I n s i d e surface Outside s u r f a c e I n n e r tubes mid-hei .ght Inside surface Outside s u r f a c e I n n e r tubes o u t l e t I n s i d e surface Outside s u r f a c e Outer tubes i n l e t I n s i d e surface Outside s u r f a c e Outer tubes o u t l e t Inside surface Outside s u r f a c e

(OF)

‘Oh (psi)

1203 1157

cuL

“r

(psi)

m ‘ (psi)

‘Oh (psi)

(psi)

“r

Pm+Q+F (psi)

3sm (psi)

-161

6504 5584

17,500 24,150

(psi)

- 147

285 285

5850 8050

-6655 +4923

-6329 +5423

-171

414 414

9850 14000

-6252 +4706

-5957 +5 104

-133 -171

6119 4933

29,550 42,000

55 55

-119 181

541 541

15050 16400

-7136 +3603

-6164 +5341

-119 181

7017 5522

45,150 49,200

-486 -486

135 135

62 1 621

15050 16400

-8032 +4513

-7144 +6331

-119 -181

7913 6521

45,150 49,200

-842 -842

135 135

--113 184 --19850

977 977

17650 18100

-8965 +2623

-6970 +6272

--51 194

8914 6466

52,950 54,300

-230 230

55 55

1088

--358 359

55 55

1062 1020

-486 -486

1065 1016 924 880

1133

Secondary and C y c l i c P m

147 -161

- 133

108

Calculated Stresses i n S h e l l s .

The stresses calculated for the

s h e l l s o f the exchanger are given i n Table A . 5 , and the calculated

stress i n t e n s i t i e s for the s h e l l s are compared with the allowable s t r e s s i n t e n s i t i e s i n Table A . 6 .

Table A.5. ~~~

Calculated Stress for Shells of Case-B Primary Exchanger ~

P'L Locat ion Top outer shell Inside surface Outside surface Top separator s h e l l Inside surface Outside surface Mid-height inner s h e l l Inside surface Outside surface

P'h

Par

F'h

@hS,C

(psi)

+3235 +3235

+6470 +6470

-194 -0

-12939 -12939

+194 +194

+12024 -12024

+3607 -3607

+2546 +2546

+5092 +SO92

-194

+lo186 +lo186

-194 -194

-9238 +9238

-2771 +2771

+3762 +3762

+7524 +7524

-166 -173

-0

Table A.6. Calculated Stress I n t e n s i t i e s for S h e l l s o f Case-B Primary Exchanger Compared With Allowable Stress I n t e n s i t i e s Location Top outer s h e l l Inside surface Outside surface Top separator s h e l l Inside surface Outside surface Mid-height inner s h e l l Inside surface Outside surface

(Fin

111

(psi)

850 850

6664 6470

18,750 18,750

-2668 -9945

+15259 -8789

-194

850 850

5286 5092

18,750 18,750

+12313 +17855

-6692 +11784

-194

1111 1111

7690 7697

12,000 12,000

(Psi)

(psi) .

-0

(psi)

17,927 9,945

56,250 56,250

19,005 17,855

56,250 56,250

110

GCalculated Stresses i n Tube Sheets.

The tube sheet thicknesses,

maximum calculated stresses, and allowable s t r e s s e s are tabulated below.

Tube Sheet Top outer annulus Top inner annulus Lower

Thickness (in.) 1.5 2.5 3.5

Maximum Calculated Stress (psi)

<17,000 <3,500 <10,000

Allowable Stress

0 17,000 3,500 10,000

111

Appendix B CALCULATIONS FOR BLANKET-SALT HEAT EXCHANGER

Heat-Transfer and Pressure-Drop Calculations for Case B

Each of the four modules in the heat-exchange system has one blanketsalt exchanger to transfer heat from the blanket-salt system to the coolantsalt system. The heat load per unit, Q is 9.471 x lo7 Btu/hr. t'

The conditions stipulated for the tube-side blanket salt were an inlet ,

temperature of 1250째F, an outlet temperature of 1150째F, a temperature drop of 100째F, and a maximum pressure drop across the exchanger of 90 psig. Other criteria given for the blanket salt were the flow rate through the core tubes, W tc = 4.3 x l@ lb/hr and the total velocity, Vt = 10.5 ft/sec. Certain properties of the blanket salt vary almost linearly with temperature and pressure.

Therefore, average conditions may be used without

serious error. These average conditions for the blanket salt were

1. 2. 3. 4.

density, p = 277 lb/fP ; viscosity,

p =

38 lb/hr*ft;

thermal conductivity, k = 1.5 Btu/hr*ft?*OF per ft; and specific heat, C = 0.22 Btu/lb*'F. P The conditions stipulated for the shell-side coolant salt were an

inlet temperature of llll째F, an outlet temperature of 1125'F,

a tempera-

ture increase of 14OF, and a maximum pressure drop across the exchanger of 20 psig. The mass flo ate of the cold fluid, Wcf, was given as 1.685 x lo7 lb/hr.

The

age conditions given for the coolant salt

were

1. 2.

125 lb/fP, = 12 lb/hr*ft,

p = p

k = 1.3 Btu/hr*ft?-OF per ft, and 4. C = 0.41 Btu/lb*'F. P The material to be used for the tubing was specified as Hastelloy N, 3.

and the tubes were to have an outside diameter of 0.375 in. and a wall thickness of 0.035 in. A triangular pitch was chosen and set at 0.8125 in.

112 The b a f f l e cut, Fw, o r t h e r a t i o o f open a r e a o f t h e b a f f l e t o t h e c r o s s s e c t i o n a l area o f t h e s h e l l required f o r t h e tubes was e s t a b l i s h e d as 0.45.

Geometry of Exchanger The reverse-flow exchanger h a s a 22-in.-OD

pipe i n t h e c e n t e r through

which coolant s a l t from t h e primary f u e l - s a l t - t o - c o o l a n t - s a l t enters the shell. t h e tubes.

exchanger

This c e n t r a l pipe i s surrounded by an annular a r e a f o r

Two tube passes with an equal number of tubes i n each pass are

used a g a i n s t a single-pass s h e l l w i t h d i s k and doughnut b a f f l e s .

The

geometry of t h i s arrangement was c a l c u l a t e d as follows, and t h e terms used i n t h e equations are defined i n Appendix F. The number o f tubes i n o u t e r pass, n a’ = number of tubes i n i n n e r pass, n C W

4.3 x 106(144) = 810 (10.5) (0.7854) (0.305p (277) (3600)

810 + 810)(0.866)(0.8125)a The a r e a required f o r tubes = ( 144

= 6.43

fp.

The a r e a o f t h e c e n t e r pipe ~ ( 0 . 7 8 5 4 ) ( 1 . 8 3 3 ) ~= 2.64 f p . The t o t a l cross-sectional a r e a = 6.43 + 2.64 = 9.07 f p . The diameter o f the s h e l l ,

(

0.7854

)”

= (11.548)l/”

= 3.398 f t o r 40.776 i n .

The diameter o f t h e doughnut hole, 2.64

Dd =

[

+ (0.45)6.43]1/a 0.7854

= 2.654 f t o r 31.852 i n .

The diameter of t h e disk, [9.07 DOD

(0.45)6.43]1/a 0.7854

= 2.804 f t o r 33.653 in.

bd’ L

113 Heat Transfer C o e f f i c i e n t I n s i d e Tubes The Reynolds number w a s c a l c u l a t e d from given data,

NRe Using E q .

(0.305)(10.5) (277) (3600) = 7ooo (12) 38

diVTPi =-=

discussed i n Chapter 3, t h e h e a t t r a n s f e r c o e f f i c i e n t i n s i d e

t h e tubes, h

= 0.000065

k di

-43 (

(~,)1

= 0.000065( '~.~0:2)(7000)

~ ~ .4~ 1 0 1.43

(

0.221.5x 38

)

= 0.00384(314,000)1.99 =

2400 Btu/hr* f$

The thermal r e s i s t a n c e i n s i d e t h e tubes,

0.375 Ri = (2400)(0.305) = 5.12 x

lod

Thermal Resistance of Tube Wall The thermal conductivity of Hastelloy N tubing i n t h e temperature 0

range of c a l c u l a t i o n i s 11.6 B t u / h r * f $ * F p e r f t .

Therefore, t h e thermal

r e s i s t a n c e of t h e tube w a l l , d

do I n

%= -

(0.375) I n

0 375 0-305= 2.8

Z(12) 11.6

10-4

Shell-Side Heat Transfer C o e f f i c i e n t and P r e s s u r e Drop e number of b a f f l e s determines t h e , h e a t t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes, ho, and t h e s h e l l - s i d e

p r e s s u r e drop, AP.

Using E q s . 3 through 11 discussed i n Chapter 3,

114

Wwhich were developed from the work of Bergelin et al,l,"

to determine

the shell-side coefficient in cross-flow exchangers, the outside film resistance was calculated for various assumed baffle spacings.

A curve

of outside film resistance versus baffle spacing was then plotted from The average cross-sectional flow area,

the data developed.

s, per

foot of baffle spacing, X, was first determined for rows of tubes in the cross-flow section.

= 5 c

(2.654

2.804)[,

- (0.9)0.375 (0 .8 125) 3

= (3.1416) (2.729)(0.487)

= 4,175 f$/ft

The cross-sectional flow area in the window section was then determined.

1620) [0.866(0.8125)a A" - 0*45 1 L = 2.335 f P

- 0.7854(0.375)a

The Prandtl number was calculated from given data,

>"I3=

1.3

= 2.43

To calculate the outside film resistance, Ro, various baffle spacings, X, were assumed.

For X = 0.74 ft,

the average cross-sectional flow area, ,

AB = 4.175(0.74)

= 3.09 ft?;

the mass velocity of the cross-flow section, GB

- 1*685 lo7 3.09

= 5.45 x 106 lb/hr*f$;

the mass velocity of the window section,

- 1e685 Gw

2.335

lo7 = 7.22 x 106 lb/hr-f?;

' 0 . P. Bergelin, G. A. Brown, and A. P. Colburn, "Heat Transfer and Fluid Friction During Flow Across Banks of Tubes -V: A Study of a Cylindrical Baffled Exchanger Without Internal Leakage," Trans. ASME, 76: 841850 (1954).

"0. P. Bergelin, K. J. Bell, aod M. D. Leighton, "Heat Transfer and Fluid Priction During Flow Across Banks of Tubes -VI: The Effect of Internal Leakages Within Segmentally Baffled Exchangers," Trans. ASME, 80: 53-60 (1958).

115 t h e mean mass v e l o c i t y , Gm

= 6.275 x

106 l b / h r * f t ? ;

the Reynolds number f o r the cross-flow section,

106

0.375)5.45 x (NRe)B = ( 12 (12)

= 14,200;

t h e Reynolds number f o r the window section,

(NRe)w

(0.375)6.275 1 2 (12)

106

= 16,350;

t h e heat t r a n s f e r f a c t o r , J = 0.346/(NRe)0

*383 ;

t h e heat t r a n f e r f a c t o r f o r t h e cross-flow section, JB = 0.00895;

t h e heat t r a n s f e r f a c t o r f o r the window section, Jw = 0.0085;

t h e heat t r a n s f e r c o e f f i c i e n t f o r the cross-flow section, hB = J

= 0.00895 (0.41)5.45

B (Npr>

x 106

2.43

= 8230 Btu/hr' f I? 'OF;

t h e h e a t t r a n s f e r c o e f f i c i e n t f o r t h e window section,

t h e h e a t t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes, h 0 = [hB(l

2Fw) + hw(2Fw)IBh,

where 3 - leakage c o r r e c t i o n f a c t o $ i 2 because t h e h e a t h t r a n s f e r equations a r e based on no leakage = 0.8, = [8230(0.1)

+ 9000(0.9)]0.8

= 7140 Btu/hr* fI? -OF; and

t h e thermal r e s i s t a n c e of t h e o u t s i d e f i l m , R

= 1.40 x

10-4.

When X = 1.48 f t , AB

= 4.175(1.48)

= 6.18

ft?,

116 GB =

1.685 lo7 = 2.73 x 106 lb/hr*f$, 6.18

Gw = 7.22 x l@

lb/hr*f$,

Gm = 4.44 x 106 lb/hr*f$, 0.37512.73 x 106 = 7100, (N~e)B = ( 12(12) (0.375)4.44 x 106 = 11,500, (NRe)w 12 (12) JB = 0.0117, Jw = 0.00972,

'06 = 5390 Btu/hr-f$.OF,

hB = 0.0117 (0*41)2*73 2.43

= [0.1(5390)

+ 0.9(7280)]0.8

= 5670 Btu/hr=ft?*OF, and

R, = 1.76 x 104. When X = 2.22 ft, = 4.175(2.22)

GB =

= 9.27 f?,

1*685 9.27 lo7 = 1.82 x 106 lb/hr*f$,

Gw = 7.22 x 106 lb/hr*f$, Gm = 3.62 x 106 lb/hr*fP,

- 0.37511.82 - ( 12(12)

x 106 = 4740,

0.375)3.62 (NRe)w = ( 12(12) JB = 0.0136,

x 106 = 9430,

(NRe)B

Jw = 0.0105, hB = 0.0136 (0*41)1'82 2.43

hW = 0.0105 (0*41)3*62 2.43 h

[0.1(4180)

+ 0.9(6410)]0.8

Ro = 2.02 x lo4.

= 4180 Btu/hr*f$ *OF, = 6410 Btu/hr*f$*OF, = 4950 Btu/hr*f$*'F,

and

6 I

117 When X = 2.96 ft, AB = 4.175(2.96) GB GW

fP,

= 12.36 =

12.36

1.365 x 106 lb/hr*ft?,

7.22 x 106 lb/hr*ft?,

= 3.14 x 106 lb/hr*ft?,

(NRe) B

- j0.375)1.365 x 106 12(12)

= 3550,

0.375)3.14 x 106 = 8180, (NRe)w = ( 12(12) JB = 0.01525, Jw = 0.0111, hB = 0,01525 (0*41)1*365 2.43

h = [0.1(3510) 0

IO6 = 3510 Btu/hr'ft?*OF,

+ 0.9(5890)]0.8

= 4520 Btu/hr'ft?=OF, and

R = 2.21 x 10-4. 0

These calculated values of the thermal resistance of the outside film were then plotted as a function of the baffle spacing, and the resulting curve is shown in Fig. B.l. From the curve in Fig. B.l, it can be quickly determined whether the baffle spacing is limited by thermal stress in the tube wall or by the allowable shell-side pressure drop. Because of thermal stress, the With this informaximum temperature drop'across the tube wall is 46'F. mation and other given data, Ro can be calculated.

Ro = Rt

- Ri - % = 0.00085 - 0.000512 - 0.00028

By extrapolati

h-l 2

s on the curve in Fig. B.l for Ro = 0.00006,

the baffle spacing limited by the temperature drop across the tube wall is approximately 3 in.

A rough approximation of the pressure drop based

ti22

2.c

1.8

1.6

1.4

r 0

1.2

I I

I I I

1.0

I I

I I I I I I

0.6

I 0.4

1 I

BAFFLE SPACING ( f t )

Fig. B.l.

Outside Film Resistance Versus Baffle Spacing.

on this 3-in. spacing is in excess of the desired maximum of 20 psi. Therefore, the baffle spacing will be limited by the pressure drop, and it will exceed 3 in. With the limiting factor for the baffle spacing established, some mathematical relationships between the baffle spacing, the outside film resistance, and the shell-side pressure drop were established. These were based primarily on the methods proposed by Bergelin etâ&#x20AC;&#x2122;a 1 . l ~ ~for

119 determining pressure drop in cross-flow exchangers.

The total heat

transfer rate,

[ APs =

N[2 + 0.6(rw)]

(2) a

(N + 1) (rB) (0.6) 144 x 64.4

144 x 64.4

L = X(N + 1)

(=)I? Gxm

From Eq. B.l,

Q, Various factors of this equation can be evaluated from known data, AtLm =

(thi

tco)

tci) = (1250

1125) (1150 125

1111)

In 39

= 73,9OF

A correction factor must be applied to a one-shell-pass two-tubepass exchanger to account for the deviation from a strictly counterflow situation.

This correction factor is represented by g a m a in Eq. B.l.

To determine gamma, the capacity and effectiveness ratios most be calculated first.

The capacity ratio,

- tho - 1250 - 1150 100 1125 - 1111 - 14 co

a = thi t

--=

tci

7s15

The effectiveness ratio, tco

B = thi From data published pass exchanger. U =

The

0.1

hapmana 7 = 0.95 for a one eat transfer coeffici

R +Ri+% 0

tci - - 14 = 139 tci

hell-pass two-tubet , u, of Eq. B-1,

1 (5.12 + 2.8)104

3 A . J. Chapman, Fig.12.9 in Chapter 12, "Heat Transfer by Combined Conduction and Convection," Heat Transfer, MacMillan, New York, 1960.

