Heat Flow Velocity in Solids: Factors & Impact

[R Power]

Hello

With what velocity does heat flow in solids? What are the factors on which its velocity depends?

Thnx

 

[Dadface]

Hello

With what velocity does heat flow in solids? What are the factors on which its velocity depends?

Thnx

What are your ideas?

 

[R Power]

Simply speaking I don't know.
I have never seen any problem or articles relating velocity of heat flow and its analysis. There is a whole subject of heat transfer but that is related to rate of heat flow or amount of heat flow per unit time but doesn't deal with velocity.
So, the question just poped up in my mind while studying rate of heat transfer.

 

[Dadface]

It might help if you looked up "thermal conductivity".In steady state the heat flow through all sections of the material equalises so what is meant by velocity of heat flow?Before steady state is reached the heat travels with a speed that depends on the structure of the material and the temperature gradient(plus other factors?) and so I think the speed reduces exponentially with time tending to zero as steady state is approached.Perhaps you want the velocity of the free electrons and or phonons that conduct the heat and in the former case it might be helpful to look up the analogy between thermal and electrical conduction.Anyway,I think it's an interesting question and I think I'm rambling a bit.I will come back later,possibly after some googling.

 

[R Power]

Take an iron rod and heat it from one end. Put each one of your hands at the respective ends of rod. You feel instantly feet hotter at the end you heat but it will take some time for you to feel hotter at the other end. This time is the time heat has taken to flow. That means heat has a flow velocity. It is this velocity I am talking about and not of free electrons on anything. I was just wondering why aren't any articles or texts written on velocity of heat.
Secondly, thermal conductivity relates to allowance by a material of how much heat per unit time travels.

 

[russ_watters]

Take an iron rod and heat it from one end. Put each one of your hands at the respective ends of rod. You feel instantly feet hotter at the end you heat but it will take some time for you to feel hotter at the other end. This time is the time heat has taken to flow. That means heat has a flow velocity. It is this velocity I am talking about and not of free electrons on anything. I was just wondering why aren't any articles or texts written on velocity of heat.

The speed at which any physical disturbance can propagate in a material is the speed of sound. So the time it takes for the temperature to start changing at the other side would be based on the speed of sound.

However, in heat transfer I don't recall ever seeing an equation that deals with this because you can't heat an object up fast enough for that to matter. So the equations assume instantaenous propagation. When you apply heat to one side - according to the equations - you set up a thermal gradient (curved) across the entire object.

[edit] Ehh - I hated heat transfer in school. I'm not completely certain I'm right about how it is dealt with.

 

[gmax137]

google or wiki 'thermal diffusivity'

it is the ratio of conductivity to volumetric heat capacity

basically, it tells you how fast heat is conducted through a material

 

[R Power]

google or wiki 'thermal diffusivity'

it is the ratio of conductivity to volumetric heat capacity

basically, it tells you how fast heat is conducted through a material

Yeah, thermal diffusivity tells us how fast heat diffuses or flows through a material. I think this is answer to my question but it contradicts Russ' answer of speed of sound.

 

[russ_watters]

It doesn't appear to me that that contradicts me - why do you think it does?

 

[Studiot]

Velocity of heat flow?

Bit unspecific don't you think?

There is a difference between the rate at which temperature changes at a given remote point from heat input and the rate at which heat energy is transferred within a body.

Yes indeed there are equations, Fourier was the first to study these and the solution to the diffusion equation for heat flow bears his name.

However heat transfer is a complicated subject because there are several transfer mechanisms available (conduction, convection, radiation), depending upon the transfer path. Heat transfer rates depend upon

The shape of the transfer medium (that is why we have cooling fins)

The material of the transfer medium

The conditions at the surface of the transfer medium

The method of heat input

The physical state of the transfer medium (solid, liquid, gas)

and probably a few things I haven't though of.

The subject is of particular interest to chemical and mechanical engineers.

 

[russ_watters]

The reason I hated heat transfer is that I am not good at partial differential equations. Here's the equation that describes [what I think is] the main issue in this thread: http://en.wikipedia.org/wiki/Heat_equation

Consider the following two questions:
1. If heat is applied to one end of a metal bar, how long does it take before the other end feels *any* change in temperature?
2. If heat is applied to one end of a metal bar, how long does it take before the temperature at the other end rises by 1C?

