A stirred-tank heat exchanger with a bypass stream is shown in Fig. E. with the available control valves. The possible manipulated variables are: mass flow rate w₂, valve stem positions x_c and x₃, and f, the fraction of mass flow rate w₁ that bypasses the tank before being added to the exit stream. Using the information given here, do the following

(a) Derive a dynamic model of the stirred-tank system. Define any additional symbols that you introduce.

(b) Perform the degrees of freedom analysis to determine the degrees of freedom that are available. Allocate the degrees of freedom by specifying manipulated variables and variables that are determined by the environment.

(c) Select controlled variables and briefly justify your choice.

(d) Suppose that only T₄ and h are to be controlled by using W₂ and f as the manipulated variables. (Valve stem positions, x_c and x₃, are held constant.) Derive an expression for the relative gain array for this control configuration.

(e) It has been proposed that w₂ be replaced by in the control problem of (d). Briefly analyze this proposal. (It is not necessary to perform another RGA analysis.)

Available information

(i) The tank is perfectly mixed, and the temperature changes are relatively small so that constant physical properties can be assumed. Mass flow rates are denoted by w₁ to W₄ and temperatures by T₁ to T₄.

(ii) The exit flow rate W₃ depends on the pressures upstream and downstream of the control valve, and the valve stem position x₃. The following empiric-al relation is available where C₁ and C₂ are constants:

(iii) The overall heal transfer coefficient for the cooling coil U depends on the velocity of the coolant in the line and hence on the valve stem position x_c, according to the relation below where C₃ is a constant:

(iv) The pump on the bypass line operates in the flat part? of the pump curve so that the mass how rate, fw₁, depends only on the control valve.

(v) The foLlowing process variables remain cons lant T₁, w₁, T₂, and T_c

Transcribed Image Text:

r-- X2 T2 (1 -/) w1 Te T4 fwr fw 1 w3 = x3(Ch - C2 fwi) U - C3xe