120

Substituting the values for AtLm, 7 , and U in Eq. B.l, L Ro + (7.92 x 1 0 9

9.471 x lo7 x 12 - (3.1416)(0.375)(0.95)(73.9)(1620) = 0.85 x loQ

L = 0.85 x laQ(R ) 0

+ 6.74

(B. la)

Values of r

and rw were then determined for substitution into B The average number of restrictions to flow in the cross-flow

Eq. B . 2 . area,

DOD

- Dd

‘B = 1.866 p

- 12.804 2.654)(12) - (1.866) (0.8 125)

= 1.19

The average number of restrictions to flow in the window area, ( ’ s

2(1.866

= 5.6

(Dd p)

- DCP’

- 2.804)

+ (2.654 (3.732) (0.8125)

lZC(3.398

- 1.833)1

Because the equation for pressure drop is based on no leakage, Bp in From the work reported by

Eq. B.2 is a leakage correction factor. Bergelin et al., its value is 0.52.

Combining Eqs. B.l and B.2 and

substituting the values for rB, rw, and BPI L (1.19)0.6 AP =[(d(144)64.4(=/

i3>d ‘1

1” (% - ’)2 (144)64.4 + (0.6)(5.6) GB + [$ - 1)(2.40 io4)(&) +

= LE (0.32

lo*)(=)

0.52

125

(B.2a)

Equations B.la and B.2a and the curve in Fig. B.l were then used to det m i n e a baffle spacing at which the shell-side pressure drop was within the maximum allowable of 20 psi.

An even number of baffles was

desired for smooth flow through the shell with a minimum number of stagnant areas.

By approximation, four baffles should give a pressure

drop within the allowable.

Assuming four baffles, the baffle spacing

was chosen so that the ratio L/X = 5. from Fig. €5.1, Ro = 1.82 x lo4. L = (0.85 x 1@)(1.82

For baffle spacing, X = 1.65 ft

Substituting in Eq. B.la, x

lo4)

+ 6.74 = 8.287 ft.

121 Therefore;

For s u b s t i t u t i o n i n Eq. 3.2a, values of GB and Gm were c a l c u l a t e d . AB

= ( 4 . 1 7 5 ) ( 1 . 6 5 ) = 6.89 f t ? ,

GB GW

= 7.22 x

- 1 * 6 8 5 lo7= 6.89

2.446 x 106 l b / h r * f ?

lb/hr-ft?,

G = (G G m B w

= [(2.446 x 106)(7.22 x

= 4.2 x 1 8 l b / h r * f t ?

lo")]'/'

There fore,

+ 13.07

= 0.74

= 13.81 p s i

I f we add f i v e cross-flow areas t o account f o r t h e cross-flow areas a t t h e entrance and e x i t of t h e s h e l l ,

AP = lO(0.32 x 10'6)(4.62 x = 1.48

+ 13.07

lo5) +

= 14.55 p s i

4 ( 2 . 4 0 x 1 0 4 ) ( 1 . 3 6 1 x 106)

Tube-Side Pressure Drop The p r e s s u r e drop of t h e blanket s a l t through t h e tubes w a s c a l c u l a t e d by using t h e Darcy e q ~ a t i o n . ~The Reynolds number was c a l c u l a t e d t o d e t e r mine t h e f r i c t i o n f a c t o r , f .

NRe =

divTp P

( 0 . 3 0 5 ) ( 1 0 . 5 ) ( 2 7 7 ) (3600) = 7ooo (12) ( 3 8 )

f = 0.035

&J. H. Perry, Ed., New York, 1950. .

Chemical Engineer's Handbook, 3rd ed., M e G r a w - H i l l ,

122 From t h e c a l c u l a t e d length, t h e t o t a l length of t h e tubes, including tube-sheet thickness,

+ 0.5

L = 2(8.287)

= 17.074 f t

From t h e Darcy equation,

VTâ&#x20AC;? P (144) 64.4

(

0.035(17.074)12 0.305

= 90.65 p s i

(10*5)a277 = (27.512) (3.295)

(144) 64.4

Temperature of Blanket S a l t a t End of F i r s t Pass Determination of the temperature between tube passes i s a t r i a l and e r r o r procedure.

By using t h e r e l a t i o n s h i p s given below and sub-

s t i t u t i n g given and c a l c u l a t e d data, t h e t r i a l and e r r o r c a l c u l a t i o n s

were simplified.

The t o t a l heat t r a n s f e r r a t e , Qt= ( t h i

- t

)w t ccp .

The o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t , U =

1 (1.82 + 7.92)104

= 1027 Btu/hr*ft?*OF

and t h e surface area of t h e first pass

81On(9)8.287

= 659 ft?

. 1236

- - 1111

the 125

- 1111 the ,

123

Assume t h a t t h e temperature of t h e blanket s a l t a t t h e end of t h e f i r s t pass, the = 1184.4'F. 65.6 = 0.680 Assume t

51*6 = 65.95 In 1.703

51*8 = 65.84 In 1.708

= 1184.2'F,

65.8 = 0.680 The value f o r the of 1184.2'F

i s a c l o s e enough approximation.

S t r e s s Analysis f o r Case B

The stress a n a l y s i s f o r the b l a n k e t - s a l t h e a t exchanger involved a determination of t h e s t r e s s e s produced i n t h e tubes, the s h e l l , and t h e tube sheets.

The s t r e s s e s produced i n t h e tubes are

primary membrane s t r e s s e s caused by pressure,

secondary s t r e s s e s caused by t h e temperature g r a d i e n t across t h e w a l l of t h e tube,

d i s c o n t i n u i t y stresses a t t h e junction o f t h e tube and t h e tube sheet, and

secondary s t r e s s e s caused by t h e d i f f e r e n c e i n growth between t h e tubes and t h e s h e l l .

The stresses produced i n t h e s h e l l a r e 1.

primary membrane stresses caused by pressure and

d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n of t h e s h e l l and tube sheet.

For t h i s exchanger, t h e s t r e s s e s i n t h e tube s h e e t s a r e those produced by pressure.

The shear s t r e s s theory of f a i l u r e w a s used as t h e f a i l u r e

c r i t e r i o n , and t h e s t r e s s e s were c l a s s i f i e d and t h e limits of stress i n t e n s i t y w e r e determined i n accordance with Section I11 of t h e ASME Boiler and Pressure Vessel Code. -The coolant s a l t e n t e r s t h e b l a n k e t - s a l t exchanger a t t h e bottom through t h e 22- in. -OD c e n t r a

pipe a t a temperature of l l l l 째 F and a

p r e s s u r e of 145 p s i , t r a v e l s t o t h e top of t h e pipe and e n t e r s t h e s h e l l s i d e of t h e exchanger a t a pressure of 138 p s i , c i r c u l a t e s through 0

t h e s h e l l and leaves t h e exchanger a t a temperature of 1125 F and a

124

wp r e s s u r e of 129 p s i .

The ,ubes through which t h e b-anket salt i s circu-

l a t e d are 0.375-in.-OD

ones w i t h a w a l l t h i c k n e s s of 0.035 in. and a

tube-sheet-to-tube-sheet

length o f 8.287 f t .

There are 810 s t r a i g h t

tubes i n t h e i n n e r annulus and 810 s t r a i g h t tubes i n t h e o u t e r annulus of t h e 40.78-in.-ID

The tubes are arranged i n a t r i a n g u l a r a r r a y with

shell.

a p i t c h of 0.8125 i n .

The blanket s a l t e n t e r s t h e t u b e s a t a temperature

of 1250째F and a p r e s s u r e of 111 p s i , flows down and reaches t h e f l o a t i n g head a t a temperature o f 1184'F

and a p r e s s u r e of 83 p s i , flows up and

leaves the exchanger a t a temperature of 1150째F and a p r e s s u r e o f 20 p s i .

S t r e s s e s i n Tubes The analyses o f t h e stresses i n t h e tubes of t h e b l a n k e t - s a l t

exchanger c o n s i s t e d o f a determination o f t h e four types of s t r e s s e s previously mentioned.

The terms u s e d i n t h e s e c a l c u l a t i o n s are defined

i n Appendix F. Primary Membrane S t r e s s .

The hoop stress,

where

a = i n s i d e r a d i u s of tube = 0.1525 in., b = o u t s i d e r a d i u s of tube = 0.1857 in.,

and

r = r a d i u s of tube a t p o i n t where stresses are being i n v e s t i g a t e d . Therefore, h '

= 1.954Pi

- 2.955P0

To o b t a i n a maximum value, use r = a. h '

= 4.913Pi

0.0688(Pi

- Po)

+ ~p Then,

- 5.913P0 .

The l o n g i t u d i n a l stress w a s assumed t o be caused by t h e p r e s s u r e drop i n t h e tubes.

Therefore, t h e l o n g i t u d i n a l stress,

= 1.954~9

0.4)

125

. The s h e l l - s i d e and tube-side p r e s s u r e s at t h e p e r t i n e n t l o c a t i o n s

and t h e p r e s s u r e drops i n t h e tubes a r e t a b u l a t e d below. Shell-Side Pressure Locat i o n I n n e r tubes Inlet Outlet Outer tubes Inlet Outlet

Tube- Side Pressure

Pressure Drop i n Tubes

(psi)

138 129

111 83

129 138

83 20

Using t h i s information and Eqs. B.4 and B.5,

(psi)

t h e hoop and l o n g i t u d i n a l

stress components i n t h e tubes were c a l c u l a t e d f o r t h e s e l o c a t i o n s .

Locat i o n Inner tubes Inlet Outlet Outer tubes Inlet Outlet

Pah

PUL

(psi)

-271

- 355

--718 355

123 123

Secondary S t r e s s e s Caused by At Across Tube Wall.

The stress com-

ponents a t t h e i n s i d e s u r f a c e of t h e tubes t h a t are caused by t h e temperat u r e g r a d i e n t a c r o s s t h e tube w a l l are t h e l o n g i t u d i n a l s t r e s s , t h e hoop stress

and

' Atah*

where k = a constant.

The stress components a t t h e i n s i d e s u r f a c e of t h e

tubes, Qt'L

-- ht'h

= kAt(4.22)

The stress components a t t h e o u t e r s u r f a c e of t h e tubes,

= kAt(0.189)

126 The constant in Eqs. B.6 and B.7,

where = the coefficient of thermal expansion,

= modulus of elasticity, and

IJ =

Poisson's ratio. CXFl

k = 2(1

- 0.3)(0.207)

= 3.45m

To determine the physical constants a! and E, the average temperature of the tube,

av T'

must be determined. R

- W+ R t

= t

av T

(ti

R 1* + RW + R o

- to)

For the applicable locations, R. = 5.12 x 104, 1

Rw = 2.8 x 104, Ro = 1.82 x 104. Therefore, t = t av T o

+ 0.3306(ti

- to)

The temperature drop in the tubes, AtT, must also be determined to solve Eqs. B.6 and B.7. R

AtT

(ti

RW to) = Ri . + Rw + Ro (ti

For all applicable locations,

RW Ri + Rw + R0

0.287

Therefore, AtT = 0.287(ti

- to)

- to) .

127 The inlet and outlet temperatures for the pertinent tube locations, the average temperatures, and the values of CX and E at these locations are tabulated below. ti Location Inner tubes Inlet Outlet Outer tubes Inlet Outlet

AtT

avtT

tw x 106 (in. /in. 'OF)

E* x 10"

(psi)

(OF)

(OF1

(OF)

1250 1184

1111 1125

40 17

1157 1145

7.7 7.7

25.8 25.8

1184 1150

1125 1111

17 11

1145 1124

7.7

25.8 25.8

7.7

Using this information in Eqs. B.6 and B.7, the longitudinal and hoop stress components on the inside and outside surfaces of the tubes were calculated for the pertinent locations. The calculated values are tabulated below. Inside Surface of Tubes Location Inner tubes Inlet Outlet Outer tubes Inlet Outlet

(psi)

A t ' L (psi)

A t ' h (psi)

AtL'

685 685

-6028 -2562

+5179 +2201

685 685

-2562 - 1658

-2562 1658

+2201 +1424

Ath'

Outside Surface of Tubes

(psi)

Tube Discontinuity Stresses. Assume that the tube sheet is very rigid with respect to the tube so that there is no deflection or rotation of the joint at the junction. The eflection produced by the discontinuity moment M and the discontinuity force F must be equal to the deflection produced by the pressure load and the slope at the junction

must

remain zero.

The deflection caused by pressure,

*Values for a and E taken from data reported by R. B. Lindauer, "Revisions to MSRE Design Data Sheets, Issue No. 9," Internal Document, Oak Ridge National Laboratory, June 24, 1964.

. 128

From t h e d a t a published by T i m ~ s h e n k o , ~ t h e d e f l e c t i o n caused by F, F

aF=m and t h e d e f l e c t i o n cuased by M, M

%=-=, where ET3 = 12(1

- 3)

and

S e t t i n g the d e f l e c t i o n caused by pressure equal t o t h e d e f l e c t i o n caused

by M and F,

t h a t is,

The slope produced by F must be counteracted by M;6 F - 2 M = O .

S u b s t i t u t i n g f o r D and solving t h e s e l a s t two equations f o r M and F, M=-

and F = P

I -

= 16.66

The l o n g i t u d i n a l stress component caused by t h e moment,'

MuL =

6M 1 8=

= 8.824P

The hoop s t r e s s component caused by t h e force,

- - 2Pr = - T

9.714P

sS. Timshenko, pp. 12, 126, and 127 i n Strength of Materials, P a r t 11, 3rd ed., Van Nostrand, New York, 1956.

'R. J. b a r k , Cases 10 and 11, p. 271 i n Formulas f o r S t r e s s e s and S t r a i n , 3rd ed., McGraw H i l l , New York, 1954.

129 t h e hoop s t r e s s component caused by t h e moment,

M h'

har = Pr = 4.587P; and T

t h e secondary hoop s t r e s s component caused by t h e moment, Mh'

sec

) = 2.647P

These stress components were c a l c u l a t e d by using t h e d i f f e r e n t i a l p r e s s u r e s a t t h e p e r t i n e n t l o c a t i o n s i n t h e exchanger.

These p r e s s u r e s

and t h e r e s u l t i n g s t r e s s e s f o r t h e l o c a t i o n s a r e t a b u l a t e d below. Differential Pressure (psi)

I n n e r tubes Inlet Outlet Outer tubes Inlet Outlet

a ML ' (psi)

a M h' (psi)

M'hsec (psi)

Fh ' (psi1

-27 46

238 406

124 -211

72 122

-262 -447

-46 -118

40 6 1041

-211 -541

122 312

-447 -1146

astress a t t h e i n n e r s u r f a c e of t h e tubes i s compressive and stress a t t h e o u t e r s u r f a c e of t h e tubes i s t e n s i l e . Buckling of Tubes Caused by Pressure. i s g r e a t e r than t h e i

Where t h e e x t e r n a l p r e s s u r e

i n t e r n a l pressure, t h e s t r e s s on t h e tube w a l l must

be checked t o a v e r t p o s s i b l e collapse of the tubes.

F r o m Fig. UG-31 i n

Section V I 1 1 of t h e ASME B o i l e r and Pressure Vessel Code.

The minimum design stress required t o match t h e design p r e s s u r e d i f f e r e n t i a l a t t h e p e r t i n e n t l o c a t i o n s and the maximum allowable design s t r e s s e s f o r Hastelloy N are t a b u l a t e d below.

130

kidMinimum Design S t r e s s

AP Locat ion

(psi)

Inner tubes Inlet Outlet Outer tubes Inlet Outlet

- 27

a000

-46

<loo0

-46 -118

--1200

<loo0

Sm f o r t 8v T

Hastelloy N (psi)

1157 1145

8000 8800

1145 1124

8800 10600

From t h e s e data, i t does not appear l i k e l y that the tubes w i l l c o l l a p s e because of e x t e r n a l pressure. L a t e r a l Buckling and Secondary S t r e s s e s Caused By Restrained Tube Since the average temperature of the o u t e r tubes i s d i f f e r e n t

Growth.

than t h e average temperature of t h e i n n e r tubes, d i f f e r e n t i a l growth s t r e s s e s w i l l occur.

pi -

+ EoLo

EiLi =

where t&

= u n r e s t r a i n e d thermal growth of l e n g t h of tubes,

E = s t r a i n on tubes required by compatability condition,

L = length of tubes, sub sub

= i n n e r tubes,

and

= o u t e r tubes. E

-- â&#x20AC;&#x2122; U

-E

where

u E

= normal stress and

= modulus o f e l a s t i c i t y

of tubes.

