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If air of a certain temperature blows through a car radiator (or a computer case), what effect will air humidity (non-condensing) have on cooling rate?

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The cooling effect of air will depend in the flow rate and thermal conductivity of air. The latter is in turn a function of temperature, pressure and water content. Counter-intuitively, if we keep pressure constant, the most water the less thermal conductivity, and the decrease in thermal conductivity is more significant for higher temperatures, as shown by the following graph:

enter image description here

A more detailed explanation of the causes of this behavior can be found at the original source of the image: electronics-cooling.com

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The fins don't have moisture so no difference from evaporative cooling there.

Air with higher humidity has a higher capacitance. As humid air passes from the front the back it will adsorb the same amount of heat and temperature at the trailing edge will be lower than the dry air. So humid air will maintain a greater temperature difference which results in more heat transfer. You should never have condensing as the air is being heated.

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@Camilo Rada's answer is good, but only addresses purely conductive heat transfer (or the assumption that $\Delta T$ is constant). For convective heat transfer, you also have to factor in heat capacity as well.

Looking at Fourier's heat law $q = k \Delta T$, we've factored in the $k$ (thermal conductivity), but not changes in $\Delta T$. When a working fluid has a higher thermal capacitance, $\Delta T$ increases more slowly, thus having a higher over all $q$.

The thermal capacitance of water is discussed in this Physics SE answer, but is given by $c_p = 1.002 + 1.802H$ (kJ/kg°C), for $H$ being the percent humidity.

Thus, the answer is "it depends", based on the working temperature of the fluid (which determines the variance in $k$ with respect to humidity), and how long the fluid is in contact with the heat exchanger. If we assume a fluid working temperature in the neighborhood of 40-50 °C, then the maximum difference in $k$ with respect to humidity is around 4%, where as the change in thermal capacitance is around 80%. Obviously it'd be better to actually calculate out what the difference in heat transfer would be for a specific problem setup, but based on the above, I'd say humidity will probably be a net benefit to heat transfer, despite the reduced thermal conductivity.

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