The equation for linear transient heat transfer is dT/dt = Alpha * d2T/dx² with Alpha equals heat conductivity / (heat capacity * density). The SI unit is m²/s. There is no temperature involved. If you increase the temperature difference you increase the heat flux but you cannot increase the speed by which the heat progresses. The driving force is the heat conductivity. If you use the Binder Schmidt scheme you find out that the waypoints for the heat progress are proportional to the heat conductivity.

http://www.egi.kth.se/courses/4a1601/files/lab3ht.pdf

Ibrahim,

You are in a foggy place.

Turn on a bright light. Does the distance over which the brightness fall by a certain factor depend on how bright your light is?

No.

James Garry, it depends, if there is some particles in air that obstruct light through it then the distance is dependent on the light intensity.

 I am not sure that the original heat transfer problem and the divergence of a light beam are qualitatively of the same nature. I have always considered the term, "penetration distance", to be associated with the effect of a sinusoidal variation of the temperature at a solid boundary. Then the penetration distance corresponds to a depth beyond which no temperature variation is felt. This distance depends on the frequency of the variation and the thermal diffusivity of the solid. In the example using light, there is the inverse square law operating, and if there are no contaminants which scatter and absorb the light, then its penetration distance is effectively infinite. In addition, the above temperature variations decay exponentially, whereas the light intensity decays as r^{-2}.

Ibrahim,

I had mentioned that the air was foggy (water droplets suspended in the air, much as in a cloud). So, 'yes' I had presumed the presence of scattering bodies.

The *distance* over which a given light's intensity will fall by a certain *factor* is dependent (naturally) on the particle density and it is independent (broadly) of the light's intensity.

Andrew: I'm aware of the qualitative differences between scattering and conduction - I was seeking to show a real-world situation in which a penetration depth is independent of the magnitude of the forcing function. I had assumed that people were familiar with cycling in fog - versus driving in fog.

Let's not get bogged down by the details - I had in my mind a spotlight in a scattering medium (hence no inverse-square relationship) - it was simply an attempt to demonstrate the situation without resorting to 'the equations show that it must be so'.

A question has been raised about the qualitative similarity between penetration of heat and light. The supporting arguments are based on a quantitative measure for their propagation. I like James' answer for the simplicity of  the analogy that a young research student would immediately appreciate. The answer by Klaus is a helpful one too. Incidentally, the penetration depth of a sinusoidal wave is also infinite. For practical purposes, we can arbitrarily take, as in the case of the boundary layer, a 99% attenuation. Another approach is to consider a depth corresponding to a small Fourier number beyond which the temperature variations could be ignored. It may be recalled that Fourier number too depends on L^{-2}. Therefore, it would appear that the analogy of James is not all that far off the mark.   

Hi James, Many thanks - I now see what you mean and why you chose the light example. Indeed I was unaware of the lack of dependence on the amplitude in your illustration.

I didn't mention it earlier, but the penetration depth in the thermal problem is also independent of the amplitude of the surface temperature variations - and it may be said to apply roughly to diurnal variations in the surface temperature of the ground. Hence permafrost? Maybe I need to check some numbers. Also, the Stokes layer in fluid mechanics has absolutely identical properties, but then it is also mathematically identical to the thermal problem. And again, my suspicion is that, if one has a long series/sequence of masses and springs (they could be hanging from a fixed point, or else lying in a straight line on a frictionless table) and the one at the end is moved sinusoidally in time, then there will be a penetration depth effect here too, one which could well be extended to continuous elastic solids.

These are all very interesting problems with some practical interest.

Ibrahim: one simple answer to your question comes from the mathematical observation that the true temperature may be scaled mathematically to one which lies between 0 and 1 without needing to rescale the distance. Then, because Fourier's equation is linear, its solution then has exactly the same time-dependence as the non-scaled one, apart from the amplitude. This applies when the diffusivity is constant. However, my conclusion will need to modified if the thermal conductivity and/or heat capacity vary with temperature.

D. Andrew S. Rees , please i do not understand this statement of you answer,

"penetration distance", to be associated with the effect of a sinusoidal variation of the temperature at a solid boundary.

Hi Ibrahim,

My use of the phrase is in the context of a semi-infinite solid having the ambient temperature T_0. The plane surface of this solid is subjected to the oscillating temperature: T=T_0 + A*cos(omega*t). Fourier's equation applies and it is in the form,

T_t = alpha* T_zz

where alpha is a constant thermal diffusivity. The analytical solution of this system is

T = T_0 + A * exp(-z/L) * cos[(z/L)-omega*t).

Here, L=sqrt{2*alpha/omega) is what I am calling the penetration depth, because the exp(-z/L) term shows that the effect of the surface boundary temperature variations decay exponentially into the solid. [In some contexts L might be called the e-folding distance.]

Given the definition of L, we can see that the penetration depth increases as the thermal diffusivity increases, which one can understand on physical grounds because a large diffusivity means more rapid conduction. We can also see that the penetration depth decreases as the period of oscillation increases.

I am interested to know what you mean by penetration depth. I suspect that it is different from mine.