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I'd like to know if it's possible to calculate the heat and mass transfer rates from a small volume to the environment.

Let's assume there is a box with a small opening at the top. The air inside the box is hot and humid (65 °C and 100% relative humidity). The air outside the box is cooler and drier (25 °C and 50% relative humidity).

How should I proceed? I've tried using Fick's law of diffusion but the rate of transfer was too low. There must be something I am missing. I've thought about free convection but it does not fit for my situation I guess. Could Boussinesq equation be used?

Thanks in advance for any help.

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For modelling the rate of air flow through the opening, empirical relationships in the ASHRAE Fundamentals Handbook seem relevant here.

Volumetric flowrate

Equation 37 on page 25.13 of the 1997 version, section "Flow Caused by Thermal Forces" may be useful for calculating flow rate:

$$Q=C_D \cdot A \cdot \sqrt{2 \cdot g \cdot \Delta H_{NPL} \cdot (T_i - T_o)/T_i} $$

where:

$Q$ : airflow rate, [$\frac{m^3}{s}$]

$C_D$ : discharge coefficient for opening [$-$]

$A$ : Area of opening, [$m^2$]

$g$ : gravitational constant, $9.81 \space \frac{m}{s^2}$

$\Delta H_{NPL}$ : height from midpoint of lower opening to NPL (Neutral Pressure Level, "the height at which the interior and exterior pressures are equal"), [$m$]

$T_i$ : indoor temperature, [$K$] (assuming $T_i>T_o$)

$T_o$ : outdoor temperature, [$K$] (assuming $T_i>T_o$)

The value for $C_D$ that takes into account interfacial mixing of the bidirectional flow of air through the opening is Equation 38:

$$C_D={0.40}+{0.0045}|T_i - T_o|$$

Density

Density can be calculated from equations 11, 22, and 27.

Specific Enthalpy

The volumetric flow rate $Q$ and density, combined with the enthalpy difference ($65^{\circ}C$ @ 100% RH vs. $25^{\circ}C$ @ 50% RH) of two points on this psychrometric chart on page 6.11, should permit you to calculate heat transfer out of the building via this natural convection air flow, assuming air pressure is near $101.325 \space {kPa}$.

If air pressure is not close to $101.325 \space {kPa}$, then the set of equations referenced by the Situation 3 table of the "NUMERICAL CALCULATION OF MOIST AIR PROPERTIES" section on page page 6.10 can be used instead to calculate specific enthalpies as a function of dry-bulb temperature $t$, Relative humidity $\psi$, and absolute pressure $p$.

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Just as a very rough start, one could assume a convective column of hot air with a diameter of a bit smaller than the box opening that accelerates up.

This column can be assumed randomly to take advantage of the buoyancy of a layer 2*d thick, d being the diameter of the opening, for its propellant force. The convection formula is $$ q = h_c A dT $$

But establishing $ \ h_c$ is not easy.

This will be our model. we can measure the temperature above this opening at different heights and adjust the parameters of our assumption

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