Comparing FDS results to fire spill-plume calculations in BRE Annex D

3.1. Calculation 1: Mass Flow Rate at Spill Edge

Following the procedure in Annex D, we first calculate the mass flow rate of smoky gases passing from the shop to the spill edge. Using BRE equation 5.7:

where:

\(M_{w}\) is the mass flow rate of smoky gases passing through a vertical opening (kg/sec)

\(C_{e}\) is the entrainment coefficient (0.34 kg/sec-m5/2 for small rooms)

\(P\) is the perimeter of the fire (m)

\(W\) is the width of the opening (m)

\(h\) is the height of the top of opening above the floor (m),

\(C_{d}\) is the coefficient of discharge for the opening (1.0 for no downstand),

The calculated result is a mass flow rate of 29.2 kg/s. The FDS results give 36.3 kg/s, 25% larger than the calculation.

3.2. Calculation 2: Depth of Smoke Layer at Spill Edge

We calculate the depth of the smoke layer at the spill edge using BRE equation 5.11:

where:

\(D_{B}\) is flowing smoke layer depth under the balcony (m)

\(C_{d}\) is the coefficient of discharge for the opening (1.0 for no downstand)

\(M_{B}\) is the mass flow rate under the balcony, calculated in Section 3.1 (kg/sec)

\(T_{l}\) is the mass-weighted average absolute temperature of gas layer (K)

\(\theta_{l}\) is the temperature rise above ambient in the gas layer (K)

\(W_{B}\) is the balcony channel width (m)

\(T_{0}\) is the absolute ambient temperature (K)

The calculated result is a layer depth of 1.19 m. The FDS results give a layer depth at the center of the balcony edge of 1.63 m, 37% larger than the calculation.

3.3. Calculation 3: Entrained Flow in Spill Plume

The BRE document provides an approach in Annex E, but also notes that a simpler calculation using the Thomas method (equation 6.5) with the Poreh method (equation 6.6) of calculating \(\rho\) gives the same result, so we use the Thomas/Poreh approach. First calculate \(\rho\) using BRE equation 6.6 and 6.4:

where:

\(C_{e}\) is the entrainment coefficient (0.44 for a free plume),

\(\rho_{0}\) is the density of ambient air (kg/m³),

\(L\) is the length of spill edge (m),

\(Q_{w}\) is the convective heat flux (kW).

Knowing \(\rho\), we then calculate the mass flow rate in the plume using BRE equation 6.5:

where:

\(M_{1}\) is the mass flow of smoky gases at height \(h_{b}\) (kg/s),

\(\rho\) is the density of warm gases at height \(h_{b}\) (kg/m³),

\(c\) is the specific heat of air (kJ/kg-K),

\(\rho\) is the height of the virtual source below void edge (m), and

\(h_{b}\) is the height of rise of thermal plume above void edge (m).

The calculated result is a layer depth of 1.19 m. The FDS results give a layer depth at the center of the balcony edge of 1.63 m, 37% larger than the calculation.

\(h\) is the height of the top of opening above the floor (m),

A comparison of the Thomas/Poreh calculation (described in BRE) and FDS results for mass flow in the plume is shown in Figure 4. The shopping mall ceiling height is 15 m and the shop balcony height is 5 m, so the distance from the balcony edge to the ceiling is 10 m. The Thomas/Poreh calculation does not account for ceiling height, while the FDS calculation includes the hot gas layer at the ceiling. This layer affects flow about 2-3 m below the ceiling, shown by the velocity vectors in Figure 5.

Figure 5 shows a slice of the model with the velocity vectors for the FDS calculation at 100 sec. You can see the formation of layers at the balcony and shopping mall ceiling. Note that the FDS solution is dynamic and varies with time. This is a typical result.