HEAT TRANSFER For

Mechanical Engineering By

www.thegateacademy.com

Syllabus

Heat Transfer

Syllabus for Heat Transfer Modes of heat transfer, one dimensional heat conduction, resistance concept, electrical analogy, unsteady heat conduction, fins; dimensionless parameters in free and forced convective heat transfer, various correlations for heat transfer in flow over flat plates and through pipes; thermal boundary layer; effect of turbulence; radiative heat transfer, black and grey surfaces, shape factors, network analysis; heat exchanger performance, LMTD and NTU methods.

Analysis of GATE Papers (Heat Transfer) Year

Percentage of marks

2013

10.00

2012

6.00

2011

4.00

2010

2.00

2009

9.00

2008

6.00

2007

8.00

2006

4.67

2005

6.67

Overall Percentage

6.26%

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com

Content

Heat Transfer

CONTENTS

#1.

#2.

#3.

Chapters

Page No.

Conduction

1 - 50

1–2 2–8 9 – 11 11 – 13 13 – 17 18 – 30 31 – 34 34 – 38 39 39 – 50

Introduction One Dimensional Heat Conduction Unsteady Heat Conduction Critical Radius of Insulation Heat Transfer Through Fins Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

Convection

51 - 97

51 – 52 52 – 56 56 – 66 67 – 68 68 – 71 72 – 85 86 – 88 88 – 90 91 91 – 97

Introduction Convection Fundamentals Forced Convection Nusselt Numbers Natural Convection Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

Radiation

98-137

98 98 – 100 100 – 102 102 – 104 104 – 110 111 – 123

Introduction Blackbody Radiation Radiative Properties The View Factor Radiation Heat Transfer Solved Examples

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page i

Content

#4.

Assignment 1 Assignment 2 Answer Keys Explanations

Heat Transfer

124 – 128 128 – 129 130 130 – 137

Heat Exchanger

138-168

138 138 – 139 139 – 140 140 – 143 143 – 147 148 – 158 159 – 160 160 – 161 162 162 – 168

Introduction Types of Heat Exchangers The Overall Heat Transfer Coefficient Analysis of Heat Exchanger The Effectiveness – NUT Method Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

Module Test

169 – 184

Test Questions

169– 177

Answer Keys

178

Explanations

178 – 184

Reference Books

185

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page ii

Chapter 1

Heat Transfer

CHAPTER 1 Conduction Introduction Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. Conduction can take place in solids, liquids, or gases. Conduction is due to the collisions and diffusion of the molecules during their random motion. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons. The rate of heat conduction through a medium depends on the geometry of the medium, its thickness and the material of the medium, as well as the temperature difference across the medium.

Figure 1: Heat conduction through a large plane wall of thickness

and area A

Consider steady heat conduction through a large plane wall of thickness x and area A, as shown in Figure 1. The temperature difference across the wall is . The rate of heat conduction through a plane layer is proportional to the temperature difference across the layer and the heat transfer area, but is inversely proportional to the thickness of the layer. That is,

or, ̇

W

Where the constant of proportionality k is the thermal conductivity of the material, which is a measure, of the ability of a material to conduct heat in the limiting case of , the equation above reduces to the differential form ̇ THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com Page 1

Chapter 1

Heat Transfer

Which is called Fourier's law of heat conduction after J. Fourier, who expressed it first in his heat transfer text in 1822. Here dT/dx is the temperature gradient, which is the slope of the temperature curve on a T-x diagram (the rate of change of T with x). at location x. The relation above indicates that the rate of heat conduction in a direction is proportional to the temperature gradient in that direction. Heat is conducted in the direction of decreasing temperature and the temperature gradient becomes negative when temperature decreases with increasing x.

Thermal Conductivity The rate of conduction heat transfer under steady conditions can also be viewed as the defining equation for thermal conductivity. Thus the thermal conductivity of a material can be defined as the rate of heat transfer through a unit thickness of the material per unit area per unit temperature difference. The thermal conductivity of a material is a measure of the ability of the material to conduct heat. A high value for thermal conductivity indicates that the material is a good heat conductor and a low value indicates that the material is a poor heat conductor or insulator.

