Question

Elementary principles of chemical engineering 4th edition problem 10.17

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(dot)m3/min. Scale has built up on the inside walls of the tank, so that the effective volume Vm3 of the tank is unknown. At time t =0, a mass m0kg 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 ofV,C,andV(dot),assumingthatthetank 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. Without doing any calculations, sketch a plot of C versus t, labeling the value of C at t =0 and the asymptotic value at t approaches infinity.

(b) Integrate the balance to prove that

C(t )= (m0/V )exp (V(dot)t/V)

(c) Suppose the flow rate through the mixer is V_ 30:0 m3 =min and that the following data are taken:

Time from injection, t(min)	1	2	3	4
C * 10^3 (kg/m3)	0.223	0.050	0.011	0.002

(For example, at t 1 min, C 0:223 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 Vm3 from the slope of your plot.

(d) A solution of a radioactive element with a fairly short half-life (see Problem 10.16) is often used as a tracer for applications like the one in this problem. The advantage of doing so is that the concentration of the tracer at the outlet can be measured with a sensitive radiation detector mounted outside the exit pipe rather than having to draw fluid samples from the pipe and analyze them. What is a potential drawback of radiotracers? Why is it important that the half-life of the tracer be neither too short nor too long?

Answer 1