A NEW MODEL FOR ESTIMATING NEUTRAL PLANE IN FIRE SITUATION

A NEW MODEL FOR ESTIMATING NEUTRAL PLANE IN FIRE SITUATION JY Zhang¹,*, Jane WZ Lu² and R Huo¹ 1 PhD student, State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, 230026, China 2 Department of Building and Construction, City University of Hong Kong HKSAR, China ABSTRACT A new model is proposed for predicting the location of neutral plane in fire situation. The shaft space is divided into two-zones (fire zone and inner space of shaft) in the model. In each zone, the temperature is assumed uniform. For validating the proposed two-zone model, a computational fluid dynamics (CFD) code, which is named FDS based on large eddy simulation (LES) developed by the Building and Fire Research Laboratory, National Institute of Standards and Technology, USA, is applied. It is shown that the location of neutral plane is above the mid-height of building, and the simulation results are uniform with the two-zone model approximately. Under fuel-control condition, the value of neutral plane height as a percentage of ceiling height is independent of the building height, the HRR of fire, the dimension of ventilation, the dimension of fire zone, and the mode of ventilation. Its value is found between 0.5 and 0.6. INDEX TERMS smoke movement; neutral plane; CFD; stack effect; INTRODUCTION The buoyant force causes an upward flow within the building shafts such as stairwells and elevator shafts. This is known as stack effect or chimney effect. However, when the air is hot outside, a downward flow of air occurs in an air-conditioned building. Such reverse stack effect is illustrated in Figure 1. Shaft Atria Inner part of building Neutral plane Normal stack effect Reverse stack effect Figure1. Normal stack effect and reverse stack effect (Arrows indicate the direction of air flow) There is a horizontal plane where the pressure inside a building or other space equals to the outside, which is regarded as neutral plane. There is no pressure difference between the space and its surroundings at the neutral plane, and no horizontal flow on it as well. Determination of the location of the neutral plane is essential to the evaluation of flow due to stack effect. Analytic equations have been developed for calculating the neutral plane for a few simple cases, e.g., leakage openings. However, the combinations of leakage openings in building shafts and compartments could be more complex and need to pay more attention. A number of computer programs have been developed that allow analyzing both airflow and smoke distribution inside buildings. Said (Said 1988) provides an overview of such models, which are intended for analysis of smoke movement or for design of smoke control systems. While such programs can be used to analyze stack effect, they do not calculate the location of the neutral plane. For a shaft connected to outside by a number of openings, the location of the neutral plane can be found by simultaneously solving a set of algebraic equations for conservation of mass for the shaft, mass flows through * Corresponding author email: zjy@mail.ustc.edu.cn 3343

connections, hydrostatic pressure inside the shaft, and hydrostatic pressure outside the shaft. This kind of general solution is presented by Klote [2], including a computer program, i.e., STACK, for such application. This program is regarded as a tool to analyze stack effect and to help engineers develop insight into relationships within stack effect, building openings, and temperatures. General discussion on stack effect assumes temperature to be uniform throughout the space. By this assumption, Klote (Klote 1991) educed a formula for calculating the neutral plane, proved the height of neutral plane is less than the half of space height, and reckoned that the vertical temperature is not important to the neutral plane. However, due to strong air-entrainment and fire source, the location of neutral plane could be changed. Even it is possibly above the mid-height of the building. Therefore, the detailed study on location of neutral plane under fire situations is necessary and important. MATHEMATICAL MODELS Common considerations on neutral plane (Klote Model) Shaft connects to buildings via some openings on each floor actually. Such case can be described by continuous opening model. The velocity pattern induced by normal stack effect for a space connected to the outside via continuous openings with a constant width from bottom to top of a space is illustrated in Figure 2. The opening extends from the bottom to the top of the space. The ratio of the neutral plane height to the ceiling height is expressed by: 1/3 Hn/ Hc = 1/[1 + ( Tc / T ) ] (1) Where, H n is the height of neutral