# How to add 2 other equations which depends on Tp change in the ODE.

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Faridatul Ain Binti Mohd Rosdan el 22 de Nov. de 2021
Respondida: Nipun el 23 de Mayo de 2024 a las 5:37
These two equations (psatv =exp (A-B/(Tp+C))*133.322 and Cvs = (psatv*Nl)/(1000*R*Tp) have parameter Tp that solves by ode3 equation. My questions is how to write the coding for these two equations that depends on the solution of the ode3.
My full coding so far:
clc;
clear;
close all;
%parameters
R = 8.314; %gas constant J/mol K - the temperature must be absolute!
Tp = 298.15; %Initial temperature of the droplet/particle K
Nl = 18.015; %Molecular weight of the water, g/mol
A = 18.3036; %Antoine constant
B = 3816.44; %Antoine constant
C = -46.13; %Antoine constantTps = 304.4; %Temperature of particle at surface, Tps=Twb Are you sure?
Dp = 0.00152; %Diameter of droplet/particle, m
rp = Dp/2; %Radius of the droplet/particle, m
vp = 0.75; %Velocity of droplet/particle, m/s
rhog = 1.028; %Density of drying gas, kg/m3
ug = 2.052*10^-5; %Viscosity of drying gas, kg/m s
Dv = 2.78*10^-5; %Vapour air diffusion coefficient, m2/s
Cvi = 0.01024; %Vapour concentration in the drying gas, kg/m3
kg = 0.03; %Thermal conductivity of drying gas, W/m K
cpp = 4009; %Specific heat capacity of the droplet/particle, J/kg K
Tg = 343.15; %Temperature of drying gas, K
hfg = 2.33*10^6; %Latent heat of vapourisation, J/kg
rhod = 1052; %Density of the droplet, kg/m3
rhol = 997; %Density of the water, kg/m3
Dls = 1.5*10^-10; %Diffusivity of water into solid,m2/s
%Eqns
psatv =exp (A-B/(Tp+C))*133.322;
Cvs = (psatv*Nl)/(1000*R*Tp);
Re = (Dp*vp*rhog)/ug;
Sc = ug/(rhog*Dv);
Pr = cpp*ug/kg;
kc = (2+(0.6*Re^0.5*Sc^1/3))*Dv/Dp;
alpha = (2+(0.6*Re^0.5*Pr^1/3))*kg/Dp;
mp = rhod*(4*pi*(rp)^3)/3;
%Define the equations
%ode1 = @(t,y) -4*pi*rp^2*kc*(Cvs-Cvi);
%ode2 = @(t,y) ode1/(4*pi*rhol*rp^2);
%ode3 = @(t,y)(-4*pi*rp^2*alpha*t*(Tp-Tg)-hfg*(ode1))/(mp*cpp);
%f=[ode1;ode2;ode3]
format long
%f= @(t,y) [-4*pi*rp^2*kc*(Cvs-Cvi);
% (-4*pi*rp^2*kc*(Cvs-Cvi))/(4*pi*rhol*rp^2);
% (-4*pi*rp^2*alpha*(t)*(Tp-Tg)-hfg*(-4*pi*rp^2*kc*(Cvs-Cvi)))/(mp*cpp)]
% The functions
%y(1)=ml
%y(2)=rp
%y(3)=Tp
f= @(t,y) [-4*pi*(rp)^2*kc*(Cvs-Cvi);
(-4*pi*(rp)^2*kc*(Cvs-Cvi))./(4*pi*rhol*(y(2))^2);
(-4*pi*(rp)^2*alpha*(t)*(y(3)-Tg)-hfg*(-4*pi*(rp)^2*kc*(Cvs-Cvi)))/(mp*cpp)]
% time discretization
dt = 0.1;
t0 = 0;
tf = 100;
h=dt;
t = t0:dt:tf;
tspan=[t0 tf];
%Define Initial conditions
cond1 = 1.93e-6;
cond2 = 0.00076;
cond3 = 298.15;
y0=[cond1;cond2;cond3];% the initial value of the vector y
%Call the Euler function
[ T, Y ] = EULER( f, t0, tf, y0, h);
disp([' ' 'time' ' ' 'Ml (Y(1))' ' ' 'rp (Y(2))' ' ' 'Tp (Y(3))'])
disp([T(:) Y(:,1) Y(:,2) Y(:,3)])
figure(1)
plot(t,Y(:,1), 'r', 'linewidth', 2)
title('Droplet Mass versus Time')
xlabel('Time,t')
ylabel('Mass,kg')
grid on
figure(2)
plot(t,Y(:,2), 'g', 'linewidth', 2)
xlabel('Time,t')
ylabel('Droplet size,m')
grid on
figure (3)
plot(t,Y(:,3), 'b', 'linewidth', 2)
title('Temperature versus time')
xlabel('time,t')
ylabel('Temperature')
grid on
EULER
function [ T, Y ] = EULER( f, t0, tf, y0, h)
% [ T, Y ] = EULER( f, t0, tf, y0, h)
% uses Euler's method to solve a system of differential equations
% It's evaluated by f Starting at time instant t0 with initial condition,
% x0, a solution is approximated at time tf moving in time step h.
% T contains the evaluation times and
% Rows of Y contain the corresponding estimates for y(t).
T = t0 : h : tf;
Y = zeros(length(T),length(y0));
Y(1,:) = y0';
for i = 2 : length(T)
Y(i,:) = Y(i-1,:)+f(T(i-1),Y(i-1,:))'*h;
end
end
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### Respuestas (1)

Nipun el 23 de Mayo de 2024 a las 5:37
Hi Faridatul,
I understand that you are trying to solve a system of ordinary differential equations where the parameters psatv and Cvs depend on the solution of the ODE, specifically the temperature Tp.
I recommend using a MATLAB function such as "ode45" instead of the custom "Euler" method for better accuracy.
For more information on "ode45" MATLAB function, refer to the following MathWorks documentation: https://in.mathworks.com/help/matlab/ref/ode45.html#d126e1117222
Hope this helps.
Regards,
Nipun
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