A tracer is used to characterize the degree of mixing in a continuous stirred tank. Water enters and leaves the mixer at a rate of V (m3/min). Scale has built up on the inside walls of the tank, so that the effective volume V (m3) of the tank is unknown. At time t = 0, a mass m0 (kg) of the tracer is injected into the tank and the tracer concentration in the outlet stream, C (kg/m3), is monitored.
(a) Write a differential balance on the tracer in the tank in terms of V. C. and v, assuming that the tank contents are perfectly mixed, and convert the balance into an equation for dC/dt. Provide an initial condition, assuming that the injection is rapid enough so that all of the tracer may be considered to be in the tank at t = 0.
(b) Integrate the balance to prove that C(t) = (m0/V) exp (—vt/V)
(c) Suppose the flow rate through the mixer is v = 30.0m3/min and that the following data are taken:
(For example, at t = 1 min, C = 0.223 x 10-3 kg/m3) Verify graphically that the tank is functioning as a perfect mixer—that is, that the expression of part (b) fits the data—and determine the effective volume V(m3) from the slope of yourplot.