Therefore, uiLi

â&#x20AC;&#x153;OL0 p i - ? =t OLo + E .

where n = t h e number o f tubes. n u

-- i i - n0

131

- -"iLi - E - tA

"i"iLo

Lo + noE

For t h e Case-B b l a n k e t - s a l t exchanger, tAL i

= 8.287(12)(7.7 = 0.828 i n . ,

t ALo

x 10'")(1081)

and

= 8.287(12)(7.7

x 10'6)(1065)

= 0.815 i n .

The s t r e s s e s , 'i

0 -

- 0.815) 8.287j12

25.8 x l e ( 0 . 8 2 8 [810(8.287) 810 +

1622 p s i (compression), and

1622 p s i (tension)

The f r e e space between b a f f l e s i s 1.65 f t , and t h e maximum d i s t a n c e between lateral supports on any one tube i s 2.3 f t . alone ,=

YI? E1

Pcc = uCcA = -

%c =

r12A

For t h e a x i a l load

where

ucc

= t h e c r i t i c a l compressive s t r e s s , p s i ,

I = c r o s s - s e c t i o n a l moment of i n e r t i a , in.4, L = length of tubes between loading p o i n t s , in.,

A = cross-sectional a r

'cc

and

of tubes,' in.a

3 (25.8 x 106) (0.000546) = 4828 psi (2.3)" (144) (0.03738)

Therefore, i t does not appear l i k e l y t h a t buckling of t h e tubes w i l l occur because of b u i l t - i n r e s t r a i n t .

132

Summary o f Calculated S t r e s s e s i n Tubes.

The v a l u e s c a l c u l a t e d f o r

U-

t h e stresses t h a t are uniform a c r o s s t h e tube w a l l and t h e i r l o c a t i o n s are given i n Table B . l .

The c a l c u l a t e d stresses on t h e i n s i d e and o u t s i d e

s u r f a c e s o f t h e t u b e s a r e given i n Table B.2,

and t h e c a l c u l a t e d stress

i n t e n s i t i e s f o r t h e tubes are given i n Table B.3. Table B . l .

Calculated S t r e s s e s Uniform Across Tube Wall PL '

Loca t ion Inner tubes Inlet Outlet Outer tubes Inlet Outlet

Table B.2.

Ph '

(psi)

aP'L

Fh'

(psi)

55 55

-271 355

-46 -46

-262 -447

123 123

-718

-46 -46

--447 1146

- 355

(psi)

Calculated S t r e s s e s on I n s i d e and Outside Surfaces of Tubes

Location I n n e r tube i n l e t Inside surface Outside s u r f a c e Inner tube o u t l e t Inside surface Outside s u r f a c e Outer tube i n l e t Inside surface Outside s u r f a c e Outer tube o u t l e t Inside surface Outside s u r f a c e

At'h

AtL '

Aâ&#x201A;Ź%

(psi)

-124 -124

-72 +7 2

-6028 +5178

-6028 +5179

1622 -- 1622

-406 +406

-211 -211

122 +122

-2562 +220 1

-- 1622 1622

-406 +406

-211 -211

122 +122

-2562 +220 1

-2562 +2201

+1622 +1622

-1041 +lo41

-541 -541

-312 +312

1658 +1424

M h'

ML' (psi)

(psi)

-238 +238

M'hsec

1658 +1424

+1622 +1622

..... .....

Table B.3.

Calculated S t r e s s I n t e n s i t i e s f o r Tubes

Primary

Inner tube i a l e t I n s i d e surfaceOutside sureace Inner tube o u t l e t I n s i d e surface Outside surface Outer tube i n l e t I n s i d e surface Outside surface Outer tube o u t l e t I n s i d e surface Outside surface

(psi)

m ' (psi)

8000 9600

-6757 -4593

-7879 +3804

-111 -138

7768 4455

24,000 28,800

410 410

8000 9600

-3697 +1310

-4581 +994

-83 -129

4498 1181

24,000 28,800

-83 -129

478 478

8000 9600

-3697 +1310

-1269 +4306

-83 -129

3614 4435

24,000 28,800

- -13820

841 841

10000 11400

-4375 -669

-1000 +4164

20 -138

4355 4833

30,000 34,200

-111 138

326 326

55 55

- 129

-83

123 123 123 123

1157 1137

-271 -271

1154 1137

-355 -355

1137

-355 -355

1130 1118

-718 -718

(psi)

55 55

(psi)

(Pm + Q

(psi)

pm (psi)

'Oh (psi)

(psi)

"r (psi)

(OF)

Location

Cyclic

CUL

(psi)

w w

134 Stresses i n Shell The stresses c a l c u l a t e d i n t h e s h e l l w e r e t h e primary membrane

stresses caused by p r e s s u r e and t h e d i s c o n t i n u i t y stresses a t t h e junct i o n s of t h e s h e l l and tube sheet. Primary Membrane S t r e s s e s .

The hoop stress component caused by

pressure,

and t h e l o n g i t u d i n a l s t r e s s component,

- -APD

PL '

For t h e o u t e r s h e l l ,

Pub -

138(41*78) = 2882 p s i , and 2(1)

For t h e i n n e r s h e l l o r d e l i v e r y tube,

L'P Discontinuity S t r e s s e s .

- -360 - -

180 p s i

I n t h e determination of t h e d i s c o n t i n u i t y

s t r e s s e s a t t h e j u n c t i o n s of t h e s h e l l and tube sheet, i t w a s assumed t h a t t h e tube sheet i s very r i g i d with r e s p e c t t o t h e s h e l l .

Therefore, t h e

d e f l e c t i o n and r o t a t i o n of t h e s h e l l at t h e j o i n t are zero.

The r e a c t i o n s

on t h e s h e l l from t h e tube s h e e t are t h e moment, M = P/2A2,

and t h e force,

F = P/h.

The constant,

For t h e o u t e r s h e l l

= 1.813[(41178)1r/a

= 0.279

135 Therefore,

138

= 2(0.779)

= 886 in.-lb/in.,

F = - - 138 0.279

and

495 lb/in.

The secondary 1,ongitudinal stress, 6M

ML '

= ygr = 5316 psi (tensile outside,

compressive inside)

The secondary hoop stress, M'hsec

uMuL= 1595 psi (tensile inside, compressive outside)

Mh'

h'F

- - 2fi2 =

2(886)(0*779) = 138 psi (tensile uniform) 1

- -FAD - - 495(0*279)(41*78)

= 5770 psi (compressive uniform)

The inner shell is a tube that will be greatly affected by anchorage and final design, and the secondary discontinuity stresses in the inner shell were not included in this analysis. Summary of Calculated Stresses in Shell.

The calculated values of

the stresses in the shells are tabulated below, and the stress intensities calculated for the shells are given in Table B.4. ?

For the outer shell,

h'P

= 2882 psi, = 1441 psi

M'L

= 5316 psi (tensile inside, compressive outside)

= 1595 psi (tensile inside, compressive outside)

M'hsec = -1-138 psi

h'F

= -5770 psi

For the inner shell, p ~ = h 360 psi

180 psi

136 Table B.4.

Calculated S t r e s s I n t e n s i t i e s f o r S h e l l s Outside S h e l l I n s i d e Surface Outside Surface

Temperature, Pr imary Xuh, p s i

Inner Shell

1111

2882

360

XuL, p s i

1441

180

Cur, p s i

- 138

1441

pm, P s i

3020

2882

180 12000

SmY P s i Cyclic Cub, p s i

12000

- 1155

-4345

XuL, p s i

+6758

-3875

Cur, p s i

- 138

7913

4345

36000

Pm +

psi

3Sm, p s i

-0

S t r e s s e s i n Tube Sheets The loads on t h e tube s h e e t s are those caused by t h e p r e s s u r e d i f f e r e n t i a l and d i f f e r e n t i a l expansion of t h e tubes i n r e l a t i o n t o t h e tube sheets.

I n t h e C a s e - B design f o r t h e b l a n k e t - s a l t exchanger, a

f l o a t i n g lower head i s used t o accommodate t h e d i f f e r e n t i a l expansion. Therefore, only t h e pressure stresses need be considered i n t h i s a n a l y s i s

of t h e tube s h e e t s f o r t h e b l a n k e t - s a l t exchanger. Lower Annular Tube Sheet. pressure, ap = 46 p s i .

The area where t h i s pressure acts,

= z (40.782 A

= 926

I n t h e lower tube sheet, t h e d i f f e r e n t i a l

2P)

- 2(810) ~(0.375)â&#x20AC;&#x2122;

179 = 747 in.a

Therefore, t h e e f f e c t i v e pressure,

747 AP = 46 = 37 p s i . 926

13 7 The ratio of the average ligament stress in the tube sheets (strengthened and stiffened by the rolled-in tubes) to the stress in the flat unperforated plates of like dimensions equals the inverse of the ligament câ&#x201A;Źficicncy if the inside diameter of the tubes is used as the diameter of the penetrations.' That is, the inverse of the ligament efficiency, I

@=%=A, P'di

where 'av'

= average ligament stress in the tube sheet, cy

= stress in unperforated plate,

p = tube pitch, and

= inside diameter of tube.

0.8125 = 1.6 0.305

= 0.8125

= 2.0 (used value)

To determine the thickness required for the lower annulus tube

sheet ,8

For A,/%

= 40.78122 = 1.85,

f3 = 0.152; and at a temperature of 1184'F,

the allowable stress, Sm = 6500 psi.

Therefore,

T = 1.7 in. = 2 in. (used value) Upper Annular Tube Sheet.

The thickness of the upper tube sheet

was determined for the outer annulus portion where the acting pressure differential,

AP = 138

20 = 118 psi

' 5 . F. Harvey, p. 106 in Pressure Vessel Design, D. Van Nostrand Co., New Jersey, 1963.

*R. J. Roark, p. 237 in Formulas for Stresses and Strains, 4th ed., McGraw Hill, New York, 1965.

138 The area where t h i s pressure a c t s ,

-A = Z[(40.75)2 -

(32)2]

- 8 1 O s (4O . 375)

= 411.87 in.'

Therefore, t h e e f f e c t i v e pressure d i f f e r e n t i a l ,

For Ao/\

= 40.78132 = 1.27, f3 2: 0.03; and a t a temperature of 1150째F,

t h e allowable stress, S = 8500 p s i . m tube sheet,

Therefore, t h e thickness of t h e

(97) (40.78)2 f = 2(0.03)8500 T

= 1.14 in.

-,.,1 118 i n .

Checking, t h e thickness required f o r t h e i n n e r annulus p o r t i o n w a s determined.

I f t h e d i f f e r e n t i a l pressure of 27 p s i i s considered t h e

e f f e c t i v e pressure, A

= 32/22 = 1.45 and $ = 0.0824.

t u r e of 1250째F, t h e allowable stress, S = 4500 p s i . m thickness of t h e tube sheet, (27) (32)2 f = 2(0.0824) 4500

A t a tempera-

Therefore, t h e

= 1.01 i n .

T 2: 1 in.

These c a l c u l a t e d thicknesses are dependent upon t h e s c r o l l of t h e pump plenum providing the necessary support f o r t h e tube sheet.

If t h i s

i s not t h e case, a much t h i c k e r sheet w i l l be required, and an approximation of t h i s thickness was calculated.

A t Ao/\

2, f3 = 0.1815.

With

a working d i f f e r e n t i a l pressure of 97 p s i , the allowable s t r e s s , Sm = 4500 psi.

The thickness, 2(0.1815)(97)(40.78)2 4500

I?=

= 13.012

in.

T = 3.61 in. = 4.00

i n , (value used).

I t

139

;L, The u s e of 4-in.-thick

material f o r t h e upper tube s h e e t matches the

assumption of a r i g i d tube sheet t h a t w a s made f o r t h e s h e l l s e c t i o n because some support from t h e pump plenum w i l l indeed be present. Summary of Calculated S t r e s s e s i n Tube Sheets.

The s t r e s s e s and

thicknesses c a l c u l a t e d f o r t h e upper and lower tube s h e e t s are t a b u l a t e d below.

Tube Sheet I

I -

Upper Lower

Thickness (in.) 4

Allowable Stress

Maximum Calculated Stress

(psi)

8500 6500

4500 C6500

140 Appendix C CALCULATIONS FOR BOILER- SUPERHEATER EXCHANGER

Heat-Transfer and Pressure-Drop C a l c u l a t i o n s

Since t h e h e a t balances, h e a t - t r a n s f e r equations, and t h e pressuredrop equations had t o be s a t i s f i e d f o r each increment of t h e U-tube ones h e l l - p a s s one-tube-pass exchanger and f o r t h e e n t i r e exchanger, an i t e r a t i v e procedure was programmed f o r t h e CDC 1604 computer t o perform t h e necessary c a l c u l a t i o n s .

The o u t l i n e o f t h e computer program used i s given

below i n i t s stepped sequence. 1. Divide t h e t o t a l h e a t t o be t r a n s f e r r e d i n t o N equal increments. 2.

Assume t h e number of tubes.

S t a r t a t t h e h o t end of t h e exchanger.

Assume a b a f f l e spacing.

C a l c u l a t e t h e h e a t t r a n s f e r c o e f f i c i e n t on t h e e x t e r i o r of t h e

tubes by using t h e method proposed by Bergelin e t a1.l)' I

Assume t h e p r e s s u r e drop through t h e increment o f tube length.

Determine t h e temperatures a t t h e end of t h e increment from a

h e a t balance. ,

I ~

Determine t h e p h y s i c a l p r o p e r t i e s o f t h e s u p e r c r i t i c a l f l u i d

a t t h e end of t h e increment. 9.

C a l c u l a t e t h e h e a t t r a n s f e r c o e f f i c i e n t on t h e i n s i d e of t h e

tube by using t h e method of Swenson e t al.3

lo. P. Bergelin, G. A. Brown, and A. P. Colburn, "Heat Transfer and A Study of a CylindriFluid F r i c t i o n During Flow Across Banks o f Tubes -V: c a l Baffled Exchanger Without I n t e r n a l Leakage," Trans. ASME, 76: 841850 (1954). 20. P. Bergelin, K. J. Bell, and M. D. Leighton, "Heat Transfer and The E f f e c t of InFluid F r i c t i o n During Flow Across Banks o f Tubes -VI: t e r n a l Leakages Within Segmentally Baffled Exchangers," Trans. ASME, 80: 53-60 (1958).

3H. S. Swenson, C. R. Kakarala, and J. R. Carver, "Heat Transfer t o S e r i e s C , J. Heat S u p e r c r i t i c a l Water- i n Smooth-Bore Tubes," Trans. A&, Transfer, 87(4) : 477-484 (November 1965).

141 10.

C a l c u l a t e t h e tube wall r e s i s t a n c e using t h e thermal conductivity

of Hastelloy N corresponding t o t h e average w a l l temperature.

11. C a l c u l a t e t h e o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t f o r t h i s increment. 12.

C a l c u l a t e t h e temperature d r i v i n g f o r c e f o r t h e increment.

13.

C a l c u l a t e t h e length of t h e increment using t h e h e a t t r a n s f e r r e d

p e r increment from Step 1, t h e number of tubes from Step 2, t h e o v e r a l l c o e f f i c i e n t from Step 11, and t h e temperature d r i v i n g f o r c e from Step 12.

14.

C a l c u l a t e t h e p r e s s u r e drop i n s i d e t h e tube f o r the increment.

15.

Compare t h e c a l c u l a t e d p r e s s u r e drop i n Step 14 t o t h e value

assumed i n Step 6. r e t u r n t o Step 6.

I f they do n o t agree, a d j u s t the assumed v a l u e and When t h e two values agree w i t h i n a s p e c i f i e d tolerance,

continue t o t h e next s t e p . 16.

C a l c u l a t e t h e allowable temperature a c r o s s t h e tube w a l l based

on thermal stress c o n s i d e r a t i o n s using t h e allowable stress corresponding t o t h e maximum w a l l temperature f o r t h e increment. 17.

C a l c u l a t e t h e a c t u a l temperature drop a c r o s s t h e tube w a l l

based on t h e h e a t t r a n s f e r c o e f f i c i e n t s . 7

18.

Compare t h e w a l l temperature drops i n Steps 16 and 17.

I f they

do n o t agree, a d j u s t t h e assumed b a f f l e spacing and r e t u r n t o Step 4. When t h e two values agree w i t h i n a s p e c i f i e d tolerance, continue t o t h e next s t e p . 19.

Repeat Steps 6 through 15 u n t i l t h e summation o f t h e tube l e n g t h s

f o r t h e increments c a l c u l a t e d equals t h e c a l c u l a t e d b a f f l e spacing. 20.

Repeat Steps 4 through 19 u n t i l a l l N increments have been c a l -

culated. 21.

Sum t h e p r e s s u r e drops through t h e tubes f o r N increments.

t h e t o t a l tube p r e s s u r e

exceeds t h e s p e c i f i e d p r e s s u r e drop, a d j u s t

t h e number o f tubes and r e t u r n t o Step 2.

