Avionics cooling question A fan is used to cool avionics located inside a box with adiabatic walls. If the mass flow rate into the box is $m$ kg/sec and its temperature coming in is $T_\textrm{in}$, the temperature of the air coming out of the box is
$$ T_\textrm{out} =  T_\textrm{in} + \frac{q}{m C_p} $$
where $q$ is the power given out by the avionics (Watts) and $C_p$ is the heat capacity of air.
My question : How come $T_\textrm{out}$ does not depend on the thermal conductivity of the air ? Would this expression still be true for a fluid with thermal conductivity $k = 0$ ?
 A: Your equation is a good approximation when the air flow is sufficiently turbulent that the mixing rapidly distributes the heat throughout the air.
As a general rule all but very slow air flows are turbulent, and unless the equipment inside the box is very smooth there will be turbulent eddies formed as the air passes over the equipment. The turbulence mixes the hot air near the surface of the equipment with the colder air farther away, and this is the mechanism that distributes the heat through the air. Conduction plays no significant role in the heat transfer so the thermal conductivity of the air does not affect the cooling.
A: It is true that the outlet temperature of the air is not significantly affected by the thermal conductivity of the air.  However, the temperature that the avionics will rise to in order to discharge the heat load to the cooling air is definitely related to the thermal conductivity of the air.  The minimum temperature that the electronics can attain can approach the exit air temperature; but the actual temperature of the avionics can be much higher than that.  A simple analysis of the heat transfer between the avionics and the air indicates that the electronics temperature can be approximated by the equation:  $$T_E=T_{in}+\frac{Q}{\dot{m}C_p\left(1-\exp\left(-\frac{hA}{\dot{m}C_p}\right)\right)}$$where h is the heat transfer coefficient and A is the heat transfer area between the electronics and the air.  At infinite h, the electronics temperature approaches the exit air temperature.  But at realistic values of h, which depends strongly on the thermal conductivity of the air, the electronics temperature can be considerably higher than the exit air temperature.
So in summary, the change in air temperature is only one consideration in the design of the avionics cooling system (even if it is not related to the thermal conductivity).  The more important consideration is the heat transfer coefficient between the air and electronics, which is intimately related to the thermal conductivity of the air and which determines the operating temperature of the avionics.
