A stirred-tank heat exchanger with a bypass stream is

Chapter 18, Problem 18.17

(choose chapter or problem)

A stirred-tank heat exchanger with a bypass stream is shown in Fig. E18.17 with the available control valves. The possible manipulated variables are mass flow rate w2, valve stem positions Xc 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) Determine the degrees of freedom for control 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 hare to be controlled by using x2 and f as the manipulated variables. (Valve stem positions, Xc and x3, are held constant.) Derive an expression for the relative gain array for this control configuration. (e) It has been proposed that x2 be replaced by x3 in the control problem of (d). Briefly analyze this proposal. (It is not necessary to perform another RGA analysis.) Tl 0-f) wl wl Xc--1 Tc fwl r--r 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 empirical relation is available where C1 and C2 are constants: (iii) The overall heat transfer coefficient for the cooling coil U depends on the velocity of the coolant in the line and hence on the valve stem position Xc, according to the relation below where C3 is a constant: U = C3Xc (iv) The pump on the bypass line operates in the "fiat part" of the pump curve so that the mass flow rate, fw1, depends only on the control valve. (v) The following process variables remain constant: T1, w1, Tz, and Tc.