Parameters for Natural Ventilation

The parameter Cd is central for the simulation of natural ventilation using the multi-zone model (mzm) and can be determined according to By og Byg (SBi), Direction 202 (In Danish). The inlet air handling unit constant Ka are used to calculate the penetration depth for the inlet air through a opening. Ka can be determined according to the Danvak basic book, chapter 7 (In Danish).

Cd - Discharge coefficient

The value can be determined according to By og Byg (SBi), Direction 202, page 70-71 (typical values range from 0.62 to 0.70):

Resistance, contraction and discharge coefficients

The volume flow through an opening depends not only on the pressure difference but also on friction, contraction and opening area.

The friction is characterized by the resistance factor ζ. If the resistance factor is known, the flow speed coefficient can be determined from:

\[ C_v = \frac{1}{\sqrt{1+\zeta}} \tag{1} \]

The discharge coefficient is found (according to equation 9.3):

\[ C_d = C_v C_k = \frac{C_k}{\sqrt{1+\zeta}} \tag{2} \]

For a normal ventilation opening without ducts, the friction is limited and corresponds to a resistance value of 0.05–0.1. The contraction coefficient is 0.6–0.7 if the opening has sharp edges, and it approaches 1.0 if the opening has rounded edges.

For more complicated openings, the friction and discharge coefficient can be determined from the average flow velocity, measured as a function of the pressure difference across the opening. The average flow speed v_m in the opening is determined from:

\[ v_m = v_k \left( \frac{A_k}{A} \right) = v_k C_k = \frac{q_v}{A} \tag{3} \]

From equation 9.1, 9.23 and 9.31 the following correspondence between resistance, pressure difference and air velocity can be found:

\[ \Delta p = ½ \rho \left( \frac{v_k}{C_v} \right)^2 = ½ \rho v_{k}^2 (1+\zeta) = ½ \rho \frac{v_m^2 (1+\zeta)}{C_k^2} \]

or:

\[ 1 + \zeta = \frac{2 \Delta p}{\rho} \cdot \frac{1}{v_k^2} = \frac{2 \Delta p}{\rho} \left( \frac{C_k}{v_m} \right)^2 \tag{4} \]

For the discharge coefficient is found by inserting in equation 9.32 in equation 9.30:

\[ C_d = C_k \frac{v_k}{\sqrt{\frac{2 \Delta p}{\rho}}} = \frac{v_m}{\sqrt{\frac{2 \Delta p}{\rho}}} \tag{5} \]

From the volume flow, by inserting equation 9.31 into equations 9.32 and 9.33, the following relationship between resistance value, discharge coefficient and volume flow can be found:

\[ 1 + \zeta = \frac{2 \Delta p}{\rho} \left( \frac{C_k A}{q_v} \right)^2 \tag{6} \]

\[ C_d = \frac{1}{\sqrt{\frac{2 \Delta p}{\rho}}} \cdot \frac{q_v}{A} \tag{7} \]

For WinDoor openings, where the volume flow can be controlled by an operable part of the WinDoor, the literature sometimes gives large resistance numbers that vary with the degree of opening, with the largest resistance number occurring at the smallest opening. The high, varying values are caused by the fact that the resistance number in these cases includes not only friction but also remaining dynamic pressure in the air beam, the contraction coefficient, and the reduction in air volume through the fully open WinDoor.

For WinDoor openings, a discharge coefficient C_d = 0.7 is often assumed.

For inlet and outlet openings the discharge coefficient is determined from a coupling between pressure difference and average air velocity. The outlet opening consists, for example, of a grid toward the room, an outlet duct, and a grid in the chimney. With a measured average air velocity of 1.0 m/s and a pressure difference of 11 Pa, C_d is found from equation 9.33:

\[ C_d = \frac{1.0}{\sqrt{\frac{2 \cdot 11}{1.2}}} = 0.23 \]

Inlet openings, where the air passes by a radiator, have the same average air velocity at the same pressure difference as outlet openings and thus the same discharge coefficient.

Read more in By og Byg (SBi) Direction 202 (In Danish).

Inlet air handling unit constant Ka

The value is determined according to Danvak basic book, chapter 7 (In Danish) where further explanations and constants can be found in the chapter Luftfordeling i rum (Air distribution in rooms) section Luftstråler (Air beams). The inlet air handling unit constant Ka are used to calculate the penetration depth for the inlet air through a opening.

Read more in Danvak basic book (In Danish).