http://irc.nrc-cnrc.gc.ca FIERAdetection Model (DRM) heory Report IRC-IR-841 Yager, B.; Kashef, A.; Bénichou, N.; Hadjisophocleous, G. January 2002
able of Contents able of Contents... i List of Figures... i Nomenclature... ii 1. Introduction... 1 2. Model Methodology...1 2.1 Plume in lower layer... 2 2.2 Plume passing through smoke layer interface... 4 2.3 Plume in upper smoke layer... 5 2.4 Ceiling jet... 6 2.4.1 Horizontal ceiling jet flow... 6 2.4.2 Plume/ceiling impingement zone... 9 2.5 Convection to smoke layer... 9 2.6 Response of detectors... 10 3. Solution Procedure... 11 4. References... 13 List of Figures Figure 1: ransport of smoke to ceiling... 2 Figure 2: Cooper s ceiling jet velocity profile... 7 Figure 3: Cooper's ceiling jet temperature profile... 8 Figure 4: DRM methodology... 12 i
Nomenclature A ceil = Area of ceiling surface (m 2 ) b = Plume width (m) C p = Specific heat of air (kj/kg K) C = Plume flow constant = 9.115 [6] D eff = Source effective diameter (m) g = Gravitational constant, 9.81 m/s 2 H = Height of ceiling above virtual origin of plume (m) m layer = Mass of smoke layer gasses (kg) m& = Mass flow of the gasses in the plume (kg/s) ent n = Number of time steps P atm = Atmospheric pressure (Pa) &Q = otal heat release rate (kw) &Q c = Convective heat release rate (kw) Q & c,2 = heat release rate of imaginary source at interface height Q & * = Non-dimensional convective heat release rate Q & intf = Heat release (kw) * Q & intf,1 = Non-dimensional heat release rate of real source * Q & intf,2 = Non-dimensional heat release rate of imaginary source in smoke layer r = Plume radius (m) R = Ideal gas constant (N m / kg K) RI = Response time index of detector or sprinkler link (m ½ s ½ ) t 0 = Initial time (s) t d = otal transport lag (s) t d,cj = ransport lag in ceiling jet (s) t d,plume = ransport lag in plume (s) Δt = imestep length (s) CEIL = emperature of the surface of the ceiling (K) CJ = emperature within ceiling jet (K) cp = Plume centerline temperature (K) det = Detector temperature (K) flow = emperature of the ceiling jet or plume flow at the detector location (K) MAX = Maximum temperature of the gasses within the ceiling jet (K) plume = emperature within plume (K) = Ambient temperature of the smoke layer (K) = Ambient air temperature (K) Δ = Difference between maximum ceiling jet and ambient temperatures (K) Δ* o = Normalized Δ U plume = Velocity within plume (m/s) U cp = Plume centerline velocity (m/s) V CJ = Velocity within ceiling jet at distance z below ceiling (m/s) V flow = Velocity of the ceiling jet or plume flow at the detector location (m/s) = Maximum horizontal velocity of flow within ceiling jet (m/s) V MAX ii
z = Distance above source (m) Z = Distance below surface of ceiling (m) z 0 = Location above source of virtual origin (m) z intf = Height of the interface above the source (m) z intf,pseudo = Distance below the smoke layer interface of the imaginary source virtual origin (m) Δz = Increase in smoke layer depth (m) β = Plume flow constant = 0.913 [6] δ = Effective depth of ceiling jet (m) ρ = Gas density of smoke layer (kg/m 3 ) ρ = Ambient air density (kg/m 3 ) ζ = Ratio of smoke layer temperature to ambient temperature θ = CEIL MAX iii
1. Introduction As Canada and other countries move from prescriptive-based building codes to performance/objective-based codes, new design tools are needed to demonstrate that compliance with these new codes has been achieved. One such tool is the computer model FiRECAM, which has been developed over the past decade by the Fire Risk Management Program of the Institute for Research in Construction at the National Research Council of Canada (NRC). FiRECAM is a computer model for evaluating fire protection systems in residential and office buildings that can be used to compare the expected safety and cost of candidate fire protection options. o evaluate fire protection systems in light industrial buildings such as aircraft hangars and warehouses, a new computer model is being developed. his model is based on a framework that allows designers to establish objectives, select fire scenarios that may occur in the building and evaluate the impact of each of the selected scenarios on life safety, property protection and business interruption. he new computer model is called FIERAsystem, which stands for Fire Evaluation and Risk Assessment system. FIERAsystem uses time-dependent deterministic and probabilistic models to evaluate the impact of selected fire scenarios on life, property and business interruption. he main FIERAsystem submodels calculate fire development, smoke