Problem 1
place:
Consider the design of a continuous-stirred tank reactor (CSTR) for the chlorination of benzene (Fig. 1) with the following reactions taking
place:
$$C_6H_6 + Cl_2 \xrightarrow{k_1} C_6H_5Cl + HCl$$
$$C_6H_5Cl + Cl_2 \xrightarrow{k_2} C_6H_4Cl_2 + HCl$$
where the rate constants $k_1$ and $k_2$ [$h^{-1}$] are known and the feed flow rate $F_1$ [kmol/h] and reactor volume $V$ [$m^3$] will be design variables.
Figure 1: The continuous-stirred tank reactor for the chlorination of benzene.
For simplicity, the reactions will be first-order (no dependence on $Cl_2$) with respect to benzene
(A) and chlorobenzene (B). The molar volumes of each specie are: $V_A = 8.937 \times 10^{-2} m^3/kmol$, $V_B = 1.018 \times 10^{-1} m^3/kmol$, and $V_C = 1.13 \times 10^{-1} m^3/kmol$ with
dichlorobenzene as C. The feed is considered to be pure A (ignoring $Cl_2$) and therefore has a mole fraction of $y_{A,0} = 1.0$. The rate constants are $k_1 = 0.40 h^{-1}$ and $k_2 =
0.055 h^{-1}$. The steady-state model equations are:
$$y_{A,0}F_1 = y_A F_2 + r_1 V \qquad (1)$$
$$y_{B,0}F_1 = y_B F_2 + (r_1 - r_2) V \qquad (2)$$
$$y_{C,0}F_1 = y_C F_2 - r_2 V \qquad (3)$$
$$y_A + y_B + y_C = 1 \qquad (4)$$
where $y_{i,0}$ is the mole fraction of $i \in \{A, B, C\}$ in the feed, $y_i$ is the mole fraction of $i \in \{A, B, C\}$
in the tank, and $r_1$ and $r_2$ are the reaction rates [kmol/($m^3$h)] defined as:
$$r_1 = k_1 y_A / (V_A y_A + V_B y_B + V_C y_C), \quad r_2 = k_2 y_B / (V_A y_A + V_B y_B + V_C y_C),$$
and we are using the definition of the variable vector $x \in R^5$:
$$x = (y_A, y_B, y_C, F_2, V)$$
a) Let's suppose we want to minimize the reactor volume and ensure that we produce at least 22kmol/h of B. Further, from another study, we found that the
residence time for the reactor must be at least 475 seconds. For this system, the residence time is defined as $\tau = V / F_2 = V / (V_A y_A + V_B y_B + V_C y_C)$.
Formulate the constrained optimization problem.
b) Write a MATLAB program to solve the optimization problem from Part a using fmincon with the SQP solver. Solve the optimization problem. What are the active
inequality constraints, if any, and their respective Lagrange multipliers?
c) Now, let's assume we're given an existing reactor with volume $10 m^3$ and we want to maximize the concentration of B in the outlet stream. Formulate the new
optimization problem using the original productivity and residence time constraints from Part a.
d) Solve the optimization problem from Part c. What are the active inequality constraints, if any, and their respective Lagrange multipliers?
