Process dynamics and control 4th edition seborg solutions manual

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Chapter 9 © 9.1

a)

Flow rate transmitter:  15 psig - 3 psig  qm(psig)=   (q gpm - 0 gpm)  3 psig  400 gpm-0 gpm 

 psig  =  0.03  q(gpm)  3 psig gpm   Pressure transmitter:  20 mA - 4 mA  Pm(mA)=   ( p in.Hg  10 in.Hg)  4 mA  30 in.Hg - 10 in.Hg   mA  =  0.8  p(in.Hg)  4 mA in.Hg   Level transmitter:

 5 VDC - 1 VDC  hm(VDC)=   (h(m) - 0.5m)  1 VDC  10 m - 0.5 m  VDC   =  0.421  h(m)  0.789 VDC m   Concentration transmitter:  10 VDC - 1 VDC  Cm(VDC)=   (C (g/L)-3 g/L)+1 VDC  20 g/L - 3 g/L 

 VDC  =  0.529  C (g/L)  0.59VDC g/L   b)

The gains, zeros and spans are: Flow

Pressure

Level

Concentration

0.03 psig/gpm

0.8 mA/in.Hg

0.421 VDC/m

0.529 VDC/g/L

Zero

0 gpm

10 in.Hg

0.5 m

3 g/L

Span

400 gpm

20 in.Hg

9.5 m

17 g/L

Gain

9.2 [Type here]

9-1 [Type here]

a)

The safest conditions are achieved by the lowest temperatures and pressures in the flash vessel.

VALVE 1.- Fail close (air-to-open) VALVE 2.- Fail open (air-to-close) VALVE 3.- Fail open (air-to-close) VALVE 4.- Fail open (air-to-close) VALVE 5.- Fail close (air-to-open) Setting valve 1 as fail close prevents more heat from going to flash drum and setting valve 3 as fail open to allow the steam chest to drain. Setting valve 3 as fail open prevents pressure build up in the vessel. Valve 4 should be failopen to evacuate the system and help keep pressure low. Valve 5 should be fail-close to prevent any additional pressure build-up.

b) Vapor flow to downstream equipment can cause a hazardous situation VALVE 1.- Fail close (air-to-open) VALVE 2.- Fail open (air-to-close) VALVE 3.- Fail close (air-to-open) VALVE 4.- Fail open (air-to-close) VALVE 5.- Fail close (air-to-open) Setting valve 1 as fail close (air-to-open) prevents more heat from entering flash drum and minimizes future vapor production. Setting valve 2 as fail open (air-to-close) will allow the steam chest to be evacuated, setting valve 3 as fail close (air-to-open) prevents vapor from escaping the vessel. Setting valve 4 as fail open (air-to-close) allows liquid to leave, preventing vapor build up. Setting valve 4 as fail close (air-to-open) prevents pressure buildup. c) Liquid flow to downstream equipment can cause a hazardous situation VALVE 1.- Fail close (air-to-open) VALVE 2.- Fail open (air-to-close) VALVE 3.- Fail open (air-to-close) VALVE 4.- Fail close (air-to-open) VALVE 5.- Fail close (air-to-open) Set valve 1 as fail close to prevent all the liquid from being vaporized (This would cause the flash drum to overheat). Setting valve 2 as fail open will allow the steam chest to be evacuated. Setting valve 3 as fail open prevents pressure buildup in drum. Setting valve 4 as fail close prevents liquid from escaping. Setting valve 5 as fail close prevents liquid build-up in drum

9-2

9.3

Note: This exercise is best understood after the material in Ch. 11 has been considered. a)

Changing the span of the temperature transmitter will change its steady-state gain, according to Eq. 9-1. Because the performance of the closed-loop system depends on the gains of each individual element (cf. Chapter 11), closed-loop stability could be adversely affected.

b)

Changing the zero of a transmitted does not affect its gain. Thus, this change will not affect closed-loop stability.

c)

Changing the control valve trim will change the (local) steady-state gain of the control valve, dq/dp. Because the performance of the closed-loop system depends on the gains of each individual element (cf. Chapter 11), closed-loop stability could be adversely affected

d)

For this process, changing the feed flow rate could affect both its steady-state gain and its dynamic characteristics (e.g., time constant and time delay).

