Fire Dynamics Calculations: Method

Fire Dynamics Calculations

The tool brings together the closed-form methods of the U.S. Nuclear Regulatory Commission's handbook NUREG-1805, Fire Dynamics Tools (FDTs), as revised by its Supplement 1 spreadsheets, for the fire-hazard questions a building fire-protection engineer meets first: how hot the gas layer in a room becomes, how large a pool fire is and how high its flame reaches, how much heat radiates to a target, when a sprinkler or detector operates, whether the room flashes over, how soon a cable or a lining ignites, how fast the smoke fills the room and clouds the exit signs, and how long the occupants need to leave. Each analysis is one chapter of the handbook; the equations are printed with the substituted numbers, the chapter and equation numbers are named, and the material, fuel, cable and smoke tables of the handbook fill the property fields. The scope is compartment and pool-fire dynamics only: the handbook's explosion-overpressure and hydrocarbon-fireball chapters are vapour-cloud consequence analysis, process-safety work outside this tool.

Units and conventions

Lengths, areas, masses, the ventilation flow and the wind speed go through the unit menus; temperatures are plain numbers in °F (USCS) or °C (SI), heat release rates in kW, heat fluxes in kW/m², and the thermal properties are in the SI units of the handbook tables in both systems, as the NRC's English-unit spreadsheets keep them. Ambient air density is ρ_a = 353/T_a kg/m³ (T_a in K), the handbook's ideal-gas value. The results are the handbook's: the printed example problems of the chapters built are reproduced to their printed digits by the tool's test, and the Supplement 1 spreadsheets' cached values to six figures.

Hot gas layer temperature and smoke layer height (chapter 2)

For a room with a door or window the McCaffrey, Quintiere and Harkleroad correlation gives the layer temperature rise ΔT_g = 6.85 [Q̇²/(A_v√h_v A_T h_k)]^1/3, a fit to 100 room fires with the fire away from the walls; the effective heat transfer coefficient is h_k = √(kρc/t) while the lining is still absorbing heat and k/δ after the thermal penetration time t_p = (ρc/k)(δ/2)². With mechanical exhaust the Foote, Pagni and Alvares correlation ΔT_g/T_a = 0.63 [Q̇/(ṁc_pT_a)]^0.72 [h_kA_T/(ṁc_p)]^−0.36 and the Deal and Beyler energy balance ΔT_g = Q̇/(ṁc_p + h_kA_T) (with its own 0.4 √(kρc/t) coefficient, valid to 2,000 s) are both printed. For a closed room whose leaks prevent a pressure rise Beyler's non-steady balance gives ΔT_g = (2K₂/K₁²)(K₁√t − 1 + e^−K₁√t). The smoke layer descends by the Yamana and Tanaka filling equation z = [2kQ̇^1/3t/(3A_c) + h_c^−2/3]^−3/2 with k = 0.076/ρ_g; it ignores the smoke leaving the vent, so below the top of the vent the height is not credited and the handbook's "smoke exiting vent" caution is repeated. The table and chart follow the spreadsheet's schedule to 60 minutes. Limits: conventional rooms, a steady fire away from the walls, layer temperatures to about 600 °C (beyond that the room is at flashover), no ceiling vents.

Pool fire (chapter 3)

A liquid spill of area A burns as a circular pool of diameter D = √(4A/π) at Q̇ = m″ΔH_c,eff(1 − e^−kβD)A, Babrauskas's large-pool burning rate corrected for small diameters; the regression rate ν = m″/ρ gives the burning duration t_b = 4V/(πD²ν); the flame height is Heskestad's H_f = 0.235Q̇^2/5 − 1.02D (a mean height; a negative value means the fire is too weak for its diameter) and Thomas's H_f = 42D[m″/(ρ_a√(gD))]^0.61. The fuel table is the handbook's Table 3-4 (butane to lube oil); a cable tray or a pallet stack is entered as a custom fuel with its bench-scale burning rate. The pool is assumed to involve instantly and fully, with no growth period.

