solving 1D heat equation with constant heat flux (boundary condition)

Hello everyone, i am trying to solve the 1-dimensional heat equation under the boundary condition of a constant heat flux (unequal zero). The heat flux is on the left and on the right bound and is representing the heat input into the material through convective heat transfer. I am using pdepe to solve the heat equation and with dirichlet boundary conditions it is working. To represent the heat flux i was thinking of a Neumann boundary condition, but i can't figure out how to contribute the value of the heat flux into the boundary condition. For a heat flux of zero (isolation) the code would look like this (leaving out the declaration of the parameters):

function u0=IC(x)
u0=T_0;
end

function [pl,ql,pr,qr]=BC(xl,ul,xr,ur,t)
ql=1;
pl=0;
qr=1;
pr=0;
end

function [c,f,s]=DGL(x,t,u,DuDx)
c=1;
f=a*DuDx;
s=0;
end

% solution
x=linspace(0,L,10);
t=linspace(0,t_End,10);
sol=pdepe(m,@PDE,@IC,@BC,x,t);
u=sol(:,:,1);
uu=sol(end,:,1);
end

Now the heat flux being unequal zero, my idea was to modify pl and pr in dependence of the heat flux density, but i can't find a connection. Has anyone an idea?

Regards, Malte

Answer 1

[Torsten]

What is the boundary condition you are trying to implement ?

lambda*dT/dn = alpha*(T-T_ext) (n: outer normal) ?

Best wishes

Torsten.

[MalteSe]

Yes i think that is the boundary condition i am trying to implement. At first i thought, that it would be something like:

alpha*(T-T_ext)=rho*c_p*dT/dt*dx

As the heat transfer causes the material to heat up, but i couldn't solve that equation due to the dt.

With your equation i get:

q=1 and p= -alpha*(T-T_ext)/(rho*c_p)

which gives me feasible results when comparing heat transfer with the "heating up" of the material, so thanks a lot for your answer!

But why is the equation i tried to implement wrong? Because technically the process at the border is heat transfer rather than heat conduction...

Cheers, Malte

[Torsten]

q = 1, p = -alpha*(T-T_ext) at the left boundary
q = 1, p =  alpha*(T-T_ext) at the right boundary.

if in DGL, you set

f = lambda*DuDx;
c = rho*cp;
s = 0;

Best wishes

Torsten.