function xprime=colamod(t,X,U)
%
% colamod - This is a nonlinear model of a distillation column with
% NT-1 theoretical stages including a reboiler (stage 1) plus a
% total condenser ("stage" NT). The liquid flow dynamics are
% modelled by a simple linear relationship.
% Model assumptions: Two components (binary separation); constant
% relative volatility; no vapor holdup; one feed and two products;
% constant molar flows (same vapor flow on all stages);
% total condenser
%
% The model is based on column A in Skogestad and Postlethwaite
% (1996). The model has 82 states.
%
% Inputs: t - time in [min].
% X - State, the first 41 states are compositions of light
% component A with reboiler/bottom stage as X(1) and
% condenser as X(41). State X(42)is holdup in reboiler/
% bottom stage and X(82) is hold-up in condenser.
% U(1) - reflux L,
% U(2) - boilup V,
% U(3) - top or distillate product flow D,
% U(4) - bottom product flow B,
% U(5) - feed rate F,
% U(6) - feed composition, zF.
% U(7) - feed liquid fraction, qF.
%
% Outputs: xprime - vector with time derivative of all the states
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%------------------------------------------------------------
% The following data need to be changed for a new column.
% These data are for "colmn A".
% Number of stages (including reboiler and total condenser:
global NT
% Location of feed stage (stages are counted from the bottom):
NF= floor(NT/2) ;
% Relative volatility
alpha=1.5;
% Nominal liquid holdups
M0(1)=0.5; % Nominal reboiler holdup (kmol)
i=2:NT-1; M0(i)=0.5*ones(1,NT-2);% Nominal stage (tray) holdups (kmol)
M0(NT)=0.5; % Nominal condenser holdup (kmol)
% Data for linearized liquid flow dynamics (does not apply to reboiler and condenser):
taul=0.063; % time constant for liquid dynamics (min)
F0=1; % Nominal feed rate (kmol/min)
qF0 = 1; % Nominal fraction of liquid in feed
L0=2.70629; % Nominal reflux flow (from steady-state data)
L0b=L0 + qF0*F0; % Nominal liquid flow below feed (kmol/min)
lambda=0; % Effect of vapor flow on liquid flow ("K2-effect")
V0=3.20629;V0t=V0+(1-qF0)*F0;% Nominal vapor flows - only needed if lambda is nonzero
% End data which need to be changed
%------------------------------------------------------------
% Splitting the states
x=X(1:NT)'; % Liquid composition from btm to top
M=X(NT+1:2*NT)'; % Liquid hold up from btm to top
% Inputs and disturbances
LT = U(1); % Reflux
VB = U(2); % Boilup
D = U(3); % Distillate
B = U(4); % Bottoms
F = U(5); % Feedrate
zF = U(6); % Feed composition
qF = U(7); % Feed liquid fraction
% THE MODEL
% Vapor-liquid equilibria
i=1:NT-1; y(i)=alpha*x(i)./(1+(alpha-1)*x(i));
% Vapor Flows assuming constant molar flows
i=1:NT-1; V(i)=VB*ones(1,NT-1);
i=NF:NT-1; V(i)=V(i) + (1-qF)*F;
% Liquid flows assuming linearized tray hydraulics with time constant taul
% Also includes coefficient lambda for effect of vapor flow ("K2-effect").
i=2:NF; L(i) = L0b + (M(i)-M0(i))./taul + lambda.*(V(i-1)-V0);
i=NF+1:NT-1; L(i) = L0 + (M(i)-M0(i))./taul + lambda.*(V(i-1)-V0t);
L(NT)=LT;
% Time derivatives from material balances for
% 1) total holdup and 2) component holdup
% Column
i=2:NT-1;
dMdt(i) = L(i+1) - L(i) + V(i-1) - V(i);
dMxdt(i)= L(i+1).*x(i+1) - L(i).*x(i) + V(i-1).*y(i-1) - V(i).*y(i);
% Correction for feed at the feed stage
% The feed is assumed to be mixed into the feed stage
dMdt(NF) = dMdt(NF) + F;
dMxdt(NF)= dMxdt(NF) + F*zF;
% Reboiler (assumed to be an equilibrium stage)
dMdt(1) = L(2) - V(1) - B;
dMxdt(1)= L(2)*x(2) - V(1)*y(1) - B*x(1);
% Total condenser (no equilibrium stage)
dMdt(NT) = V(NT-1) - LT - D;
dMxdt(NT)= V(NT-1)*y(NT-1) - LT*x(NT) - D*x(NT);
% Compute the derivative for the mole fractions from d(Mx) = x dM + M dx
i=1:NT;
dxdt(i) = (dMxdt(i) - x(i).*dMdt(i) )./M(i);
% Output
xprime=[dxdt';dMdt'];