Thermal Diffusivity The product , which is frequently encountered in heat transfer analysis, is called the heat capacity of a material. Another material property that appears in the transient heat conduction analysis is the thermal diffusivity. Which represents how fast heat diffuses through a material and is defined as ⁄ Note that the thermal conductivity k represents how well a material conducts heat, and the heat capacity represents how much energy a material stores per unit volume. Therefore, the thermal diffusivity of a material can be viewed as the ratio of the heat conducted through the material to the heat stored per unit volume. A material that has a high thermal conductivity or a low heal capacity will obviously have a large thermal diffusivity. The larger the thermal diffusivity. The faster the propagation of heat into the medium. A small value of thermal diffusivity means that heat is mostly absorbed by the material and a small amount of heat will be conducted further.

One Dimensional Heat Conduction Heat transfer has direction as well as magnitude. The rate of heat conduction in a specified direction is proportional to the temperature gradient, which is the change in temperature per unit length in that direction. Heat conduction in a medium, in general, is three-dimensional and time dependent. That is, and the temperature in a medium varies with position as well as time. Heat conduction in a medium is said to be steady when the temperature does not vary with time, and unsteady or transient when it does. Heat conduction in a medium is said to THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com Page 2

Chapter 1

Heat Transfer

be one-dimensional when conduction is significant in one dimension only and negligible in the other two dimensions, two-dimensional when conduction in the third dimension is negligible and three-dimensional when conduction in all dimensions is significant. The governing differential equation in such systems in rectangular, cylindrical and spherical coordinate systems is derived in below section. Rectangular Coordinates Consider a small rectangular element of length , width and height , as shown in Figure 2. Assume the density of the body is and the specific heat is C, an energy balance on this element during a small time interval can be expressed as

Figure 2: Three-dimensional heat conduction through a rectangular volume element

(

+ xy

(

z

x

, y

xy z z

(

,

(

y

,

Noting that the volume of the element is the change in the energy content of the element and the rate of heat generation within the element can be expressed as x y z ̇

̇ x y z

Substituting into equation we get ̇

̇

̇

̇

̇

̇

̇ x y z

x y z

Dividing by x y z gives ̇ y z

̇ ̇

x

x z

̇ y

̇ x y

̇ z

̇

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com Page 3

Chapter 1

Heat Transfer

Nothing that the heat transfer area of the element for heat conduction in the directions are respectively and taking the limit as and yields (

*

(

*

(

S ̇

̇

̇

̇

̇

̇

̇

̇

̇

*

̇

v

v

F

’

w

.

(

*

(

*

(

*

(

*

(

*

(

*

In the constant case of constant thermal conductivity ̇

Where the property is again the thermal diffusivity of the materials and above equation is known as the Fourier-Biot equation and it reduces to these forms under specified conditions: 1.

Steady-State: (called Poisson equation) ̇

2.

Transient, no heat generation: (called the Diffusion equation)

3.

Steady state, no heat generation: (called the Laplace equation)

Note that in the special case of one dimensional heat transfer in the derivatives with respect to y and z drop out.

direction, the

Cylindrical Coordinates The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates, shown in Figure 3, by following the THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com Page 4

Chapter 1

Heat Transfer

steps just outlined. It can also be obtained directly by coordinate transformation using the following relations between the coordinates of a point in rectangular and cylindrical coordinate systems

Figure 3: A differential volume element in cylindrical coordinates.

After lengthy manipulations we obtain (

*

(

*

(

*

̇

Spherical Coordinates The general heat conduction equations in spherical coordinates can be obtained from an energy balance on a volume element in spherical coordinates, shown in Figure 4, by following the steps outlined above. It can also be obtained directly by coordinate transformation using the following relations between the coordinates of a point in rectangular and spherical coordinate systems

Again after lengthy manipulations, we obtain (

*

(

*

(

*

̇

Conduction through a cylindrical wall For a cylinder at steady state, with no internal heat generation, the equation becomes

O

b

B. ’

w

b

equation as

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com Page 5

Chapter 1

Q=(

Heat Transfer

where

Comparing the above equation to that of heat transfer through a wall Q = KA (T1 – T2 Where

δ

(T1 – T2) / (r2 – r1)

is the logarithmic mean area = (A2 – A1) / log (A2 / A1)

The above equations are applicable to any general heat conduction problem. The one dimensional heat conduction is out particular area of interest as they result in ordinary differential equations.

Figure 4: A differential volume element in spherical coordinates. Conduction through sphere Steady state, one dimensional with no heat generation equation in spherical co-ordinates is (

*

O

b [

B. ’ w

b

equation as

]

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacamy.com © Copyright reserved. Web: www.thegateacademy.com Page 6