I'm asking myself a question related to temperature integration. First is possible to integrate a temperature distribution over the time and over a certain volume? If yes, these resulting integrals which physical quantity are representing? For example, the integral of the temperature over a volume can represent the Internal heat energy divided by the specific heat and the density of the volume (assumed both constant)? While the temperature integral over time can be assumed as the opposite of the heat flux density?
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Yes, you can integrate a temperature field $T(x,y,z,t)$ over all variables the same way you could integrate any other function. However, that does not mean that said integral must have a direct physical interpretation. In this case, the units would be $\mathrm{K \cdot m^3 \cdot s}$, which do not have any physical significance that I know of.
However, if you integrate temperature times specific heat times density, you would get the internal energy of the volume as this post describes.
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$\begingroup$ What is really interesting in this comment "...which do not have any physical significance..." that when said spatial integral is divided by the volume or as a surface integral by the surface area, both of a fixed size body, resulting in an average temperature, we are supposed to believe that all of the sudden it becomes a meaningful quantity. $\endgroup$ Sep 13 at 17:34
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$\begingroup$ @hyportnex That would just be the definition of the average of a function over some volume. $\endgroup$– agaminonSep 13 at 17:50
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$\begingroup$ exactly, so why would that average be any more meaningful quantity for a fixed size/shape object than just its integral? It will not. $\endgroup$ Sep 13 at 18:14
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$\begingroup$ @hyportnex I disagree. The lone integral scales with volume and therefore two identical temperature fields over identical shapes but with different volumes would yield different values, instead of the same value that one would expect for an average. It is true that averages are not uniquely defined, though. Craig Bohren in his book "Fundamentals of atmospheric radiation" discusses this concept lucidly, I believe in chapter 1. $\endgroup$– agaminonSep 13 at 18:19
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$\begingroup$ You mean this? Bohren: "The emissivity of the present atmosphere is, say, 0.8, which if substituted in Eq. (1.72) with $S = 240Wm^{−2}$ yields $Te = 289K(16◦C)$. We note that this is not the global average temperature. In the first place, there is no such thing as the average temperature (or the average anything) but rather infinitely many possible averages depending on the function of temperature averaged and how it is weighted. And what exactly does Te correspond to? Is it the temperature of the ground? If so, this is not the air temperature, etc.". Very well said. $\endgroup$ Sep 13 at 18:45
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Temperature is an intensive property. Integration makes physical sense when applied to extensive properties, such as mass or energy. The time integral of the mass that flows out of a pipe is the total mass that flows over the corresponding time. The time integral of the pressure, however, is not physically meaningful because pressure is intensive, and adding pressures, while mathematically permissible, does not mean much.
We can turn the time integral of an intensive property to something meaningful by dividing it by the time interval of the integration limits. The result is the mean temperature over that time interval. I used time here as an example, but integration over space works the same way.
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$\begingroup$ Integrating intensive quantities can make sense in some cases; for example, the surface integral of pressure is the total force. $\endgroup$– JavierSep 17 at 15:20
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$\begingroup$ Yes, but only integration with respect to the particular variable that turned an extensive variable it intensive. With temperature we don;t have the same luxury... $\endgroup$– ThemisSep 17 at 15:46