movement through a building, time of failure of building elements and occupant response and evacuation. In addition, there are submodels dealing with the effectiveness of fire suppression systems and the response of fire departments. In the event of a fire, the fire plume rises directly above the burning fuel and impinges on the ceiling. Upon impinging, the hot gases in the plume turn and form a horizontal flow under the ceiling moving away from the source fire. he detectors (heat and smoke) and sprinkles installed near the ceiling surface become submerged in the formed hot gas layer. he response of this hardware provides the basis for the fire protection of the building. his report describes the theoretical framework of the detector response model (DRM) of FIERAsystem. he detection model predicts the time to heat detector activation as a result of a fire in a compartment. Wherever possible, theory and correlations commonly used are incorporated in the model. For this reason, the model presented here is similar in many ways to models such as LAVEN [1], FPEOOL [2], DEAC-QS [3] and ASME [4]. 2. Model Methodology his model incorporates a number of regions that the smoke and products of combustion move through as they travel to and along the surface of the ceiling. It also considers a number of processes associated with the smoke flow. hese regions and processes are treated separately and form the various components of the model. hey are (Figure 1): Plume in lower layer; Plume passing through smoke layer interface; Plume in upper smoke layer; Plume/ceiling impingement zone; 1
Horizontal ceiling jet flow; and Smoke layer filling. he gas properties and processes associated with the flow through each region are calculated in a time dependent fashion. he calculation process is marched through time, with initial time, t 0, time increment, Δt, and number of time steps, n, specified as inputs. he details of the time marching process are further explained and illustrated in the following sections. Ceiling jet Smoke layer Z Plume in smoke layer cp plume Plume at interface layer Entrained air Plume in lower layer Zo Real source Virtual source Figure 1: ransport of smoke to ceiling 2.1 Plume in lower layer In order to provide flexibility for modeling a wide range of fire scenarios, DRM uses the assumption of a quasi-steady heat release rate. he heat release rate, &Q, and the corresponding effective diameter of the fire source, Deff, are input for each time step. he quasi-steady assumption alone is not always accurate for fires that have a heat release rate that changes rapidly with time. his is because the plume generated by a fire with a distinct heat release rate will take time to reach a steady state. In order to take this fact into consideration, Mower s transport lag [9], t d, is introduced. he time lag is explained in more detail in the detector response section. he plume characteristics (temperature, velocity and entrainment) can be calculated using Heskestad's correlations [5] and assuming the initial conditions in the compartment space to be the ambient temperature and pressure. It should be noted that those correlations are applicable 2
in the area above the fire source level up to the smoke layer interface. Also, since those correlations were derived for a plume produced by a point source, a correction was made to represent a real area source plume. his was done by introducing a virtual source or virtual origin located a distance z 0 from the real source (Figure 1). he location of the virtual source is determined such that the plume originating at a point will have identical entrainment characteristics to the real plume. his distance is calculated using Heskestad's plume correlations, which extrapolate the centerline temperature of the plume to the point where its centerline temperature, cp, approaches ambient temperature, : 25 z = 0. 083Q& / 102. D eff (1) 0 where: z 0 &Q D eff = location of virtual origin (m); = total heat release rate (kw); and = real fire source diameter (or effective diameter for noncircular fire source) 4Area = (m). π Heskestad s equations are used to predict the plume centerline temperature and velocity, cp and U cp respectively: cp =. gc pρ 91 2 2 13 / Q& 23 / c 53 / 0 ( z z ) + (2) U cp 13 / g 13 = 34. Q& c z z Cp ( 0 ) ρ / 13 / (3) where: &Q c = convective heat release rate (kw); C p = specific heat of air (kj/kg K); g = gravitational constant, 9.81 m/s 2; z = distance above source (m); ρ = ambient air density (kg/m 3 ); and = ambient air temperature (K). he plume width, b, is defined here as the radius at which the local temperature rise ( plume ) decreases to half the plume centerline temperature rise (i.e. 0.50 ( cp )). he plume width may be calculated from the following relationship [5]: 12 / b = 012. ( / ) ( z z ) cp 0 (4) 3