When t h e c a l c u l a t e d tube

p r e s s u r e drop a g r e e s w i t h t h e s p e c i f i e d drop, continue t o t h e next s t e p . 22.

C a l c u l a t e t h e p r e s s u r e drop i n t h e s h e l l of the exchanger and

compare t h i s with the s p e c i f i e d s h e l l - s i d e p r e s s u r e drop.

(a)

If t h e s p e c i f i e d p r e s s u r e drop exceeds t h e c a l c u l a t e d p r e s s u r e drop,

t h e b a f f l e spacing i s l i m i t e d by thermal s t r e s s c o n s i d e r a t i o n s and t h e problem i s f i n i s h e d .

142

LiI f t h e c a l c u l a t e d p r e s s u r e drop exceeds t h e s p e c i f i e d p r e s s u r e drop,

(b)

each b a f f l e spacing i s increased and Steps 4 through 2 1 (excluding Steps 16, 17, and 18) are repeated u n t i l the two v a l u e s agree.

t h i s case, t h e b a f f l e spacing i s l i m i t e d by t h e s h e l l - s i d e p r e s s u r e drop and t h e problem i s f i n i s h e d . 23.

Write o u t t h e f i n a l answers.

The computer input d a t a a r e given i n Table C . l , computer c a l c u l a t i o n s a r e given i n Table C.2, one case are given i n Table C.3.

t h e terms used i n t h e

t h e computer input d a t a f o r

The computer output d a t a f o r t h e b a f f l e s

f o r one case are given i n Table C.4,

the output d a t a f o r tube increments

f o r one case are given i n Table C.5, and t h e computer output d a t a f o r t h e o v e r a l l exchanger f o r one case are given i n Table C.6. Table C . l . Card

Columns

Computer-Input Data f o r Boiler-Superheater Exchanger Format inches inches inches

1- 10 11-20 2 1-30 31-40

F10.5 F10.5 F10.5 I10

Tube o u t s i d e diameter Tube w a l l thickness Tube p i t c h Number of increments

1-10 11-20 2 1- 30 31-40

F1O.l F10.1 F1O.l F10.1

I n l e t s a l t temperature E x i t s a l t temperature I n l e t steam temperature E x i t steam temperature

F OF OF 0 F

1-10 11-20 21-30 31-40

F1O.l F1O.l F10.1 F10.5

I n l e t steam p r e s s u r e E x i t steam p r e s s u r e Allowable s h e l l p r e s s u r e drop Bergelin leakage f a c t o r f o r p r e s s u r e drop

psi psi Psi

1-20 21-40 41-60 61-70

E20.6 E20.6 E20.6 F10.5

Steam flow r a t e S a l t flow rate Total h e a t t r a n s f e r r e d Bergelin leakage f a c t o r f o r heat transfer

lb/hr lb/hr Btu/hr

1-10 11-20 2 1-30 31-40

F10.5 F10.5 F10.5 F10.5

Specific heat f o r salt Thermal conductivity f o r s a l t Viscosity f o r salt Density f o r s a l t

Btu/lb OF 0 Btu/ft*hr* F lb/ft *hr lb/fe

1-10 11-20 21-30

F10.5 F10.5 F10.5

Window opening F i r s t guess a t tube length F i r s t guess a t b a f f l e spacing

fraction ft ft

143 Table C.2.

AREAX

heat transfer area for length SBS, ft? heat transfer area for length S X , ft?

BLFH

by-pass leakage factor for heat transfer

BLFP

by-pass leakage factor for pressure

baffle spacing, ft

CPH

specific heat, hot fluid

DELP

tube AP for given increment, psi

DELPS DELPSA

calculated AP for 1 baffle in shell allowable shell AP, psi

DELPTA

allowable tube AP, psi

DELTW

calculated At across tube wal1,OF

DELTWA

allowable At across tube wall, OF density, hot fluid, lb/ft?

AREAB

DENH DS

DTLME DTO I IBK

Terms Used in Computer Calculations for Boiler-Superheater Exchanger

ID of shell, in. 0 log mean At, F outside diameter increment position number increment number at baffle

baffle position number

MP N NUMT

index of baffle spacing dependence number of increments number of tubes, total

pitch of tubes, in.

pressure of cold fluid

PC1

inlet pressure of cold fluid, psi

PC2

outlet pressure of cold fluid, psi fractional window cut

QT RI Ro

total heat transfer rate, Btu/hr thermal resistance inside film for particular increment thermal resistance, outside film for particular baffle

thermal resistance, total for particular increment

thermal resistance, wall

. 144 Table C.2 (continued) SBS

t o t a l b a f f l e space length, f t

SDPS

t o t a l p r e s s u r e drop i n s h e l l , p s i

SDPT

t o t a l p r e s s u r e drop i n tubes, p s i

t o t a l tube length, f t

temperature, o f cold f l u i d ,

TCH

thermal c o n d u c t i v i t y o f h o t f l u i d , Btu/(hr*fi? *OF/ft)

TC1

temperature o f cold f l u i d a t i n l e t , OF

TC2

temperature o f cold f l u i d a t o u t l e t , 0

temperature o f h o t f l u i d ,

THK

tube w a l l thickness, in.

TH1

temperature o f h o t f l u i d a t i n l e t ,

. ,-

F 0

TH2

temperature of hot f l u i d a t o u t l e t ,

UEQB

o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t , AREAB, Btu/(hr"F=f@)

UEQX

o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t , AREAX, Btu/(hr*OF*f?)

VISH

v i s c o s i t y of h o t f l u i d , l b / h r * f t

flow r a t e of cold f l u i d , l b / h r

flow r a t e o f h o t f l u i d , l b / h r

tube length f o r increment, f t

145

Table C.4. Computer OUtvut Data for Baffles for One Case SWELL PRES D R O P l . I # I T I N @ 8AFFLE SPACING wra J= J=

2 3

IBKt I DELTU. 60r96846 c3Sf 9093476 l B K = 13 DELTW= 93.15944 RS= 9,93476 18Kr 2A OELTWS 126.62404

OELTWg I 18.29501 RS= .I9656

DELTWAZ 87.34793 DELPS= a63753

Rdr

e90033

DELTWAS 121.52259 DELPSZ 1.25107

RO=

t000S3

ROr

000014

DELTWAr 130,34932

DELTUAS 153,24864 DEEPS= 16.54442

146 Table C.5.

Computer Output &it8 for Tube Increments for One Case

ii3

TCz 993. 4

1 e

1 I

7 152 I074 i66

TC= 968; Rf= 00 T C t 961.

i I

7 2

7 -2 I

1 I

41779 1% 13

THii092.Q '57 RWr . o Ill76 '84 1076 i97 1077

TC= 910.9938

TC= 397.9552 RIS

0000J7

T C = 591.4359 RIf

000037

I 47 PC13649 e1339 Rtr . a 0 I 47

.743a 3,2539 I XIS ,7299 XIS

DELP.

-F I PCr3656. e HT= .oo 1

PC=3L558

'636 47 190 I 46 I 85

XI'

,6696 2,72467 XI= 6559 DELrf 2162691 XI= ,6446 UELPZ

1651 46

XIf DELPI

06343 2.46581

'079 i15

i' e 1 i'C A 'C

R 'C R 'C

R T 3

. e

' U

ii 186 4

186

i e L: 723. I= e o i.~ ca Rf

i726 1008 1768

;

i C a: 721. 186 i4 186 ti7 I86

18 i.~ C R fC 1

C I

1 C

187 16 167 9 I87 i2

T = .O a i eC 71 3. i= e o

I81 16

I87

T Ca

712.

i3 5

PCIS~S6

e7 r 797

i -c

. 'I

i7 i i I BiI zi e

t i r e

ZTi

DELPS.

I '290

PC=375 RT' b

'I I

r '95 r i75

149 Table C.5 (continued)

Table C.6.

NUMTr 349 S X r 68rBfl760 AREAX ? 2 9 1 5 * 0 0 1 9 3 AREAB~2905e54091

Computer Output Data for Overall Exchanger for One Case OS% 17.1646 S D P T 8 166135626 UEOX.3032i6S359 O T L W E I 167,12037 UEOR~1035~99605

SRS.

63r61070

sllPS=

58,10013

The computer program w a s run f o r s e v e r a l cases t o i n v e s t i g a t e a l l t h e parameters, and "hand" c a l c u l a t i o n s were made t o check t h e accuracy of t h e computer program.

These hand c a l c u l a t i o n s involved t h e detennina-

t i o n o f t h e h e a t t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes f o r t h e f i r s t b a f f l e spacing, t h e s h e l l - s i d e p r e s s u r e drop f o r t h e f i r s t b a f f l e spacing, t h e thermal r e s i s t a n c e of t h e tube w a l l f o r t h e f i r s t increment, t h e length of t h e f i r s t increment, and t h e p r e s s u r e drop i n s i d e t h e tubes f o r t h e f i r s t increment.

The terms used i n t h e s e c a l c u l a t i o n s a r e defined

i n Appendix F.

Heat-Transfer C o e f f i c i e n t Outside Tubes f o r F i r s t B a f f l e Spacing Assume t h a t t h e number of tubes, n, i n t h e b o i l e r - s u p e r h e a t e r exchanger i s 349, and t h e c r o s s - s e c t i o n a l a r e a of t h e s h e l l , 349

Ss = g $ 0 . 8 6 6 ) ( 0 . 8 7 5 ) "

= 1.607 ft? o r 231.4 in."

The diameter of t h e s h e l l , d

= [ ( 4 / ~ ) ( 1 . 6 0 7 ) ] ~ =/ ~1.4304 f t o r 17.16 i n .

The f r a c t i o n a l window cut, F = 0 . 4 0 , and t h e c r o s s - s e c t i o n a l area of t h e

u .

window,

S = 0 . 4 0 ( 2 3 1 . 4 ) = 92.56 in." W

150

Therefore,

dZ = S

92*56 = 0.3143 (17.16)a

From t h e d a t a published by Perry,*

Window h e i g h t = o.4213 d S

Window height = 0.4213(17.16)

= 7.23 i n .

To determine t h e length o f t h e chord formed by t h e edge of t h e window a c r o s s t h e c i r c u l a r area, t h e included angle of t h e chord must be determined. 8 = 1/2 t h e included angle.

= 80.946'

The length of t h e chord = Z(8.58)sin 8 = 16.95 in. spacing, X = 9.935 f t ,

16.95 + 17.16 2 (12)

(9.935)

10.8750.875-

O o 5 0 ) = 6.051 f ? ,

349n(0*5)2)s ,

= (231.4

3*6625 IO6 = 6.053 x 105 l b / h r * f ? , 6.051

306625 '06 = 8.096 x 106 l b / h r * f @ , 0.4524

= r(6.053 x 1@)(8.096 x

= 0.4524

106)]1/2

(NRe)B =

0.5(6.053 x 105) = 2102, 12 (12)

(NRe)

0.5(2.2137 x 106) = 7686, 12( 12)

When t h e b a f f l e

= 2.2137 x 106 l b / h r * f P ,

JB = 0.0186, J

= 0.0113,

4J. H. Perry, Ed., p. 32 i n Chemical Engineer's Handbook, 3rd ed., McGraw H i l l , New York, 1950.

151

: w I

(Npr)2/3

[ 0.41( 12))a I3= 2.43, 1.3

h 0 = b.2(1900) 1

RO=3006=

+ 0.8(4221)]0.8

= 3006 Btu/hr*ft?*OF, and

3.33 x 10".

From t h e computer c a l c u l a t i o n s , t h e thermal r e s i s t a n c e o u t s i d e t h e tubes, R

lo4

= 3.3 x

Shell-Side Pressure Drop f o r F i r s t B a f f l e Spacing The number of cross-flow r e s t r i c t i o n s , 2(8.58

- 7.23)

rB -- 0.866(0.875)

= 3.56

and t h e number of window r e s t r i c t i o n s ,

- 1 = 8.54 .

7.23

rw -- 0.866(0.875)

From t h e d a t a published by Bergelin e t a1.,2 t h e leakage f a c t o r f o r b a f f l e d exchangers i s 0.52.

Therefore, t h e s h e l l - s i d e p r e s s u r e drop f o r

t h e f i r s t b a f f l e spacing, 0.6(3.56)

D s=, 0.52

[

= 0.631 p s i

[ 6 o o ' , ~ o'@I~2

144(64.4)(125)

0.5

E2 +

(0.6)8.54](

3600

144(64.4) (125) 2137

From t h e computer c a l c u l a t i o n s , aPs = 0.638 p s i .

Thermal Resistance of Tube Wall f o r F i r s t Increment The temperature o f t h e tube w a l l f o r t h e f i r s t increment of t h e The thermal exchanger i s approximately 1000 + 0.5(125) = 1062'F.

152 conductivity of the tube wall,

10.6 Btu/hr-ft?*OF per ft.

Therefore,

the thermal resistance of the tube wall, d

do In

0 7

0.5 0.5 ln- 0.346

i - 2( 12) (10.5) = 7.23 x 10"

From the computer calculations,

. 7.2 x lo4

Heat-Transfer Coefficient Inside Tubes for First Increment The heat transfer rate,

= O.OOl(4.1275

x 108) = 4.1275 x 106 Btu/hr

The temperature of the hot fluid leaving the first increment, thf = 1125

4*1275 = 1122.252O~ , (3.6625 x 106)0.41

and the average temperature of the hot fluid in the first increment, t

av hf

1125 + 1122.252 = 2

1123.630F

The change in enthalpy of the supercritical fluid,

4*1275 '06 = 6.52 Btu/lb AH = 6.3312 x 105

and the enthalpy of the supercritical fluid at the end of the first increment, H = 1421

6.52 = 1414.48 Btu/lb.

The temperature of the

supercritical fluid at the end of the first increment, t = 992'F,

and

the average temperature of the supercritical fluid in the first increment, av

t =

992 + 1000 = 2

9960F

The mass velocity of the fluid inside the tubes,

- 6*3312 105)144 GT - (349(2) (0.346)a

2.778 x 106 lb/hr*f$

153 Then Assume t h a t t h e temperature drop across t h e insiL2 f i l m i s 37'F. t h e temperature of t h e i n s i d e surface, ti = 996 + 37 = 1033OF. Using

Eq. 13 discussed i n Chapter 3, t h e heat t r a n s f e r c o e f f i c i e n t i n s i d e t h e tubes,

- 0.00459(12)(0.064)

hi -

x 108) [0.346(2.778 12(0.075)

0.346

1' = 3422 Btu/hr*fid*'F

-- 1417.8)0.075 996)0.064 I

1445.6 (1033

0.1977 O (0.2071)

O *613

231

The thermal r e s i s t a n c e i n s i d e t h e tubes,

0.50 4.22 3422(0.346) =

104

and t h e t o t a l thermal r e s i s t a n c e i n t h e f i r s t increment, Rt

= (4.22

7.23

3.33)10'4

= 14.78 x

Therefore, t h e c a l c u l a t e d temperature drop across t h e i n s i d e film, At =

4*22 (1123.63 14.78

996) = 36.4OF

This i s s u f f i c i e n t l y c l o s e t o t h e assumed temperature drop a c r o s s t h e i n s i d e f i l m of 37OF. and Rt = 14.8 x

From the computer c a l c u l a t i o n s , Ri = 4.3 x 1 0 4

lo4.

Length of F i r s t Increment The length o f t h e f i r s t increment,

a = ( 4.1275 x

106)(14.8 x

3 4 9 4O- gd5 (1123.63 From t h e computer c a l c u l a t i o n s ,

lo4)

- 996)

= 1.056 f t .

154

Pressure Drop I n s i d e Tubes f o r F i r s t Increment The Reynolds number f o r t h e tubes,

and t h e f r i c t i o n f a c t o r , f = 0.0029.

APT =

4(0.0029)(1.056)(12) 0.346

= 5.39 p s i

Therefore,

f 2.7783600x 106 1

0.1977 144(64.4)

From t h e computer c a l c u l a t i o n s , APT = 5.37 p s i .

S t r e s s Analysis f o r Case B

I n t h e v e r t i c a l U-tube one-shell-pass

one-tube-pass b o i l e r - s u p e r h e a t e r

exchanger, t h e s u p e r c r i t i c a l f l u i d e n t e r s the head a t a temperature o f 700째F and a p r e s s u r e of 3766.4 p s i , flows through t h e tubes, and l e a v e s t h e I

opposite high-pressure head a t a temperature of 1000째F and a p r e s s u r e of 3600 p s i .

The s u p e r c r i t i c a l f l u i d entrance p r e s s u r e used i n t h i s a n a l y s i s

w a s 3800 p s i , which has a more conservative e f f e c t on t h e a n a l y s i s .

Be-

cause of t h e s e high temperatures and pressures, t h e m a t e r i a l chosen for t h e exchanger was Hastelloy N.

The high-pressure head h a s an i n s i d e

r a d i u s of 12 in. and a w a l l thickness of 4.5 i n .

