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dsm_diffusion_controlled_growth.m
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dsm_diffusion_controlled_growth.m
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% ----------------------------------------------------------------------- %
% __ __ _______ _ ____ _ _ _______ _____ %
% | \/ | /\|__ __| | /\ | _ \| || | /\|__ __| __ \ %
% | \ / | / \ | | | | / \ | |_) | || |_ / \ | | | |__) | %
% | |\/| | / /\ \ | | | | / /\ \ | _ <|__ _/ /\ \ | | | ___/ %
% | | | |/ ____ \| | | |____ / ____ \| |_) | | |/ ____ \| | | | %
% |_| |_/_/ \_|_| |______/_/ \_|____/ |_/_/ \_|_| |_| %
% %
% ----------------------------------------------------------------------- %
% %
% Author: Alberto Cuoci <alberto.cuoci@polimi.it> %
% CRECK Modeling Group <http://creckmodeling.chem.polimi.it> %
% Department of Chemistry, Materials and Chemical Engineering %
% Politecnico di Milano %
% P.zza Leonardo da Vinci 32, 20133 Milano %
% %
% ----------------------------------------------------------------------- %
% %
% This file is part of Matlab4ATP framework. %
% %
% License %
% %
% Copyright(C) 2022 Alberto Cuoci %
% Matlab4ATP is free software: you can redistribute it and/or %
% modify it under the terms of the GNU General Public License as %
% published by the Free Software Foundation, either version 3 of the %
% License, or (at your option) any later version. %
% %
% Matlab4CFDofRF is distributed in the hope that it will be useful, %
% but WITHOUT ANY WARRANTY; without even the implied warranty of %
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %
% GNU General Public License for more details. %
% %
% You should have received a copy of the GNU General Public License %
% along with Matlab4ATP. If not, see <http://www.gnu.org/licenses/>. %
% %
%-------------------------------------------------------------------------%
% %
% Code: Population Balance Equation (PBE) with growth term only %
% Solved using the Discrete Sectional Method %
% R. McGraw (1997) Description of Aerosol Dynamics by the %
% Quadrature Method of Moments, Aerosol Science and Technology, %
% 27:2, 255-265 (1997), DOI: 10.1080/02786829708965471 %
% %
% ----------------------------------------------------------------------- %
close all;
clear variables;
% Initial distribution: f=a*r^2*exp(-b*r)
% r = particle radius (mum)
% a and b: distribution parameters
a = 0.108; % (1/mum/cm3)
b = 0.60; % (1/mum)
% Growth rate: phi(r) = kG/r
kG = 0.78; % growth rate constant (mum2/s)
% Domain of integration
rMax = 30; % maximum radius (mum)
M = 500; % number of intervals
tf = 20; % maximum time (s)
% Initial distribution (#/cm3/mum)
r = 0:rMax/M:rMax;
fIn = fInitial(r,a,b);
% Number of particles (per unit of volume) in each interval (#/cm3)
NIn = zeros(M,1);
for i=1:M
NIn(i) = fIn(i+1)*(r(i+1)-r(i));
end
% Solution of discrete section equations
[t, N] = ode15s(@ODESystem, 0:0.25:tf, NIn, [], r, kG);
ntimes = length(t);
% Reconstruction of density function (#/cm3/mum)
f = zeros(ntimes, M+1);
for i=1:ntimes
f(i, 2:end) = N(i,:)./(r(2:end)-r(1:end-1));
end
% Plot solution
figure; hold on;
plot(r, fIn, 'r');
plot(r, f(end,:), 'b');
plot(r, fAnalytical(r, tf, kG, a, b), 'g');
xlabel('r (micron)');
ylabel('f (#/micron/cm^3)');
legend('initial', 'numerical', 'analytical');
hold off;
% Calculation of moments
m0(1) = Moment(r, fIn, 0);
m0(2) = Moment(r, fIn, 1);
mf(1) = Moment(r, f(end,:), 0);
mf(2) = Moment(r, f(end,:), 1);
fprintf('Time=0 s: Ntot(#/cm3)=%f Rm(micron)=%f\n', m0(1), m0(2));
fprintf('Time=20 s: Ntot(#/cm3)=%f Rm(micron)=%f\n', mf(1), mf(2));
% Evolution of mean radius
% Analytical results
muAnalytical1 = zeros(length(t),1);
muAnalytical2 = zeros(length(t),1);
for k=1:length(t)
muAnalytical1(k) = AnalyticalMoments(1, r, t(k), kG, a, b);
muAnalytical2(k) = AnalyticalMoments(2, r, t(k), kG, a, b);
end
% Numerical results
m1 = zeros(length(t),1);
m2 = zeros(length(t),1);
for i=1:length(t)
m1(i) = Moment(r, f(i,:), 1);
m2(i) = Moment(r, f(i,:), 2);
end
figure; hold on;
yyaxis left; ylabel('mean radius (micron)');
plot(t, m1);
plot(t, muAnalytical1, 'b--');
yyaxis right; ylabel('std deviation (micron)');
plot(t, sqrt(m2-m1.^2));
plot(t, sqrt(muAnalytical2-muAnalytical1.^2), 'r--');
xlabel('time (s)'); hold off;
legend('mean radius', 'mean radius (analytical)', 'std deviation', 'std deviation (analytical)');
%% Dynamic evolution of density function
video_name = 'dsm.mp4';
videompg4 = VideoWriter(video_name, 'MPEG-4');
open(videompg4);
figure;
for k=1:length(t)
hold off;
plot(r, fAnalytical(r, t(k), kG, a, b), 'g');
hold on;
plot(r, f(k,:), 'b');
hold on;
xlabel('r (\mum)'); ylabel('f (#/micron/cm3'); title('time=20 s');
xlim([0 20]); ylim([0 0.6]);
legend('numerical','analytical');
frame = getframe(gcf);
writeVideo(videompg4, frame);
end
close(videompg4);
%% Equations
function dN = ODESystem(~, N, r, kG)
M = length(N);
f = zeros(M+1,1);
for i=1:M
f(i+1) = N(i)/(r(i+1)-r(i));
end
dN = zeros(M,1);
dN(1) = -f(2)*Ldot(kG, r(2));
for i=2:M-1
dN(i) = -f(i+1)*Ldot(kG, r(i+1)) + f(i)*Ldot(kG, r(i));
end
dN(M) = f(M)*Ldot(kG, r(M));
end
%% Evaluation of moments
function m = Moment(r, f, order)
np = length(r);
m = 0;
for i=1:np-1
deltar = r(i+1)-r(i);
I = 0.50*(f(i+1)*r(i+1)^order+f(i)*r(i)^order);
m = m + deltar*I;
end
end
%% Growth rate function
function Lprime = Ldot(kG, r)
Lprime = kG/r;
end
%% Initial solution
function f = fInitial(r, a, b)
f = a*(r.^2).*exp(-b*r);
end
%% Analytical solution
function f = fAnalytical(r, t, kG, a, b)
f = zeros(length(r),1);
for i=1:length(r)
arg = r(i)^2-2*kG*t;
if (arg > 0)
f(i) = r(i)/sqrt(arg).*fInitial(sqrt(arg), a, b);
end
end
end
%% Moments of the analytical solution
function muk = AnalyticalMoments(k, r, t, kG, a, b)
muk = 0.;
for j=1:length(r)-1
deltar = r(j+1)-r(j);
Ic = 0.50*(r(j+1)^(k)*fAnalytical(r(j+1), t, kG, a, b)+r(j)^(k)*fAnalytical(r(j), t, kG, a, b));
muk = muk + Ic*deltar;
end
end