Can the heat equation apply to gases?

[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

 

[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.

 

[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.

 

[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/dp/0073529362/?tag=pfamazon01-20.

 

[Studiot]

convection also technically follows Fourier's Law.

Are you sure about that?

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 C_p and C_v and other thermodynamic matters to consider.

 

[mikeph]

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

 

[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.

 

[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.

 

[timthereaper]

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

 

[mikeph]

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.

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.

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

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.

 

[Studiot]

Someone else is also studying this problem here.

The replies there are instructive.

https://www.physicsforums.com/showthread.php?t=514292

No there is no single equation for a more complicated question, there is a process.

 

[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.

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.

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.

 

[boneh3ad]

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

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.

With a solid you only have one constant to consider - the thermal conductivity.

When you transport heat by moving the molecules themselves you have C_p and C_v 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:

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

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

 

[Studiot]

boneh3ad

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

 

[boneh3ad]

boneh3ad

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

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.

 

[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 realize 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.

 

[Studiot]

boneh3ad

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

mikeyw

I would add
Transport Phenomenon
by
Bird, Stewart and Lightfoot

to the reading list.

go well

 

[boneh3ad]

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.

 

[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.

you need to simultaneously solve the Navier-Stokes equations along with the energy equation

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.

 

[timthereaper]

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

 

[boneh3ad]

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.

Let me provide you with the quote by the OP. It didn't ask about the specific application, but whether or not the heat equation applied to gases. It does. That is what I answered.

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

 

[Studiot]

Let me provide you with the quote by the OP. It didn't ask about the specific application, but whether or not the heat equation applied to gases. It does. That is what I answered.

Fair enough, but lots of things apply to gasses and here is a further quote from mikey.

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

Edit
And in my post #8 I gave various versions of the energy equation, applicable to different situations.

You have provided a more general one.

All that remains for us to help mikey choose the appropriate one is for mikey to tell us the application.

If, as I wondered, mike is actually looking for an approximating function for a numerical mesh, then none of those so far proposed are appropriate.

 

[boneh3ad]

Fair enough, but lots of things apply to gasses and here is a further quote from mikey.

That was a later quote after you guys asked him. I was trying to clear up all the confusion at the beginning, though it seems I just threw the thread into a greater state of disarray. Oops.

 

[Studiot]

No, your input is very welcome. This is a fine discussion

Please also see my edit to my last post.

 

[mikeph]

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.

I tried to make myself clear, I have no specific problem that needs to be solved, I just want to learn about the mathematics of temperature distributions in gases, rather than solids. Maybe it's just the way I learn is different to the way you learn, I don't think specific examples will help, so I don't want to mislead anyone by suggesting one. I want a general overview of what processed affect the temperature evolution in a gas.

 

[timthereaper]

I think the main problem with asking such a general question is that there are so many techniques out there that would do what you're looking for. It really depends on how granular you want the details. I guess the most general you can get is analyzing the gas in a lumped-model and considering the convection outside and doing a power analysis. To get more information, you can use more sophisticated heat transfer techniques, boneh3ad suggested a solution if you want the most detail that fluid mechanics can give. If the gas is too rarefied that it can't be considered as a continuum, then you would have to use statistical mechanics techniques. In general, numerical solutions can give more detail than analytical, but you have to make sure that your numerical algorithms are correct. There are a whole host of techniques that can be applied that give different amounts of information and the one you use depends on the problem you're solving.

 

[boneh3ad]

In general, numerical solutions can give more detail than analytical, but you have to make sure that your numerical algorithms are correct.

That is absolutely false. Numerical solutions give essentially a discrete set of solution points, but it is very possible to lose information in there. A true analytical solution gives you a continuous solution across the entire solution space. The problem is, many if not most problems can't be solved analytically.

 

[Studiot]

You can get easily get bogged down in calculation complexity or you can try things the easy (engineer's) way as in this thread.

https://www.physicsforums.com/showthread.php?t=515471