MikeyW said:
I've never known this but the equation only seems to contain a conduction term so I assume it can only apply to solids. Is there a similar equation for the time-evolution of temperature fields in gases, where convection is also considered? (how about radiation? although that sounds like it will be extremely complicated).
Thanks
timthereaper said:
While Fourier's Law is applied mainly to conduction, convection also technically follows Fourier's Law. If you look at the original derivation of the heat equation, it uses thermal diffusivity properties for solids. You could possibly rederive the heat equation starting with gases and take convection into account. Defining the boundary conditions would be harder as well, but I think it's doable. I'm not sure the classic heat equation would work because radiative heat transfer is more complex.
Studiot said:
With a solid you only have one constant to consider - the thermal conductivity.
When you transport heat by moving the molecules themselves you have Cp and Cv and other thermodynamic matters to consider.
Fourier's law applies equally (and is used equally) for convection and conduction. In fact, convection is actually a combination of heat diffusion (or conduction) and advection. All materials - solids, liquids, gases, plasmas - have a thermal conductivity, \kappa, that can be used with Fourier's law. The difference is that for a fluid in motion, Fourier's law must be combined with the fluid motion.
Typically, convection is taught in undergraduate heat transfer courses empirically, that is, using an empirical convection coefficient for a given system. Slightly more advanced classes will then move into the more useful, but still empirical nondimensional numbers such as the Nusselt number or Grashof number to name a couple. However, if you really want to get the true heat transfer properties in a convective system, you need to simultaneously solve the Navier-Stokes equations along with the energy equation and you can get instantaneous heat transfer anywhere and any time in the flow. You can even add in a radiation term if you would like. Of course, solving those directly often requires a supercomputer, so the empirical correlations are usually used in practice to get a good engineering estimate.
Take a quick look at the canonical form of the energy equation for fluids:
<br />
\frac{Dh}{Dt} = \frac{Dp}{Dt} + \mathrm{div}(\kappa \nabla T) + \Phi<br />
That \kappa \nabla T term is in fact Fourier's law. It is the diffusion term in the energy equation.
Just for clarity, in the above equation,
\frac{D}{Dt} is the total derivative
h is the enthalpy
p is the pressure
\kappa is the thermal conductivity
\Phi = \tau^{\prime}_{ij}\frac{\partial u_i}{\partial x_j} is the dissipation term
