A blending tank that provides nearly perfect mixing is connected to a downstream unit by a long transfer pipe. The blending tank operates dynamically like a first-order process.
The mixing characteristics of the transfer pipe, on the other hand, are somewhere between plug flow (no mixing) and perfectly mixed. A test is made on the transfer pipe that shows that it operates as if the total volume of the pipe were split into five equal-sized perfectly stirred tanks connected in series.
The process (tank plus transfer pipe) has the following characteristics:
\begin{aligned}V_{ tank } &=2 m ^{3} \\V_{\text {pipe }} &=0.1 m ^{3} \\q_{\text {total }} &=1 m ^{3} / min\end{aligned}
where q_{\text {total }} represents the sum of all flow rates into the process.
(a) Using the information provided above, what would be the most accurate transfer function C_{\text {out }}^{\prime}(s) / C_{\text {in }}^{\prime}(s) for the process (tank plus transfer pipe) that you can develop? Note: c_{ in } and c_{\text {out }} are inlet and exit concentrations.
(b) For these particular values of volumes and flow rate, what approximate (low-order) transfer function could be used to represent the dynamics of this process?
(c) What can you conclude concerning the need to model the transfer pipe’s mixing characteristics very accurately for this particular process?
(d) Simulate approximate and full-order model responses to a step change in c_{\text {in }}.
