  ### Question 28676  Verified

Process Control

1. Both tanks are perfectly mixed.

2. Both tanks operate at constant volume, V and V.

3. The fluid density P is constant.

4. In the first tank, component A is consumed according to:

RA = -kCA

where k is a known constant. There is no chemical reaction in the second tank.Write

a. Write dynamic component A balances for both tanks. (Do not write any energy balances.)Eliminate total mass balances (if you wrote them), q2, and the rate equation, so that you onlyhave two equations for:

\frac{d c_{A 2}}{d t}=f_{2}\left(q, q_{3}, c_{A 1}, c_{A 2}, c_{A 3}\right)

b. Perform a degree of freedom analysis on your model. The variables should include:

\frac{d c_{A 1}}{d t}=f_{1}\left(q, c_{A i}, c_{A 1}\right)

q, q_{3}, c_{A i}, c_{A 1}, c_{A 2}, c_{A 3} \text {. Determine the model inputs and model outputs. Also assign DVs }

and MVs.

c. Linearize your model equations, and convert them to perturbation variables.

d. Take the Laplace transform of the linearized equation for dc,, (t)/dt from part c. You do not need to take the Laplace transform of the equation for dc', (t)/dtAlA2

e. Show that:

c_{A 1}^{\prime}(s)=G_{1}(s) c_{A i}^{\prime}(s)+G_{2}(s) q^{\prime}(s)

f. For G2 (s), find the steady-state gain and all time constants.

g. In part c, you linearized the model equations. These can be written in the state-space form:

\dot{x}=\left[\begin{array}{l} \dot{c}_{A 1}^{\prime} \\ \dot{c}_{A 2}^{\prime} \end{array}\right]=A x+B u+E d

Find the matrix A.

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