# Is there a heat transfer equation that takes conduction, convection, radiation and dT/dt into account?

I see equations that take 2 or 3 of the listed parameters into account but I haven't been able to find one that is that complete. I'm seeking to solve this equation using matlab for a simulation project to study how the heat dissipates on a microheater given an electrical power or current as input and see how quickly the temperature rises, and so forth. But first, I need the proper equation.

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The general form of the heat equation would be

$$\frac{\partial T}{\partial t} + (\vec{u} \cdot \nabla) T =a \nabla^2T + S$$

The first term is the time derivative, the second term convection, the third term diffusion of heat and the last term is the source term, which can be anything.

However, normally I would expect radiation to come into the equation via the boundary conditions as some kind of heat flux, and not as a source term.

Radiation is often described by the Stefan-Bolzman law, e.g.

$$\text{heat flux} \propto T^4$$

which is a boundary condition.

In terms of modelling, one normally linearized this term, e.g $4T_0^3 (T-T_0)$.

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I don't see how the second term takes convection, usually I see a convection coefficient "h" times u. – user17338 Oct 24 '13 at 22:18
@user17338 In that case, it is also a boundary condition. If there is flow in the interior of the domain, you will need this second term. – Bernhard Oct 25 '13 at 5:53

The nature of 'complete' heat equation will depend on your domain. If I understand correctly, you want to model the temperature rise/heat dissipation in a microheater (a solid?) when it is electrically powered. The heat transfer inside the heater in this case will be solely due to diffusion. Convection/radiation will not be a part of heat transfer internal to the heater. However, both these mechanisms would appear at the heater boundary i.e the boundary conditions. So the governing equations would be:

1) $$\frac{\partial T}{\partial t} =a \nabla^2T + S$$ in the heater interior. The term S represents heat generated per unit volume.

2) $$-k\nabla T =h (T - T_0) + \sigma\epsilon(T^4 - T_0^4)$$ at the boundary.

You will need to know the heat transfer coefficient 'h' and emissivity in the problem. As a simplification (since the heater is 'micro') you can assume the heater to be a 'lumped' system so that you can neglect heat diffusion inside the heater. The first equation would then simply become

3) $$\frac{\partial T}{\partial t} = S$$

Equation 2 remains unchanged. Be careful about the units and dimensions on 'S', the heat source...

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