# How to do integration of this equation and plot graph between T and y

7 views (last 30 days)
Aayush Meena on 3 Jul 2021
Commented: Alan Stevens on 8 Jul 2021 Here
variables are t' and n only
u=0.020;
x=0.025;
y=(0:0.0001:0.010); we have to vary y
z=0;
t=5;
neta= 0.6;
P=200;
C=500;
l=0.008;
Alpha=17e-6;
ro= 1000;
r=sqrt((x-(u*(5-t)))^2+y^2);

Alan Stevens on 3 Jul 2021
Like this perhaps:
u=0.020;
x=0.025;
z=0;
t=5;
eta= 0.6;
P=200;
C=500;
ro= 1000;
l=0.008;
Alpha=17e-6;
r = 0.01; % Guessed value as you haven't specified r
k = eta*P/(pi*ro*C*l);
y= (0:0.001:0.010);
DT = zeros(1,numel(y));
for i = 1:numel(y)
f1 = @(tp) (4*Alpha*(t-tp)+r^2).^-1;
f2 = @(tp) exp(-((x - u*(t-tp)).^2+y(i).^2).*f1(tp));
DeltaTfn = @(tp) f1(tp).*f2(tp).*(1 + 2*f3(tp)); % You set z to zero, so the cosine term is jusr 1
DT(i) = k*integral(DeltaTfn,0,t);
end
plot(y,DT),grid
xlabel('y'), ylabel('\Delta T') disp(DT(numel(y)))
40.3366
function s = f3(tp)
t = 5;
Alpha=17e-6;
l=0.008;
s = 0;
nmax = 10; % Adjust this until you get convergence
for n=1:nmax
s=exp(-Alpha*n^2*pi^2*(t-tp)/l^2) + s;
end
end
Alan Stevens on 7 Jul 2021
Like this:
u=0.020;
x=0.025;
z=0:0.0002:0.002;
t=5;
eta= 0.6;
P=200;
C=500;
ro= 1000;
l=0.008;
Alpha=17e-6;
r = 0.01; % Guessed value as you haven't specified r
k = eta*P/(pi*ro*C*l);
y= 0.005; %(0:0.001:0.010); Arbitrary constant - use your own value
DT = zeros(1,numel(z));
for i = 1:numel(z) %%%%%%%%%% y goes to z
f1 = @(tp) (4*Alpha*(t-tp)+r^2).^-1;
f2 = @(tp) exp(-((x - u*(t-tp)).^2+y.^2).*f1(tp)); %%%%%%% take the (i) off y
DeltaTfn = @(tp) f1(tp).*f2(tp).*(1 + 2*f3(tp,z(i))); % pass z(i) to function f3
DT(i) = k*integral(DeltaTfn,0,t);
end
plot(z,DT),grid
xlabel('z'), ylabel('\Delta T') disp(DT(numel(z)))
58.9313
function s = f3(tp,z) %%%%%%%
t = 5;
Alpha=17e-6;
l=0.008;
s = 0;
nmax = 10; % Adjust this until you get convergence
for n=1:nmax
s=exp(-Alpha*n^2*pi^2*(t-tp)/l^2)*cos(n*pi*z/l) + s; %%%%%%%
end
end

Aayush Meena on 8 Jul 2021
What if we vary x(x=0:0.0001:0.050) and take y and z as constants.
Thanks
Alan Stevens on 8 Jul 2021
Like so:
u=0.020;
x=0:0.0001:0.050; %%%%%%%%%%%%%
z=0.002; %%%%%%%%%%%%%
t=5;
eta= 0.6;
P=200;
C=500;
ro= 1000;
l=0.008;
Alpha=17e-6;
r = 0.01; % Guessed value as you haven't specified r
k = eta*P/(pi*ro*C*l);
y= 0.005; %(0:0.001:0.010); Arbitrary constant - use your own value
DT = zeros(1,numel(x));
for i = 1:numel(x) %%%%%%%%%% y goes to z
f1 = @(tp) (4*Alpha*(t-tp)+r^2).^-1;
f2 = @(tp) exp(-((x(i) - u*(t-tp)).^2+y.^2).*f1(tp));
DeltaTfn = @(tp) f1(tp).*f2(tp).*(1 + 2*f3(tp,z)); % pass z to function f3
DT(i) = k*integral(DeltaTfn,0,t);
end
plot(x,DT),grid
xlabel('x'), ylabel('\Delta T') disp(DT(numel(x)))
46.9846
function s = f3(tp,z) %%%%%%%
t = 5;
Alpha=17e-6;
l=0.008;
s = 0;
nmax = 10; % Adjust this until you get convergence
for n=1:nmax
s=exp(-Alpha*n^2*pi^2*(t-tp)/l^2)*cos(n*pi*z/l) + s; %%%%%%%
end
end
That's pretty much every combination of x, y and z dealt with now!

R2018b

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