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Derive Transient Balance Equations

Exercise 1: Mixer Transient Species Balance

There are two inlet streams to a well-mixed and constant volume vessel. The flow rate (qi) and concentration (ci) for all streams (i=1,2,3). Follow the procedure to build a dynamic model in the modeling introduction. For the model equations, use a transient mass and species balance as shown in balance equations. State any assumptions that are needed. Simplify the equations by eliminating any unnecessary terms and express the final equations in terms of the quantities on the diagram. There is no need to simulate the dynamic response, only derive the form of the equations that will be simulated in later exercises.

Exercise 2: Mixer Transient Energy Balance

With only one inlet and one outlet stream, derive an expression for the temperature in a vessel. The energy balance should consider heat loss due to convective heat transfer to ambient air at temperature Ta and heat transfer to the cooling jacket fluid at temperature Tc. There is shaft work `W_s` and an exothermic chemical reaction. Reduce this energy balance by eliminating any terms and simplifying the expression. The liquid heat capacity and density are constant for all streams.

Solution 2 Help

An energy balance for this application starts with the balance equation for enthalpy, h.

$$m\,c_p\frac{dT}{dt} = \dot m_{in} c_p \left( T_{in} - T_{ref} \right) - \dot m_{out} c_p \left( T_{out} - T_{ref} \right) + UA \left(T_\infty-T\right) + W_s + r\,V$$

Enthalpy is related to temperature as m cp (T-Tref) where cp is the heat capacity. With a constant reference temperature (Tref), `m=\rho V`, and `q=q_1=q_2`, this reduces to the following.

$$\rho V \, c_p \frac{dT}{dt} = q \, V \, c_p \left( T_{in} - T_{out} \right) + UA_{c} \left( T_c-T \right) + UA_{a} \left( T_a-T \right) + W_s + r\,V$$