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What does absolute humidity do in a large space with a temperature gradient?

Situation: Enclosed space, approximately 8 meters in height. The space is nearly leak-proof, essentially no air being transferred between the inside and the outside. There is a significant temperature difference between the floor and the ceiling. The temperature of the air is around 35 degrees C at the floor, and around 85 degrees C at the ceiling. There is no liquid water in the space, and no condensation occurs. There is some air movement in the space, but it is minimal and only caused through natural methods, no forced air movements with fans.

I am aware that if there was no temperature gradient the absolute humidity (gr/m^3) would stay the same and relative humidity would change depending on the temperature (for this instance also assuming that the outer envelope is not entirely leak-proof, so the pressure stays the same with the increase of temperature). But what happens when there is a temperature gradient in the space? My initial thought would be that absolute humidity would be the same at every height, making it easy to calculate the relative humidity at each height. But my knowledge on this subject is limited, and I have not been able to find any information to confirm or contradict my thought.

The following question seems to come closest to my own question, albeit with two connected rooms and not a single space with a temperature gradient: Which is the same between two connected rooms, relative or absolute humidity?

The answer provided by dmckee states “ Thus the absolute fraction of water will be the same at both ends and the relative humidity will vary.” But the response from the user asking the questions ends with “…temperatures mean different mass densities as well. If this approach isn't wrong, then both absolute and relative humidity’s in the two bottles will be different.”

So I am uncertain what this actually means for my situation. Even if the absolute humidity is not the same at different heights, can somebody provide with some information about the absolute humidity in the space? Would the absolute humidity at the ceiling be higher, the same or lower than the absolute humidity near the floor and would there be some way of calculating this?

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This looks like an engineering-course homework problem, but which happens to resemble a meteorology problem (maybe the other way around). I'll address it from the 'meteo' standpoint.

First, you state that "some air movement in the space, but it is minimal and only caused through natural methods." So for this temperature difference one appearst to have first ruled out turbulent motion, which is suppressed by the stable stratification (increasing temperature with height, which leads to a seemingly infinitely positive gradient Richardson number).

To handle the entropy arguments (which Cppg started to theoretically address above from a statistical mechanics perspective), I'd recommend considering this using potential temperature, i.e. "meteorology's entropy variable". In particular, look at the virtual potential temperature $\theta_v$, which depends on temperature and water vapor mixing ratio $q_v$. Via this and the equation of state (ideal gas law) for air and water vapor, you can then find a profile $\theta_v(z)$ and also $q_v(z)$. Consult Chapters 8-9 in Wyngaard's book Turbulence in the Atmosphere (2010) for more details. If you'd like to relate this to and finish Cppg's explanation, consult also Bohren's Atmospheric Thermodynamics.

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The exact answer would require the combinatorial computations of entropy of both systems in conjunction with one another, to obtain an equation for the partition function of both room's momentum space. The helmhotz energy could then be obtained for the resonance between both rooms.

$$ F = -k_{B}T_{1}T_{2}\ln(\mathcal{Z}) = U_{T} + P(T_{1})V(S_{1}) + S_{2}T_{2} $$

Subject to Gibbs partition thereom.

However a macroscopic and generalised estimation would be to use a diffusivity or chemical potential coefficient in the phase space expansion of the heat equation.

The chemical potential is seen a potential gradient, either electromagnetic and gravitation of a system of particles within the field. e.g

$ \mu(x) = \int_{0}^{L} e^{mgx} .dx $

Similarly, Log normal distributions could then be used to estimate the variance in the humidity, which have local points at the certain variances and expand around the mean associated with that path length.

Fortunately, thermodynamically, a boltzmann distribution for water would reveal a comparable velocity to path length ratio in a room of water molecules at gas phase, even at low temperatures. Therefore a high temperature at the cieling of a room would act to move more molecules against the gravitation chemical potential.

This allows us to safely say that a thermal density reading with a thermal-hygrometer at the centre of the room is accurate.

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