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 V̇ (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 V̇, 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. 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 → 1.

(b) Integrate the balance to prove that

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

(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 V(m³) from the slope of your plot.

(d) A solution of a radioactive element with a fairly short half-life  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?

Transcribed Image Text:

C(t) = (mo/V)exp(-Vt/V)