Because the performance of the closed-loop system depends on the gains of each individual element (cf. Chapter 11), closed-loop stability could be adversely affected.

9.4

Starting from Eq. 9-7: q Cv  Pv Nf (l ) gs

(1)

The pressure drop in the valve is: Pv  P  Ps where Ps  Kq 2

(2) (3)

Solve for K by plugging in the nominal values of q and Ps . First, convert the nominal value of q into units of m3/h to match the metric units version of N (the parameter N = 0.0865 m3/h(Kpa)1/2 when q has units of m3/h and pressure has units of KPa).

9-3

qd  0.6 m3 / min  36 m3 / h Psd  200 kPa Psd 200 kPa  2  0.154 kPa/(m3 /h) 2 2 qd 36 (m3 /h)2 Now substitute (3) into (2) to get an expression for Pv in terms of q.

K

Pv  P  Kq 2 (4) Substitute (4) into (1) to get: q Cv  (5) P  Kq 2 Nf (l ) gs The problem specifies that qd should be 2/3 of qmax (where qmax is the flow rate through the valve when the valve is fully open).

2 qd  qmax 3 3 3 qmax  qd  36 m3 /h 2 2 3 qmax  54m /h Now find the Cv that will give qmax = 54 m3/h. Substitute q = qmax and f (l)=1 (valve fully open) into (5). qmax Cv  P  Kqmax 2 N gs Now that all of the variables on the right hand side of the equation are known, plug in to solve for Cv. kPa m3 P  450 kPa, K  0.154 3 , N  0.0865 , (m / h) 2 h(kPa)1/2

g s  1.2, qmax  54 m3 /h

54

Cv  3

0.0865

m h(kPa)1/2

m3 h

450kPa  0.154

m3 h   m3 m3 1/2 0.0865 0.88(kPa) 0.076   h(kPa)1/2 h Cv  710.5 54

m3 h

kPa 54 2 (m 3 /h) 2 3 2 (m /h) 1.2 54

9-4

9.5 Let Pv/Ps = 0.33 at the nominal q  320 gpm

Ps = Pb+ Po = 40 + 1.953  10-4 q2 Pv= P - Ps = (1 –2.44  10-6 q2)PDE – (40 + 1.953  10-4 q2) (1 - 2.44  10 -6  320 2 )PDE - (40 + 1.953  10 -4  320 2 )  0.33 (40 + 1.953  10 -4  320 2 ) PDE = 106.4 psi Let qd= q  320 gpm For the rated Cv, the valve is completely open at 110% qd i.e., at 352 gpm or the upper limit of 350 gpm 

1

 p  2 Cv  q  v   gs  

1

 (1  2.44 106  3502 )106.4  (40  1.953  104  3502 )  2 Cv  350   0.9  

Then using Eq. 9-27,

 q  66.4  4.55  10  4 q 2  1 / 2     ln  0.9 101.6    l  1 ln 50 The plot of the valve characteristic is shown in Figure S9.5. From the plot of the valve characteristic for the rated Cv of 101.6, it is evident that the characteristic is reasonably linear in the operating region 250  q  350.

The pumping cost could be further reduced by lowering PDE to a value that would make Pv/Ps = 0.25 at q  320 gpm. Then PDE = 100 and for qd = 320 gpm, the rated Cv = 133.5. However, as the plot shows, the valve characteristic for this design is only slightly more nonlinear in the operating region. Hence, the selected valve coefficient is Cv = 133.5.

9-5

400

350

300

250

q (gpm) 200

150

------- Cv = 101.6 100

- - - - Cv = 133.5 50

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

l (valve lift)

Figure S9.5. Control valve characteristics.

9.6 a)

b)

There are three control valves. The selection of air-to-close vs. air-to-open is based on safety considerations: i.

