Air flow | EBE clim

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:

$P_1 - P_2 = \frac{\rho Q_v^2}{2S^2C_d^2}$

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 $V$ .

$P = P_{atm} + C_p \frac{1}{2} \rho V^2$

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

$\rho = \frac{353}{T}$

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 $z$ :

$P_e(z) - P_i(z) = \left( P_e(0)-\rho_e \, g \, z\right) - \left( P_i(0)-\rho_i \, g \, z \right) = P_e(0)-P_i(0)-\rho_e \, g \, z \, \frac{T_i-T_e}{T_i}$

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

$z_n = \dfrac{P_e(0)-P_i(0)}{g(\rho_i-\rho_e)}$

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 $z$ :

$V(z) = \sqrt{2\, \frac{P_1(z)-P_2(z)}{\rho}}$

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:

$Q_{m,12} = C_d \int_0^{z_n} \rho_1 \, V(z) \, w \, \mathrm{d}z$

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

$Q_{m,21} = C_d \int_{z_n}^{H} \rho_2 \, V(z) \, w \, \mathrm{d}z$

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