Radiant heat flux to a target (chapter 5)

The point-source model radiates the fraction χ_r of Q̇ isotropically from the flame centre, q″ = χ_rQ̇/(4πR²), R = L + D/2; it is within 5 % of the better models only for L/D > 2.5 and overstates the flux near the fire. The solid-flame model treats the flame as a cylinder of diameter D and Heskestad height radiating with Shokri and Beyler's emissive power E = 58 × 10^−0.00823D kW/m² (data from kerosene, fuel oil, gasoline, JP-4, JP-5 and LNG pools of 1 to 50 m); the target sees it through the view factor of a vertical cylinder, the horizontal and vertical factors combined as √(F_H² + F_V²) for a target at ground level, or the sum of the factors of the cylinders below and above the target height for a vertical target above ground (a flame below the target gives the upper cylinder a negative height, which the handbook spreadsheet and the tool treat as a negative factor). In wind the flame is tilted by Mudan's correlation, cos θ = (u*)^−1/2 for u* = u_w/(gm″D/ρ_a)^1/3 above 1, shortened by Thomas's wind flame height, and seen through the tilted-cylinder view factors of Mudan (1984), with a = H_f/r and b = R/r as in Mudan's formula and the handbook's ground-level spreadsheet; the handbook's above-ground wind spreadsheet writes a₁ = 2H₁/r, twice that value, which the tool does not follow. The tilted formulas reduce exactly to the vertical-cylinder ones at zero tilt, and both were checked against a numerical integration of the view factor.

Plume and ceiling jet (chapters 9 and 10)

Above the mean flame height Heskestad's plume correlation gives the centreline temperature rise 9.1[T_a/(gc_p²ρ_a²)]^1/3Q̇_c^2/3(z − z₀)^−5/3 and velocity 3.4[g/(c_pρ_aT_a)]^1/3Q̇_c^1/3(z − z₀)^−1/3, measured from the virtual origin z₀ = −1.02D + 0.083Q̇^2/5, with the convective part Q̇_c = χ_cQ̇. Where the plume strikes an unconfined flat ceiling it turns into Alpert's ceiling jet: T_jet − T_a = 16.9Q̇^2/3/H^5/3 within r/H ≤ 0.18 and 5.38(Q̇/r)^2/3/H outside, u_jet = 0.96(Q̇/H)^1/3 within r/H ≤ 0.15 and 0.195Q̇^1/3H^1/2/r^5/6 outside. The Supplement 1 spreadsheets use the total Q̇ in these and the 0.18 bound; the original 2004 sheets used Q̇_c and 0.15, which is why some printed 2004 example answers differ. The correlations assume a steady fire away from walls, a smooth ceiling and no hot layer yet under it.

Sprinkler, heat detector and smoke detector activation (chapters 10 to 12)

A heat-sensitive element is a lumped mass heated by convection in the ceiling jet, so its time to reach the rating is t = (RTI/√u_jet) ln[(T_jet − T_a)/(T_jet − T_act)]; if the jet never reaches the rating the device does not operate. The response time index of a listed sprinkler should be the manufacturer's; the generic values of Madrzykowski (standard link 130, standard bulb 235, quick response 34 to 42 (m-s)^1/2) and the generic temperature ratings are offered as defaults, and the RTI of a listed heat detector follows from its UL spacing and rating by the handbook's Table 12-2. A smoke detector has no rating; the handbook offers three estimates, which the tool prints together: the same RTI model with a low RTI and a small temperature rise (Alpert), the transport lags of the plume and ceiling jet (Mowrer), and the NFPA 92B temperature-rise form of Milke with its stratification check H_max = 74Q̇_c^2/5/ΔT_f→c^3/5 in ft, Btu/s and °F. The worst position of a device in a square array is r = S/√2.

Flashover, post-flashover temperature and fire severity (chapter 13)

Flashover (a layer at 500 to 600 °C or 15 to 20 kW/m² on the floor) needs a fire of at least Q̇_FO = 610√(h_kA_TA_v√h_v) by MQH with the lining heated through (h_k = k/δ), 750A_v√h_v by Babrauskas, or 7.8A_T + 378A_v√h_v by Thomas; the lowest is the conservative threshold. After flashover Law's correlation of the CIB tests gives the maximum temperature 6000(1 − e^−0.1Ω)/√Ω °C with Ω = (A_T − A_v)/(A_v√h_v); the handbook evaluates Ω with its A_T already net of the opening, so the opening is deducted twice, and the tool keeps that evaluation so that its values match the handbook's. Law's fire severity turns the fuel load into its wood equivalent L_eq = LΔH_c/ΔH_c,wood and the equivalent standard-furnace exposure t_f = KL_eq/√(A_vA_T) minutes with K = 1; it should not be used for openings under 0.4 m².