There are 349 tubes

i n s i d e the s h e l l and they have an o u t s i d e diameter of 0.5 in., t h i c k n e s s of 0.077 in.,

and are on 7/8-in.

a wall

center lines.

The coolant s a l t e n t e r s t h e s h e l l i n l e t below t h e head a t a temperat u r e of 1125OF and a p r e s s u r e of 251.5 p s i , flows around t h e tubes through t h e 18.25-in.-ID

s h e l l , and l e a v e s t h e exchanger a t a temperature of

850째F and a p r e s s u r e of 193.4 p s i .

The r e l a t i v e h e a t t r a n s f e r r e s i s t a n c e s

a t t h e s u p e r c r i t i c a l - f l u i d t u b e - i n l e t s i d e o f t h e exchanger and a t t h e tube-outlet s i d e are t a b u l a t e d below.

Resistances of S u p e r c r i t i c a l Fluid Tube-Inlgt End ( h r *f @ F/Btu) Ri = 1.3 x

lo4 lo4

= 8.8 x

= 1 . 4 x 10-4

W 0

Resistances of S u p e r c r i t i c a l Flu Tube-Ou t k e t End ( h r =f P F/Btu) Ri = 4 . 3 x

Rw = 7.2 x Ro = 3 . 3 x

lo4 loR loA

The stress a n a l y s i s f o r t h e boiler-superheater exchanger involved

a determination of t h e s t r e s s e s produced i n t h e tubes, t h e s h e l l , t h e tube sheets, and i n t h e high-pressure head.

The s t r e s s e s developed i n

t h e tubes t h a t were considered i n t h i s a n a l y s i s are

1. primary membrane s t r e s s e s caused by pressure, 2.

secondary s t r e s s e s caused by t h e temperature g r a d i e n t a c r o s s t h e tube w a l l , and

d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n of t h e tube and tube sheet.

The s t r e s s e s i n t h e s h e l l t h a t were considered i n t h i s a n a l y s i s a r e

1. primary membrane s t r e s s e s caused by p r e s s u r e and 2.

d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n of t h e s h e l l and tube sheet.

The tube s h e e t thickness required f o r t h e allowable s t r e s s under t h e e x i s t i n g temperature and p r e s s u r e conditions was determined, and t h e maximum primary membrane stresses were c a l c u l a t e d f o r t h e high-pressure head. The maximum shear s t r e s s theory of f a i l u r e was used as t h e f a i l u r e c r i t e r i o n , and t h e l i m i t s of stress i n t e n s i t y were determined i n accordance w i t h Section I11 of t h e ASME B o i l e r and Pressure Vessel Code.

The terms

used i n t h e s e c a l c u l a t i o n s are defined i n Appendix F.

S t r e s s e s i n Tubes The s t r e s s e s i n the and d i s c o n t i n u i t y s t r e s s e

bes may be grouped as pressure, temperature,

156 Pressure Stresses.

From page 208 i n Ref. 5, t h e l o n g i t u d i n a l stress

component caused by pressure,

where b = o u t s i d e r a d i u s o f tube = 0.25 in., a = i n s i d e r a d i u s of tube = 0.173 in., Pi = 3800 p s i , ,

Po = 193.4 p s i . P"L

3800(0.02993) (193.4)0.0625 0.03257

= 3121 psi

(tensile)

The hoop stress component caused by pressure,

A t t h e i n s i d e surface of t h e tubes, = 3121

(0.0625) + 3607(0.02993) 0.02993(0.03257)

= 6436 p s i ( t e n s i l e ) ,

and

a t t h e o u t s i d e surface of t h e tubes,

pah = 3121

3607 (0.02993) (0.06251 0.0625(0.03257)

= 6436 p s i ( t e n s i l e )

The r a d i a l stress component a t t h e i n s i d e s u r f a c e o f t h e tubes,

= 3800 p s i (compressive),

and

a t t h e o u t s i d e surface of t h e tubes,

= 193.4 p s i (compressive)

Timoshenko, Strength o f Materials, P a r t 11, 3rd ed., Van Nostrand, New York, 1956.

'S.

157 From page 63 i n Ref. 6, t h e stresses i n t h e

Temperature S t r e s s e s .

tube caused by t h e temperature gradient,

where r i s t h e i n n e r o r o u t e r r a d i u s o p p o s i t e t h a t a t which t h e stress i s being computed.

A t t h e tube-side i n l e t , t h e maximum temperature

g r a d i e n t a c r o s s t h e tube w a l l , Atmax

- - 8*8(850 - 11.5

700) = 115'F

The average temperature of t h e tube w a l l = 700

tav

+ 11.5 5 * 7 (150)

= 774.4OF

For t h i s temperature, t h e c o e f f i c i e n t of thermal expansion, a! = 8.1 x 10" in./in:'F, and t h e modulus of e l a s t i c i t y , E = 27.8 x 106 p s i . Therefore, At'L

8.1(27.8)(115 1.4(0.37) )

= -49,992(1

2p - 0.03257

(0.37)l

- 22.729) .

A t t h e o u t s i d e s u r f a c e o f t h e tube w a l l , r = a = 0.173 and

AtaL -

= -16,008 p s i (compressive)

A t t h e i n s i d e s u r f a c e of t h e tube w a l l , r = b = 0.25 and

L ' t A

-h ' t A

= 20,997 p s i ( t e n s i l e )

Discontinuity Stresses

A t t h e j u n c t i o n of t h e thick-walled tube

and t h e tube sheet, t h e d e f l e c t i o n o f t h e o u t s i d e s u r f a c e o f t h e tube caused by i n t e r n a l p r e s s u r e

age 276 i n Ref. 71, (C.

6J. F. Harvey, P r e s s u r e Vessel Design, Van Nostrand, New J e r s e y , 1963. 7 R . J. Roark, Formulas f o r S t r e s s e s and S t r a i n , 3rd ed., McGraw H i l l , New York, 1954.

158 To maintain c o n t i n u i t y a t t h e tube sheet junction, t h e d e f l e c t i o n and

slope a t t h e j u n c t i o n must be zero.

From Case 11 on page 271 of Ref. 7,

t h e d e f l e c t i o n caused by a moment, M

%=2Dha-, and t h e d e f l e c t i o n caused by a force,

where

D =

EP 12(1

- 3) â&#x20AC;&#x2122;

The slope a t t h e j u n c t i o n must a l s o be zero.

Therefore, t h e change i n

slope caused by t h e f o r c e must be compensated f o r by t h e a c t i o n of t h e moment; t h a t is, from page 271 i n Ref. 7,

To s a t i s f y t h e c o n t i n u i t y requirements, Eqs. C.2 and C.3 must equal Eq. C.1,

S u b s t i t u t i n g f o r F,

7R. J. b a r k , Formulas f o r S t r e s s e s and S t r a i n , 3rd ed., McGraw H i l l , New York, 1954.

159 or

and F = 2AM. From page 271 i n Ref. 7 , the s t r e s s e s caused by M a r e l o n g i t u d i n a l stresses,

ML '

hoop s t r e s s e s caused by lengthening of the circumference,

and t h e hoop s t r e s s e s caused by d i s t o r t i o n ,

From page 27 i n Ref. 5, t h e s t r e s s e s caused by t h e force due t o shortening of t h e circumference,

For t h e boiler-superheater exchanger, b = 0.25 in., a = 0.173 in., T = 0.077 in.,

= 0.3,

and

a + b 2

= 0.2115 in. = r.

3(1 0.09) A = [(0.2115)2(0.77)2]

Therefore, M = 12.59 in.-lb/in.,

and

F = 253.56 l b / i n . The hoop s t r e s s e s a t t h e j u n c t i o n a r e Fh' ; I

M h'

- -2F

A r = 14,027 p s i (compressive) and

--= 2m2r

7017 p s i ( t e n s i l e ) .

1 .

= 10.07 i n .

160 The longitudinal stresses at the junction, - - 6M = 12,741 psi (tensile inside compressive outside) L'M - P

The secondary hoop stress, h'M

%aL

= 3822 psi (tensile inside compressive

= 0.3(12,741)

sec

outside)

Stresses in Shell The stresses in the shell are the primary membrane stresses and the discontinuity stresses at the junction of the shell and tube sheet. Primary Membrane Stresses. =

PD 2T

The hoop stress,

= 251*5(18*25) = 6120 psi (tensile)

2(0.375)

and the longitudinal stress,

L'P

- -PD - 4T

Discontinuity Stresses.

= 3060 psi (tensile)

If it is assumed that the tube sheet is

very rigid with respect to the shell, the resulting moment and force applied to the shell at the junction are M = P/2A2 and F = Pfi where

Therefore, the longitudinal stress in the shell at the junction, = 6M = 11,108 psi (tensile inner surface, compressive

M'L

outer surface)

The hoop stresses in the shell at the junction,

#h h'M

% Azr -PT r =

= 6120 psi (tensile)

uMuL = 3332 psi (tensile inner surface, compressive

sec

outer surface), and h'F

- -2F - T

Ar = 12,240 psi (compressive)

ied

161 S t r e s s i n Tube Sheet The thickness required f o r t h e tube sheet,' T = - -Zdi(p\l/2 !

2 \S!

where T = thickness of tube sheet, in., di

= i n s i d e diameter of t h e pressure p a r t , i n . ,

P = pressure, p s i , S = t h e allowable s t r e s s , p s i , and

Z = a constant.

For S = 16,600 p s i a t 1000째F, Z = 1, and T = 9.125(0.4785)

= 4.366 i n .

S t r e s s i n High-pressure Head The s p h e r i c a l high-pressure head has an i n s i d e radius of 12 i n . and

a wall thickness of 4.5 i n .

The maximum primary membrane s t r e s s e s caused

by t h e i n t e r n a l pressure,

where a = 12 in.,

b = 16.5 i n . and P = 3800 p s i . There fore, P'h

POL

= 5463 p s i ( t e n s i l e ) ,

and Pur = 3800 p s i (compressive). 8Tubular Exchanger Manufacturers Association, Inc., p. 24 i n Standards of Tubular Exchanger Manufacturers, 4 t h ed., New York, 1959.

162

Summary of Calculated Stresses Stresses in Tubes.

The stresses calculated for the tubes are tabu-

lated below.

Stress

Inside Surface

Outside Surface

PL ' P'h par

ML '

M h' M'hsec Fah

3121

10043 -3800 12741

6436

7017

3822

-3822

- 14027

+21000

- 16000 - 16000

27855

-20396

36862

-25620

+12000

L ' t A

Ath' "h

L'=

- 3800

Car

-193.4 12741

-193.4

Therefore, the primary membrane stress intensity,

= 13,843 psi for the inside surface, and

p = 6629 psi for the outside surface. m

The allowable stress intensity,

= P

allowable

= 16,000 psi at 1036'F

for inside surface,

= 13,000 psi at llOO째F for outside surface.

The calculated primary membrane stress intensity plus the secondary stress intensity

Pm + Q = 40,662 psi for inside surface, = 25,427 psi for outside surface.

The allowable primary membrane stress intensity plus the secondary stress intensity, P m + Q = m'

= 48,000 psi at 1036OF for inside surface,

= 39,000 psi at llOO째F for outside surface.

163

S t r e s s e s i n Shell.

The s t r e s s e s c a l c u l a t e d f o r t h e s h e l l are

tabulated below. Inside Surface (psi)

Stress PL ' P'h

(psi)

3060

6120

11108 6120

0 -11108 6120

-3332

- 3332

-251.5

par ML'

M h'

Outside Surface

- 12240 - 3332

- 12240

Fh '

3332

14168 -251.5

CaL

-8084 0

The maximum primary membrane stress i n t e n s i t y i n t h e s h e l l , Pm (max) = 6372 p s i . The allowable stress i n t e n s i t y , Sm = 10,500 p s i a t 1125OF. The maximum c a l c u l a t e d Pm + Q = 14,420 p s i , and t h e allowable Pm + Q = 3sm = 31,500 psi at 1125OF.

S t r e s s i n Tube Sheet.

The maximum stress c a l c u l a t e d f o r t h e 4.75-in.-

t h i c k . t u b e s h e e t i s less than 16,600 p s i .

The allowable stress a t a

temperature of 1000째F = 16,600 p s i . S t r e s s e s i n High-pressure Head. P'L

Ph'

The maximum primary membrane stresses,

= 5463 p s i ( t e n s i l e ) .

The r a d i a l stress, ur = -3800 P s i , and therefore, Pm = 5463

9263 p s i .

The allowable stress, Sm = 16,600 p s i a t 1000째F.

(-3800) =

. 164 Appendix D CALCULATIONS FOR STEAM REHEATER EXCHANGER

The steam reheater exchanger transfers heat from the coolant salt to the exhaust steam from the high-pressure turbine.

The coolant salt

or hot fluid is on the shell side of the exchanger, and the steam or cold fluid passes through the tubes of the exchanger.

Heat-Transfer and Pressure-Drop Calculations

The criteria governing the design for the reheater exchanger stipulated that the coolant salt enter the exchanger at a temperature of 1125'F and leave the exchanger at a temperature of 850째F, a temperature drop of 275OF.

The flow rate for the coolant salt, fh'

Oo8'

lo'' = 1.1 x ldj lb/hr

Certain properties vary with temperature in a manner that is close to being linear.

Therefore, average conditions may be used without serious

error except in calculating pressure loss at the entrance and exit.

The

average conditions given for the coolant salt are tabulated below. Density, p = 125 lb/ft? Viscosity, p = 12 lb/hr*ft Thermal conductivity, k = 1.3 Btu/hr*f?*OF per ft Specific heat, C = 0.41 Btu/lb-'F P Prandtl number, N = 3.785 Pr The conditions stipulated for the steam in the exchanger are tabulated below.

Temperature, F Pressure, psia Specific volume, ft? /lb Enthalpy, Btu/lb

Inlet

Outlet

Chanze

650 580 1.0498 1322.3

1000 540 1.5707 1518.6

+350

-40 +O .5209

+196.3

165 The flow rate for the steam, wcf

- 5*50

'06

= 0.63 x 106 lb/hr

The average conditions given for the steam in the exchanger were 1. viscosity, p = 0.062 lb/hr'ft, per ft, 2. thermal conductivity, k = 0.036 Btu/hr*f?"F 3.

specific volume, v = 1.3103 ft?/lb,

and

4. NPr = 1.0. c

The material to be used for the tubing was specified as Hastelloy N, and the tubes were to have an outside diameter of 3/4 in. and a wall thickness of 0.035 in. From this information, the pertinent diameters and areas tabulated below were calculated. do = 0.0625 ft = 0.05667 ft di Ai = 0.002522 f? per tube a = 0.1964 f?per ft of tube length 0

ai = 0.178 fi? per ft of tube length a = 0.187 f? per ft of tube length m The selection of the type exchanger to be used to reheat the exhaust steam was based on the data published by Kays and L0ndon.l Our study of the principles discus.sed led to the selection of a straight counterflow unit as being the best suited to meet the requirements for this particular application. The design for this vertical shell-and-tube counterflow exchanger with disk and doughnut baffles is illustrated in Fig. 8 of Chapter 7. The inside diameter of the shell is 28 in., and the exchanger contains 628 tubes on a triangular pitch of 1 in.

Heat-Transfer Calculations Heat Transfer Coefficient Inside Tubes. Equation 2 of Chapter 3 was used to determine the heat transfer coefficient inside the tubes. 'W. M. Kays and A. L. London, Chapter 2 in Compact Heat Exchangers, 2nd ed., McGraw Hill, New York, 1964.

166

= 0.023

.' (Npr)'

;i-(NRe)O

The Reynolds number,

d,G

where

0*63 '06 = 3.98 x 628 (0.002522)

105 l b / h r * f $

and

0 036 hi = 0.023 o,i5667(3.64 x = 409 Btu/hr*ft? -OF

105)'

(1)'

The v e l o c i t y o f t h e steam through t h e tubes,

v = - =GcfV 3600

3.98 x 1@(1.3101 = 145 ft/sec 3600

A modification of Eq. 5 i n

Heat-Transfer Coefficient Outside Tubes.

Chapter 3 was used t o determine t h e s h e l l - s i d e h e a t t r a n s f e r c o e f f i c i e n t f o r t h i s cross-flow exchanger w i t h no i n t e r n a l leakage and d i s k and doughnut b a f f l e s .

The opening i n t h e doughnut b a f f l e w a s sized so t h a t t h e

flow p a r a l l e l t o t h e tubes i s turbulent; t h a t is, NRe was set a t 7500 on t h e b a s i s o f t h e o u t s i d e diameter of t h e tubes.

