Find the heat rate exchanged by a cylindrical fin and find a conical fin geometry that exchanges the same heat rate

I have to select an arbitrary cylindrical geometry (fixing Radius and Lenght of the cylindrical fin) and calculate the heat rate that it exchanges with the ambient with the 1D heat transfer approximation.

All parameters are known and listed at the beginning of the Matlab code.

Once I find the heat rate exchanged by the cylindrical fin I have to find the geometry of a conical fin that exchanges the same heat rate as the previous calculated cylindrical fin.

I tried this study using both 2d and 3d PDEToolBox in MatLab.

clear all
clc

LR=0.075;               % Fin Lenght
R=0.005;                % Fin radius
P=pi*2*R;               % Fin perimeter
A=R^2*pi;               % Fin cross section (constant for cylindrical fin)
T0=373;                 % Temperature at the base of the fin
Tinf=293;               % Temperature of the air surrounding the fin
k = 17;                 % Thermal conductivity, W/(m*C)
h = 50;                 % Thermal convection coefficient 


lambdaR=sqrt((k*A)/(h*P));

model = createpde('thermal');       % solution of Fourier equation, steady state as default

R1 = [3,4,0,LR,LR,0,...     % Half section of a cylindrical fin
    0,0,R,R]';

gm = [R1];
sf = '(R1)';
ns = char('R1');
ns = ns';
[dl,bt] = decsg(gm,sf,ns);
geometryFromEdges(model,dl); 

figure(1)
pdegplot(model,'EdgeLabels','on','FaceLabels','on')
title ('domain')

axis equal

mtl = thermalProperties(model,...
    'ThermalConductivity',k);

%Boundary conditions: 1,2,3: Robin; 4: Dirichlet
thermalBC(model,'edge',[1,2],...
    'ConvectionCoefficient',0,...
    'AmbientTemperature',0); 
thermalBC(model,'Edge',[3],...
   'ConvectionCoefficient',h,...
   'AmbientTemperature',Tinf);
thermalBC(model,'Edge',4,...
   'Temperature',T0);

% Mesh generation 
generateMesh(model,'Hmax',R/10,'Hmin',R/10,...
    'GeometricOrder','quadratic','Hgrad', 1.)
figure(2)
pdeplot(model)
title ('mesh')
axis equal;

% Solutions
result = solve(model);
T = result.Temperature;
figure(3)
pdeplot(model,"XYData",T,"Contour","off","ColorMap",'jet')
axis equal
title("Steady-State Temperature")

% post processing of the results. Profile of temperature along x direction
np=1000;

for i=1:np
    x(i)=LR/(np-1)*(i-1);
    y(i)=R/2;
    Tfin_th(i)=(T0-Tinf)*cosh(LR/lambdaR*(1-x(i)/LR))/cosh(LR/lambdaR)+Tinf;
end

Tfin_num = interpolateTemperature(result,x,y);
figure(4), plot(x,Tfin_num,'-k');

fs1=16;
title ('Temperature')
xlabel ('$x (m)$','interpreter','latex','FontSize',fs1)
ylabel ('$T (K)$','interpreter','latex','FontSize',fs1)

% heat rate

Qn = -2*pi*R*evaluateHeatRate(result,"Edge",4); % [W]
Qn

Qn_th=(T0-Tinf)*sqrt(h*P*k*A)*tanh(LR/lambdaR);


% Conical fin with the same heat rate of the one above 

% Qn = h*A*(T0-Tinf)
% Find conical area that exchange Qn heat rate!

A_lat_cyl=2*pi*R*LR;
V_cyl=pi*R^2*LR;

[Jon]

You should be able to find closed form, analytical solutions to this problem, especially for the constant diameter pin. This is a standard, text book problem. No need for solving 2d or 3d PDE's. Worst case you should be able to reduce this down to solving an ordinary differential equation.

[Alberto Colombo]

Yes, I found the analitical solutions I just wanted to calculate the heat rate also with a numerical method to compare the result but the main problem is find a conical geometry that exchanges the same heat rate.

Answer 1

[Jon]

Let's say the heat exchange rate for the pin (cylindrical) fin for your specified geometry has been found to be equal to a value Qpin. Assuming you have an analytical expression (function) for your cone fin that outputs the heat exchange rate as a function of the geometry, e.g. base diameter, cone angle, .. choose the geometric parameter, p, to be varied, and assign all others to constant value. This will then give you a funtcion, let's call it Q = f(p). Now you have to find the value of p such that f(p) = Qpin. You can do this using MATLAB's fzero by rearranging to solve f(p) - Qpin = 0

[Alberto Colombo]

What is the expression to get the heat rate starting from geometrical parameters for a conical fin?

[Torsten]

I wonder how the PDE toolbox knows that in your code above, you work in cylindrical and not in cartesian coordinates.

[Alberto Colombo]

So how I should treat the problem with cylindrical coordinates, can you make a simple code example of a rod fin in cylindircal coordinates with pdetoolbox?

Thank you

[Torsten]

I have no experience with the PDE Toolbox, but my guess is that

model = createpde("thermal","steadystate-axisymmetric");

would be correct.

Now you only have to find out which axis represents r and which axis represents z.