plane above the bottom of space, and H c is the total height of space. Both T c and T are temperatures of shaft space and outside environment respectively. By the definition of normal stack effect, the ratio of space temperature to outside temperature is always greater than one ( Tc / T ). Thus, inspecting equation (1) shows that H n /H c is always less than half of building height, i.e., the neutral plane is below the mid-height of the space normally. The air with higher density flowing through the lower part of the opening needs less area than the air with lower density passing the upper portion of the opening. Two-zone model The main drawback of Klote model is the assumption that T c is a constant. Actually under fire situation, the temperature at inlet where fire source locates is much higher than others in the shaft. At the same time, temperature in the shaft above the fire floor is achieved uniform rapidly due to the strong heat transfer between inside and outside. Thus, H f is defined to be the height of fire floor shown in Figure 2, and T f is the temperature within it, which is higher than T c remarkably. Hence, a two-zone model is introduced in the study. The two-zone assumption will be validated by field model later. In two-zone model, the pressure difference between space and outside is expressed as: pc, = Kpgh( ρ ρ c) (2) Where ρ is the density of external air, ρ c is the density inside the shaft space, and h is elevation above the neutral plane. The mass flow rate, dm out, through a differential section, dh, of the opening above the neutral plane is: dm = KCW 2ρ p dh= KCW 2 ρ ghk ( ρ ρ ) dh (3) out o c c, o c p c Where C, W, and pc, are the flow coefficient, width of opening, and the pressure difference between the space and the surroundings. The flow coefficient C is dimensionless and generally in the range of 0.6 to 0.7 for most flow paths in buildings. To obtain the mass flow rate from the space, Eq. (3) is integrated from the neutral plane (h=0) to the top of the building (h=h c - H n ) expressed as below: 2 3/2 m out= KCW o ( Hc Hn) 2 ρcgk p( ρ ρc) (4) 3 3344

Upward flow Shaft Space T c Neutral Plane T H c 20m Atria Air Entrainment H n H f z y y x x 3m 1.5m Fire Source Figure 2. Normal stack effect in fire situations (The horizontal arrows indicate velocity of air movement, and the gray domain indicates fire zone) It is supposed that the temperature in the shaft space (T c ) is uniform except for the fire floor, represented by T f. So, in a similar manner, an expression for the mass flow rate into the space can be described as: 2 3/2 3/2 min = KCW o [ H f 2 ρfgkp( ρ ρf ) + ( Hn H f) 2 ρ gk p( ρ ρc)] (5) 3 Where Figure 3. Sketch map of shaft and atria ρ f is the density of gas at the bottom of shaft which connects the fire floor. For steady flow, the mass flow rate leaving the shaft space equals to that entering the space. Equalizing Eqs, (4) and (5), substituting the perfect gas equation ( ρ = p/ RT ), yielding: 3/2 3/2 3/2 ( Hc H n) ( H f ) ( Tc / Tf ) ( Tf T )/( Tc T ) ( Hn H f) Tc/ T = + (6) From Eq. (6), it is found that the location of neutral plane is determined by T c, T f, and H f, and the fire source plays an important role. The ratios of neutral plane height as a percentage of ceiling height (H n /H c ) can be more than half of building height under fire situations. Eq. (6) is non-linear type, so a Newton iterative program is developed to solve it. NUMERICAL SIMULATIONS Numerical Model Large eddy simulation (LES) model directly solves the transient behavior of large-scale turbulent motion and reduces computing time by approximating small-scale activity. The LES model directly calculates large-scale turbulent motion, which tends to have the greatest impact on turbulent transport, and addresses smaller-scale turbulent motion (those eddies smaller than the grid-cell size chosen) with a sub-grid-scale eddy viscosity model(mcgrattan et al.1994). Because Navier-Stokes equations are not time averaged, this method is inherently time dependent. The LES model is therefore best suited to solving transient flow problems, but results can be averaged over time for comparing with steady-state experimental data. The LES simulation has traditionally been very time consuming, but a very fast solver can be used if the solution is limited to regular geometries. This technique has been employed to develop the simplified model, (FDS) (fire dynamics simulator) (McGrattan et al.2000). 3345