The d i s k

b a f f l e was

s i z e d so t h a t t h e flow around i t p a r a l l e l t o t h e tubes i s turbulent, and t h e b a f f l e s were spaced so t h a t the flow of t h e coolant s a l t outward from t h e c e n t r a l p a r a l l e l - f l o w region i s t h e same as t h e flow i n t h e p a r a l l e l flow regions. To determine t h e diameter o f t h e hole i n t h e doughnut b a f f l e , dG

o w = 7500 v

0 .0625Gw 12

when

Gw w'

-- 75000 = 0.0625

1*44

1.44 x

106 l b / h r * f ? and

'06 = 3.2 f t / s e c .

125(3600)

167 The flow a r e a through t h e window region,

- a r e a of tubes Ird doughnut hole)2 - nd - - - 0.764

AW = area of doughnut opening =

~r ( r a d i u s

where n = t h e number of tubes through t h e doughnut opening. The a r e a d a s s o c i a t e d with each tube i n t r i a n g u l a r p i t c h can be expressed a s 0.866p".

Therefore,

Ir(radius)2 = nd(0.866$) and nd(0.866p2)

ndo2 - nd - 0.764 4

0.764(144) 0.866 0.442

'lo - 0.424

( r a d i u s doughnut hole)'

ff?

259 tubes

259 (0.866) 3.14

r a d i u s doughnut h o l e = 8.46 i n . and Dd = 16.92 i n .

The maximum number of r e s t r i c t i o n s o r tubes i n t h e cross-flow region through t h e i n n e r window,

16.92 = 9.07 i w - 2(0.933)

The number used w a s 9.

The b a f f l e spacing w a s based on t h e a r b i t r a r y requirement t h a t GB = G

o r AB = AW

Where X = t h e b a f f l e spacing,

A -W- - zDd X

- ndo

when n i s t h e number of tubes i n a band around t h e edge of t h e doughnut opening.

The band width i s equal t o t h e p i t c h of t h e tubes, and t h e a r e a

168 o f the band i s approximately arDdp. i n g t h e number of t h e tubes,

An approximate equation f o r determin-

Then

-X = ID W

and t h e b a f f l e spacing, A

X =

I nDd\l

d - L] 0.9D

0.764 (144) 3.14(16.92)(0.167)

= 12.4 in.

To determine the o u t s i d e diameter of the d i s k b a f f l e s , the flow a r e a o u t s i d e t h e disk, Aw = c r o s s - s e c t i o n a l a r e a o f t h e s h e l l minus t h e a r e a of t h e d i s k minus t h e c r o s s - s e c t i o n a l a r e a of t h e tubes i n t h e o u t e r window region. 14

AW =

“(i?) -

n ( r a d i u s disk)’

- n o w“doa (T) .

The number of tubes i n t h e o u t e r window region, 2

= 628

n ow

-7 ‘\!)disk 866 = 628 -

14);

144x 0.866 i sk ?

. Therefore,

= 40.747

- o.49(i);isk]

0.49(qJadisk

= 6.747

0.764 -Y r ’

,r-.

169 a

= 1.028

r a d i u s d i s k = 1.014 f t = 12.17 i n . diameter d i s k = 2 4 . 3 i n . The maximum number of r e s t r i c t i o n s i n the cross-flow region o f t h e o u t e r window, r

- 28 - 0 . 9 3 3 2( 24)* 3 = 1 . 9 8 (use 2 )

Where t h e Reynolds number = 7500, t h e h e a t t r a n s f e r f a c t o r f o r t h e

cross-flow area, 1

JB = 0.346(NRe)-0*38a = 0.0115

and

Since we a r b i t r a r i l y set GB = G W

, the

bracketed f r a c t i o n i n t h e above

equation becomes 1 and

Therefore, t h e s h e l l - s i d e heat t r a n s f e r c o e f f i c i e n t ,

= 2800 Btufhr- f ? *OF

i s assumed t o be t h e same f o r c r o s s low through t h e windows of t h e b a f f l e s . leakage flow a r e a

The average

170 according t o Bergelin e t a1.,2 t h i s makes i t necessary t o multiply t h e heat t r a n s f e r c o e f f i c i e n t by a c o r r e c t i o n f a c t o r of 0.8. h

= 0.8(2800)

= 2240 Btu/hr*f$-째F

Therefore,

The log mean temperature drop across

Determination of Tube Length. t h e exchanger,

- 650) -

850 AtLrn = ( = 160째F

= - =Qt At-

(1125

200 125

..1

1000)

-- 75 - 0.470

1*245 IOe = 7.78 x 105 Btu/hr*OF 160

Lrn

-1 -

Uat

1 hiainL

+ -kamL +T

1 hoaonL

and solving f o r L,

Uat

L=--

n [hiii

where 1

- =L

-k a T m

10.035\ ! 12-1 =

= 0.00150

0.187

1 7*78 628 105[40S(0.178)

= 21.70 f t

-a kT m

+ -

+&] , 0 0

. +

0*00150

2240(0.1964)

This tube-sheet-to-tube-sheet

. tube length was increased t o 22.1 f t

t o allow room f o r t h e i n l e t and o u t l e t p i p e s on t h e s h e l l s i d e o f t h e exchanger. i

This minute change i n length was neglected i n t h e pressure-

drop c a l c u l a t i o n s , but i t was used t o c a l c u l a t e t h e o v e r a l l h e a t t r a n s f e r coefficient.

The t o t a l o u t s i d e h e a t t r a n s f e r s u r f a c e a r e a of t h e tubes,

' 0 . P. Bergelin, K. J. B e l l , and M. B. Leighton, "Heat Transfer and Fluid F r i c t i o n During Flow Across Banks of Tubes -VI: The E f f e c t of I n t e r n a l Leakages Within Segmentally Baffled Exchangers," Trans. ASME, 80: 53-60 (1958).

171

ii w

= nrtd

= 628rt(0.0625)(22.1) = 2725 f ?

There fore, 272521 = 7.78 x 1O5

U = 285 Btu/hr'f?

Pressure-Drop C a l c u l a t i o n s From t h e work published by Perry,3

P r e s s u r e Drop I n s i d e Tubes.

the

entrance loss,

(

Mi = 0 . 4 1 25

vr" - ::.)2gC(144)

The v e l o c i t y of t h e steam a t t h e i n l e t ,

and

-*it - ss

628 (0.002522) = 1.584 = 0.37 4.28

Therefore, t h e entrance loss, mi

0.4(1.25 0.37)(115.5)2 = 64.4(1.0498) (144) = 0.4825 p s i

The k i n e t i c energy loss,

ci Gcf2(vco 2gc(3600)2 ( 1 4 4 )

(3.98 x 1@)2(0.5209) 64.4(3600)2 ( 1 4 4 )

= 0.686 p s i The f r i c t i o n loss, 4fLv'l?p Mi = 2g d ( 1 4 4 ) c i

0 . 0 1 5 ( 2 1 . 7 ) (145)2 ( 0 . 7 6 3 ) 6 4 . 4 ( 0 . 0 5 6 6 7 ) (144)

= 9.93 p s i .

3 J . H. Perry, E d i t o r , Chemical Engineer's Handbook, 3rd ed., McGraw H i l l , New York, 1950. i

172 The e x i t l o s s ,

a - 0 . W 2 (145 1.5707 1.3103)

2(144) (32.2) (1.5707)

0.825 p s i

The t o t a l pressure drop i n s i d e t h e tubes i s t h e sum of t h e entrance, k i n e t i c energy, f r i c t i o n , and e x i t l o s s e s . psi

Entrance l o s s K i n e t i c energy l o s s Friction loss Exit loss Total mi Pressure Drop Outside Tubes.

0.482 0.686 9.930 0.825 11.923 o r 12.0 p s i

The number o f b a f f l e s on t h e s h e l l

s i d e of t h e exchanger,

N =

2 1.7 (12) - l = 2 0 , 12.4

and t h e number of cross-flow r e s t r i c t i o n s ,

24*3 l 6 a 9 0.933(2)

= 3.97 (use 4)

The pressure drop caused by t h e cross-flow r e s t r i c t i o n s 0

= 6.96 psi

The

pressure l o s s through t h e i n n e r window region o r t h e h o l e s i n

t h e doughnut b a f f l e s ,

= 10[2 + 0.6(9)]0.138

= 10.2 p s i

The pressure l o s s i n t h e o u t e r window region around t h e d i s k baffles,

173

APo = lO(2 + 0.6rw) = 10[2

VB”P 2g, (144)

0.6(2)]0.138

= 4.42 p s i

The t o t a l s h e l l - s i d e pressure drop equals t h e sum of t h e pressure losses. psi

Cross-flow l o s s Inner window l o s s Outer window l o s s Total APo

6.96 10.20 4.42 21.58 p s i

Using t h e f a c t o r of 0.53 from t h e d a t a published by Bergelin e t al.,a

the

t o t a l s h e l l - s i d e pressure drop,

APo = 21.58(0.53)

= 11.4 p s i

S t r e s s Analysis f o r Case B

The oper t i n g p r e s s u r e s of t h e coolant s a l t were r a i s e d f o r t h e Case-B design of t h e heat-exchange system.

The coolant s a l t e n t e r s t h e steam r e h e a t e r exchanger a t t h e same temperature (1125OF) but a t a pres-

s u r e of 208.5 p s i , and i t leaves t h e exchanger a t t h e same temperature (850’F)

but a t a pressure of 197.1 p s i .

The steam e n t r y and e x i t temperat u r e s and p r e s s u r e s a r e t h e same f o r Case B a s they a r e f o r Case A. The steam e n t e r s t h e exchanger a t a temperature of 650°F and a p r e s s u r e of 580 p s i , and i t leaves t h e p r e s s u r e of 568 p s i .

xchanger a t a temperature of l00O0F and a

The same type v e r t i c a l shell-and-tube counter flow

exchanger with d i s k and doughnut b a f f l e s i s used f o r Case B.

The s h e l l of t h e exchanger has an i n s i d e diameter of 28 in. and a w a l l thickness of 0.5 i n .

There are 628 tubes w i t h an o u t s i d e diameter o f 3/4 i n . and a

w a l l thickness of 0.035 i n , arranged i n a t r i a n g u l a r a r r a y on a 1-in. p i t c h i n t h e exchanger. The stress a n a l y s i s f o r t h e C a s e - B exchanger involved a determination of t h e s t r e s s e s produced i n t h e tubes, s h e l l , and t h e tube sheets.

The shear s t r e s s theory of f a i l u r e was used a s t h e f a i l u r e c r i t e r i o n , and t h e

174 stresses were c l a s s i f i e d and t h e l i m i t s of i n t e n s i t y were determined i n accordance w i t h Section I11 of t h e ASME B o i l e r and P r e s s u r e Vessel Code.

Stresses i n Tubes

The stresses produced i n t h e tubes are

primary membrane stresses caused by t h e steam pressure,

secondary stresses caused by the temperature g r a d i e n t a c r o s s the

w a l l of t h e tube,

d i s c o n t i n u i t y s t r e s s e s a t the j u n c t i o n of t h e tube and tube sheet, and

secondary s t r e s s e s caused by t h e d i f f e r e n c e i n growth between t h e tubes and t h e s h e l l . Primary Membrane Stresses. Pdi h'

The hoop stress component,

3880 =

= 2(0.035)

3769 p s i ( t e n s i l e )

Assume a l o n g i t u d i n a l stress caused by t h e p r e s s u r e drop only.

The

l o n g i t u d i n a l stress a t t h e p o i n t where t h e steam e n t e r s t h e tubes, Pdi

L' i

-- - 4T - -1 20= 4(0.035)

58.2 p s i ( t e n s i l e )

Secondary S t r e s s e s Caused by Temperature Gradient.

. The secondary

s t r e s s e s caused by t h e temperature g r a d i e n t across t h e tube w a l l are l o n g i t u d i n a l , &aL, and hoop, Atah* From page 63 of Ref. 4, t h e stresses a t t h e i n s i d e s u r f a c e of t h e tube caused by t h e temperature gradient,

and t h e stresses a t t h e o u t e r s u r f a c e of t h e tube,

where k = a constant.

A t t h e steam e n t r y i n t o t h e tubes, t h e relative

*J. F, Harvey, Pressure Vessel Design, D. Van Nostrand Company, New Jersey, 1963.

175

h e a t t r a n s f e r r e s i s t a n c e i n s i d e t h e tube, Ri = 86; t h e r e s i s t a n c e a c r o s s

t h e tube w a l l ,

14; and t h e r e s i s t a n c e o u t s i d e t h e tube, Ro = 51.

Therefore, t h e temperature g r a d i e n t a c r o s s t h e tube a t t h e steam entrance AtT =

l4 (200) 151

= 18.5OF

The i n s i d e r a d i u s of t h e tube, a = 0.34 i n . , t h e tube, b = 0.375 i n .

and t h e o u t s i d e r a d i u s of

A t t h e p o i n t where t h e steam e n t e r s t h e tubes,

= t h e mean temperature o f t h e tube = 773'F, tm E = t h e modulus o f e l a s t i c i t y = 27.8 x 106 p s i , CII =

k =

c o e f f i c i e n t o f thermal expansion = 8.1 x 033

2(1

- u)(ln

b/a)

in./in.*

8.1 x 10*(27.8 x 106) = 1641 1.4(0.098)

F, and

The stresses a t t h e i n s i d e s u r f a c e of t h e tube, At'L

- At'h

= 1641(18.5)b

- ::i2

(0.098)]

= -3154 p s i ;

and t h e stresses a t t h e o u t s i d e s u r f a c e o f t h e tube, At'L

At'h

= 1641(18.5) 11 = 2845 p s i

Discontinuity Stresses.

:::$2(0.098)l

Since t h e tube s h e e t i s very r i g i d w i t h

r e s p e c t t o t h e tube, assume t h a t t h e r e i s no d e f l e c t i o n o r r o t a t i o n o f t h e j o i n t a t t h e j u n c t i o n of t h e tube and tube sheet.

Therefore, t h e

d e f l e c t i o n produced by the moment M and the f o r c e F must be equal t o t h e d e f l e c t i o n , A, produced by t h e p r e s s u r e load, and t h e s l o p e a t t h e j u n c t i o n must remain zero.

The d e f l e c t i o n caused by d i f f e r e n t i a l pressure,

From pages 12, 126, and 127 i n Ref. 5, t h e d e f l e c t i o n s caused by F and M y

5 S . Timoshenko, S t r e n g t h o f Materials, P a r t 11, 3rd ed., D. Van Nostrand Co., New York, 1956.

176

and

% = - 2Dh2 ’ where

D =

h = S e t t i n g t h e d e f l e c t i o n caused by pressure equal t o t h e d e f l e c t i o n caused by M and F,

-PI? -

F 2

- AM~

The slope produced by F must be counteracted by M; t h a t is,

Substituting for

and solving t h e s e l a s t two equations f o r M and F,

M = P ~ a n d F =P - . h 2A The o u t s i d e diameter of t h e tube i s 0.75 in., and t h e mean diameter, d

= 0.715 in.

t h e w a l l thickness i s 0.035 in.,

Therefore,

The moment,

388 = 2(132.05)

= 1.468 in.-lb,

and t h e force, F =

388 = 33.74 11.5

From page 271 i n Ref. 6 , t h e hoop s t r e s s e s , 2F T hr

F‘h

M h‘

- 2 A2r - T

Z(33.74) (11.5) (0.3575)

= -7926 psi,and

(0.035)

= 2(1.468)(132.05)(0.3575)

= 3960 psi

(0.035)

6R. J. b a r k , Formulas f o r S t r e s s e s and S t r a i n , 3rd ed., McGraw H i l l , New York, 1954.

177

The longitudinal stresses, L'M

-- 6M

97190 psi (tensile inside, compressive (0.035) outside) . =

The secondary hoop stresses, h'M

= uMoL = 0.3(7190)

= 2157 psi (tensile inside, compressive

outside)

Secondary Stresses Caused by Growth Difference. Unrestrained differential growth arises from two sources:

(1) temperature differences

between the tubes and the shell and (2) pressure differences between the tubes and the shell. The differential growth caused by temperature differences was calculated first. The mean temperature of a tube, tm(tube)

= t

+ -93 151

steam

(tsalt

tsteam)

where the fraction 93/151 represents the ratio of the heat transfer resistance of the steam side plus one-half the tube wall resistance to the total resistance. Therefore, the mean temperature of a tube at the steam inlet, tm( tube)

= 650

91 (200) +151

= 773OF ;

and the mean temperature of a tube at the steam outlet, tm( tube)

= 1000

93 (125) +151

= 1077OF

Thus, the average temperature of the tubes is 925OF, and the average

unrestrained tube expansion per inch of length,

= 6.216 x 10"

x lo4)

= 855(7.27

in./fn.

It was assumed that the temperature of the shell is the same as the temperature of the coolant salt flowing through it. The average temperature of the shell is and the average unrestrained growth of the shell per inch of length, tES = (988

70)70ag,8

= 6.793 x

lo9

= 918(7.4

in./in.

loA)

178 Therefore, the unit unrestrained differential growth caused by tempera-

ture differences, =

â&#x201A;Ź

t s

- T't

= 6.793 x 10-

= 0.577 x lo9

6.216 x 10"

in./in.

