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Numerical simulation on cooling of the fire-induced air flow by sprinkler water sprays
Fire Safety Journal 17 (1991) 263-290
Numerical Simulation on Cooling of the Fire-induced Air Flow by Sprinkler Water Sprays W. K. C h o w & N. K. F o n g Department of BuildingServicesEngineering, Hong Kong Polytechnic,Hung Hom, Hong Kong (Received 29 May 1989; accepted 10 September 1990)
ABSTRACT By the air drag and cooling effect of water droplets, the cooling of the fire-induced hot air flow by sprinkler water sprays is studied by using a three-dimensional field modelling technique. This gives a 'microscopic' description of visualizing how the sprinkler interacts with a smoke layer. Macroscopic parameters, such as drag-to-buoyancy ratio, convective cooling rate, etc., can be computed accordingly. With such a model, it is possible for the fire-safety engineers to design proper location, type and performance of sprinkler systems. Since experimental data for validating the model are absent, it is concluded that measurement from full-scale burning tests in a sprinklered compartment is needed.
NOTATION Coefficient of the linear equation Buoyancy of smoke B Downward air drag D Air drag components (kg/ms 2) Ox, Oy, Oz Gravity (m/s 2) g Enthalpy of air (kJ/kg) h Turbulent kinetic energy (m2/s 3) k Nu Nusselt number of water drops Pressure of air (N/m 2) P 263 Fire Safety Journal 0379-7112/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland ai
264
W. K. Chow, N. K. Fong
U, /3, W
Air velocity components (m/s)
E
Turbulent energy dissipation Effective viscosity (kg/ms) Turbulent viscosity of air (kg/ms) Laminar viscosity of air (kg/ms) Density of air (kg/m 3) Variables (u, v, w, P, h, k or e)
~etI ~,~i
#l P
1 INTRODUCTION Past statistical records on fires illustrate that sprinklered buildings have a much lower risk of having uncontrolled fires. 1,2 This leads to a reduction of the insurance premium for buildings installed with automatic sprinkler systems. However, the fire extinguishing mechanism of water sprays is not yet clearly understood) Out of the many possible outcomes, interaction of a water spray with the smoke layer is an important issue for study. 4-8 This is because the fire-induced buoyant smoke layer might be cooled down significantly by the water spray. Therefore, it could lose buoyancy and fall to a lower level. The occurrence of 'smoke logging' reduces the efficiency of smoke extraction systems, causing hazards to building occupants. The situation becomes very significant for atrium buildings. 9,1° Secondly, it is interesting to know whether the sprinkler water spray obtained under certain operating conditions is able to confine a fire. With such knowledge, the minimum amount of water can be discharged from an appropriate sprinkler head in order not to 'overflood' the compartment and cause water damage. A good understanding of the interaction between the fire-induced flow and sprinkler water spray enables engineers to improve the present design for sprinkler layout, H by applying a reasonable amount of water, and giving guidance for modifying the physical features of the sprinkler head itself. The sprinkler system can then be designed to perform more efficiently in controlling a fire. Solving this problem is very difficult, however, because many physical phenomena such as direct cooling of the smoke, air entrainment into the spray and water evaporation ~2'~3 have all to be considered concurrently. To the best of our present knowledge, there is still no good theory which is capable of explaining how the sprinkler water spray interacts with a smoke layer realistically. Papers which have appeared in the literature include those on the stability analysis of the smoke layer, by investigating the drag-to-buoyancy ratio, by e.g. Bullen4 and Morgan, 5 sprinkler-induced air flow by Alpert, 14"1~study of the interac-
Numerical simulation on the cooling of the ]ire-induced airflow
265
tion between a ceiling jet and a sprinkler water spray, using the particle-source-in-cell method, 16 interaction of smoke venting by using a simple zone model with ceiling jet by Hinkley, 7'8 the simple heat extraction model by Chow and Fong, 17 and the more recent works with a two-phase field model by Hoffmann et al. TM None of these have satisfactorily indicated a clear picture of the interaction of a smoke layer with the sprinkler water spray inside an enclosure. Previously, the resultant air convective flow and temperature fields induced by a fire plume can be simulated successfully from the existing fire field 19-25 models with minicomputers (such as a V A X 8500 system). Alpert ~4"~5 and the current authors have investigated the effect of a sprinkler water curtain on the fire-induced flow field by using twodimensional simulation (Chow, W. K & Fong, N. K., 1988, unpublished). It is the objective of the present article to investigate how sprinkler water sprays will interact with a fire-induced thermal statified layer by using a similar technique, but extended to a three-dimensional picture. The predicted result is useful as the aerodynamics would indicate the smoke movement pattern. Other macroscopic parameters, such as convective heat extraction by the water spray and drag-tobuoyancy ratio of the hot air layer, 4'5 can all be computed under different sprinkler operating conditions. Section 2 gives the background theory; Section 3 describes the numerical experiments performed; Section 4 shows the results achieved; and the paper ends in Section 5 with conclusions. Although there are no satisfactory experimental data available for validating the results, the capability of applying the fire field model to solving this problem is illustrated.
