Pdepe- Spacial Discretization has failed. Trying to solve a heat transfer equation and I'm confused on how to definte the boundary conditions for my case

Hello,

I'm trying to solve a heat transfer equation for a heat exchanger modeled as a flat plate which has a constant temperature at one end, with no directly applied heat flux, and I want to get an output graphic of the temperature change through the material over time, and I'm getting the following error:

Error using pdepe

Spatial discretization has failed. Discretization supports only parabolic and elliptic equations, with flux term involving spatial derivative.

I was trying to do this with PDEPE but I'm unsure of how to define the boundary conditions for this case.

Please see the below:

L = 0.0155; %15.5mm

global rho cp k q

phi = 0.1;

rho=(phi*2700)+((1-phi)*712); %kg/m^3

cp=(phi*900)+((1-phi)*2200); %J/kg-K

k=(phi*80)+((1-phi)*0.36); %W/m-K

q = 73E3;

x = linspace(0,L,500);

t = linspace(0,200,500);

m = 0;

sol = pdepe(m,@PCMpde,@PCMic,@heatbc,x,t);

figure(1);

colormap hot

imagesc(x,t,sol)

colorbar

xlabel('x')

ylabel('time')

function [c,f,s] = PCMpde(x,t,u,dudx)

global rho cp k

c = rho*cp;

f = k*dudx;

s=0;

end

function u0 = PCMic(x) %Temperature is 'u'

u0 = 22; %Initial Temperature

end

function [pl,ql,pr,qr] = heatbc(xl,ul,xr,ur,t)

global q

pl = 40;

ql = 1;

pr = ur;

qr = 1;

end

I'm unsure if I'm able to do this with the above, as I believe I need to have a definite boundary condition for the temperature at the far end of the heat exchanger, but I thought this would be able to solve for that in the steady state as I don't want to limit the temperature at the distance L.

Answer 1

[Torsten]

At the moment, your boundary conditions read

k*dT/dx = -40 at x = 0
k*dT/dx = -T  at x = L

I don't think this is what you try to set.

To set the boundary conditions, p and q have to be chosen such that

p + q*f = 0

where f = k*dT/dx in your case.

Choose pl, ql at x = 0 and pr, qr at x = L appropriately.

[David Argueta]

I'm confused on how I would choose these appropriately for my case, as I'm unsure of what the temperature at x=L would be, however I want to specify that at x=L, my heat flux, q , would be zero as I want to have a sealed/adiabatic end.

I'd like to have an applied temperature of 40 at x=0, while the initial temperature for all oher points would be u0 of 22 at time=0, which lead me to believe that qr should be 0 at x=L.

Would you be able to explain this as I think I've misunderstood how to apply this in the form of p+q*f = 0

Thank you

[Torsten]

It should be obvious that

pl = ul - 40;
ql = 0;
pr = 0;
qr = 1;

for the boundary conditions you want to set since

(T-40) + 0*k*dT/dx = 0 at x = 0
0      + 1*k*dT/dx = 0 at x = L.