The unit unrestrained differential growth caused by pressure differThe unit unrestrained longitudinal expansion

ences was calculated next.

of the shell caused by pressure,

where q = the longitudinal stress caused by pressure and

= the hoop stress caused by pressure.

PES

200 (28 (0.5 2(0.5)(26.: x 106)

= 0.042 x 10"

- 0.3)

in./in.

Since the longitudinal stresses in the tubes are only 194 psi, they were assumed to be zero.

The unit unrestrained longitudinal expansion of

the tubes caused by pressure, PET -

uPdo

-0.3 (388) (0.68) x 106)

-2TE = 2(0.035)(27

= -0.0419 x lo3

in./in.

Therefore, the unit unrestrained differential growth caused by pressure,

= s'0

- PET = (0.042 x 103) -

= 0.0839 x lo9

(-0.0419 x 10")

in./in.;

and the total unrestrained differential growth, "total

=tAâ&#x201A;Ź+#k = (0.577

+ 0.0839)109

= 0.6609 x lo9

in./in.

Actually, the growth is restrained and the load induced in the shell by the total differential growth must equal the load induced in

179 t h e tubes.

The sum of t h e d e f l e c t i o n s o f t h e tubes and of t h e s h e l l

caused by t h e induced load must equal t h e d e f l e c t i o n caused by t h e d i f f e r e n t i a l growth; t h a t is,

Fs = FT

- - +1 -

Aâ&#x201A;Ź

%ET The a r e a of a 3/4-in.-OD 0.0786 in,a

AsEs

tube with a w a l l thickness of 0.035 i n . i s

Therefore, t h e t o t a l a r e a of t h e tubes,

628(0.0786) = 49.4 in.2

The a r e a of t h e s h e l l ,

= n(28.50)(0.5)

= 44.77 in.2

t h e modulus of e l a s t i c i t y of t h e tube,

A t a temperature of 925'F,

ET = 27 x 106 p s i ; and a t a temperature of 988'F,

t h e modulus of

e l a s t i c i t y of t h e s h e l l , Es = 26.7 x 106 p s i . FT = Fs

0.6609 x 1 49.4(27 x

106)

= 4.16 x 1O5 l b

lo9

+ 44.77(26.71

x 106)

The l o n g i t u d i n a l s t r e s s e s i n t h e tubes caused by t h e r e s t r a i n e d d i f f e r e n t i a l growth, LkUL

4*16 49.4

8440 p s i

Stresses i n Shell The s t r e s s e s produced i n t h e s h e l l a r e

1. primary membrane s t r e s s e s caused by t h e p r e s u r e of t h e coolant s a l t , 2.

secondary s t r e s s e s ca

d by t h e d i f f e r e n c e i n growth between t h e

tubes and t h e

d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n of t h e s h e l l and t h e tube sheet.

180

The primary membrane stresses are the

Primary Membrane Stresses. hoop stress,

and the lonitudinal stress, - - PDS =

2919 psi

uL - 41: Differantial Growth Stresses.

The longitudinal stress in the shell

caused by the difference in growth between the tubes and the shell, AE

‘L

= - 4.16 x 105 = -9300 psi 44.77

It was assumed that the tube sheet is very rigid with respect to the shell. Therefore, the deflection and rotation Discontinuity Stresses.

of the shell at the junction of the shell and tube sheet are zero.

moment and force,

The

P P M = -2 and F = 2A A ’ where

12(1D E Pg)]’/’ =

= 0.482

From page 271 in Ref. 3, the longitudinal stresses,

MuL =

60- 10,784 2 (0.232); 1 -

psi (tensile inside, compressive outside) ;

and the hoop stresses, = %uh

= 0.3(10784)

= 3235 psi (tensile inside, compressive

M‘hsec M‘h

-w

F‘h

outside),

r = 208*5(14*25L= 5942 psi (tensile), 0.5

- -2nr - - 2(208*5)(14*251 T

0.5

= 11,884 psi (compressive).

181

S t r e s s e s i n Tube Sheet The load on t h e tube sheet comes from two sources:

(1) t h a t caused

by p r e s s u r e and (2) t h a t caused by t h e d i f f e r e n t i a l growth between t h e tubes and t h e s h e l l .

Assume t h a t t h e load caused by d i f f e r e n t i a l growth

can be converted i n t o an equivalent pressure,

& e

t o t a l force

= tube-sheet a r e a

4*16 IO5 = 675 p s i (14)â&#x20AC;&#x2122; YI

fore, t h e t o t a l e f f e c t i v e pressure, 378

192.1 = 378 p s i . There675 = 1063 p s i . From page 24

The n e t a c t u a l p r e s s u r e on t h e tube sheet, 570 c

of Ref. 7, based on bending t h e tube sheet thickness,

where

Ds = i n s i d e diameter of pressure p a r t f o r s t a t i o n a r y

i n t e g r a l tube

sheet,

P = pressure, p s i , S = allowable working stress = 10,500 p s i a t a temperature of 1125OF, and

k = a constant = 1 f o r T/D

T =

< 0.02.

= 14(0.03162) m(efâ&#x20AC;&#x2122;2 2 10500

= 4.43 i n .

A tube s h e e t t h i c k n e s s of 4.5 i n . was used.

Summary of Calculated S t r e s s e s The maximum shear stress theory of f a i l u r e was used as t h e f a i l u r e c r i t e r i o n , a n d - t h e s t r e s s e s were c l a s s i f i e d and l i m i t s of stress

7Tubular Exchanger Manufacturers Association, Inc., Standards of Tubular Exchanger Manufacturers, 4 t h ed., New York, 1959.

182

i n t e n s i t y were determined i n accordance with Section I11 o f t h e ASME Boiler and Pressure Vessel Code. Calculated S t r e s s e s i n Tubes.

The c a l c u l a t e d s t r e s s e s i n t h e tubes

a r e tabulated below. S t r e s s on I n s i d e Surface

S t r e s s on Outside Surface (psi)

(psi)

Type

58.2

PL '

3769

P'h

- 580

-208.5

8440

-7926

7926

3960

2157

7190

7 190

-3154

+2845

-3154

+2845

12534

4153

- 1167

+491

Pur h'L

Fh' e h

h'M

sec

L'M At'L At'h CaL

"h "r

- 580

-208.5

The c a l c u l a t e d primary membrane stress i n t e n s i t y i n the tubes, Pm = 3769

(-580) = 4349 p s i .

The allowable Pm a t a temperature o f 1077OF = 14,500 p s i .

The c a l c u l a t e d

primary membrane stress i n t e n s i t y p l u s the secondary stress i n t e n s i t y , Pm +

- 'min = 12534 - (-1167) =

umax

A t a temperature of 1077'F S

= 14,500 p s i .

t h e allowable stress i n t e n s i t y f o r Hastelloy N,

The allowable P rn P m

= 13,701 p s i

+ Q f o r t h e tubes,

= 3s = 43,500 p s i m

183 Calculated S t r e s s e s i n Shell.

The s t r e s s e s c a l c u l a t e d f o r t h e

s h e l l are tabulated below. S t r e s s On I n s i d e Surface (PS 5)

5Pe P‘L P‘h

S t r e s s On Outside Surface (PS i)

2919

5838

-9300

A€ ‘L

F‘h

-9300

- 11884

5942

3235

-3235

10784

440 3

- 1 7165

3131

-3339

M h‘

M‘L

cuL “h

-208.5

“r

The c a l c u l a t e d primary stress i n t e n s i t y i n t h e s h e l l , P = 5838 m

A t a temperature of 1125’F,

(-208.5)

= 6046.5 p s i

t h e allowable Pm = 10,600 p s i .

The calcu-

l a t e d primary p l u s secondary s t r e s s i n t e n s i t y i n t h e s h e l l , m

= ‘max

= 0

- ‘min (-17165) = 17,165 p s i

A t a temperature of 1125 F t h e allowable stress i n t e n s i t y , Sm = 10,600 p s i . The allowable Pm + Q = 3Sm = 31,800 p s i Calculat,ed S t r e s s i n Tube Sheet. i n t h e tube sheet

< 10,500

The maximum c a l c u l a t e d stress

p s i , and t h e allowable stress = 10,500 p s i .

184 Appendix E CALCULATIONS FOR REHEAT- STEAM PREHEATER

Heat i s t r a n s f e r r e d from t h r o t t l e steam o r s u p e r c r i t i c a l f l u i d i n

t h e reheat-steam p r e h e a t e r t o h e a t t h e exhaust steam from t h e highp r e s s u r e t u r b i n e before i t e n t e r s t h e r e h e a t e r .

The exchanger s e l e c t e d

i s a one-tube-pass one-shell-pass counterflow type with U-tubes and a

U-shell,

as i l l u s t r a t e d i n Fig. 9 of Chapter 8.

There a r e 603 tubes

with an o u t s i d e diameter o f 3/8 i n . and a w a l l thickness o f 0.065 i n . on

a t r i a n g u l a r p i t c h o f 3/4 i n . i n t h e exchanger, and no b a f f l e s are used. The s u p e r c r i t i c a l f l u i d flowing through t h e tubes e n t e r s t h e

exchanger a t a temperature of 1000째F and leaves a t a temperature o f 869OF. The reheat steam e n t e r s t h e s h e l l s i d e of t h e exchanger a t a temperature of 551째F and l e a v e s a t a temperature of 650째F.

The terms used i n t h e

following c a l c u l a t i o n s a r e defined i n Appendix F.

Heat-Transfer and Pressure-Drop C a l c u l a t i o n s

The p r o p e r t i e s of the s u p e r c r i t i c a l f l u i d flowing through t h e tubes vary almost l i n e a r l y because the f l u i d does not temperature range.

p a s s through t h e c r i t i c a l

Therefore, average c o n d i t i o n s were used, and t h e

i n l e t , o u t l e t , and average values o r change of t h e p r o p e r t i e s of t h e s u p e r c r i t i c a l f l u i d t h a t t h e s e c a l c u l a t i o n s were based on are t a b u l a t e d below.

1421.O 0.1988 0.76 0.078 0.066

Outlet 869 3550 1306.8 0.1620 1.07 0.074 0.066

0.90

1.2

Inlet 0

Temperature, F Pressure, p s i Enthalpy, Btu/lb S p e c i f i c volume, f t? / l b S p e c i f i c heat, Btu/lb*OF Viscosity, Lb/hr* f t Thermal conductivity, Btu/hr*f?*OF p e r f t P r a n d t l number

1000

3600

Change o r Average At = 131

AP = 50 AH = 114.2 Vav = Cp av k v = kav =

0.18 = 0.91 0.076 0.066

185 The flow r a t e of t h e s u p e r c r i t i c a l f l u i d i n t h e tubes, 'hf

2*94 8

and t h e t o t a l heat t r a n s f e r

Qt =

'06

= 3.68 x

105 l b / h r

rate,

WhpH

= 3.68 x W ( 1 1 4 . 2 )

= 4.21 x

lo7

Btu/hr

The flow rate of t h e reheat steam on t h e s h e l l s i d e of t h e exchanger,

Wcf

= 6.31 x

106 lb/hr.

The i n l e t , o u t l e t , and average values o r change

of t h e p r o p e r t i e s of t h e reheat steam t h a t t h e s e c a l c u l a t i o n s were based on a r e t a b u l a t e d below.

Temperature, F Pressure, p s i Enthalpy, Btu/lb S p e c i f i c volume, f $ l b S p e c i f i c h e a t , Btu/lb*OF Viscosity, l b / h r * f t Thermal condugtivity, Btu/hr'fl?* F p e r f t P r a n d t l number

Inlet

Outlet

551 600 1256.2 0.8768 0.72 0.048 0.029

650 580 1322.8 1.0498 0.60 0 .OS4 0.032

1.19

1.01

Change o r Average

A t = 99 20 AH = 66.6 vav = 0.9633 Cp av = 0.66 pav = 0.051 kav = 0.0305 hp =

Heat Transfer C a l c u l a t i o n s The log mean temperature d i f f e r e n c e f o r t h e exchanger,

The h e a t t r a n s f e r s u r f a c e a r e a s were e s t a b l i s h e d from t h e physical d a t a f o r t h e tubes. = 0.065 in.,

a n d ' t h e i n s i d e diameter = 0.245 i n .

= 0.0313 f t ,

d . = 0.0204 f t , 1

The tube o u t s i d e diameter = 0.375 in., w a l l thickness Therefore,

186 a0 =

0.0982 f e / f t ,

Xdo =

a i = Pdi = 0.0641 € $ / f t , a a

+ a

and

= 0.0812 f ? / f t .

Heat Transfer C o e f f i c i e n t I n s i d e Tubes.

From Eq. 2 i n Chapter 3, the

heat t r a n s f e r c o e f f i c i e n t i n s i d e t h e tubes, hi = 0.023

k r(NRe)O

(Npr)O

Based on the i n s i d e diameter of t h e tube, t h e Reynolds number, NRe =

diGT Ir

where di = 0.0204 ft, p = 0.076,

and

-- -'hf

Ait

The t o t a l i n s i d e flow area, Ait

= n

ndi2 = 603(0.000327) 4

= 0.197 f t "

Therefore,

= 3*68 '05 = 1.868 x 108 l b / h r * f t ? 0.197

and NRe

- 0.0204(1.868 x 108) = 5.01 0.076

105

The h e a t t r a n s f e r c o e f f i c i e n t i n s i d e the tubes, hi = 0.023

::i84

(3.63 x 1@)(1.0197)

= 2750 Btu/hr-fP-'F

The v e l o c i t y of the s u p e r c r i t i c a l f l u i d through the tubes,

1.868 x 3600

106

(0.18) = 93.5 € t / s e c

187 The i n s i d e diameter of

Heat Transfer C o e f f i c i e n t Outside Tubes. the s h e l l , Ds = 20.25 i n . ,

and since t h e r e a r e no b a f f l e s used i n the

exchanger,

s - s

'(20.25)a [z) 144

603(0.000767)

= 1.773 f e

and t h e s h e l l - s i d e mass v e l o c i t y ,

- 6*31 lg = 3.56 x 105 l b / h r * f $ 1.773

I f Eq. 2 of Chapter 3 i s used, t h e heat t r a n s f e r c o e f f i c i e n t o u t s i d e t h e tubes,

h0 = 0.023 ~ ( N ~ 08~(N)P rO 10 4 e where

k = 0.035 and (Npr)'

= (1.10)'

= 1.039.

The equivalent diameter, 'e

4(1.773) 603(0.0982)

0.25)= + ~ ( 212

64.4

= 0.110 f t

and t h e Reynolds number,

(N,)'

= 0.023

= 5.1

'i,")",'

104

(5.1 x 1@)(1.039)

= 338 3tu/hrDft?*OF

However, i f Eq. 12 of Chapter 3 i s used, t h e s h e l l - s i d e heat t r a n s f e r coefficient, h0

d = O . l 6 kd0 (*) = 0.16

60.6

(-a)

~0'33

0.0305 0.0313(3.56 x 18))' 0.0313 0.051

[

= 257 Btu/hr'ft? *OF

(1.032)

w *

The more conservative value given by Eq. 12 w a s used. LenRth of Tubes.

To determine t h e length of t h e tubes required f o r

t h e exchanger,

- - -Qt - 4*21

lo7= 1.266 x 105 Btu/hr*OF 333

- AtLm

-1 Uat

1 hoaonL

+-a,nL 94 +

and n

% = 5 = 0*065 12(12)

= 4.51 x

hiai

lo4.

1 L = 1.266 x 105 60 3 10.0982 (257) = 10.68 f t

+-+A), Rw

where

1 hiainL

4.51 x lo4 0.0812

0.0641(2750)

This a c t i v e length of 10.68 f t w a s increased by 2.52 f t t o allow room f o r t h e s h e l l - s i d e i n l e t and o u t l e t pipes, giving a t o t a l tube-sheetto-tube-sheet

tube length of 13.2 f t .

The o v e r a l l h e a t t r a n s f e r c o e f f i c -

i e n t was c a l c u l a t e d on t h e b a s i s of t h e t o t a l o u t s i d e h e a t t r a n s f e r surface a r e a o f t h e tubes, a

= nndoL = 603n 0.0313(13.2) = 781 ftZ

There fore, 781U = 1.266 x

105

U = 162 Btu/hr*f$

The d i s t r i b u t i o n of f i l m r e s i s t a n c e s and temperature drops a t t h e hot end of t h e exchanger are tabulated below.

