Mathematically, it obviously is, but I would like to have the physical justification also, and I’m stuck on one certain step.
The solution to the CSTR problem begins like this:
$f’(t) = \frac{qw – qf(t)}{V}$
where $f(t)$ is the concentration at time $t$, $q$ is the flow rate, $w$ is the concentration of the inlet flow, and $V$ is the volume.
Applying this to the situation of a cooling body, it is easy to identify $qw$ as the amount of “coolness” arriving, but I am unable to justify the amount of “hotness leaving” as $qf(t)$, because the “mixing” present in the CSTR problem seems absent in the case of a cooling body. It seems like the cooling body is losing heat much like an onion might be peeled. So, a picture or analogy that would justify the $qf(t)$ term would be appreciated.
Background: CSTR stands for “Continuous Stirred Tank Reactor”, which is an abbreviation for “Continuous Flow Stirred Tank Reactor”. Finding an expression for the concentration in the tank at time $t$ is a textbook example in Differential Equations.
Regards,
Mike Jones
22.May.2011 (Beijing time)