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multiple_effect.m
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multiple_effect.m
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% Multiple Effect Evaporator Solver %
% Giovanni Correra 03/2024 %
clear variables
close all
clc
OPTIONS = optimset('Display','off','MaxIter',1e20,'MaxFunEvals',1e20...
,'TolFun',1e-10,'Algorithm','levenberg-marquardt');
% ----------------------------- Data ----------------------------------- %
TF = 95 + 273.15; % (K) %
omF = 0.05; % (kg/kg) %
TS = 110 + 273.15; % (K) %
omf = 0.3; % (kg/kg) %
Tf = 80 + 273.15; % (K) %
F = 98*1000/3600; % (kg/s) %
% ---------------------------- Solution -------------------------------- %
N = 6;
n0 = first_guess(TF,Tf,F,N,omF,omf);
n = fsolve(@(n) parameters(n,F,TF,omF,TS,omf,Tf,N),n0,OPTIONS);
[T,V,L,om,A,U,S,dT,np,logf,err] = data(n,F,omF,TS,omf,Tf,TF,N);
SE = sum(V)/S(N);
Atot = N*A;
Pop = zeros(1,N);
for i = 1 : N
P0 = saturation(T(i));
P = fsolve(@(P) boiling_point(P,T(i)-273.15,om(i)),P0,OPTIONS);
Pop(i) = P;
end
% ------------------------- Post - processing -------------------------- %
postprocessing(V,L,om,A,U,S,np,F,omF,TS,TF,N,SE,Atot,T,Pop,dT);
% ---------------------------- Functions ------------------------------- %
function n0 = first_guess(TF,Tf,F,N,omF,omf)
n01 = zeros(1,N-1);
n02 = zeros(1,N-1);
Lf = F*omF/omf;
L0 = (F-Lf)/(N);
n02(1) = F-L0;
for i = 1 : N-1
n01(i) = TF + i*(Tf-TF)/(N-1);
n02(i) = F - i*L0;
end
n0 = [n01,n02];
end
function np = parameters(n,F,TF,omF,TS,omf,Tf,N)
% Mass Balances %
Lf = F*omF/omf;
L = [n(1,N:2*(N-1)),Lf];
F = [F,L(1,1:N-1)];
V = zeros(1,N);
om = zeros(1,N);
for i = 1 : N
V(i) = F(i) - L(i);
end
om(1) = omF*F(1)/L(1);
for i = 1 : N-1
om(i+1) = om(i)*F(i+1)/L(i+1);
end
om(N) = omf;
% Energy Balances %
T = [n(1,1:N-1),Tf];
U = zeros(1,N);
for i = 1 : N
U(i) = transfer_coefficient(T(i),om(i));
end
hL = zeros(1,N);
hV = zeros(1,N);
for i = 1 : N
hL(i) = liquid_h(T(i),om(i));
hV(i) = vapour_h(T(i));
end
hF0 = liquid_h(TF,omF);
hF = [hF0,hL(1,1:N-1)];
Ts = [T(1,2:N),TS];
lambda = zeros(1,N);
for i = 1 : N
lambda(i) = latent_heat(Ts(i));
end
S = zeros(1,N);
S(N) = (L(N)*hL(N)+V(N)*hV(N)-L(N-1)*hL(N-1))/lambda(N);
for i = 1 : N-1
S(i) = V(i+1);
end
A = S(N)*lambda(N)*1000/(U(N)*(Ts(N)-T(N)));
% Solving equations %
np1 = zeros(1,N-1);
np2 = zeros(1,N-1);
for i = 1 : (N-1)
np1(i) = F(i)*hF(i) + S(i)*lambda(i) - L(i)*hL(i) - V(i)*hV(i);
np2(i) = U(i)*A*(Ts(i)-T(i))*1e-3 - S(i)*lambda(i);
end
np = [np1,np2];
% Correlations %
function U = transfer_coefficient(T,om)
U = (4.086*(T-273.15)+72.6)/om;
end
function hL = liquid_h(T,om)
hL = (4.184-2.9337*om)*(T-273.15);
end
function hV = vapour_h(T)
hV = exp((64.87678+11.76476*(log(647.096/T))^0.35 ...
-11.94431*(647.096/T)^2+6.29015*(647.096/T)^3 ...
-0.99893*(647.096/T)^4)^0.5);
end
function lambda = latent_heat(T)
lambda = 2500.304 -2.2521025*(T-273.15) - ...
0.021465847*(T-273.15)^1.5 + ...
3.1750136e-4*(T-273.15)^2.5 - ...
