.IDEAS.Airflow.Multizone.DoorOpen

Information

Model for bi-directional air flow through a large opening such as a door.

In this model, the air flow is composed of two components, a one-directional bulk air flow due to static pressure difference in the adjoining two thermal zones, and a two-directional airflow due to temperature-induced differences in density of the air in the two thermal zones. Although turbulent air flow is a nonlinear phenomenon, the model is based on the simplifying assumption that these two air flow rates can be superposed. (Superposition is only exact for laminar flow.) This assumption is made because it leads to a simple model and because there is significant uncertainty and assumptions anyway in such simplified a model for bidirectional flow through a door.

Main equations

The air flow rate due to static pressure difference is

V̇_ab,p = C_D w h (2/ρ₀)^0.5 Δp^m,

where V̇ is the volumetric air flow rate, C_D is the discharge coefficient, w and h are the width and height of the opening, ρ₀ is the mass density at the medium default pressure, temperature and humidity, m is the flow exponent and Δp = p_a - p_b is the static pressure difference between the thermal zones. For this model explanation, we will assume p_a > p_b. For turbulent flow, m=1/2 and for laminar flow m=1.

The air flow rate due to temperature difference in the thermal zones is V̇_ab,t for flow from thermal zone a to b, and V̇_ba,t for air flow rate from thermal zone b to a. The model has two air flow paths to allow bi-directional air flow. The mass flow rates at these two air flow paths are

ṁ_a1 = ρ₀   (+V̇_ab,p/2 +   V̇_ab,t),

and, similarly,

V̇_ba = ρ₀   (-V̇_ab,p/2 +   V̇_ba,t),

where we simplified the calculation by using the density ρ₀. To calculate V̇_ba,t, we again use the density ρ₀ and because of this simplification, we can write

ṁ_ab,t = -ṁ_ba,t = ρ₀   V̇_ab,t = -ρ₀   V̇_ba,t,

from which follows that the neutral height, e.g., the height where the air flow rate due to flow induced by temperature difference is zero, is at h/2. Hence,

V̇_ab,t = C_D ∫₀^h/2 w v(z) dz,

where v(z) is the velocity at height z. From the Bernoulli equation, we obtain

v(z) = (2 g z Δρ ⁄ ρ₀)^1/2.

The density difference can be written as

Δρ = ρ_a-ρ_b ≈ ρ₀ (T_b - T_a) ⁄ T₀,

where we used ρ_a = p₀ /(R T_a) and T_a T_b ≈ T₀². Substituting this expression into the integral and integrating from 0 to z yields

V̇_ab,t = 1⁄3 C_D w h (g h ⁄ (R T₀ ρ₀))^1/2 Δp^1/2.

The above equation is equivalent to (6) in Brown and Solvason (1962).

Main assumptions

The main assumptions are as follows:

• The air flow rates due to static pressure difference and due to temperature-difference can be superposed.

• For buoyancy-driven air flow, a constant density can be used to convert air volume flow rate to air mass flow rate.

From these assumptions follows that the neutral height for buoyancy-driven air flow is at half of the height of the opening.

Notes

For a more detailed model, use IDEAS.Airflow.Multizone.DoorDiscretizedOpen.

References

• Brown, W.G. and K. R. Solvason. Natural Convection through rectangular openings in partitions - 1. Int. Journal of Heat and Mass Transfer. Vol. 5, p. 859-868. 1962. doi:10.1016/0017-9310(62)90184-9.
  Also available at https://nrc-publications.canada.ca/eng/view/ft/?id=081c0ace-7c31-449c-9b3b-e6c14864b196.

Revisions

• January 22, 2020, by Michael Wetter:
  Revised buoyancy-driven flow based on Brown and Solvason (1962).
• January 19, 2020, by Klaas De Jonge:
  Revised influence of stack effect.
• October 6, 2020, by Michael Wetter:
  First implementation for #1353.