These are two very different questions. The wiki tells you how to answer the second question and assumes that the answer to the first question is zero. The OP appears to be asking the first question, but may also not be realizing that the two are in fact different questions.

The wiki actually has an animation showing how temperature changes along the length of a metal bar over time.

 

[Mapes]

Russ and Studiot have already given great answers; I'll just add to why one can't find any articles or books on "heat velocity." It's because conductive heat transfer occurs by diffusion, which has no well-defined velocity. As an example, consider a temperature change that propagates down an iron rod, as the original poster described. If you make the rod twice as long, the same temperature change at the end takes not twice as long (which would be the case if velocity were constant), but four times as long (since time scales as $L^2/D$, where D is the thermal diffusivity, also written $\alpha=k/c\rho$, where k is the thermal conductivity, c is the specific heat, and $\rho$ is the density). So it's not productive to develop any theory or equations on "heat velocity" (except to acknowledge the upper bound of the speed of sound, as Russ describes).

 

[R Power]

Now I have doubt in thermal diffusivity as answer also.
Diffusivity is the ratio of thermal conductivity to heat capacity. Now lower the heat capacity, less heat will be required to raise the temp. so more heat will be available to flow, this will just increase the heat flow rate and decrease the time required to increase temperature.

Consider the following two questions:
1. If heat is applied to one end of a metal bar, how long does it take before the other end feels *any* change in temperature?
2. If heat is applied to one end of a metal bar, how long does it take before the temperature at the other end rises by 1C?

Yes, I am asking the first question.

 

[Mapes]

The time for an infinitesimal temperature increase will be the length of the bar divided by the speed of sound.

 

[R Power]

The time for an infinitesimal temperature increase will be the length of the bar divided by the speed of sound.

So, speed of sound is the answer. Thnx to everyone.

 

[Studiot]

The time for an infinitesimal temperature increase will be the length of the bar divided by the speed of sound.

This is surely a gross oversimplification.

It implies that the temperature rise at the remote end is independent on the shape of the bar, which is in direct contravention of Fourier's Law.

Suppose part of the centre of the bar were drawn out to infinitesimal thickness.

What would then be the rate of temperature increase at the remote end?

 

[Mapes]

It implies that the temperature rise at the remote end is independent on the shape of the bar, which is in direct contravention of Fourier's Law.

I wouldn't say it implies that. It is compatible with the idea that a temperature change will not propagate any faster that the speed of sound, and with the idea that, in theory, you can come arbitrarily close to a time delay of $L/v$ (where v is the material's speed of sound and L is the bar length) to achieve a given temperature change at the far end, given that you're allowed to increase the instantaneous temperature change at the near end. I think these are pretty reasonable statements. But it doesn't say anything about the rate of temperature increase, and I agree that this rate will depend on geometry.

 

[Studiot]

Yes, I am asking the first question.

So yes the time indicated by the speed of sound in the material is the answer.

But, forgive my confusion, the original question requested a velocity, which is a rate not a time interval.

 

[Integral]

Velocity is not a easy thing to talk about in this problem. There is a "wavefront" of change propagating through the material that is described by the http://en.wikipedia.org/wiki/Erfc" . This means mathematically that information concerning the coming change reaches the far end INSTANTLY. So the change at any point begins very slowly and it remains slow changing until the steep wavefront passes at which time the temperature quickly rises to near the final temp then slowly increases to the final temp.

So how do you define a velocity here? The initial small change moves at a very high rate while the main part of the change lags far behind. This is not a simple dynamics problem and you need to be very careful about trying to make it one.

(Ref Churchill Operational Mathematics )

 

[R Power]

The initial small change moves at a very high rate while the main part of the change lags far behind.

How much high rate? Is there a certain value you know for a given material? This is what I want to enquire to which I have been answered with speed of sound.

 

[CraigH]

Yeah, thermal diffusivity tells us how fast heat diffuses or flows through a material. I think this is answer to my question but it contradicts Russ' answer of speed of sound.