Heskestad [5] proposed the following equations, assuming Gaussian profiles, to calculate the local plume temperature, plume, and gas velocity, U plume, at radius, r, within the plume: plume 2 ( ) 1.2 cp e + = r b (5) 2 r 1.2b U = U e (6) plume As the fire products in the plume rise towards the ceiling, ambient cool air is entrained into the plume at lower heights. he mass flow rate of fire products, m& ent (kg/s), at height above the fire source z, can be expressed as: cp 1/ 3 2 2 / 3 gρ 1/ 3 = ( ) 5 / 3 2.9Q& c m& 0.196 Q& ent c z z0 1+ (7) C p ( ) ( ) 2 3 3 g 1/ C 2 / z z 5 / p ρ 0 For normal atmospheric conditions and typical fuels, the plume mass flow rate at the mean flame height, (z-z o ) in Equation 7 can be rewritten as: 3 [ ] 5 / 3 3 ( z z ) 5 / 1+ 0.027Q& 2 / ( z ) m& & (8) 1/ 3 ent = 0.071Q c 0 c z0 2.2 Plume passing through smoke layer interface A fire plume acts as a pumping mechanism of mass and enthalpy to the upper portion of the compartment. As a result, two distinct zones are formed. he upper hot zone, or smoke layer, consists of the products of combustion plus the air entrained in the plume as it moves upwards. As the plume crosses the interface at the lower boundary of the smoke layer, the hot plume encounters a decrease in buoyancy because of the reduced temperature difference between the hot layer gases and the plume gases. he reduced buoyancy is treated by locating an imaginary source near the smoke layer interface, with virtual origin slightly below the level of the interface [7]. he heat release rate of the real fire source determined at the position of interface between the upper and lower layers can be normalized using: * Q& Q& intf,1 = (9) ρ C gz p 5 intf Where: z intf = height of the interface above the source. 4
his non-dimensional heat release rate below the interface is converted to an equivalent heat release rate located within the ambient conditions just above the smoke layer interface: 3 ( 1+ C Q& ) *2 / 3 / 2 * ntf,1 1 & i intf,2 = (10) ζc C Q Where: C = plume flow constant = 9.115 [6]; ζ = / = ambient temperature within smoke layer. he virtual origin s distance below the smoke layer interface is: z intf,pseudo = Q& *1/ 3 intf,2 ζq& * C 2 / 5 intf,1 3 z ntf [( 1)( 1) ] *2 / i ζ β + + ζq& C intf,2 (11) Where: β = plume flow constant = 0.913 [6]. And finally, the convective fraction of heat release rate of the imaginary source is: Q& = Q& ρ C gz (12) c,2 * intf,2 P 5 intf,pseudo Where: ρ = density of gases in smoke layer. By locating this imaginary source at the interface, the plume mass flow and energy transfer will remain continuous for the plume passing through the interface. he reduced buoyancy of the plume gases is taken into account by using the same equations as for the plume in the lower layer, except with ambient conditions now set to the ambient conditions of the smoke layer. 2.3 Plume in upper smoke layer With the source relocated to the height of the smoke layer interface and ambient conditions representative of the gas properties within the smoke layer, the same equations outlined for the plume in the lower layer (Equations 1-8) will be used for the plume in the upper layer. he substitutions necessary to accomplish this include: ρ = ρ = ambient density within smoke layer; &Q c = Q & c,2 = heat release rate of imaginary source at interface height; z z 0 = z z z = distance above virtual origin of imaginary source at interface; and intf 0 5
= = ambient temperature within smoke layer. his allows the calculation of the plume properties from the smoke layer interface to the plume/ceiling impingement zone. he procedure used to implement these equations is identical to that of the plume in the lower layer of the compartment, and identified in previous sections. 2.4 Ceiling jet he objective of this model is to predict the time to activation of a heat-actuated device. herefore, the most relevant calculated quantities are the thermal response of the device to a flow of hot gases and the properties associated with this flow. his section deals with the properties of the flow of the hot gases below the ceiling. he thermal response will be discussed in the next section. here are two distinct regions of the flow for a plume impinging on a ceiling. he first region is a plume/ceiling impingement zone that generally assumed to extend from the centerline of the plume, outward in the radial direction along the ceiling surface to the point where the radial distance is about 20% of the ceiling height. he second region forms outside the impingement region and is characterized by a relatively smooth and horizontal radial flow of the smoke and plume gases. Both of these two regions are discussed in the following subsections. 