Chapter 4, Problem 4.6E

A stirred-tank blending system can be described by a first-order transfer function between the exitcomposition x and the inlet composition $X_1$ (both are mass fractions of solute):
   $\frac{X^{\prime }\left(s\right)}{{X^{\prime }}_i^{}\left(s\right)}=\frac{K}{\tau s+1}$
where $K=0.6$ (dimensionless) and $T=10\min$ . When the blending system is at steady state $\left(\bar{x}=0.3\right)$ , thedynamic behavior is tested by quickly adding a large amount of a radioactive tracer, thus approximating animpulse function with magnitude 1.5.
a. Calculate the exit composition response $x\left(t\right)$ using Laplace transforms and sketch $x\left(t\right)$ . Based on thisanalytical expression, what is the value of $x\left(0\right)$ ?
b. Using the Initial Value Theorem of Section 3.4, determine the value of $x\left(0\right)$ .
c. If the process is initially at a steady state with $\bar{x}=0.3$ , what is the value of $x\left(0\right)$ ?
d. Compare your answer for parts (a)-(c) and briefly discuss any differences.