Heat Equation: Understand Propagation of Heat

[Clausius2]

Well, this is my question. It's something like a philosophical thinking.

Look at this problem of the Heat Equation:

$$\frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2}$$

with: 0<x<L

$$T(x,0)=g(x)$$ and

$$T(0,t)=T_1$$

$$T(L,t)=T_o$$

This is a parabolical PDE equation. It has only one characteristic line:

$$\frac{dx}{dt}=\infty$$

So that, only information happened along the entire domain and in past times affects to what will happen in the future. It has sense. But the difference with hyperbolic equations, like Wave Equation is there is not any physical velocity of propagation of the phenomena. I will explain that:

A Wave has a finite velocity in any medium (except in a perfect liquid). One perturbation produced in some place takes a time away for being propagated to another zone.

On the other hand, in a change of time of $$dt$$ in Heat Equation, the initial distribution g(x) is smoothed and every point of the domain feels the heat flux instantaneously.

This seems rather disappointing, isn't it?. I don't know neither instantaneous interaction or instantaneous effect. With Heat Equation, the effects appear to be transmitted with infinite velocity. In fact, nobody talks never about velocity propagation of the heat.

What do you think about this strange effect?
Or am I missing something?.

 

[Tide]

You've discovered one of the dirty little secrets of physics! This is typical of diffusion type equations. Obviously, you cannot have atoms or molecules or electrons zipping along at infinite velocities but it's an artifact of the approximation. For example, one could work with the full Boltzmann equation or something similar but that is way too difficult to deal with for most situations. In the relatively rare instances where it matters people often invoke "flux limiters" or simply choose an arbitrary cuttoff such as the speed of light.

It is somewhat akin to the problem of using a Maxwellian distribution for particle velocities and including the range of velocities all the way to infinity. It's simply convenient and the errors are relatively small with the number of particles out further than a few dozen standard deviations being minuscule.

BTW - that should be a partial derivative with respect to time on the left hand side of your heat equation!

 

[Clausius2]

You've discovered one of the dirty little secrets of physics! This is typical of diffusion type equations. Obviously, you cannot have atoms or molecules or electrons zipping along at infinite velocities but it's an artifact of the approximation. For example, one could work with the full Boltzmann equation or something similar but that is way too difficult to deal with for most situations. In the relatively rare instances where it matters people often invoke "flux limiters" or simply choose an arbitrary cuttoff such as the speed of light.

It is somewhat akin to the problem of using a Maxwellian distribution for particle velocities and including the range of velocities all the way to infinity. It's simply convenient and the errors are relatively small with the number of particles out further than a few dozen standard deviations being minuscule.

BTW - that should be a partial derivative with respect to time on the left hand side of your heat equation!

Ooppps! Thanks. I've corrected the equation.

Well, you said that's an artifact of approximation. In fact, you agree with me that a perturbation of the temperature in some point of the domain, cannot be physically felt instantaneously by further points. Yes, one can say something about light velocity, but it does not appear neither in the Wave Equation nor in the Heat Equation (also surprising, isn't it?).

But what you told me is a bit vague. As you said, is it possible I haven't got any way of obtaining some idea of how fast the heat is propagated inside my flow field?. I mean, something like a Heat Speed.

Is not there any physicist who had thought about a Heat Wave? It seems the more probably mechanism of heat transfer, like waves traveling in a liquid or solid. In some way, the Heat Equation has an internal misconception with this, as I think.

 

[MiGUi]

Classically we use the infinite speed of information transmission to improve our models. Think in mechanic's rigid solid. When we apply a force in an extreme, that force is transmited instanteneously to the other extreme of the rigid solid, so it begins to move instantaneously. That can't happen in SR and that's why solid rigid don't exists in relativity.

This, can be applied to your question. The flux is transmited instantaneously? False, it has an infinite speed, but we would not have the L too big to have to consider relativistic effects on it!

MiGUi

 

[Clausius2]

Classically we use the infinite speed of information transmission to improve our models. Think in mechanic's rigid solid. When we apply a force in an extreme, that force is transmited instanteneously to the other extreme of the rigid solid, so it begins to move instantaneously. That can't happen in SR and that's why solid rigid don't exists in relativity.

This, can be applied to your question. The flux is transmited instantaneously? False, it has an infinite speed, but we would not have the L too big to have to consider relativistic effects on it!

MiGUi

In Vibration Engineering or Acoustic Engineering, the finite deformation rate of the model, and also the finite propagation of the sound are present. The rigid body is only used by physicist or so. For us, nothing is rigid.