For the FDS model, the governing equations of fluid flow are written in a form suitable for low Mach number applications (Rehm and Baum 1978). Sometimes, this form of equations is referred to as weakly compressible or thermally expandable to emphasize that the divergence of gas mixture is governed only by the introduction of heat or a change in the composition of the mixture. In a large eddy simulation, the grid resolution is not fine enough to capture the mixing processes at all relevant scales, so a sub-grid scale model for the viscosity is applied. FDS relies on the Smagorinsky(Smagorinsky 1963) model for the viscosity. There have been numerous refinements of the original Smagorinsky model (Deardorff 1972, Germano et al. 1991, Lilly 1992), which has been difficulty to apply to airflow in rooms. Such refinement is mainly due to the lack of precision in most large-scale experimental data as well as to the dominant influence of large-scale resolvable eddies in fire plumes and ventilation jets (Baum et al.1997). The Smagorinsky model produces satisfactory results for most large-scale applications in which boundary layers are not well resolved. Case descriptions The simulation case is shown in Figure 3. It consists of shaft space and atria. The dimensions of each floor are uniform. Each floor contains two doors. One connects the shaft and the atria; another opens to outside. The dimensions of two doors are the same with 1.8m (high) 0.8m (width). The fire source was set at the floor level in the atria with fixed power levels of 90kW, 45kW, and 22.5kW respectively. Ambient temperature was 293 K (20 C). To study the influence of ventilation air supply to neutral plane, two different modes of ventilations are chosen under the same ventilation factor. The sketch map is shown in Figure 4. (a) (b) Opening to outside Opening to outside Figure 4. Two ventilation modes in atria on floor one (a) Side ventilation; (b) face ventilation On neutral plane, the pressure difference between inside and outside is zero, so the horizontal velocity is zero. In the simulations, the horizontal velocity of air movement is calculated to determine the location of neutral plane on each door, which connects the shaft space and the atria. A thermocouple tree with eight thermocouples implemented in FDS code is used to estimate the temperature on the centerline of shaft space, which can be used to validate the assumption of two-zone model. The location of thermocouples is at the same height as that of velocity calculation points. RESULTS In simulations, the sizes of grid cells in atria and shaft are 0.1m 0.1m 0.1m and compressed to 0.05m 0.05m 0.05m adjacent to fire source. Some quantitative results are shown here for analyzing characteristics of neutral plane and other relevant parameters. Figure 5 shows the temperature distributions under different modes of ventilation. As assumed, the temperature in the shaft space is approximately uniform except the fire domain and keeps constant under the same HRR with side ventilation. However, under the face ventilation, the temperature at floor two is lower than that in higher floors obviously (see Figure 6). This is due to the advantageous opening condition in face ventilation. The hot plume from the fire source spreads along the side of shaft, and leads a turbulent flow above floor three. Thus, the temperature above floor three is steady approximately. In Eq. (6), the Newton iterative program is used to solve the location of neutral plane (H n ). The principle of Newton iterative method was introduced in Igor s paper (Moret,Igor 1987). After simulating the horizontal velocity, the location of neutral plane is obtained by linear-approach, where the horizontal velocity is zero. Table 1 shows the simulation results of neutral plane under the different conditions. 3346

(a) (b) Normalized shaft height 1.0 0.8 0.6 0.4 0.2 Side ventilation with lowest HRR Side ventilation with lower HRR Side ventilation with normal HRR Face ventilation with lower HRR Face ventilation with normal HRR 0.0 20 25 30 35 40 45 50 Temperature ( o C) Figure 5. Effect of ventilation mode and HRR(Normal HRR is 90kW, lower HRR is 45 kw, and the lowest HRR is 22.5kW) Figure 6. Isotherm in simulation 1 and 2 (a) Normal HRR (b) Lower HRR Table 1. The comparison of ε (H n / H c ) in Klote model, two-zone (TZ) model and CFD results No. Mode of ventilation HRR (Kw) Dimension of opening H c H f T c (K) T f (K) ε 1 (Klote) ε 2 (TZ) ε 3 (CFD) 1 Face 90 0.8m 1.8m 20 2.5 302.23 321.40 0.4988 0.5238 0.5628 2 Face 45 0.8m 1.8m 20 2.5 298.87 311.25 0.4983 0.5235 0.5653 3 Face 90 0.8m 1.8m 15 2.5 303.61 318.45 0.4970 0.5294 0.5507 4 Face 45 0.8m 1.8m 15 2.5 299.62 310.16 0.4981 0.5274 0.5520 5 Face 90 1.0m 1.8m 15 2.5 303.08 317.48 0.4972 0.5291 0.5443 6 Face 90 0.4m 1.8m 15 2.5 308.76 339.30 0.4956 0.5255 0.5675 7 Face 90 0.2m 1.8m 15 2.5 314.95 356.61 0.4940 0.5264 0.5670 8 Face 90 0.1m 1.8m 15 2.5 321.23 352.87 0.4923 0.5314 0.5718 9 Side 90 0.8m 1.8m 20 2.5 302.91 