2.4.1 Horizontal ceiling jet flow Cooper s correlations for the temperatures and velocities of a flow below a smooth unobstructed surface [1] are used to predict the properties of the ceiling jet at different distances below the ceiling. hese velocities and temperatures profiles are assumed to have quadratic shapes, with maximum velocity and temperature at z = 0.23δ ( Figure 2 and Figure 3). V V CJ MAX 1/ 7 8 Z Z 1, 7 0.23δ 8( 0.23δ ) = 2 0.23 Z cosh ( ) acosh 2 1, 0.77 0.23δ Z 0 1 0.23δ Z > 1 0.23δ (13) Where: V CJ = Velocity within ceiling jet at distance Z below ceiling; V MAX = Maximum horizontal velocity of flow within ceiling jet; Z = Distance below surface of ceiling; and δ = Effective depth of ceiling jet, ie. V CJ (δ ) = 1/2 V MAX. and, 6
CJ MAX θ + 2 1 = V / V CJ Z 0.23δ ( θ ) ( 1 θ ) MAX 2 Z, 0.23δ Z 0 1 0.23δ Z, > 1 0.23δ (14) Where: CJ = emperature within ceiling jet at a distance z below ceiling; θ = CEIL ; MAX CEIL = emperature of the surface of the ceiling; = Ambient temperature of the smoke layer; and = Maximum temperature of the gases within the ceiling jet. MAX V CJ /V max 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1.0 1 2 Z/0.23δ 3 4 5 6 7 4.3 Z Figure 2: Cooper s ceiling jet velocity profile 7
( CJ - )/( MAX - ) Z/0.23 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 1.0 θ s >1 4.3 Z Figure 3: Cooper's ceiling jet temperature profile Cooper s Equations [1] for the maximum velocity within the ceiling jet and the depth of the ceiling jet are: V MAX = 0.85g 1/ 2 H 1/ 1.1 1/ 2 3 r Q& intf (15) H 0.9 r δ = 0.1 H (16) H he maximum temperature of the flow within the ceiling jet, MAX, and the temperature of the smoke layer,, must be obtained before the correlation stated above can be used. he temperature of the smoke layer is obtained in the following sections and MAX can be approximated using the correlation of Heskestad and Delichatsios [8]: Δ Δ / * 0 = = 0.188 + 0. 313 2 / 3 * ( Q& ) Where: Δ = MAX ; H = Height of ceiling above virtual origin of plume; Q & * = Q & /(ρ C P g 1/2 H 5/2 ); and r H 4 / 3 (17) 8
r = Plume radius. he above equation assumes that a smoke layer has not yet developed. If the smoke layer has developed, the non-dimensional source strength, Q& *, should be replaced by that of the * imaginary source located just above the smoke layer interface, Q & int f,2. Also, the ambient temperature,, should be replaced by the temperature of the smoke layer,. And finally, the height of ceiling above the virtual origin of the plume, H, should be set to the height of the ceiling above the virtual origin of the plume in the upper hot layer. 2.4.2 Plume/ceiling impingement zone he gases within the plume ceiling impingement zone are difficult to model. his region is where the upward, vertical flow of the plume changes to a radial horizontal flow, modeled by the ceiling jet equations in the previous section. he flow in this region changes direction 90 o and changes into flow against a flat surface with a boundary layer. he modeling approach for this region assumes that the temperature is the greater of either the ceiling jet or plume temperatures for a particular radius, r, and distance below the ceiling, Z flow = max[ CJ ( r, Z), plume( r, Z)] (18) 2.5 Convection to smoke layer Assuming that the smoke from the plume mixes perfectly into the upper layer, the temperature and density of the gases within the layer for the next time step, i+1 and ρ, i+1 respectively, are calculated from [10]: = 0.5Q& + c, i+ 1, i mlayercp (19) P atm ρ, i+ 1 = (20) R, i+ 1 he increase in layer depth, Δz, can then be calculated: where: A ceil = Area of ceiling surface. Δz m = & ρ 9 ent Δt A ceil (21)
