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An electrically heated, stirred-tank system is shown in
Chapter 11, Problem 11.8(choose chapter or problem)
An electrically heated, stirred-tank system is shown in Fig. E11.8. Using the given information, do the following: (a) Draw a block diagram for the case where T3 is the controlled variable and voltage signal V2 is the manipulated variable. Derive an expression for each transfer function. (b) Repeat part (a) using V1 as the manipulated variable. (c) Which of these two control configurations would provide better control? Justify your answer. Available Information 1. The volume of liquid in each tank is kept constant using an overflow line. 2. Temperature To is constant. 3. A 0.75-gal/min decrease in q0 ultimately makes T1 increase by 3 F. Two-thirds of this total temperature change occurs in 12 min. This change in q0 ultimately results in a 5 F increase in T3. 4. A change in V1 from 10 to 12 volts ultimately causes T1 to change from 70 to 78 F. A similar test for V2 causes T3 to change from 85 to 90 F. The apparent time constant for these tests is 10 min. 5. A step change in Tz produces a transient response in T3 that is essentially complete in 50 ( = T min). 6. The thermocouple output is amplified to give V3 = 0.15T3 + 5, where V3 [=]volts and T3 [ =] F. 7. The pipe connecting the two tanks has a mean residence time of30 s
Questions & Answers
QUESTION:
An electrically heated, stirred-tank system is shown in Fig. E11.8. Using the given information, do the following: (a) Draw a block diagram for the case where T3 is the controlled variable and voltage signal V2 is the manipulated variable. Derive an expression for each transfer function. (b) Repeat part (a) using V1 as the manipulated variable. (c) Which of these two control configurations would provide better control? Justify your answer. Available Information 1. The volume of liquid in each tank is kept constant using an overflow line. 2. Temperature To is constant. 3. A 0.75-gal/min decrease in q0 ultimately makes T1 increase by 3 F. Two-thirds of this total temperature change occurs in 12 min. This change in q0 ultimately results in a 5 F increase in T3. 4. A change in V1 from 10 to 12 volts ultimately causes T1 to change from 70 to 78 F. A similar test for V2 causes T3 to change from 85 to 90 F. The apparent time constant for these tests is 10 min. 5. A step change in Tz produces a transient response in T3 that is essentially complete in 50 ( = T min). 6. The thermocouple output is amplified to give V3 = 0.15T3 + 5, where V3 [=]volts and T3 [ =] F. 7. The pipe connecting the two tanks has a mean residence time of30 s
ANSWER:Problem 11.8An electrically heated, stirred-tank system is shown in Fig. E11.8. Using the given information,do the following:(a) Draw a block diagram for the case where T is the contr3lled variable and voltage signal V 2is the manipulated variable. Derive an expression for each transfer function.(b) Repeat part (a) using V as t1 manipulated variable.(c) Which of these two control configurations would provide better control Justify your answer.Available Information1. The volume of liquid in each tank is kept constant using an overflow line.2. Temperature T is c0stant.3. A 0.75-gal/min decrease in q ultimat0ly makes T increase by 31°F. Two-thirds of this totaltemperature change occurs in 12 min. This change in q ultimately resu0ts in a 5 °F increase inT34. A change in V from 10 to 12 volts ultimately causes T to change from 70 to 78 °F. A similar 1 1test for V causes T to change from 85 to 90 °F. The apparent time constant for these tests is 2 310 min.5. A step change in T produces a transient response in T that is essentially complete in 50 ( = 2 3 min).6. The thermocouple output is amplified to give V = 0.15T + 3, where V3 3 [=] volts and T 3 [=]°F.7. The pipe connecting the two tanks has a mean residence time of30 s Step-by-step solution Step 1 of 7 ^(a)Consider the following informations. 1. The outlets of both tanks have flow rate q at all times. 0 2. The temperature T is co0tant. 3. The energy balance is a first-order transfer function between T and Q . 1 0Consider the following expression. T (t) 12 = 1 e 1 T )Substitute 2 for T(t), 3 for T() and 12 for t 2 12 = 1 e 1 3\n 1= 10.9 minConsider the general transfer function, G(s) = K s+1Here, Input change K = Output changeWrite the transfer functions with the given values. 3°F T (s) 0.75 gpm 1 = Q 0s) 10.9s+1 T1(s) 4 Q (s)= 10.9s+1 ……..(1) 0And (53)°F T 1s) 0.75 gpm Q (s) = s+1 0 2 T1(s) 2.67 Q0(s)= 2s+1………(2)