# Can the heat equation apply to gases?

1. Jul 19, 2011

### mikeph

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

2. Jul 19, 2011

### timthereaper

The heat equation was derived from Fourier's Law. Fourier's Law really just states that the heat flow is proportional to the temperature gradient. 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.

3. Jul 19, 2011

### mikeph

How about a numerical approach. Could I split a region into subregions and treat each subregion as a black body with its own temperature? From that I could find the net change in radiation power in and out (from neighbouring regions), and apply Fourier's law across each boundary?

Do you know if there's any commonly accepted work done on this? The wikipedia article is very vague and all its references are online books or ones that aren't in the library.

4. Jul 19, 2011

### timthereaper

I'm not an expert with heat transfer, but I'm sure you can do some sort of numerical lumped-model transient nodal analysis with convection and conduction. I think that unless the radiative heat transfer is signficant when compared with the conduction-convection heat transfer, you shouldn't have to consider it. Modeling radiation heat transfer would entail Monte Carlo methods or something like that, which are somewhat complicated and might be unnecessary.

A great introductory text on heat transfer modes and anaylsis techniques is https://www.amazon.com/Heat-Transfe...362/ref=sr_1_2?ie=UTF8&qid=1311095676&sr=8-2".

Last edited by a moderator: Apr 26, 2017
5. Jul 19, 2011

### Studiot

Mikey, you need to provide more details.

Are you talking about a sealed volume of gas and is phase change envisioned?
Is the gas still or flowing?

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.

6. Jul 19, 2011

### mikeph

Sealed volume with flowing gas. But I could make the concession that it's isobaric.

7. Jul 19, 2011

### timthereaper

Good call Studiot. Maybe I said something that's untrue, but I guess I should explain my line of thinking:

Things we know already: The differential form of Fourier's law is that heat flow is proportional to the temperature gradient and it's used to model conduction flow. It's applicable anywhere inside the solid.

Things I was thinking: Conduction models heat flow across a solid-fluid interface and it's states that the heat flow is again proportional to the temperature gradient across the interface. It's only valid across the interface (i.e. no convection happens from the center of the solid to the fluid, only happens from outside of solid to fluid). The proportionality constant is also a bit variable (h depends on so many factors).

It mimics Fourier's Law, so I figured you could derive a similar heat equation to the classic one using convection.

8. Jul 19, 2011

### Studiot

If by a sealed volume but the gas is flowing you mean heat transfer by gas flowing in through a pipe heat exchanger of fixed volume you need the flow version of Bernouilli's equation.

If by a sealed volume you mean the space between say double glazing panes the calculation is totally different.

If by sealed volume with gas flow you mean the heat pipe that is commonly used these days in laptop computers the calculation is different again.

Without sufficient information no detailed progress can be made, only generalisations are possible.

9. Jul 19, 2011

### timthereaper

How does the gas flow if the volume is sealed? Convection currents?

10. Jul 20, 2011

### mikeph

Surely there is a general equation to describe heat transfer in all these cases, and you only require more information to chose a limited form of this general equation? The basics of heat transfer should not depend on the shape of the volume. I'd like to investigate the basics, not solve a limited case.

Boundaries can move (eg. taylor couette flow), heat sources inside it can move, for example. Or perhaps my initial conditions just involve some sort of vorticity which would take some time to die down.

11. Jul 20, 2011

### Studiot

12. Jul 20, 2011

### timthereaper

I guess I'm still not understanding the setup. I'm picturing a sealed cylindrical container with gas inside of it and I'm guessing this isn't correct. You mentioned Taylor-Couette flow, which would apply if you're doing an analysis on the gas as it moves through the control volume, like in a section of pipe.

There is no general equation to study the heat flow for any given situation. The problem is that heat transfer itself is complex. There are whole books devoted to techniques of studying heat transfer and people in industry that only deal with it. As a professor told me once, even the best measurements for calculating specific coefficients for problems can still have 100% error (depending on other factors). Also, shape affects heat transfer problems a lot. Unfortunately, for all but the simplest systems, you have to go through a process. There is no magic formula.

13. Jul 20, 2011

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:

$$\frac{Dh}{Dt} = \frac{Dp}{Dt} + \mathrm{div}(\kappa \nabla T) + \Phi$$

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

14. Jul 20, 2011

### Studiot

I would be more than a little interested to see your equation applied to my heatpipe example.

15. Jul 20, 2011

So would I, but I highly doubt it could be done by hand, and it would depend on the heat pipe. I remember during undergrad I did a project where we designed a heatsink for a CPU and my group did some heat pipes but we used the empirical correlations. They are usually a really good estimate over a wide range of conditions. For the OP, I would suggest picking up a heat transfer book such as Incropera and DeWitt or Mills. They would be helpful. I don't remember all of those relations because it has been a while since I took that class and I don't use it much these days.

16. Jul 20, 2011

### mikeph

Ah Incropera that book always comes up to haunt me. I spent a long time trying to follow a paper that referenced it, turns out it was a nonsense reference and I was wasting my time! I'll check these books out, thanks.

I realise it's a huge area, just wanted some sort of fundamental overview so I can figure out for myself what is necessary and what parts I can simplify.

17. Jul 20, 2011

### Studiot

The point is that my heat pipe is a phase change refrigeration system.

mikeyw

Transport Phenomenon
by
Bird, Stewart and Lightfoot

go well

18. Jul 20, 2011

Well in theory, the Navier-Stokes equations can handle phase changes. Two-phase flows are an active area of research. I am not familiar with two-phase flows personally though.

19. Jul 20, 2011

### Studiot

It has been said, both here and in the other thread I linked to, and by several posters, that there is a process involved.

I subscribe to that view.

I don't call this one equation I call this a process.

I have not, in fact, suggested that heat transfer is not proportional to temperature difference (gradient).
Rather I have suggested that it is one (small) part of the overall set of equations.

It may be that mikey wants to create a numerical model (FE or other) within the gas itself and is looking for a suitable function to apply to the grid.

But not being a mind reader I can't tell.

20. Jul 20, 2011

### timthereaper

I agree. The information given by the OP is about the bare minimum needed to start a discussion.