Stefan–Boltzmann law: Which applies? Stagnation temperature or static temperature?

[kmarinas86]

http://en.wikipedia.org/wiki/Stagnation_temperature

In thermodynamics and fluid mechanics, stagnation temperature is the temperature at a stagnation point in a fluid flow. At a stagnation point the speed of the fluid is zero and all of the kinetic energy has been converted to internal energy (adiabatically) and is added to the local static enthalpy. In incompressible fluid flow, and in isentropic compressible flow, the stagnation temperature is equal to the total temperature at all points on the streamline leading to the stagnation point.
===Adiabatic===
Stagnation temperature can be derived from the [[First Law of Thermodynamics]]. Applying the Steady Flow Energy Equation
<ref>Van Wylen and Sonntag, ''Fundamentals of Classical Thermodynamics'', equation 5.50</ref> and ignoring the work, heat and gravitational potential energy terms, we have:

:$h_0 = h + \frac{V^2}{2}\,$

where:

:$h_0 =\,$ stagnation (or total) enthalpy at a stagnation point

:$h =\,$ static enthalpy at any other point on the stagnation streamline

:$V =\,$ velocity at that other point on the streamline

Substituting for enthalpy by assuming a constant specific heat capacity at constant pressure ($h = C_p T$) we have:

:$T_0 = T + \frac{V^2}{2C_p}\,$

or

:$\frac{T_0}{T} = 1+\frac{\gamma-1}{2}M^2\,$

where:

:$C_p =\,$ [[Heat capacity ratio|specific heat]] at constant pressure

:$T_0 =\,$ stagnation (or total) temperature at a stagnation point

:$T =\,$ temperature (also known as static temperature) at any other point on the stagnation streamline

:$V = \,$ velocity at that other point on the streamline

:$M =\,$ Mach number at that other point on the streamline

:$\gamma =\,$ [[Heat capacity ratio|Ratio of Specific Heats]] ($C_p/C_v$), 1.4 for air

It appears to me that if we have two fluids which are in "heat conduction" contact with each other, then the body with the highest stagnation temperature will transfer heat to the body with the lowest stagnation temperature.

However, it is possible for the fluid having the higher stagnation temperature to also have a lower static temperature, provided that it has a higher velocity (with respect to the chosen frame of reference). Now, if one were to remove the ability for heat to transfer by means of conduction and also that of convection, then any heat between them must be radiation. The radiative emissions would flow from one fluid to the other according to which medium has highest temperature from the point of view of Stefan–Boltzmann law. Is the temperature relevant to heat transfer by radiation, in this case, the static temperature and not the stagnation temperature? If that is so, doesn't it suggest that one can reverse heat flow between the two fluids (as evaluated by the arbitrary inertial observer) by simply adding or removing means of conducting heat between them so as long as the two fluids have relative motion?

 

[eljose79]

Thank you for bringing up this interesting topic of stagnation temperature and its relation to heat transfer. I would like to clarify a few points and address your questions.

Firstly, the concept of stagnation temperature is primarily used in the study of fluid mechanics and thermodynamics, where it is defined as the temperature at a stagnation point in a fluid flow. It is not directly related to the transfer of heat between two bodies. However, as you correctly pointed out, the difference in stagnation temperature between two fluids can affect the direction of heat transfer between them.

In the case of two fluids in heat conduction contact, the body with the higher stagnation temperature will indeed transfer heat to the body with the lower stagnation temperature. This is because the higher stagnation temperature indicates a higher internal energy of the fluid, which can be transferred to the lower energy fluid through heat conduction. This is true for both gases and liquids.

In the case of two fluids in heat convection contact, the direction of heat transfer will depend on the relative velocities of the two fluids. As you mentioned, the fluid with the higher stagnation temperature can also have a lower static temperature if it has a higher velocity. In this case, the heat transfer will be from the fluid with the higher velocity to the one with the lower velocity. This is because the higher velocity fluid has a higher kinetic energy, which can be transferred to the lower energy fluid through heat convection.

Now, regarding your question about the role of static temperature and stagnation temperature in heat transfer by radiation, it is important to note that radiation heat transfer is primarily dependent on the temperature of the emitting body. The temperature used in the Stefan-Boltzmann law is the absolute temperature, which is equivalent to the static temperature in the case of a fluid at rest. However, in the case of moving fluids, the stagnation temperature may also play a role as it represents the total energy of the fluid, including the kinetic energy.

In conclusion, the stagnation temperature is a useful concept in understanding fluid flow, but it is not directly related to heat transfer. The direction of heat transfer between two fluids will depend on various factors such as their relative velocities and temperatures. I hope this helps clarify any confusion and thank you for your interest in this topic.