But you're telling me the ultimate vehicle of Heat transmission is the Light. I do think the ultimate vehicle of all events is the Light. But there are another effects in which the transport velocity of some magnitude is very much slower than light velocity (like sound propagation). I wonder if such Heat Speed is not derivable and is present and smaller than light velocity.

When you just put a saucepan on the hotplate when you are cooking, is the high temperature of the fire being felt in the boundary of the saucepan instantaneously?. Which is that velocity of propagation?. Light velocity? One smaller than light velocity?.

The Heat Equation states the boundary will sense the puntual fire instantaneously. This is part of the model, Migui, as you said. But one can use the Wave Equation to study the propagation of a sound wave inside a solid. And that propagation is not too slow, God knows it!.

In part, I think the Heat Equation does not admit any discontinous solution (am I right?). So that, any effect of propagation of a hypothetic Heat Wave is smoothed by the laplacian operator.

On the other hand, Wave equation does admit discontinous solutions (am I right?), so that any discontinuity can be propagated as a wave front.

 

[MiGUi]

The transmission of heat deppends on several factors: thermal conductivity, difference of temperature, and so. We can only say about a speed of heat if it is radiated, but when we use convection or conduction, its not a good way to explain the heat as a wave.

When you use a teaspoon to move the café or else, if the teaspoon is made of Ag you can sense the heat very fast and with a plastic one you may not sense anything. But actually, the flux is another thing. We use it to explain why the heat goes in one direction and not in other, and where the temperature will grow faster.

What I wanted to say before, is that we can use that if the size of our system is small, we can use the speed of light as infinite, and that will be a very good approximation.

 

[ReyChiquito]

This is a parabolical PDE equation. It has only one characteristic line:

$$\frac{dx}{dt}=\infty$$

Is that so? I thought only hyperbolic equations had characteristic lines... I am confused...

 

[Integral]

I am a bit fuzzy on the details, but I recall doing solutions of the heat equation/diffusion problems which showed disturbances traveling through the media as a erfc wave front. I do not recall the details of the solution, need to dig into my old notes.

 

[Clausius2]

Is that so? I thought only hyperbolic equations had characteristic lines... I am confused...

Hi ReyGrandecito!

Yes. Take a look at any PDEs book and the difference between hyperbolic, parabolic and elliptic behavior. Parabolic equations have only one characteristic line. The field of temperatures just in the next step of time depends of what happens in the entire domain. To point that in another way, the domain of influence of any point is the entire domain.

I am a bit fuzzy on the details, but I recall doing solutions of the heat equation/diffusion problems which showed disturbances traveling through the media as a erfc wave front. I do not recall the details of the solution, need to dig into my old notes.

Maybe you are remembering the method of the Green's Function. But remeber the Green Function of the Heat was not discontinous. I don't remember well, but I think it was a Dirac's delta function modulated by an exponential factor.
On the other hand, the Green's Function for the Wave Equation was discontinous (am I right?).

 

[Integral]

Clausius,
Considering the problem you have set up, I see no real possibility of the propagation of a thermal wave, each point of the object must migrate toward the steady state temperature distribution given by:
$$T(x)= \frac {T_1 - T_0} L x + T_1$$

Frankly this is not real interesting.

Consider the following boundary conditions.

$$T(0,t) = \left \{ \begin{array}{cc}T_i , & t\leq t_1\\ T_f, & t > t_1 \end {array}\right$$

So this is a step function temperature change at one end of the bar.

Now at the other end:
$$\dot {T}(L,t) = C$$

This is a statement of constant rate of loss, we could use a Newtons law of cooling condition here.

Let us define the initial temperature distribution as:

$$T(x,0) = T_i$$

So the bar starts at a constant temperature.

The question is, how long does it take for the change to be felt at L. What is the temperature distribution at any time t?

 

[Clausius2]

Clausius,
Considering the problem you have set up, I see no real possibility of the propagation of a thermal wave, each point of the object must migrate toward the steady state temperature distribution given by:
$$T(x)= \frac {T_1 - T_0} L x + T_1$$

Frankly this is not real interesting.

Consider the following boundary conditions.

$$T(0,t) = \left \{ \begin{array}{cc}T_i , & t\leq t_1\\ T_f, & t > t_1 \end {array}\right$$

So this is a step function temperature change at one end of the bar.

Now at the other end:
$$\dot {T}(L,t) = C$$

This is a statement of constant rate of loss, we could use a Newtons law of cooling condition here.

