A perfectly stirred, constant volume tank has two input streams, both consisting of the same liquid as shown in the following figure. The temperature and flow rate of each of the streams can vary with time. The liquid has constant heat capacity $C$ and density $.w_i$ denotes mass flow rate of Steam $i.$ Assume that the International Systems $($SI$)$ of Units are used in all questions.

$($a$)$ Derive a dynamic model that will describe transient operation. Please simplify your model by eliminating any algebraic equations. Also, simplify any derivatives of products of variables if necessary.

$($b$)$ Make a degrees of freedom analysis assuming that both Streams $1$ and $2$ come from upstream units $($i$.$e$.,$ their flow rates and temperatures are known functions of time$).$