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.
The air flow rate due to static pressure difference is
V̇_{ab,p} = C_{D} w h (2/ρ_{0})^{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, ρ_{0} 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} = ρ_{0} (+V̇_{ab,p}/2 + V̇_{ab,t}),
and, similarly,
V̇_{ba} = ρ_{0} (-V̇_{ab,p}/2 + V̇_{ba,t}),
where we simplified the calculation by using the density ρ_{0}. To calculate V̇_{ba,t}, we again use the density ρ_{0} and because of this simplification, we can write
ṁ_{ab,t} = -ṁ_{ba,t} = ρ_{0} V̇_{ab,t} = -ρ_{0} 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 ⁄ ρ_{0})^{1/2}.
The density difference can be written as
Δρ = ρ_{a}-ρ_{b} ≈ ρ_{0} (T_{b} - T_{a}) ⁄ T_{0},
where we used ρ_{a} = p_{0} /(R T_{a}) and T_{a} T_{b} ≈ T_{0}^{2}. 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 T_{0} ρ_{0}))^{1/2} Δp^{1/2}.
The above equation is equivalent to (6) in Brown and Solvason (1962).
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.
For a more detailed model, use Buildings.Airflow.Multizone.DoorDiscretizedOpen.