Question

# 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 1}}{d t}=f_{1}\left(q, c_{A i}, c_{A 1}\right) \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: 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.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12  Fig: 13  Fig: 14  Fig: 15  Fig: 16  Fig: 17  Fig: 18  Fig: 19  Fig: 20  Fig: 21