Coupled conductive-convective heat transfer using FDM

[angsu119]

Now I need to solve a transient problem involving both conduction and convection.

Homework Statement

In this problem there's a thin plate being heated by two constant-temperature sources. One small source is in contact with part of the plate. Another one is separated with the plate by air.

Homework Equations

The Attempt at a Solution

I'm using Matlab, and able to solve pure conduction problem. I use the heat equation a*dT/dt=delta²dT and five point scheme. But now significant convection is coupled into the problem when air velocity is given. I'm not sure what is the corresponding equation combining both conduction and convection, and how to write the FDM form. Can anyone help me? Thanks!

 

[angsu119]

Is my question not stated clearly?

 

[piercas]

I would recommend using the conservation of energy equation to combine both conduction and convection in your problem. This equation states that the change in internal energy of a system is equal to the sum of heat transfer and work done on the system. In this case, the internal energy change is due to both conduction and convection, so we can write it as:

ρc∂T/∂t = k∇^2T + ρc(∂T/∂t)conv

where ρ is the density, c is the specific heat, k is the thermal conductivity, and (∂T/∂t)conv is the convective heat transfer term.

To solve this using FDM, you can discretize the equation in both space and time using a similar approach as you did for the pure conduction problem. This will give you a set of equations that can be solved using an iterative method such as the five-point scheme you mentioned.

In terms of incorporating the air velocity, you can use the convective heat transfer coefficient (h) to calculate the convective heat transfer term as:

(∂T/∂t)conv = h(Ts - T)

where Ts is the temperature of the surrounding air and T is the temperature of the plate. The convective heat transfer coefficient can be calculated using empirical correlations or from experimental data.

I hope this helps you solve your transient problem involving coupled conductive-convective heat transfer using FDM. Good luck!