2.8607959e-5*(T-273.15)^3;
end
end
function [T,V,L,om,A,U,S,dT,np,logf,err] = data(n,F,omF,TS,omf,Tf,TF,N)
% Mass Balances %
% Mass Balances %
Lf = F*omF/omf;
L = [n(1,N:2*(N-1)),Lf];
F = [F,L(1,1:N-1)];
V = zeros(1,N);
om = zeros(1,N);
for i = 1 : N
V(i) = F(i) - L(i);
end
om(1) = omF*F(1)/L(1);
for i = 1 : N-1
om(i+1) = om(i)*F(i+1)/L(i+1);
end
om(N) = omf;
% Energy Balances %
T = [n(1,1:N-1),Tf];
U = zeros(1,N);
for i = 1 : N
U(i) = transfer_coefficient(T(i),om(i));
end
hL = zeros(1,N);
hV = zeros(1,N);
for i = 1 : N
hL(i) = liquid_h(T(i),om(i));
hV(i) = vapour_h(T(i));
end
hF0 = liquid_h(TF,omF);
hF = [hF0,hL(1,1:N-1)];
Ts = [T(1,2:N),TS];
lambda = zeros(1,N);
for i = 1 : N
lambda(i) = latent_heat(Ts(i));
end
S = zeros(1,N);
S(N) = (L(N)*hL(N)+V(N)*hV(N)-L(N-1)*hL(N-1))/lambda(N);
for i = 1 : N-1
S(i) = V(i+1);
end
A = S(N)*lambda(N)*1000/(U(N)*(Ts(N)-T(N)));
% Solving equations %
np1 = zeros(1,N-1);
np2 = zeros(1,N-1);
for i = 1 : (N-1)
np1(i) = F(i)*hF(i) + S(i)*lambda(i) - L(i)*hL(i) - V(i)*hV(i);
np2(i) = U(i)*A*(Ts(i)-T(i))*1e-3 - S(i)*lambda(i);
end
np = [np1,np2];
% Correlations %
function U = transfer_coefficient(T,om)
U = (4.086*(T-273.15)+72.6)/om;
end
function hL = liquid_h(T,om)
hL = (4.184-2.9337*om)*(T-273.15);
end
function hV = vapour_h(T)
hV = exp((64.87678+11.76476*(log(647.096/T))^0.35 ...
-11.94431*(647.096/T)^2+6.29015*(647.096/T)^3 ...
-0.99893*(647.096/T)^4)^0.5);
end
function lambda = latent_heat(T)
lambda = 2500.304 -2.2521025*(T-273.15) - ...
0.021465847*(T-273.15)^1.5 + ...
3.1750136e-4*(T-273.15)^2.5 - ...
2.8607959e-5*(T-273.15)^3;
end
dT = Ts - T;
% Verify %
mbal1 = zeros(1,N);
mbal2 = zeros(1,N);
ebal1 = zeros(1,N);
ebal2 = zeros(1,N);
omF = [omF,om(1,1:N-1)];
for i = 1 : N
mbal1(i) = F(i)-L(i)-V(i);
mbal2(i) = F(i)*omF(i)-L(i)*om(i);
ebal1(i) = F(i)*hF(i)+S(i)*lambda(i)-L(i)*hL(i)-V(i)*hV(i);
ebal2(i) = U(i)*A*(Ts(i)-T(i))*1e-3 - S(i)*lambda(i);
end
logf = [mbal1,mbal2,ebal1,ebal2];
err = sum(abs(logf));
end
function P = saturation(T)
T = T-273.15;
P = 0.000135*T^3 - 0.00552*T^2 + 0.175918*T + 0.207651; % (kPa) %
end
function f = boiling_point(P,T,om)
Ts = 18.536*log(P) + 5.5052; % (C) %
f = T - Ts - (0.175*om^1.11)*exp(3.86*om)*(1000*P/101325)^0.2898;
end
function postprocessing(V,L,om,A,U,S,np,F,omF,TS,TF,N,SE,Atot,T,P,dT)
fprintf(['zero functions : ' repmat(' %e ',1,numel(np)) ' [-]\n'],np);
fprintf('n. effects = %.0f [-]',N)
fprintf(' A = %.2f [m2]',A)
fprintf(' A,tot = %.2f [m2]\n', Atot)
fprintf('F = %.2f [kg/s] ',F)
fprintf(' T,F = %.2f [°C]',TF-273.15)
fprintf(' om,F = %.2f %% [-]\n',omF*100)
fprintf('S = %.2f [kg/s]',S(N))
fprintf(' T,S = %.2f [°C]',TS-273.15)
fprintf(' Steam economy = %.2f\n',SE)
for i = 1 : N
fprintf('V,'); fprintf('%d',i); fprintf(' = %.2f [kg/s] ',V(i))
fprintf('L,'); fprintf('%d',i); fprintf(' = %.2f [kg/s] ',L(i))
fprintf('om,'); fprintf('%d',i); fprintf(' = %.2f %% [-] ',om(i)*100)
fprintf('T,'); fprintf('%d',i); fprintf(' = %.2f [°C] ',T(i)-273.15)
fprintf('dT,'); fprintf('%d',i); fprintf(' = %.2f [°C] ',dT(i))
fprintf('U,'); fprintf('%d',i); fprintf(' = %.2f [W/m2K] ',U(i))
fprintf('P,'); fprintf('%d',i); fprintf(' = %.2f [kPa] ',P(i))
fprintf('\n')
end
end