How to rewrite a PDE equation and set the parameters for solving the temperature changes?

I am working with heat exchange problems. I have cermic exchanger with slots for air inlet (ceramic is being heated from warmer inside air) and after period of time the direction is changing and air from the outside remove heat from the ceramic. I am modelling one tiny slot heating (square, 3mmx3mm). I have one PDE equation for solid (here: ceramic, Ts is a modeled temperature):

ro*cp*(dTs/dt)-k*((d^2Ts)/dx^2)=h*(Tg-Ts)

And other equation for the air heating (Tg is an air temperature, modeled now):

ro*cp*(dTg/dt)+u*ro*cp*(dTg/dx)=h*(Ts-Tg)

Annotations:

t - time; x - length, m; ro - density, kg/m^3; cp - specific heat,J/(kgK); h - convection coefficient,W/(m^2K); k - thermal conductivity of copper, W/(mK); u - fluid velocity, m/s;

Should I manage it with PDE solver and set the parameters, i.e. for Ts:

c = ro*cp;

f = -k*DuDx;

s = h*(Ts-Tg);

?

Then, how set the boundary conditions and make Ts nad Tg as a function of the temperature? It should be evaluated for different values during the analysis.

Answer 1

[Bill Greene]

I think you can use pdepe to solve this but you need to include both the equation for Ts and the one for Tg; in your pde function, the returned variables c, f, and s will have two entries. Example 2 in the pdepe documentation shows how to handle two equations. In your two equations, do you have different values of ro and cp for ceramic and air?

Since you didn't specify your boundary conditions, it is difficult to comment but I don't understand what you mean by "make Ts nad Tg as a function of the temperature".

I should also add, if the value of u is "large", it can be challenging to solve these equations with pdepe.

[Joanna Aleksiejuk]

Thanks a lot, I just found that Example 2.

Yes, in my two equations I have different values of ro and cp for ceramic and air.

Ts will be changed with time as a solution of equation, won't it? Maybe I wrote it not correctly. In part of equation "h*(Tg-Ts)" Tg will be constant and Ts will change with time. Then, I want to input my real data as a Tg also. Finally, I want to model how the temperature of ceramic and air will change.

Do this explanation is more clear to understand?

Answer 2

[Joanna Aleksiejuk]

According to boundary conditions, for the analysis, I am considering that the square holes ale bounded symetrically on all sides of its wall and within this boundary is adiabatic, where the heat going into the boundary is equal to the energy leaving the boundary, exapmle: https://www.nuclear-power.net/nuclear-engineering/heat-transfer/thermal-conduction/heat-conduction-equation/adiabatic-boundary-thermal-symmetry/

So, in my case the above mentioned conditions will be the same:

-k*(dTs/dx)=h*(Ts-Tg) for (0,t)

and

dT/dx=0 for (L/2,t)