Physical Significance of Thermal Diffusivity

[Soumalya]

What physical interpretation can we draw from the thermophysical property of thermal diffusivity?

How might we visualize the true physical meaning of this property and relate it to its definition?

 

[Geofleur]

By performing scale analysis on the heat equation, it's possible to derive the relation $L \approx \sqrt{\kappa \tau}$, where $L$ is the location of a thermal front, $\tau$ is the time, and $\kappa$ is the thermal diffusivity. In other words, the diffusivity gives a measure of how far a thermal front will propagate during a given time interval due to conduction only.

 

[Soumalya]

By performing scale analysis on the heat equation, it's possible to derive the relation $L \approx \sqrt{\kappa \tau}$, where $L$ is the location of a thermal front, $\tau$ is the time, and $\kappa$ is the thermal diffusivity. In other words, the diffusivity gives a measure of how far a thermal front will propagate during a given time interval due to conduction only.

Well I have just started with the subject of "Heat Transfer" and as such I am yet to go through the latter portions of heat conduction analysis including transient heat conduction analysis.For the time being could you explain to me in some simple words about the meaning of a 'thermal front'?

My textbook gives a brief introduction about the property of "thermal diffusivity" in the opening introductory chapter.It says the property could be viewed as the ratio of the heat conducted through the material to the heat stored per unit volume.It also says "a small value of thermal diffusivity means that heat is mostly absorbed by the material and a small amount of heat is conducted further."

Doing a control volume analysis on an infinitesimally thin volume element (length 'dx' and cross sectional area 'A') of a plane wall of a given material we might write the heat balance equation as:

Q_in-Q_out = $$\frac{d}{dt} \$$(ΔE_thermal,CV)

where Q_in and Q_out are the rates of at which heat is being transferred into and out of the element (control volume) and ΔE_thermal,CV is the change in the thermal energy content of the element(CV).

For the control volume to reach steady state conditions from an initial transient phase the term 'Q_out' would be initially minimum for a given Q_in such that $$\frac{d}{dt} \$$(ΔE_thermal,CV) is maximum initially. As the control volume would approach steady state conditions Q_out should increase and tend to become equal to Q_in when $$\frac{d}{dt} \$$(ΔE_thermal,CV)→0 and steady state conditions are approached.

Thus initially it seems that the rate at which heat is being conducted out of the element(i.e, Q_out) is minimum and the rate at which thermal energy is being stored is maximum and it is evident that as the control volume approaches steady state conditions the rate of heat conduction out of the element i.e, Q_out increases and tends to become equal to Q_in and consequently the rate at which thermal energy is being stored decreases and tends to a magnitude of zero.

Looking at the author's statement now I wonder if he meant by the phrase "a small value of thermal diffusivity means that heat is mostly absorbed by the material and a small amount of heat is conducted further.", that during the very onset of the transient conduction phase a larger fraction of the thermal energy entering one side of the wall is stored and the rest conducted further.I assume so because the magnitude of the fraction of incoming thermal energy that is stored in the material and the other fraction that is conducted further seems to vary even during the transient phase of the conduction.Is it so?

 

[Chestermiller]

The author is just trying to give you a very rough physical interpretation. His statement is pretty much on target. If the medium is very thick, it is not just at very short time that what he says applies. Why don't you just wait until you have seen the solution to some transient heat transfer problems before you try to interpret it any further. Once you see the solution to some problems, you'll get the idea. Your time is too valuable to spend more time on it at this stage.

Chet

 

[Geofleur]

For the time being could you explain to me in some simple words about the meaning of a 'thermal front'?

Imagine a cold medium, say a cube, of temperature $T_C$ with one face held at a hot temperature, $T_H$, and the other faces insulated. Eventually the whole cube will heat up to the temperature $T_H$. On the way to this final state, a thermal front originating at the hot face will propagate into the cube. We can define the front position as the position where the temperature has just started be affected by the heat from the wall. So, for example, the position where the temperature has increase by some amount that is significant in comparison to $T_H - T_C$. The position of this front as a function of time will be given by $L \propto \sqrt{\kappa \tau}$. Even though it's only a proportionality, the proportionality constant is often not that far from one.

 

[Soumalya]

The author is just trying to give you a very rough physical interpretation. His statement is pretty much on target. If the medium is very thick, it is not just at very short time that what he says applies. Why don't you just wait until you have seen the solution to some transient heat transfer problems before you try to interpret it any further. Once you see the solution to some problems, you'll get the idea. Your time is too valuable to spend more time on it at this stage.

Chet

Thank you Sir Chet

I will be patient until I go through the chapters of transient heat conduction analysis.Maybe I come up with a better understanding then.

 

[Soumalya]

Imagine a cold medium, say a cube, of temperature $T_C$ with one face held at a hot temperature, $T_H$, and the other faces insulated. Eventually the whole cube will heat up to the temperature $T_H$. On the way to this final state, a thermal front originating at the hot face will propagate into the cube. We can define the front position as the position where the temperature has just started be affected by the heat from the wall. So, for example, the position where the temperature has increase by some amount that is significant in comparison to $T_H - T_C$. The position of this front as a function of time will be given by $L \propto \sqrt{\kappa \tau}$. Even though it's only a proportionality, the proportionality constant is often not that far from one.

Thank you...that was really helpful!
I do have a brief understanding at this stage after you explained to me about a thermal front and I could imagine it as a property of a material that indicates how fast the 'effect' of heat propagates through a medium and possibly how quickly it can attain a steady state heat conduction situation.I hope my understanding will get better after I find my way through the topic of transient heat conduction analysis.

 

[anaranjo]

Thermal diffusivity is the relationship among three properties: thermal conductivity, density and specific heat. In a conductive transient heat transfer process it determines how fast the Temperatur change inside the body.
The meaning can easily be explained with an analogie. Cooling a body is similar to empty a water tank. The speed of the water level descen (analog to the Temperature) depends on the " flow resistance" of the drain pipe (inverse analog to the Thermal conductivity) and the capacity of tank (analog to the product of density and specific heat = specific heat pro volume).