Steam control valve: Air-to-open to prevent overheating of the evaporator.

ii.

Level control valve (that adjusts liquid flow rate B): Air-to-open to prevent the steam coils from being exposed to the vapor space, which could lead the coils to being burned out.

iii.

Pressure control valve (that adjusts solvent flow rate D): Air-to-close to prevent over-pressurization of the evaporator.

For the three controllers: i.

Concentration controller: As the product concentration xB increases, we want the steam pressure, Ps to increase. Since the steam valve is air-to-open, this means that the controller output signal to the control valve (via the I/P) should increase. Thus, the controller should be direct-acting.

9-6

ii.

Level controller: As the liquid level h increases, we want the product flow rate B to increase. Since the control valve is air-to-open, this means that the controller output signal to the control valve (via the I/P) should increase. Thus, the controller should be direct-acting.

iii.

Pressure controller: As the pressure P increases, we want the solvent flow rate D to increase. Since the control valve is air-to-close, this means that the controller output signal to the control valve (via the I/P) should decrease. Thus, the controller should be reverse-acting.

9.7

Because the system dynamic behavior would be described using deviation variables, the dynamic characteristic can be analyzed by considering that the input terms (not involving x) can be considered to be constant, and thus deviations are zero. The starting form is the linear homogeneous ODE: M d 2x dx  R  Kx  0 2 gc dt dt Taking the Laplace transform gives, M  X ( s)  s 2  Rs  K   0  gc 

 M 2 R  X ( s)  s  s  1  0 K  Kgc  Calculate and ζ by comparing this equation to the standard form of the secondorder model in (5-39) (keeping in mind that gc = 32.174 lbm ft/(lbf s2)).

M  0.00965s Kg c



2  2



R 2

M R  Kg c K

gc  155.3 KM

The valve characteristics are highly overdamped and can be accurately approximated by a first-order model obtained by neglecting the d2x/dt2 term.

9-7

9.8

Configuration I: This series configuration will not be very effective because a large flow rate has to pass through a small control valve. Thus, the pressure drop will be very large and flow control will be ineffective.

Configuration II: This parallel configuration will be effective because the large control valve can be adjusted to provide the nominal flow rate, while the small control valve can be used to regulate the flow rate. If the small valve reaches its maximum or minimum value, the large valve can be adjusted slightly so that the small valve is about half open, thus allowing it to regulate flow again.

9.9

First write down the time-domain step response for a step change of 10°C. Then solve the equation to find when y(t) is equal to 5 (since the variables are in deviation variables, this represents when TM will reach 30°C). ym (t )  KM (1  e  t / ) where M  10o C,

K  1, and   10s

ym (t )  10(1  e  t /10 ) 5  10(1  e  ta /10 ) ta  6.93s

Therefore, the alarm will sound 6.93 seconds after 1:10PM.

9.10    

0.1 psig  0.5% of full scale 20 psig accuracy is unknown since the "true" pressure in the tank is unknown 0.1 psig resolution =  0.5% of full scale 20 psig

precision =

repeatability =

±0.1 psig =±0.5% of full scale 20 psig

9-8

9.11

Assume that the gain of the sensor/transmitter is unity (i.e. there is no steady-state measurement error). Then, Tm ( s ) 1  T ( s) ( s  1)(0.1s  1)

where T is the temperature being measured and Tm is the measured value. For the ramp temperature change: 0.3 T  (t) = 0.3t (C/s) , T  (s) = 2 s 1 0.3 Tm ( s)   2 ( s  1)(0.1s  1) s Tm (t )  0.00333e10t  0.333et  0.3t  0.33

The maximum error occurs as t : Maximum error = |0.3t  (0.3t  0.33)| = 0.33 C If the smaller time constant is neglected, the time domain response is slightly different for small values of t, although the maximum error (t) does not change.

T °C 6 5 4 3 2 1 0 0

5

10

15

Time s 20

Figure S9.11. Response for process temperature sensor/transmitter. Orange solid line is T’(t), and purple dashed line is T’m(t).

9-9