Let us define the initial temperature distribution as:

$$T(x,0) = T_i$$

So the bar starts at a constant temperature.

The question is, how long does it take for the change to be felt at L. What is the temperature distribution at any time t?

I don't know if you are really hoping an answer. You finished so suddenly your post. I was waiting for an answer by yourself...

To be honest, I don't undestand what you mean. But this is not a IRCchat , so I'll suppose your boundary conditions and initial conditions for t>t1. Then, the boundary temperature of x=0 changes suddenly. How long it takes the effect to be felt ar L? ----> Zero time.

In fact, there will be a front of heating, the temperature will probably be increasing progresively, with the larger peak at x=0. But I'm careful with the word front . This hypothetic wave front is not such a wave. It is not discontinous. And it's not a wave. It's an smoothed wave by the laplacian operator. Watch this imaginary game:

"Suppose you have a rope on the floor. If you strike the rope at one end, an elastic wave will be propagated towards the other end. Catch now the rope by <n> points, and get it up slowly but not uniformly. As you are getting it up from more than one point, there will be a progresive and continuous arising of each of the points."

I don't know if I've transmitted above something eatable. Sorry if you don't understand no word of what I meant.

The fact is the wave of heat exists, the temperature at the other end cannot be felt instantaneously , because the word instantaneous has no sense in physics. The problem here is if the time involved is of the order $$t_o\approx \frac{L}{c}$$ where c is the light velocity, or less than that time. If really it is of the order of $$t_o$$ then MiGUi is right, and it could be neglected if L is too small.

I will try to estimate a time of temperature transmision using Fluid Mechanics techniques in a particular problem. ...Hmmmmmm. Let me see...
and write that... then... :zzz:

 

[Clausius2]

Let's see. Let's recover the original problem I've posted:

$$\frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2}$$
$$T(x,0)=g(x)$$
$$T(0,t)=T_1$$
$$T(L,t)=T_2$$

Right. This is what a Fluid Mechanics Engineer would do:

The first term of the equation can be estimated, taking into account the boudary constraints as:

$$\frac{\partial T}{\partial t}\approx \frac{T_o}{t_o}$$ (1)

where $$T_o=(T_1+T_2)/2$$ is the characteristic Temperature and $$t_o$$ is the characteristic time of Temperature variation.

The term on the right can be estimated as:

$$\alpha \frac{\partial^2 T}{\partial x^2}\approx\alpha \frac{T_o}{L^2}$$(2)

because the heat flux suffers severe variations in distances of the order L.

Equalling (1) and (2) we obtain an estimation for the characteristic time:

$$t_o\approx \frac{L^2}{\alpha}$$

It has sense, the more diffusivity the less time is needed for reaching the steady state you was referring, Integral. In fact, the number:

$$F_o=\frac{\alpha t_o}{L^2}$$ is called the Fourier Number. When it takes the value of 1 the steady solution is practically reached.

For a bar of length L=1m, and diffusivity $$\alpha=23\cdot10^{-6}$$ m2/s (a steel bar) the time involved is $$t_o=43863$$ seconds. !

:yuck: I've done all this calculus for nothing clear... After all, we only have been able to obtain an estimation of the time needed to reach the steady state. (If you want, you can check it solving the problem analytically. Surely you'll obtain a preexponential time-dependent factor which tends to 1 when
t--->to.)

But nothing appears about the time needed to felt the presence of other temperature in the surroundings of some point. Why we can make figures with elastic and sound waves traveling and not with heat waves?

After all, we cannot estimate that time using the Heat Equation. It's an absurd. I've realized it's an absurd because Heat Equation is again parabolic, and it has no sense asking oneself about a time of propagation with that mathematical behavior.

Puzzling thinkings indeed... :uhh:

 

[ReyChiquito]

Well, this problem only reflects the lack of accuracy of the model i think. We need to trust physics (wich i do) and the causality principle states that information cannot propagate instantly, not to mention experimental evidence. So there is something wrong with the equation.

In other heat problems (other equations), the system must reach an umbral temperature before it starts to propagate.

I think this is a great example of how far we are from understanding or even modeling a "simple" physical phenomena.

 

[Integral]

I think that the heat equation works fine, the model is good. Though I still say that you have to set up a problem which will have a solution which will display a moving front of energy.

Reading some texts Operational Mathematics by Churchill, for one, I see that I have misstated my constant heat flux boundary condition. It should be:

$$U_x (L,t) = C$$ where C is a constant.

