$6\mathrm{.}21$ The jacketed vessel in Fig. E$6\mathrm{.}21$ is used to heat a liquid by means of condensing steam. The following information is available:

$($i$)$ The volume of liquid within the tank may vary, thus changing the area available for heat transfer.

$($ii$)$ Heat losses are negligible.

$($iii$)$ The tank contents are well mixed. Steam condensate is

removed from the jacket by a steam trap as soon as it has

formed.

$($iv$)$ Thermal capacitances of the tank and jacket walls are

negligible.

$($v$)$ The steam condensation pressure Ps is set by a control

valve and is not necessarily constant.

$($vi$)$ The overall heat transfer coefficient U for this system is

constant.

$($vii$)$ Flow rates $q_F$ and $q$ are independently set by external

valves and may vary.

Derive a dynamic model for this process. The model should

be simplified as much as possible. State any additional as$-$

sumptions that you make.

$($a$)$ Find transfer functions relating the two primary output

variables $h($level$)$ and $T($liquid temperature$)$ to inputs $q_F,q,$

and $T_S.$ You should obtain six separate transfer functions.

$($b$)$ Briefly interpret the form of each transfer function using

physical arguments as much as possible.

$($c$)$ Discuss qualitatively the form of the response of each

output to a step change in each input.