Outside f i l m Tube w a l l Inside film Total

Resistance (%) 77.9 10.9 11.2 100 .o

38 89 350

189 P r e s s u r e Drop C a l c u l a t i o n s On t h e b a s i s o f a tube l e n g t h of 13.2 f t , t h e p r e s s u r e drop i n t h e tubes,

= -1.07

(12.6)5.24 = 65 p s i

C a l c u l a t i o n of t h e s h e l l - s i d e p r e s s u r e drop was based on t h e a c t i v e tube l e n g t h of 10.68 f t .

This p r e s s u r e drop o u t s i d e t h e tubes,

[ 3*5&37a (0.173)

[0.0~~~0.68)

64.4(144)

= 0.183

+ (5.165)(1.015)

= 5.43 p s i

\ 3 * 5 ~ 6 ~ 0 1 0 5 ](0.9633) a

64.4(144)

S t r e s s Analysis

The m a t e r i a l s e l e c t e d f o r u s e i n t h i s exchanger was Croloy (2.25% C r , 1% Mo, and 96.75% Fe) SA 387 Grade D f o r t h e p l a t e and SA 199 Grade T22 f o r t h e tubes.

The stress a n a l y s i s c a l c u l a t i o n s were based on a

s u p e r c r i t i c a l - f l u i d i n l e t p r e s s u r e of 3600 p s i and an o u t l e t p r e s s u r e o f 3535 p s i and a steam i n l e t p r e s s u r e of 595.4 p s i and an o u t l e t p r e s s u r e of 590 p s i .

This a n a l y s i s involved t h e determination of t h e s t r e s s e s i n

t h e tubes, s h e l l , tube sheets, and the high-pressure head.

The s t r e s s e s produced i n t h e tubes a r e

1. primary membrane s t r e s s e s caused by pressure,

190 2.

secondary stresses caused by the temperature gradient across the tube wall, and

discontinuity stresses at the junction of the tube and tube sheet. From page 208 of Ref. 1, the longitudinal stress

Pressure Stresses. component,

â&#x20AC;&#x153;L =

Pis

P - H

The hoop stress component,

= 1651

(0.03515) + 3010(0.015) 9 (0.02015)

= 1651

78 76 ??

where r = the mean radius of the tube.

At the inside surface of the tube

ah = 1651 + 78 76 = 6903 psi (tension) 0.015

and at the outside surface of the tube, 78.76 â&#x20AC;&#x153;h = 1651 + 0.035i5 = 3892 psi (tension)

The radial stress component at the inside surface of the tube

= 3600 â&#x20AC;&#x153;r psi (compression) and at the outside surface of the tube ar = 590 psi (compression).

Temperature Stresses.

From page 63 in Ref. 2, the stresses in the

tube caused by the temperature gradient, lS. Timoshenko, Strength of Materials, Part 11, 3rd ed., D. Van Nostrand, New York, 1956.

2J. F. Harvey, Pressure Vessel Design, D. Van Nostrand Company, New Jersey, 1963.

19 1

where R i s t h e inner o r o u t e r r a d i u s opposite t h a t a t which t h e stress i s being computed.

The maximum temperature gradient a c r o s s t h e tube wall, A t , = 0.109(350)

= 38.15OF

The maximum average temperature of t h e wall = 650

+ 0.0835(350) = 942OF.

For t h i s temperature, t h e c o e f f i c i e n t o f thermal expansion, a! = 8.5 x

los

and t h e modulus of e l a s t i c i t y , E = 27 x 1@

in./in."F,

-=-= a 0.1225

At'L

1.5306, and I n

- At'h -

= 0.4257.

Therefore,

(8.5 x 104)(27 x 1@)38.2 2(0.7) (0.4257) = 14710(1 42.2P)

2p - 0.02015

A t t h e o u t s i d e surface of t h e tube wall, R = a = 0.1225,

At'L

At'h

= 14710[1

42.2(0.0150)]

Discontinuity S t r e s s e s .

= -7112 p s i

(0.4257)J

and

= +5399 p s i

A t t h e i n s i d e surface o f the tube wall R = b = 0.1875, =

p s i , and

and

From page 210 i n Ref. 1, t h e d e f l e c t i o n a t

t h e junction o f t h e thick-wall tube and t h e tube sheet,

When r = b,

To maintain c o n t i n u i t y a t t h e j u n c t i o n of t h e tube and tube sheet, both t h e slope and d e f l e c t i o n of t h e tube w a l l must be zero a t t h e junction. Replacing E1 with D i n Eqs. 11 and 12 on page 12 of Ref. 1,

- AM)

(E 2)

192

and

where

EP = 12(1

- $1

- a,

T = b

and

r i s assumed t o be-

a + b 2 -

i t may be seen t h a t F = 2AM.

From Eq. E.3,

Using t h i s value f o r F and

the values f o r D, r, T, and 1 l i s t e d above, from Eqs. E . l and E.2 one obtains b + a

[&?bpi

(b2 + 2 ) b P o + u(w

which reduces t o = h 2 ( a + b)

3[22bPi

- s)bPo]

(b2 + 8 ) b P o +

2(a

U(p

b)2M

- 3)bP01

For t h e preheater with a tube o u t s i d e diameter o f 0.375 in. and w a l l thickness of 0.065 in., a = 0.1225 in., value of 0.30. b

b = 0.1875 in.,

and

assumed

Therefore,

a = 0.31,

(b + a)2 = 0.0961,

(b + a)3 = 0.0298, b

B = 0.01500625,

= 0.03515625,

E? + b2 = 0.05016250,

-3

= 0.02015,

A = 12.81, and = 164.

M = 0.40923(20.2635 = 6.296 in.-lb/in.

- 5.549

0.669)

o f circumference, and

F = 2AM = 2(12.81)(6.296)

= 161.3 lb/in. of circumference.

193 The discontinuity stresses at the junction of the tube and the tube

sheet caused by the moment are the longitudinal stress component,

M ~ L=

6M ?? -

6(6.296)2 (0.065)

= 8941 psi (tension inside,

compression outside)

E ii i

the hoop stress component caused by lengthening of the circumference, Mah =

a + b ha \T)=

164(0.155)

= 2462 psi (tension)

and the hoop stress component caused by distortion, E

M'hsec

M 0L ) = 0.3(8941)

= 2682 psi (tension inside,

compression outside)

The discontinuity stress at the junction of the tube and the tube sheet caused by the force is the hoop stress component,

x a( 7 b ) = 2(161.3)(12.81) 2 (0.065) = 4927 psi (compression) .

2F pah = T

(o.155)

Stresses in Shell The stresses in the shell are

1. primary membrane stresses caused by pressure, and 2. discontinuity stresses at the junction of the shell and tube sheet. Primary Membrane Stresses.

The primary membrane stresses in the

shell caused by pressure are the hoop stress, PDS ah=-=

595*4(20*25) = 13,780 psi 2(0.0365)

and the longitudinal stress, PDs

- - --- - 6890 psi L' - 4T

194 Discontinuity Stresses.

To determine the discontinuity stresses

in the shell at the junction of the shell and tube sheet, the tube sheet was assumed to be very rigid with respect to the shell.

The resulting

moment and force applied to the shell at the junction are

M where

A =

& and F -AP ’ =

[w]”‘ =

2.73 (0.4375)2,

f (10.344)Z

y/4

= (0.1333)l

= 0.6042.

The stresses in the shell at the junction caused by the moment are the longitudinal stress component,

= ?? 6M

6(595.4 2(0.3651)(o.i375)’

25,561 psi (tensile inside, compressive out side)

the hoop stress component caused by lengthenihg of the circumference, =

M‘h

2 Aar T

= -r = 595*4(10*344)= 14,077 psi (tensile)

(0.4375)

and the hoop stress component caused by distortion, M‘hsec

= u(

u ) = 7668 psi (tensile inside, M L compressive outside)

The stress in the shell at the junction caused by the force is the hoop stress, F‘h

- -2F Ar

= 28,136 psi (compressive)

Stress in Tube Sheet From page 24 in Ref. 3, the thickness of the tube sheet,

Tubular Exchanger Manufacturets Association, Inc., Standards of Tubular Exchanger Manufacturers, 4th ed., New York, 1959.

Li f

195 where F = a constant, Di = i n s i d e diameter of t h e p r e s s u r e p a r t , in.,

P = pressure, p s i , and S = t h e allowable stress, p s i .

For F = 1 and S = 7800 p s i a t a temperature of 1000°F,

T = -21(301 2

7800

)’”

= 10.125(0.621)

= 6.288 i n .

A tube-sheet thickness of 6 . 5 i n . was used i n t h e preheater.

High-pressure Head The high-pressure head i s a thick-wall sphere with a w a l l t h i c k n e s s of 5.75 i n . and an i n s i d e r a d i u s o f 12 i n .

The maximum primary mem-

brane s t r e s s e s caused by i n t e r n a l pressure,

where

a = 12 i n . and b = 17.75 i n . Therefore,

P‘h

17.75)3 POL

= 3600[2i(17.75)3

2(1Z)a (12)3]

= 4215 p s i ( t e n s i l e )

The r a d i a l s t r e s s component

‘r

= 3600 p s i (compressive).

Summary of S t r e s s C a l c u l a t i o n s The shear s t r e s s theory of f a i l u r e was used as t h e f a i l u r e c r i t e r i o n , and t h e stresses were c l a s s i f i e d and l i m i t s of stress i n t e n s i t y were determined i n accordance with Section I11 of t h e ASME B o i l e r and P r e s s u r e Vessel Code.

I !

i i

196

Calculated Stresses in Tubes.

The calculated stresses in t..e

tubes

are tabulated below. Inside Surface (psi)

Stress

Outside Surface (psi)

1651

6903

3892

-3600

-590

8941

-8941

2462

2682

-2682

-4927

L ' t A

-7112

5399

Ath'

-7112

5399

4144

L'P h'P par

L'M h'M M'hsec h'F

- 1891

3480

cuL

- 3600

-590

Therefore, the calculated primary membrane stress intensity, P

= 6903

(-3600) = 10,503 psi

and the allowable primary stress intensity at a temperature of 961째F, Pm = Sm = 10,500 psi.

The calculated maximum primary plus secondary

stress intensity, (Pm + Q) max = 3480

(-3600) = 7080 psi

and at a temperature of 961째F, the allowable

= 3s = 31,500 psi Q'rnax m Calculated Stresses in Shell. The stresses calculated in the shell

m'(

,are tabulated on the following page.

197 I n s i d e Surface Stress

Outside Surface

(psi)

6890

13780

. -595

25561

-25561

1407 7

7668

-7668

-28136

7443

-7883

32486

- 18600

-595

PL ' P'h par ML '

M h' M'hsec Fh ' "h =L ' =Or

Therefore, the c a l c u l a t e d primary membrane s t r e s s i n t e n s i t y , Pm = 13780

- (-595)

and t h e allowable Pm = Sm = 15,000 p s i .

= 14,375 p s i

The c a l c u l a t e d maximum primary

p l u s secondary stress i n t e n s i t y (m '

Q'max

= 32,486

(-595) = 33,081 p s i

and t h e allowable (Pm + Q)max a t 650째F = 3Sm = 45,000 p s i . Calculated S t r e s s i n Tube Sheet. intensity 1 '

The maximum c a l c u l a t e d stress

i n t h e tube s h e e t i s less than 7800 p s i and t h e allowable

at l O O O O F is 7800 psi.

Calculated S t r e s s e s i n High-pressure Head.

The maximum primary

membrane stresses c a l c u l a t e d , POL

P'h

= 4215 p s i

t, or = -3600 p s i .

= 4215

(-3600) = 7815 p s i

and t h e allowable Pm a t 1000째F = Sm = 7800 p s i .

Therefore, t h e calctl-

, These v a l u e s were

judged to be in adequate agreement for the purposes of this preliminary analysis.

f L

" I

198

Appendix F NOMENCLATURE

a = heat transfer surface area, ft2/ft (heat-transfer and pressuredrop calculations) a = inside radius of tube, in. (stress analysis) A = flow area, ft2 (heat-transfer and pressure-drop calculations) As,

area of shell or tubes

, in.2

(stress analysis)

b = outside radius of tube, in. B = by-pass leakage factor

C = heat exchanger parameter Cp = specific heat d = diameter of tube

D = diameter of shell

= a constant in stress analysis

ET3 12(1

- v2)

E = modulus of elasticity, psi f = friction factor F = fractional window cut (heat-transfer and pressure-drop calculations) F = force per inch at discontinuity, lb/ n. (stress analysis) Fm, FIT, Fs = load on tubes or shell caused by differential expansion, lb (stress analysis) gc = gravitational conversion constant, lbm*ft/lbf-sec2

G = mass velocity, lb/hr*ft2 h = heat transfer coefficient, Btu/hr.ft2-*F

H J <.

= enthalpy, Btu/lb = heat transfer factor

k = thermal conductivity, Btu/hr*ft2-OF per ft (heat-transfer and pressure-drop calculations) k = a constant in stress analysis

1 = length

199

L = length of tubes, ft M = moment per inch at discontinuity, in.-lb/in. n = number of tubes

N = number of baffles = Prandtl number

NPr

= Reynolds number

p = pitch of tubes

P = pressure, psi Pm = primary membrane stress intensity, psi q = volumetric flow rate, ft"/sec

Q = heat transfer rate, Btu/hr ( heat-transfer and pressure-drop calculations) Q = secondary stress intensity, psi (stress analysis) r = number of restrictions (heat-transfer and pressure-drop calculations) r = mean radius of tube or shell, in. (stress analysis) B = thermal resistance, ft2*hrO8F/Btu (heat-transfer and pressure-drop calculations)

R = relative heat transfer resistance (stress analysis) S = cross-sectional area (heat-transfer and pressure-drop calculations) S = calculated stress intensity, psi (stress analysis) ,S = allowable stress intensity, psi t = temperature, OF T = wall thickness of tube, in. U = overall heat transfer coefficient, Btu/hr*ft2*OF v = specific volume, ft3/ib V = velocity, ft/sec W = flow rate, lb/hr X = baffle spacing

= capacity ratio

= coefficient of thermal expansion, in./in.-OF

in heat-transfer and pressure-drop calculations

â&#x201A;Ź3 = effectiveness ratio

.. n

(stress analysis)

200

%, +, + =

deflection caused by pressure, force, and moment, in.

Aâ&#x201A;Ź = differential growth, in./in. AP

= pressure difference, psi

At = temperature difference, OF AtZrn= log mean At, OF E = growth, in./in.

X = a constant in stress analysis

= log mean correction factor (heat-transfer and pressure-drop

calculations) p = viscosity, lb/hr-ft v = Poisson's ratio p =

density, lb/fts

uL, ah, ur = longitudinal, hoop, and radial stress components, psi

At'd ,

= hoop and longitudinal stress components caused by temperature differences, psi

At'L

M'h"

= hoop and longitudinal stress components due to the moment, psi

ML'

= hoop stress components due to force, psi = longitudinal stress component due to differential growth, ~e 'L psi

h'F

P'h'

P'L

= hoop and longitudinal stress components due to pressure, psi 5

Subscripts for Nomenclature Above a = annulus I

B = cross flow

, 8

ce = cold fluid at end of first pass cf = cold fluid ci = cold fluid in co = cold fluid out

cu = cold fluid at end of unbaffled section d = doughnut opening e = equivalent

20 1

he = hot f l u i d a t end of f i r s t pass hf = hot f l u i d h i = hot f l u i d i n ho = hot f l u i d o u t hu = hot f l u i d a t end of unbaffled s e c t i o n i = inside

i t = t o t a l i n s i d e flow area m = mean

o = outside OD = OD of d i s k

ow = I D of inner window P = pressure

s = shell t = total

T = tube

u = unbaffled s e c t i o n

w = window W = tube w a l l

! i

20 3 ORNL-TM- 1545

Internal Distribution 1. 2-4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

M. C. E. E. R. R. G. F. S. H. D. D. W. F. C. W. W.

Bender E. Betris S. Bettis G. Bohlmann J. Braatz B. Briggs A. Cristy L. Culler J. Ditto G. Duggan A. Dyslin E. Ferguson F. Ferguson C. Fitzpatrick H. Gabbard R. Gall R. Grimes A . G. Grindell P. N. Haubenreich H. W. Hoffman P. R. Kasten R. J. Ked1 0. A. Kelly

26. 27. 28. 29. 30. 31. 32. 33. 34. 35-36. 37. 38. 39. 40. 41. 42. 43. 44-45. 46. 47. 48-57. 58. 59.

G. R. H. R. H. E. L.

H. Llewellyn E. MacPherson E. McCoy L. Moore A. Nelms L. Nicholson C. Oakes A. M. Perry T. W. Pickel M. W. Rosenthal Dunlap Scott W. C. Stoddart R. E. Thoma J. R. Weir M. E. Whatley J. C. White W. R. Winsbro Central Research Library Document Reference Section GE&C Division Library Laboratory Records Department Laboratory Records, ORNL R. C. O R m Patent Office

External Distribution 60-74. 75.

Division of Technical Information Extension Research and Development Division