Estimate heat transfer caused by ventilation, wind and stack effect

## Video

How to predict heat transfer due to ventilation, caused by the wind or the stack effect. PDF slides

## Formulas

A small opening with an area $S$ and a discharge coefficient $C_d$ separate two rooms at the pressures $P_1$ and $P_2$. The air flow rate $Q_v$ [m$^3$/s] through the opening can be estimated by:

In case of wind, the total air pressure is the sum of the atmospheric pressure $P_{atm}$ and some dynamic pressure, which depends on a pressure coefficient $C_p$ and wind velocity $V$.

The mass density of air $\rho$ [kg/m$^3$] is roughly equal to this relation, where the temperature $T$ is in unit [K]:

The stack effect is caused by this relation between density and temperature. The two air volumes shown here have temperatures notes $T_e$ and $T_i$. The pressure difference between them is a function of the coordinate $z$:

The neutral plane is the coordinate $z_n$ at which this pressure difference is equal to zero:

## Exercise

A door of height $H=2$m and width $w=1$m separates two rooms at temperatures $T_1=15°C$ and $T_2=21°C$.

Suppose (without proving it) that the air velocity at the interface is a function of the coordinate $z$:

where $\rho$ is the mass density on the side from which air flows (which depends on whether the coordinate is above or below the neutral plane).

Integrate $V(z)$ over the height $H$ to find the total mass flow rate of air $Q_m$ [kg/s] through the door. Suppose the door is an infinite stack of small openings with a discharge coefficient $C_d=0.4212$.

Below the neutral plane, air flows from room 1 to room 2 with a total rate of:

Above the neutral plane, air flows from room 2 to room 1 with a total rate of:

The net air flow rate from 1 to 2 is $Q_{m,12}-Q_{m,21}$