This system would behave much like an infinite half slab of material, that is, x runs from 0 to infinity.

The initial condition is for the slab to be at some initial temperature.

The boundary condition at x=0 is to apply a step function change in temp at t=0.

Now I posted the problem above just to get you to think of the physical situation. What would you observe at some distance L down the slab?

Initially the Temperature appears to remain constant. After a period of time it becomes evident that the temp is slowly increasing. Then there is a brief period of rapid temperature change until something near the final temperature is reached. Fianlly there is another period of slow change as the system reaches the final temperature.

OK, this is not speculation on my part, I have done this experiment, that is what happens. Also in doing this experiment, I setup a program to model the heat equation numerically. The model predicted the same behavior. So while there is nothing in the heat equation corresponding to wave velocity there is the possibility of a moving energy front. The velocity of this front depends on the diffusivity of the material (Diffusivity has units L²T^-1) See the velocity in there?


Now the ERFC (Error function Compliment) function has the feature that the initial effects of a change are felt INSTANTLY through out the material, it just is not a very big effect, until the main wave front arrives.

I do not think what I am saying disagrees with your analysis. What you do not tell us is the MAGNITUDE of the instantaneous "message" sent through the material.

By the way some of the examples worked in the above mentioned text include an erfc solution.

 

[Clausius2]

OK, this is not speculation on my part, I have done this experiment, that is what happens. Also in doing this experiment, I setup a program to model the heat equation numerically. The model predicted the same behavior. So while there is nothing in the heat equation corresponding to wave velocity there is the possibility of a moving energy front. The velocity of this front depends on the diffusivity of the material (Diffusivity has units L²T^-1) See the velocity in there?

Thanks Integral for sharing with me and us your opinion.

I'm only going to emphasize in your word "front". A Wave front from hyperbolics equations is a discontinuity in the field. Pay attention to the word discontinuity, it's crucial. The front you talked to me hasn't got any discontinuty. It's mathematically impossible that the Heat Equation admits a discontinuity in its solution. You talked to me about a gradual adaptation of temperature, with a severe variation in time and x, but it is not a front at all, as it is understood in hyperbolic PDE's.

In fact, and answering to your problem, the equation states that in the infinity, the step change of temperature will be felt instantaneously. An infinitesimal change will ocurre at x-->infinite and t+dt. This internal error is more severe, as MiGUi said before, because it is physically imposible. At lenghts L enough large the role of the ¿¿light?? speed is not negligible at all.

Thanks again. :)

 

[Integral]

Who said anything about discontinuities? The ERFC function has a very well defined "front" with no discontinuities.

Once again, consider the problem I have set up. It is completely physical and it displays the propagation of thermal energy with time. The analytic solution of the proposed problem displays the expected behavior.

I am not sure where your analysis (looks like an Engineers approach to me! ) is going astray, but, since what you are trying to tell me, does not correspond to the physical situation or to the solutions to the heat equation there must be some misinterpretation somewhere. I still believe it to be in the magnitude of the instantaneous effects. (They are vanishingly small)

 

[Integral]

Ok, I finally found the solution I was looking for.

Warning! big PDF!
Warning! poorly scanned PDF! Perhaps you can just turn your monitor on its side?

Warning! Involved Mathematics!

Enjoy!

This is from http://home.comcast.net/~rossgr1/Math/Churchill.pdf Operational Mathematics

I think that it addresses several of your questions.

 

[Clausius2]

After have been gathering enough will power to download your pdf, I took a look at it, and it seems Churchill, you and me agree absolutely.

Churchill gives a solution in terms of error function, which is smooth and without discontinuities as you mentioned. And he says the heat flux is transmitted instantaneously all over the domain once the boundary condition at x=0 has changed.

Well, we knew that yet. I was only wondering me if an hyperboilc behavior wouldn't be more adapted to the real nature of the phenomena. I was comparing the heat propagation with an elastic or shock wave, in which discontinuities are present. As a matter of fact, if we look at the heat transfer problem with time scales such that t--->0, it is not possible at all that the heat flux is felt instantaneously over the entire domain as Churchill and you said.

To sum up,

i) the Heat Equation yields an instantaneous propagation of thermal effects, due to its parabolic behavior.

ii) the Wave Equation yields a finite velocity of propagation of some type of wave (I'm not going to specify which)

iii) All of us know that instantaneous effects (infinite propagation velocity) are impossible.

