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

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);

Answer 1

[Alan Stevens]

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

[Aayush Meena]

Thanks for your answer i really appreciate your work but r is also given in terms of x,y and tp.

[Alan Stevens]

Like this then?

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)
    r = @(tp) sqrt( (x-u*(5-tp)).^2+y(i).^2 );  
    f1 = @(tp) (4*Alpha*(t-tp)+r(tp).^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)))

   52.3394

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

[Aayush Meena]

If we want to take x and y as constant and vary z from 0 to 0.002 in the first code(when we take r= 0.01). Then how can we do that.

PLEASE REPLY AND THANKS.

[Alan Stevens]

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

Answer 2

[Aayush Meena]

What if we vary x(x=0:0.0001:0.050) and take y and z as constants.

Thanks

[Alan Stevens]

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!