2.6 Response of detectors he assumption of a quasi-steady fire was explained near the beginning of this report. It assumes that the plume and ceiling jet will be immediately established subsequent to a change in the heat release rate of a fire. In reality, the products of combustion will take time to reach the location of interest. his time lag is approximated using the transport lags developed by Mower [9]. he total transport lag, t d is the sum of a transport lag in the plume, t d,plume and a transport lag in the ceiling jet, t d,cj. = (22) t d td, plume + td, CJ he transport lag in the plume is a function of time and height. It is based on the time required for the smoke to be convected upwards through the plume. td, plume( z) = z /[1.5U plume ( z)] (23) he transport lag in the ceiling jet is based on the time required for the products of combustion to reach the point of interest on the ceiling from the centerline of the plume. td, CJ ( R) = R /[6V CJ ( r)] (24) he transport delay is used to calculate the temperature of the flow either within the plume or the ceiling jet as follows: flow ( d, plume + d, CJ r z, r) = t ( z) t ( ) (25) Each detector or sprinkler link has a response time index (RI) associated with it, which is a property of the detector and flow orientation. RI allows designers to calculate the detector temperature, det, due to the convective heat transfer at the location of the detector as follows: ( flow det ) d dt V det = RI flow (26) Where: flow V flow RI = emperature of the ceiling jet or plume flow at the detector location; = Velocity of the ceiling jet or plume flow at the detector location; and = Response time index of detector or sprinkler link. By integrating this equation over each time step, the temperature increase of the detector can be calculated. Once the detector temperature det reaches its activation limit, the detector is assumed to activate. 10
3. Solution Procedure Details of the solution procedure used by the model are shown in Figure 4. he model marches through time using a user specified time step. It first calculates the plume and ceiling jet characteristics based on the fire source data for that time step. If detectors are located within the ceiling jet or the plume impingement zones, the model calculates the detectors temperatures due to heat transfer. he model considers the different heat transfer modes for detectors located within the plume/ceiling impingement region and detectors located within the ceiling jet zone. he model loops through calculations until the time defined by the user is reached. he output of the model is the temperature of each detector and its time of activation. 11
Start Detection Response Model Set time to first time step Get fire source data for this time step Plume equations in lower layer Smoke layer depth > ceiling jet depth? Yes Calculate plume characteristics for transition through smoke layer interface Next ime step No No Plume/ceiling impingement calculations Detectors within plume/ ceiling impingement zone? No Yes Get flow characteristics at detector location Determine heat transfer to detector & detector temperature Calculate probability of activation vs. time for detector Next detector Yes Any more detectors? No Last ime Step? Yes End Detection Response Model Calculations for ceiling jet Detector within ceiling jet zone? No Yes Ger flow characteristics at detector location Determine heat transfer to detector & detector temperature Calculate probability of activation vs. time for detector Next detector Yes Any more detectors? No Unbounded ceiling? No Determine filling of layer and layer depth Calculate gas properties of layer Yes Calculate radiation to layer and new layer gas properties Figure 4: DRM methodology 12
4. References 1. Cooper, L.Y., Estimating the Environment and the Response of Sprinkler Links in Compartment Fires with Draft Curtains and Fusible Link-Actuated Ceiling Vents - Part I: heory, NBSIR 88-3734, National Bureau of Standards, Gaithersburg, Maryland, 1988 2. Deal, S., echnical Reference Guide for FPEtool Version 3.2, NISIR 5486-1, National Institute of Standards and echnology, Gaithersburg, Maryland, 1995 3. Evans, D.D. and Stroup, D.W., Methods to Calculate the Response ime of Heat and Smoke Detectors Installed Below Large Unobstructed Ceilings, NBSIR 85-3167, National Bureau of Standards, Gaithersburg, Maryland, 1985 4. Klote, J.H., Method of Predicting Smoke Movement in Atria With Application to Smoke Management, NISIR 5516, National Institute of Standards and echnology, Gaithersburg, Maryland, 1994 5. Heskestad, G., Fire Plumes, SFPE Handbook of Fire Protection Engineering, Society of Fire Protection Engineers, 2nd Ed., 1995 6. Zukoski, E.E., Kubota,. and Cetegen, B., Fire Safety Journal, 3, 107, 1981 7. Evans, D.D., Ceiling Jet Flows, SFPE Handbook of Fire Protection Engineering, Society of Fire Protection Engineers, 2nd Ed., 1995 8. Heskestad, G. and Delichatsios, M.A., he Initial Convective Flow in Fire, 17th International Symposium on Combustion, Combustion Institute, Pittsburgh, 1978. 9. Mowrer, F.W., Lag imes Associated With Fire Detection and Suppression, Fire echnology, August 1990 10. NFPA 204 Guide for Smoke and Heat Venting, 1998 Edition 13