Problems of Process Control

Problem
A thermometer having a time constant of 0. min is placed in a temperature bath and after the thermometer comes to equilibrium with the bath, the temperature of the bath is increased linearly with time at the rate of I deg C / min what is the difference between the indicated temperature and bath temperature (a) 0. 1 min (b) 10. min after the change in temperature begins. (c) What is the maximum deviation between the indicated temperature and bath temperature and when does it occurs. (d) plot the forcing function and the response on the same graph. After the long enough time buy how many minutes does the response lag the input?
Solution

Consider thermometer to be in equilibrium with temperature bath at temperature (a) the difference between the indicated temperature and bath temperature
Problem
Determine the transfer function H(s)/Q(s) for the liquid level shown in figure below Resistance R1 and R2 are linear. The flow rate from tank 3 is maintained constant at b by means of a pump; the flow rate from tank 3 is independent of head h. The tanks are non-interacting.
Solution
Balance on tank 3 gives writing the steady state equation Subtracting and writing in terms of deviation Taking Laplace transforms
We have three equations and 4 unknowns (Q(s),H(s),H1(s) and H2(s). So we can express one in terms of other. From (1) Combining equation 4,5,6
Problem
Heat transfer equipment shown in fig. consists of tow tanks, one nested inside the other. Heat is transferred by convection through the wall of inner tank.

Hold up volume of each tank is 1 ft3
The cross sectional area for heat transfer is 1 ft2
The overall heat transfer coefficient for the flow of heat between the tanks is 10 Btu/(hr)(ft2)(oF)
Heat capacity of fluid in each tank is 2 Btu/(lb)(oF)
 Density of each fluid is 50 lb/ft3 Initially the temp of feed stream to the outer tank and the contents of the outer tank are equal to 100 oF.

Contents of inner tank are initially at 100 oF. the flow of heat to the inner tank (Q) changed according to a step change from 0 to 500 Btu/hr. (a) Obtain an expression for the laplace transform of the temperature of inner tank T(s). (b) Invert T(s) and obtain T for t= 0,5,10, U
Solution
(a)For outer tank Substituting numerical values Now Ti(s) = 0, since there is no change in temp of feed stream to outer tank. Which gives For inner tank
Problem
The thermal system shown in fig P 13. 6 is controlled by PD controller. Data; w = 250 lb/min; ? = 62. 5 lb/ft3; V1 = 4 ft3, V2=5 ft3; V3=6ft3; C = 1 Btu/(lb)(°F) Change of 1 psi from the controller changes the flow rate of heat of by 500 Btu/min. the temperature of the inlet stream may vary. There is no lag in the measuring element. (a) Draw a block diagram of the control system with the appropriate transfer function in each block. Each transfer function should contain a numerical values of the parameters. b) From the block diagram, determine the overall transfer function relating the temperature in tank 3 to a change in set point. (c ) Find the offset for a unit steo change in inlet temperature if the controller gain KC is 3psi/°F of temperature error and the derivative time is 0. 5 min. Fig. (1) (b)
Problem
For the control shown, the characteristics equation is s 4 +4 s3 +6 s 2 +4 s +(1 + k) =0 (a) Determine value of k above which the system is unstable. (b) Determine the value of k for which the two of the roots are on the imaginary axis.
Solution
s 4 +4 s3 +6 s 2 +4 s +(1 + k) =0 For the system to be unstable.