Have you found the answer to your question yet? I would also like to understand why the velocity of heat flow is not something that is taught in heat transfer lectures. I've gone through the entire heat transfer section of my data book and there is nothing on velocity.

My guess is that if it is based on diffusion it is too random to define. An analogy would be that if the heat was ink, and the medium was water, what is the velocity of the ink flow when it is dropped into the water? It can not be easily calculated as there are too many variables.
I'm not sure how valid this analogy is, I'm pretty awful at thermodynamics. Also heat is nothing like a particle of ink, so I'm not sure if the spread of energy through a material is like the spread of particles through a material, since energy is a property of a thing and not an actual tangible thing.

It would be good if someone could confirm my analogy?

 

[Chestermiller]

You need to examine solutions to the transient heat conduction equation. If you have a semi-infinite slab at a constant initial temperature T₀, and you suddenly change the temperature at the bottom surface of the slab to T₁, the solution to the transient heat conduction equation for this situation is:

$$T=T_0+(T_1-T_0)erfc(\frac{x}{\sqrt{4\alpha t}})$$

where x is the distance measured up from the bottom of the slab, t is time since the surface temperature was changed, and $\alpha$ is the thermal diffusivity. This equation tells us that the location at which a specific temperature is observed within the slab increases with the square root of time. Thus, this equation can tell you the location at which the temperature has changed by only a specified small percentage of the overall temperature difference. In practice, this is often taken as a measure of the leading edge of the temperature profile.

 

[AlephZero]

You need to examine solutions to the transient heat conduction equation.

But first, you need to decide if the usual heat conduction (or diffusion) equation is just a nice approximate model of the physics. The conduction equation is parabolic PDE, and its solutions say that effects can propagate instantaneously over any distance in space, however large. I think Mr Einstein might have a problem with that

I think the OP's original question was whether a more accurate model of the physics would be a hyperbolic equation, where effects can only propagate through space at a finite speed (and if so, what is that speed).

Of course for most practical purposes, solutions of the "standard" conduction equation are sufficiently accurate that the difference doesn't matter.

But clearly radiative heat transfer travels at the same speed as any other form of EM radiation, for example...

 

[daveyrocket]

The speed of sound does not have anything to do with thermal conduction for metals. In insulating systems, thermal conduction occurs only because of the vibrations of the atoms, which also carry sound. For metals, the free electrons also conduct heat but play no role in propagating sound. This gives metals a thermal conductivity that is a hundred times larger than thermal conductivity of insulators.

 

[davidyanni10]

Also, keep in mind that while primarily phonons (upper bound speed of sound) conduct heat through insulators, and primarily free electrons conduct heat through conductors (not sure of the theoretical upper bound on velocity there), you still have radiative heat that travels at the speed of light, such as the heating of the Earth from the sun.

As another poster said, because heating is a diffuse process, it is not really productive (though interesting) to think about the velocity of an individual heat carrier.

 

[Chestermiller]

But first, you need to decide if the usual heat conduction (or diffusion) equation is just a nice approximate model of the physics. The conduction equation is parabolic PDE, and its solutions say that effects can propagate instantaneously over any distance in space, however large. I think Mr Einstein might have a problem with that

The OP's response in posting #5 seemed to clearly indicate (at least to me) that he was seeking a practical answer to his question. He appeared to be asking how long would it take for his hand to feel the effect at the other end of the bar. It is true that that the unsteady state heat conduction equation has some minor approximations involved, as does any useful model of physical systems. For example, it omits the effects of the phases of Jupiter's moons and the effect of a mosquito burping in Africa. But, over the years, it has stood the test of time in delivering accurate answers to practical problems.

 

[russ_watters]

Guys, this thread was brought back from the dead by someone posting noise (post deleted). Its best to just let it die again.

 

[Avski]

from what I can gather, hydrogen heat applied to a plate of some sort to provide heat which inter is used as thermal thrust or emission probable excites the particles in the plate material theirby causing the heat by friction. These excited particles would be the heat and would vairee depending on the plate material. Maximum heat emissions\thrust would be achived with a plate material that could handle the hydrogen internal heat sourse. What this material would be I am still unsure of at this point in time.