Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings

Fire Science and Technorogy Vol.24 No.4(2005) 165-178 165 Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings Jun-ichi Yamaguchi 1, Takeyoshi Tanaka 2 1 Technical Research Institute Obayashi Corporation, Japan 2 Disaster Prevention Research Institute, Kyoto University, Japan ABSTRACT This paper proposes simple equations for predicting the smoke filling time in the room of fire origin, which may be practically used to evaluate the performance of evacuation safety design of building. The simple equations were developed for the typical types of design fires, i.e. constant, t-square and their combination. Also these equations can be used for rooms having irregular ceilings, whose horizontal section areas change with height. Their prediction capabilities were verified by comparison with the predictions by BRI2002, a two-layer zone model often used as a tool for evacuation safety design. These simple equations are found to have prediction capability almost equivalent to BRI2002 for smoke filling in the room of origin. Keywords: Smoke Filling Time, Simple Equations, t 2 fire, Irregular Ceiling, Evacuation Safety design 1. INTRODUCTION In the context of performance-based fire safety designs of buildings, it is a common practice that the performance of evacuation safety in fire is evaluated by means of comparing the required safe egress time (RSET) with the available safe egress time (ASET), i. e. the time when fire-induced conditions within an occupied space become untenable [1]. Although heat, toxic gases, visibility etc. in smoke are involved in the untenable conditions, the assessment of the available safe egress time in the room of origin in fire safety design practices is most often based on the smoke filling time to the critical height [2]. If all the occupants have escaped from the room before the interface of the smoke layer descends to their height, it follows that they are not exposed to the smoke so it is not necessary to care complicated conditions of tenability. This is of course a bit conservative but a very convenient safety criterion for practical building designs. The smoke filling time is influenced by the configuration of the fire room and the behavior of the fire growth. It is true that the behaviors of fire growth of real world combustible items, such as chairs, wardrobes, trash baskets and beds, are complicated and differ from one to another. So their burning behaviors are experimentally investigated using furniture calorimeters based on oxygen consumption heat release rate measurement method [3],[4]. From such experimental measurements, it has been found that heat release rates at initial stage of many burning items can be empirically Received 22 July 2005 Accepted 6 February 2006

166 J.YAMAGUCHI and T.TANAKA approximated in the form of Q= α t 2 letting Q be the heat release rate, t be the time and α be a constant associated with the item [3]. This type of fire is now being used extensively as the design fires in evacuation safety and smoke control designs of actual buildings [5], [6]. The purpose of predicting fire behavior in fire safety design of a building is to verify the fire safety performance of the building under some appropriately conservative assumption but not to analyze the fire behavior as accurately as possible. What design fire is to fire safety design is just like what design load is to structural design of building. It does not necessary represent a realistic fire behavior but it is an artificial fire condition that is so determined that somewhat severer result than most of real fire cases is predicted [7]. As long as the fire behavior in a building is predicted at safer side for fire safety design, the simpler the design fire the easier the calculation for safety verification. However, too severe design fire condition may claim unbearably expensive fire safety measures. So the design fire of which the heat release rate grows proportionally to t-square at the initial stage and then levels off at the maximum heat release rate for a while until the available fuel has been depleted is often employed [8]. Currently, three types of design fires are most frequently used in fire safety design practices, i. e., constant, t-square, and their combination. Smoke filling time etc. in complex conditions have been predicted often using two layer zone smoke transport models, such as BRI2 in Japan [9], ASET[10] and CFAST[11] in USA and many others [12]. On the other hand, for the prediction of the smoke filling in a fire room with simple geometry, simple prediction equations are provided as practical FSE tools [5], [6], [8]. The advantages of such simple equations in fire safety design practices are as follows: 1) Easy to grasp the relation between design conditions and smoke filling behavior. 2) Easy to understand the effect of fire parameters, such as heat release rate, on smoke filling behavior. 3) Easy to use for designers, who are not fire experts. 4) Can be calculated promptly responding to change of design conditions In addition, simple calculation equations in general have a function as an educational tool of fundamental fire physics for designers etc. since such equations explicitly show what parameter how much affects the fire behavior. In this study, for the purpose of expanding the applicability of such simple smoke filling equations, the simple equations which can apply to several typical design fires and rooms with somewhat complex ceiling geometries were developed.

Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings 167 2. THE SMOKE FILLING TIME IN THE FIRE ROOM 2.1 e fundamental equations of smoke filling Simple equations for smoke filling have already been proposed for some simple cases [5] and validated with full-scale smoke filling experiment in a large scale space [13]. The outline of the method is summarized as follows. In order to predict the smoke filling behavior, it is, in principle, necessary to solve the conservation of mass and the conservation of heat, at least. However, to derive the simple smoke filling equation, only mass conservation is considered, i.e. (1) where, ρ and V are the density and the volume of the smoke layer, respectively. m p is the flow rate of the fire plume. As shown in Figure 1, considering a rectangular room with constant ceiling height, the volume of the smoke layer, V, can be expressed as (2) where A r is the floor area of the room, H r is the ceiling height, z is the smoke layer interface height. Using Equation 2 into Equation 1 is given as follows (3) It is assumed here that the plume mass flow rate is given by the following formula by Zukoski et. al.[14]. (4) where C m is the plume coefficient, for which Zukoski proposed C m=0.076. Incidentally, the smoke layer interface height, z, should be measured from the virtual point of heat source distance. However, the virtual point of heat source distance is ignored for convenience. Using Equation 4 into Equation 3 yields (5) Equation 5 is the basis for the following discussions in this paper. Figure 1 Schematic of calculation model

168 J.YAMAGUCHI and T.TANAKA 2.2 e heat release rate of design fire and the ceiling geometry of fire room As stated already, the design fires most frequently used in fire safety design practices of actual buildings are illustrated in Figure 2 in terms of the heat release rates. Namely, the heat release rates of such design fire sources are (a) constant with time (constant fire) (b) proportional to time-square (t 2 fire) (c) proportional to time-square in the beginning, then constant (t 2 -constant fire) (a) constant fire (b) t 2 fire (c) t 2 -constant fire Figure 2 The heat release rate of the design fire In fire safety design practices, smoke filling calculations are frequently required for not only simple rectangular rooms but also somewhat complex spaces whose horizontal section areas vary with height, atria for example. Figure 3 illustrates the ceiling geometries of fire room consideration in this paper. (A) regular ceiling space (B) irregular ceiling space Figure 3 The ceiling geometries of the fire room 2.3 e smoke filling for constant fire For a constant fire, the heat release rate of Equation 5, Q f, is substituted by the constant heat release rate Q c, i.e. (6) 2.3.1 Regular ceiling space For rooms of regular ceiling height, the smoke filling equation with a constant fire have been already available [2], i.e., the smoke layer interface height as a function of time is given by (7)

Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings 169 Reversely, the smoke filling time as a function of the smoke layer interface height is given by (8) 2.3.2 Irregular ceiling space Equation 7 and 8 can be extended to irregular ceiling space, such as a dome and a sloped ceiling atrium etc., if the space geometry is modeled into a combination of rectangular space elements as illustrated in Figure 4. It is not so difficult to derive the smoke filling equation in such a configuration as shown in Figure 4. That is, by slightly modifying the equation demonstrated in [16], the smoke filling time as a function of smoke layer interface height is given as (9) where n is number of rectangular space elements, A i is horizontal section area of rectangular space elements and H i is ceiling height of rectangular space elements. And the smoke layer interface height, z, needs to satisfy z H n. It is interesting note that Equation 9 is given by making summation of smoke filling time calculated by Equation 8 regarding each part of the space with different ceiling height. Figure 4 Dividing the room vertically Reversely, as an identical equation, the smoke layer interface height is given as a function of time as follows: (10) where the time, t, in Equation 10 needs to satisfy the following condition: (11)

170 J.YAMAGUCHI and T.TANAKA 2.4 e smoke filling for t 2 fire In case of t 2 fire, the heat release rate of Equation 5, Q f, is expressed as follows (12) where α is the fire growth rate coefficient. Using Equation 12 into Equation 5 yields (13) Equation 13 can be integrated as (14) where z 0 is the smoke layer interface height at time t 0. 2.4.1 Regular ceiling space For a simple rectangular space whose ceiling height is regular, using (t 0,z 0)=(0,H r), at the time of fire ignition, into Equation 14 yields the equation for the smoke layer interface height as a function of time as follows: (15) Reversely, the smoke filling time as a function of smoke layer interface height is as follows: (16) 2.4.2 Irregular ceiling space In an irregular ceiling space, a simple smoke filling equation with t 2 fire has not yet been proposed. This can be derived as follows: First, the smoke filling time to fill the height ceiling part, i.e., the smoke filling time to the height which changes the cross-section area, t 1, can be calculated as, using A r=a 1, H r=h 1 and z=h 2 in Equation 16 as (17) Next, the time for the smoke layer interface height descends to height between H 2 and H 3, i.e. H 3 z < H 2, is given using A r=a 1+A 2 and (t 0,z 0)=(t 1,H 2) in Equation 14 as (18)

Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings 171 Using Equation 17 into Equation 18 yields as follows: (19) where H 3 z < H 2. The similar process, Equation 17 through Equation 19, can be repeated to yield the equation for smoke filling time to arbitrary height z as (20) where z H n. Reversely, the smoke layer interface height can be given as a function of time as follows: (21) where the time, t, in Equation 21 needs to satisfy the following condition: (22) Note that the smoke filling time for t 2 fire is again given by the summation of the filling times in the spaces fictitiously divided corresponding to different ceiling height. 2.5 e smoke filling for t 2 -constant fire In this section, the simple equations in t 2 -constant fire, as shown in Figure 2(c), is considered. In this case, the heat release rate of Equation 5, Q f, is given as follows (23) where t c is the time at which the initial heat release of the growth fire reaches the constant value, i.e. (24) 2.5.1 Regular ceiling space During the initial growing stage, i.e. 0<t t c, is given by Equation 15, therefore the smoke layer interface height H c at time t c is (25)

172 J.YAMAGUCHI and T.TANAKA Figure 5 The schematic of the smoke layer interface height After the transition to constant fire, i.e. t>t c, the simple equations constant fire, Equation 7 and Equation 8, can be invoked. Using t=t-t c, H r=h c in Equation 7 yields (26) where t t c. Hence, the smoke layer interface height in Equation 26 can be obtained by calculating the smoke layer interface height at the transition time, H c, and then using Equation 26. Or we may obtain a single equation, by using Equation 25 into Equation 26, as follows: (27) Equation 26 can be solved for t to yield the smoke filling time as a function of the smoke layer interface height as follows: (28) or identically (29) 3. COMPARISON OF SIMPLE EQUATION AND BRI 2002 PREDICTIONS 3.1 Calculation conditions In order to examine the level accuracy of the simple equations for smoke filling in this paper, comparisons of prediction results are made between the simple equation and BRI2002, a smoke transport zone computer model, for 3 sample cases shown Table 1. Two types of room geometries as shown in Figure 6 (a), i.e. Type1: irregular ceiling space, and (b), Type 2: regular ceiling space, are considered. Incidentally, the reason that each room has an opening in the lower part of wall is that BRI2002 needs a certain opening for room pressure calculation. Unlike BRI2002, the smoke layer density ρ is not calculated by the simple equations, so ρ is assumed as ρ=1, in these case studies.

Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings 173 Table 1 Calculation conditions CASE 1 CASE 2 CASE 3 Heat Release Rate[kW] t 2 fire t 2 -constant fire Ceiling Geometry irregular ceiling space (Type 1) Figure 6(a) regular ceiling space (Type 2) Figure 6(b) irregular ceiling space (Type 1) Figure 6(a) (a) Type1: irregular ceiling space (b) Type 2: regular ceiling space Figure 6 The schematic of room geometries 3.2 Comparison of prediction results (1) CASE 1 Figure 7(a) and (b) shows the comparison of the predictions for CASE 1 by BRI2002 [15], which is the revised version of BRI2 smoke transport model, and the simple equations. The calculations by BRI2002 were made for two different locations of the fire source, i.e. in Area 1: high ceiling part and in Area 2: low ceiling part, of which results are shown in Figure 7(a) and (b), respectively. The smoke filling predictions by the simple equations are slightly faster than the predictions by BRI2002, particularly when the fire source is in Area 2. However, the simple equations can be said to be sufficiently accurate as tool used for smoke filling prediction in usual fire safety design practices.

174 J.YAMAGUCHI and T.TANAKA (a) The fire source location is in Area 1 (b) The fire source location is in Area 2 Figure 7 Comparison of the predictions by BRI2002 and the simple equations for CASE 1 (2) CASE 2 Figure 8 shows the comparison for CASE 2, in which t 2 -constant fire was assumed in a regular ceiling space. As can be seen in this comparison, the agreement of the predictions is very good between the simple equation and BRI2002. Figure 8 Comparison of the predictions by BRI2002 and the simple equations for CASE 2 (3) CASE 3 Figure 9 shows the comparison for CASE 3, in which t 2 -constant fire was assumed in a irregular ceiling space. The fire source location for prediction by BRI2002 is in Area1. As can be seen in this comparison, the agreement of the predictions is very good between the simple equation and BRI2002.

Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings 175 As can be seen in this comparison the agreement between the simple equation is again very satisfactory. Figure 9 Comparison of the predictions by BRI2002 and the simple equations for CASE 3 4. CONCLUSION In this paper, simple predictive equations for smoke filling were developed for t-square fire source and a t-square and constant combined fire source, i.e. t-square fire at first and then constant fire at later stage, as shown in APPENDIX A. These equations are applicable not only to spaces with regular ceiling but also with irregular ceiling. It is very interesting to note that the smoke filling time in a irregular ceiling space can be calculated by fictitiously dividing the space corresponding to the different ceiling height, and adding up the smoke filling times calculated for each of the fictitiously rooms. The comparison of the predictions by the simple equations with those by BRI2002 demonstrates that the simple equations can be useful tools for smoke filling calculation in fire safety design practices. The simple equations developed in this study are summarized in Appendix. Incidentally, unlike a computer model such as BRI2002, the simple equations cannot predict smoke layer temperature so the density has to be assumed. It is desirable for better prediction to improve the model to enable to estimate the smoke layer temperature. REFERENCES 1. The SFPE Handbook of Fire Protection Engineering, 3rd Ed. Chapter 3, p-367. 2. For example, The verification method for evacuation safety of buildings, Ministry Order No. 1441, Ministry of Construction, JAPAN, 2000 3. Quintiere, J.: Principles of Fire Behavior, Delmar Publishers, p113 4. Gross, D.: Data Sources for Parameters Used in Predictive Modeling of Fire Growth and Smoke Spread, NBSIR 85-3223, NBS, 1985 5. Tanaka, T. and Yamana, T.: Smoke Control in Large Scale Space ; (Part 1 Analytical theories for smoke control in large scale spaces), Fire Science and Technology, Vol. 5, No. 1, 1985

176 J.YAMAGUCHI and T.TANAKA 6. NFPA 92B: Smoke management System in Malls, Atria, and Large Areas, 1991 Edition, NFPA, 1991 7. Tanaka, T.: Concept and Framework of a Performance Based Fire Safety Design System for Building, J. of Applied Fire Science, Vol.3, No.4, 1993 8. Karlsson, B. and Quintiere, J. Q.: Enclosure Fire Behavior, CRC Press. 9. Tanaka, T.: A Model of Multiroom Fire Spread, NBSIR 83-2718, NBS, 1983 10. Cooper, L. Y. and Stroup, D. W.: ASET - A Computer Program for Calculating Available Safe Egress Time, Fire Safety J., 9, 1985 11. Jones, W. W., Forney, G. P., Peacock, R. D. and Reneke, P. A.: A Technical Reference for CFAST: An Engineering for Estimating Fire and Smoke Transport, TN-1431, NIST, 2000 12. Friedman, R.: A Irnational Survey of Computer models for Fire and Smoke, J. of Fire Protection Engineering, 48(3), 1992 13. Yamana, T. and Tanaka, T. : Smoke Control in Large Scale Space ; (Part2 Smoke control experiments in a large scale space), Fire Science and Technology, Vol.5, No. 1, 1985 14. Zukoski, E.E., Kubota,T., Cetegen, B., : Entrainment in Fire Plume, Fire Safety Journal, Vol.3, 107-121, 1980/1981 15. Tanaka, T., Yamada S., : BRI2002 Two Layer Smoke Transport Model, Fire Science and Technology, Vol. 23, No. 1(special issue), 2004 16. Tanaka, T., : Introduction to Fire Safety Engineering of Buildings, The Building Center of Japan, pp.232-235, 2002(in Japanese) NOMENCLATURE LISTING A i horizontal section area of rectangular space elements (m 2 ) A r horizontal section area of fire room(m 2 ) c p specific heat of air(kj/kg.k) C m plume coefficient(kg/kj 1/3.m 5/3.s 2/3 ) H c H i H r m p n Q c Q f t t c smoke layer interface height with t c(m) ceiling height of rectangular space elements (m) ceiling height of fire room(m) flow rate of fire plume(kg/s) room division number heat release rate of constant fire(kw) heat release rate(kw) time(s) Transition time from t 2 -fire to constant fire(s) V smoke layer volume(m 3 ) z smoke layer interface height(m)

Simple Equations for Predicting Smoke Filling Time in Fire Rooms with Irregular Ceilings 177 Greek α fire growth coefficient(kw/s 2 ) ρ smoke layer density(kg/m 3 ) APPENDIX The simple equations for smoke filling Ceiling geometry Type of design fire regular ceiling space irregular ceiling space constant fire t 2 fire t 2 -constant fire