In this problem, you will use your knowledge of solving systems of nonlinear equations and continuation methods to determine both the composition and the temperature distribution in an ideal staged rectification column. Recall from mass and energy balances that the operating line for a rectifying column is given by

where R_D is the reflux ratio, x_D is the concentration of the distillate, y_n+1 is the concentration of the gas entering stage n, and x_n is the concentration of the liquid exiting stage n. The concentrations refer to the lighter boiling component. We will be considering a total condensor, whereupon x₀ = y₁ = x_D. In an ideal column, the liquid composition x_n and gas composition y_n exiting stage n are assumed to be in equilibrium. We will further assume that this equilibrium is given by Raoult’s law

where species 1 is the lighter boiling component, P_i^sat is the saturation pressure of species i = 1, 2, and P is the pressure of the column. The saturation pressures (in mm Hg) are correlated using the Antoine equation,

where T_n is the temperature of stage n in centigrade A_i, B_i, and C_i are the Antoine coefficients for species i.

In mass and energy balances, you learned how to make a graphical solution to this problem using the McCabe–Thiele method on an x–y phase diagram for the lighter boiling component. In this problem, you will figure out how to make this solution numerically without creating the phase diagram in advance. As an added bonus, the numerical solution will also furnish the temperature of each equilibrium stage. To help guide your solution to the problem, there are several subtasks to complete prior to making the full calculation.

For this problem, we will consider the rectification of benzene and toluene at atmospheric pressure. The Antoine coefficients are given in the table below.

We will consider a rectifying column that produces a distllate of 99.95% pure benzene using a reflux ratio R_D = 1.95. The column contains 15 stages.

(a) Recast the problem into a pair of coupled nonlinear equations that you can use to compute x_n and T_n for given values of y_n and P.

(b) Determine the Jacobian for this 2 × 2 system.

(c) Write a MATLAB function called computeJ that computes the elements of the Jacobian. Confirm that this function produces the correct output for the values T_n = 90°C and x_n = 0.7 if you input y_n = 0.7. Report the Jacobian matrix for these values in your written submission. Save the function as an m-file in your working directory for use in later programs.

(d) Write a MATLAB function called computeR that computes the elements of the residual. Confirm that this function produces the correct output for the values T_n = 90°C and x_n = 0.7 if you input y_n = 0.7. Report the residual vector for these values. Save the function as an m-file in your working directory for use in later programs.

(e) Write a MATLAB function called NewtonRaphson that performs the Newton– Raphson method to solve your linear system. This function should call computeJ and computeR. Use the norm of δ for convergence and stop the iterations when this is less than 10−4. You are allowed to use a built-in MATLAB function to do the matrix inversion. Test your solution when y_n = 0.7 using an initial guess of T_n = 90°C and x_n = 0.7. Report the values of T_n and x_n produced by your program. Also confirm, by hand, that these values are the correct solution to the nonlinear equations. Save the function as an m-file in your working directory for use in later programs.

(f) Write a MATLAB function called rectification that executes a zero-order continuation method to determine the values of x_n, y_n, and T_n for each stage of the column. The function should call NewtonRaphson for the equilibrium calculation. The output of the function should be an N × 3 matrix with the values of x_n in the first column, y_n in the second column, and T_n in the third column. Your function should also plot each variable versus the stage number n.

Transcribed Image Text:

Yn+1 RD XD -Xn+ 1 + RD 1 + RD