Analytical and Matlab code....
BINARY BATCH DISTILLATION
The batch distillation of benzene (component 1) and toluene (component 2) mixture is being carried out at a pressure of 1.2 atm. Initially, there are 100 moles of liquid in the still, comprised of 60% benzene and 40% toluene (mole fraction basis). Calculate the amount of liquid remaining in the still when concentration of toluene reaches 80%.
Numerical Methods Solution of a system of equations comprised of ordinary differential equations and nonlinear algebraic equations. Concepts Utilized Batch distillation of an ideal binary mixture. Course Useage Separation Processes.
Problem Statement For a binary batch distillation process involving two components designated 1 and 2, the moles of liquid remaining, L, as a function of the mole fraction of the component 2, x2, can be expressed by the following equation dL L = dx 2 x 2 (k 2 − 1)
where k2 is the vapor liquid equilibrium ratio for component 2. If the system may be considered ideal, the vapor liquid equilibrium ratio can be calculated from k i = Pi / P , where Pi is the vapor pressure of component i and P is the total pressure.
A common vapor pressure model is the Antoine equation which utilizes three parameters A, B, and C for component i as given below where T is the temperature in °C. Pi = 10
B A− T +C
The temperature in the batch still follow the bubble point curve. The bubble point temperature is defined by the implicit algebraic equation which can be written using the vapor liquid equilibrium ratios as
k1 x1 + k 2 x 2 = 1 Consider a binary mixture of benzene (component 1) and toluene (component 2) which is to be considered as ideal. The Antoine equation constants for benzene are A1 = 6.90565, B1 = 1211.033 and C1 = 220.79. For toluene A2 = 6.95464, B2 = 1344.8 and C2 = 219.482 (Dean1). P is the pressure in mm and T the temperature in °C.
Solution This problem requires the simultaneous solution of an ordinary differential equation and a non-linear algebraic equation.
MATLAB does not have a function specifically
designed for this task, but it does have functions that perform each of the individual tasks. In the following program, the non-linear algebraic equation solver, FZERO, is called from within the ordinary differential equation solver, ODE45. Run the problem by issuing the command “Prob_batch”. It calls “distill.m”, which defines the distillation equation, and “vap_press.m”, which calculates the vapor pressure.
MATLAB CODE %filename Prob_batch.m % Binary Batch Distillation clear global A B C P T_guess
A = [6.90565 6.95464]; B = [1211.033 1344.8]; C = [220.79 219.482]; P = 1.2*760; % mmHg Lo = 100;
x_start = 0.40;
% moles of toluene
x_final = 0.80;% moles of toluene T_guess = (80.1+110.6)/2; % C xspan = [x_start x_final]; [x L] = ode45(‘distill’,xspan,Lo); % For Version 4, use %[x L] = ode45(‘distill’,x_start,x_final,Lo); plot(x,L,’r’) title(‘Batch Distillation’) xlabel(‘Mole Fraction of Toluene’)
ylabel(‘Moles of Liquid’) output = [x L]; save batch.dat output -ascii
%filename distill.m function dL_dx = distill(x,L) global A B C P T_guess x2 x2 = x; T = fzero(‘vap_press’,T_guess); P_i = 10.^(A-B./(T+C)); k = P_i./P; dL_dx = L/x2/(k(2)-1);
%filename vap_press.m function f = vap_press(T) global A B C P x2 x1 = 1-x2; P_i = 10.^(A-B./(T+C)); k = P_i./P; f = 1 - k(1)*x1 - k(2)*x2;