Therefore, I would say maybe exists a more accurate formulation of heat phenomena taking into account this needed discontinuity in its propagation.

The question is:
Would this formulation be worth of being calculated?

 

[dhris]

I don't see how you're coming to the conclusion that the heat travels instantaneously. See that alpha in the equation? $$L^2/\alpha$$ is a characteristic timescale for heat to redistribute itself around your domain of size L. To see that, just think of your problem with an initially very large temperature at the centre of your domain that is zero everywhere else (as in a delta function). You will find that the solution gradually becomes a wider and wider gaussian, but does not in fact transmit heat instantaneously.

dhris

 

[Clausius2]

I don't see how you're coming to the conclusion that the heat travels instantaneously. See that alpha in the equation? $$L^2/\alpha$$ is a characteristic timescale for heat to redistribute itself around your domain of size L. To see that, just think of your problem with an initially very large temperature at the centre of your domain that is zero everywhere else (as in a delta function). You will find that the solution gradually becomes a wider and wider gaussian, but does not in fact transmit heat instantaneously.

dhris

False. If you don't believe my words read the PDF of Integral, or take a look at some book of PDE's. Also, take a look at another posts of mine written here before, I have derived yet your time scale before, and I have given it a physical meaning. That meaning has nothing to do with a propagation of a wave, (I'm referring to the mathematical meaning of wave), it is only the characteristic time of propagation until the steady regime is reached.

The parabolic behavior of the Heat Equation must be a enough proof of that instantaneous effect propagation. You have mentioned the words:

"You will find that the solution gradually becomes a wider and wider gaussian"

Yes! gradually!. The word gradually has nothing to do with a wave. An elastic wave is not propagated gradually. There, some points of the domain feel suddenly an increasing of pressure. The gradual flattering of the gaussian is felt infinitesimally in each point of the domain. The gaussian function is defined, smooth and continuous all over the domain. It hasn't got any discontinuity as a wave. Therefore, the flattering is felt over the entire region. Think of it, you are not able of flattering any continuous function without modifying any point of the domain (except the boundaries).

 

[dhris]

Ok, I see what you're getting at. To get your wave-like behaviour you'll probably need to add another term in the equation that is second order in time.

 

[Clausius2]

Yes. That's what I was just trying to say.

 

[nemo_memo]

This is just a matter of modeling. For example we know that the disturbances are propagated with several wave speeds (u,u,u+a,u-a) for the compressible Navier-Stokes equations and all of them has finite values. However, if we go very low velocities (u<<1) then we can use the assumption that the disturbances can move with infinite velocity and solve the incompressible Navier-Stokes equations. But physically every one knows that the disturbances can not move with infinite velocity. However, the solutions give very good results to the physical problem. In here, you should be careful with the incompressible model and it is assumption.

This the same thing with heat equation. It is just a model and it give very good results for some problems. And in here, you also need to be careful with its assumptions. Depending on the problem, you may not use the classical equations.

 

[hunt_mat]

Look up hyperbolic diffustion equation, a good place to start is here: http://rspa.royalsocietypublishing.org/content/454/1974/1659.full.pdf

I personally disregard and paprbolic equation as being unphysical.

 

[Kidphysics]

Hi ReyGrandecito!

Yes. Take a look at any PDEs book and the difference between hyperbolic, parabolic and elliptic behavior. Parabolic equations have only one characteristic line. The field of temperatures just in the next step of time depends of what happens in the entire domain. To point that in another way, the domain of influence of any point is the entire domain.

Can anyone point me in the right direction so I can read up on the "characteristic line" it seems very very interesting!

 

[hunt_mat]

Look up method of characteristics.

 

[Kidphysics]

Look up method of characteristics.

I had and I was only able to extract this out: For a first-order PDE, the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.

These integral curves are called the characteristic curves of the original partial differential equation.

So are they just integral curves? I did not know that integral curves had the definition that the next time step depended on the entire domain.. Do I have my facts mixed or am I not interpreting something correctly?

 

[hunt_mat]

They are integral curves but when the original partial derivatives become multivalued. I posted my characteristics (basic definitions and examples) on this website somewhere if you want to take a further look.

 

[Kidphysics]

They are integral curves but when the original partial derivatives become multivalued. I posted my characteristics (basic definitions and examples) on this website somewhere if you want to take a further look.

Ah I do but you have 20 pages of posts any keywords I can put into a search function??

I've tried multivalued* characteristic line*

 

[hunt_mat]

They're called characteristics.pdf