Ignition and cable damage time (chapter 6)

A thermally thick solid under a constant flux q″_e above its critical flux ignites after a time that five correlations estimate from different data: Tewarson's t = (π/4)TRP²/(q″_e − CHF)² with the critical heat flux and thermal response parameter of the FMRC apparatus (the handbook's cable table), Mikkola and Wichman's t = (π/4)kρc(T_ig − T_a)²/(q″_e − q″_cr)², Quintiere and Harkleroad's t = [q″_cr/(bq″_e)]² from the LIFT flame-spread parameter, Janssens's t = 0.563(kρc/h_ig²)[(q″_e/q″_cr) − 1]^−1.83, and the flux-time product t = FTP/(q″_e − q″_cr)ⁿ of Toal, Silcock and Shields. They diverge near the critical flux and differ among themselves by factors; the handbook prints them side by side, as the tool does. The IEEE-383 damage criteria of the handbook (10 kW/m² or 370 °C for qualified cable, 5 kW/m² or 218 °C for unqualified cable) are stated against the exposure.

Smoke filling and visibility (chapters 2 and 18)

The smoke analysis runs the natural-ventilation layer model and inverts the filling equation for the time at which the interface reaches a critical height (head height, the lowest cable tray, a detector); below the vent top it is a lower bound. Jin's relation S = K/(α_mm_p) then gives the visibility of an object from the particulate concentration m_p = y_pM_f/V spread through the whole room (the handbook's well-mixed assumption; a layer holds the same particulate in a smaller volume), with K = 3 for reflecting signs and building components and 8 for illuminated signs, α_m = 7,578 m²/kg (37,000 ft²/lb) for flaming and 4,301 m²/kg for smoldering combustion, and the particulate yields of the handbook's Table 18-3.

Egress time (SFPE hydraulic model)

The required safe egress time is estimated by the hydraulic model of the SFPE Handbook (Nelson and Mowrer, also in NIST's FPETool), used as mathematics with every constant an entered value: on each route element the speed is S = k(1 − aD) for the density D, the specific flow F_s = SD peaks at D = 1/(2a), the effective width is the clear width less the boundary layers, and the element with the smallest flow F = F_sW_e controls the queue, so that N occupants take N/F to pass it. The tool adds the entered pre-movement time, the travel time of the route at the applied density and the queue time, a conservative sum, and compares the total with an available time when one is entered (the smoke-filling time of the smoke analysis, for instance). Occupant loads, pre-movement times, minimum widths and travel limits are the code's (NFPA 101, NFPA 130 or the building code in force) and are cited only.

Sources, checks and what is not covered

Every correlation and table comes from NUREG-1805 and its Supplement 1 (public domain), which print the equations, the property tables (compiled from the SFPE Handbook, Klote and Milke, the SFPE Engineering Guide and NFPA 72, credited there) and worked examples. The tool's test reproduces the handbook's example problems 2-1 to 2-6, 3-1 to 3-3, 5-1, 5-4, 6-1, 6-2, 9-1, 9-2, 10-1, 10-2, 11-1 to 11-3, 12-1, 13-1 to 13-3 and 18-1 to 18-3 and the cached values of the Supplement 1 spreadsheets, with every formula re-derived in an independent numpy script; the few printed answers that the revised spreadsheets no longer give (a 2004 answer computed with the convective HRR, a Celsius misprint) are recorded in the test and the references README. Not covered: the explosion, fireball, battery-room hydrogen, structural fire-resistance and THIEF cable-failure chapters of the handbook, fires against walls or in corners (chapter 4), growing (t²) fires, multi-room smoke movement, zone or field models, and the evacuation of whole buildings with merging flows.