320.07 0.4972 0.5255 0.5532 10 Side 45 0.8m 1.8m 20 2.5 299.30 307.04 0.4982 0.5287 0.5469 11 Side 22.5 0.8m 1.8m 20 2.5 296.87 300.12 0.4989 0.5318 0.5514 12 Side 90 0.8m 1.8m 15 2.5 304.37 319.97 0.4968 0.5296 0.5523 13 Side 45 0.8m 1.8m 15 2.5 300.09 308.05 0.4980 0.5321 0.5384 14 Side 22.5 0.8m 1.8m 15 2.5 297.33 300.79 0.4988 0.5359 0.5403 From Table 1, it is found that in two-zone model and CFD simulations, the H n H c values are all higher than 0.5 under all conditions, approximately uniform, and independent of height of building, HRR of fire, dimension of opening, and modes of ventilation. Although the HRR rate influences the temperature of shaft space, it does not contribute to the location of neutral plane. The reason is that the flow in shaft space is a turbulent mixing process. The temperature in shaft space is determined by the temperature in fire zone and air-entrainment. With the increase of temperature in fire zone, the air-entrainment is enhanced. Finally the steady state is obtained. The horizontal velocity distribution in a 15m-shaft is shown in Figure 7. From this figure, it is found that the air-entrainment is 3347

stronger under normal HRR than that in low HRR. From the tilted plume, it can be seen that the normal stack effect is under the fuel-controlled regime due to the strong air-entrainment. If the area of ventilation is reduced until the fire source is under ventilation-controlled, the plume will fills into the shaft space, not be drawn in. In simulation cases 3, 5, 6, 7, 8, it is noticed that with the decrease of area of ventilation, the air- entrainment is weakened, and the temperature in shaft is increased. Nevertheless, the H / H value is similar to different ventilation factor because it is under fuel-controlled. n c CONCLUSIONS The results obtained by simulations indicate that the location of neutral plane in fire situation is higher than the mid-height of building normally. Traditional (Klote) model for estimating the neutral plane is suitable for none-fire situation. The air-entrainments below neutral plane are all cold air. So the location of neutral plane is under half of the building height. In fire situation, the temperature in fire zone is much higher than that in other parts of the shaft space, and the plume with lower density flow into the shaft. Therefore, the condition that temperature in shaft space is uniform cannot be assumed here. This feature is important for the modeling of flow in vertical shaft. A two-zone model is developed in this study and shown to be in better agreement with CFD simulations in locating the neutral plane. The height of neutral plane as a percentage of ceiling height is found between 0.5 and 0.6. Under the fuel-controlled, the value of H n / H c is independent of height of building, HRR rate of fire, dimension of opening, dimension of fire zone, and modes of ventilation when, the most important parameter, air-entrainment, is limited. Does the neutral plane still exist? The answer is unknown. It proposes a challenge work to further study smoke filling into shaft space under fire situation. ACKNOWLEDGEMENT The paper was partially supported by the China NKBRSF project (No. 2001CB409604) and the Strategic Research Grant #7001463(BC), City University of Hong Kong, HKSAR. REFERENCES Baum H., McGrattan K. and Rehm R. 1997. Three-dimensional simulations of fire plume dynamics. J. Heat Transfer Society of Japan, 35, pp455 52. Deardorff J. 1972. Numerical Investigation of neutral and unstable planetary boundary layers. [J]. Atmos. Sci., 29, pp91 115. Germano M., Piomelli U., Moin P. and Cabot W. 1991. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3, pp 1760 1765. Klote JH. 1991. A General Routine for Analysis of Stack Effect, National Institute of Standards and Technology, NISTNR 4588. Lilly D. 1992. A proposed modification of the Germano subgird-scale closure method. Phys. Fluids A, 4, pp633 635. McGrattan K., Baum H., Rehm R., Hamins A. and Forney G. 2000.Fire dynamics simulator, technical reference guide. Technical Rep.NISTIR 6467, National Institute of Technology, Gaithersburg, Md. McGrattan K., Rehm R. and Baum H.1994. Fire-driven flows in enclosures. J. Comput. Phys., 110, pp 285 291. Moret Igor. 1987. On a general iterative scheme for Newton-type methods. Numerical Functional Analysis and Optimization, v 9, n 11-12, p 1115-1137. Rehm R. and Baum H. 1978. The equations of motion for thermally driven, buoyant flows. Journal of Research of the NBS, 83, pp297 308. Said MNA. 1988. A Review of Smoke control Models, ASHRAE Journal, Vol 30, No 4, pp 36-40. Smagorinsky J. 1963. General circulation experiments with primitive equations. I. The basic experiment. Monthly Weather Review, 91, pp99 164. 3348