2 THEORY The fire field generated by a thermal plume inside an enclosure can be simulated by field models. 19-25 The transient behaviour of air flow and temperature field can be predicted successfully by visualizing the plume as a heat source. Since the combustion process during a fire is very complicated, realistic treatment of combustion in a fire field model has not yet been achieved. 22 At the moment, fire can only be treated as a simple chemical reacting system. Extension by a two-fluid model 26 is possible and has been developed in studying the two-dimensional diffusion flame. Application of a laminar flamelet model (Moss, J. B. et al., 1989, pers. comm.) to fire simulation is in progress and results are to be investigated. In this paper, the smoke movement field model treats the flow as primarily a natural convection problem. However,
266
W. K. Chow, N. K. Fong
this is good enough to define the fire environment within a building at the preflashover stage, zs It is particularly useful for studying the interaction between fire-induced air flow and a water spray as the sprinkler head is expected to be actuated during the initial stage of a fire. The effects of thermal radiation have been ignored. The fire field model employed is the one developed by Chow and Leung, z3,z5 which is able to predict the transient three-dimensional air flow, temperature and pressure fields induced by a volumetric heat source. It is based on finite-difference calculations of a set of conservation equations for mass, m o m e n t u m and energy. The model is now applied in studying how a sprinkler water spray interacts with the fire induced hot air layer. The particle-source-in-cell m e t h o d 16 is used to describe the interaction between the water droplets and hot air. The flow is still visualized as 'single-phase' with those air drags and convective cooling terms appearing only in the source term in the set of m o m e n t u m and enthalpy equations. In this way, only ' m o m e n t u m ' and 'heat' coupling effects between hot air and water droplets are considered. This is reasonable, as evaporation heat lost for a water droplet is much smaller than the convective one (e.g. Beyler, z7 Kung, 12 and Chow28). The problem now becomes one of studying how an axisymmetric sprinkler flow would interact with a transient, horizontally flowing hot air stream. A three-dimensional simulation picture of the air flow is necessary, and the turbulent effect is described by a k - e model, z9 The set of equations becomes
-
ap+ a(ou)+ a(ov)+ a(ow) - -
at
ax
9y
az
0
(continuity equation)
(1)
where p is the air density, and u, v, w are the velocity components in a Cartesian coordinate system. The x - m o m e n t u m for the horizontal velocity c o m p o n e n t u is
a(pu) +
a
a--T- Tx
(puu) +
a
Yy ( o u r )
+
(puw)
a( au) a( a~) a( a.) a(a.)
= - ~ap + 7x
#°"-~x
+ 7y
~'°"
+ 7z
+O---z /~t-~-x
~'°"~
+ Tx #'--~x
ay ~U'oxx - D x
(2)
267
Numerical simulation on the cooling of the fire-induced air flow The y - m o m e n t u m for the vertical velocity c o m p o n e n t v is
a(pv) v
~
(OUr)+
at
_
a(po)
g(p
ay
s (pvv) + o (pvw)
~y
~z
p,a) +
_
+--~x
-~x
,ue. ~
"'
"eft
ay
'~-
"JV--~Z "t
as "t
"eli
--By
(3)
The z - m o m e n t u m for the velocity component w is
a(pw) ~
~
at
=-a~
o (pvw) + o (pww)
(ouw) +
-~y
7z
,9
~ "°"
ap + a (a_~xx) + ~a
(
aw + a
" e .aw ~
)
av
(ow)
+_a_a az ""~zzz - D =
(4)
Here, Dx, Dr, D= are the air-drag components experienced by the water droplets and appear only in regions containing water droplets. The equation for the turbulence kinetic energy k is
a,< a,< Ty a,< w ~a,<- -~,{a[,.~]+_~ r,.,~,
--+
u
p
+
v
+
--
kay/
\azlJ
Lay
r ao
axJ
aw]h
+ [~+~J
1 ap
.[- ~ +"'~p2 ay o)
The equation for the turbulence energy dissipation e is
a,~ --+u
at
ae cae ae 1{ a [ / . , a s ] a [.tone -] + +w~=-+ -~x v--~y az p -~x Lo, ax J ~ La~ ay J +~z[~t~z])+
p
kt
L\ax/ + \ O r / + \ O z / J
[~u ~1~ [~u ~wl~ [~o ~wl~ ~ ~
+ ~+axJ + ~ + a z J
+ ~+ayJJ
(6)
where C1 = 1.44, C 2 = 1.92, ok = 1.0, a, = 1-3, #en = #, + #1, #, = Copk2/e, CD = 0-09, and #t = 1.82 x 10 -5 kg -1 s -1.
W. K. Chow, N. K. Fong
268
The equation for the enthalpy h is
+-~x (pUhl +--~y(pvh) +-~z (pWh)=-~x e\o,
Ot /
+ --Oyt\olO [(~-2 +/~t]at/0hyh] + -~z0 [(/~_~+ la~)~zOh] + q "
(7)
4 " is a heat source expressed by the heat released rate per unit volume (W m -3) of burning fuel in regions containing the fire source. H o w e v e r , it becomes a sink term in regions containing water droplets and has to be computed from the rate of convective cooling. A description on the fire source z3,z5 has b e e n reported elsewhere and will not be r e p e a t e d here. W h e n the sprinkler head is actuated, a water spray will be discharged. In this model, the water envelope is taken to be a constant hollow cone (Wraight, H. G. & Morgan, H. P., 1986, pers. c o m m . ) with uniform distribution of water droplets. The shape is parabolic and the size depends on the physical features of the sprinkler head, water flow rate, pressure, etc. No attempt is m a d e to c o m p u t e the droplet trajectories and the water spray is, therefore, treated as an imposed water envelope. Hot air properties such as /~, v, K, Cp are calculated, using the following t e m p e r a t u r e d e p e n d e n t equations: y = y, + y2Tp + y3T~ + •4 T3
(8)
where Yi, i = 1 , . . . , 4 are empirical coefficients; y can be the values of either /~, v, K or Cp, and a list is shown in Table 1. The Reynolds n u m b e r Re of air can be calculated from the relative velocity Vre,
TABLE 1 Temperature Dependent Quantities a ~1
~2
~3
~4
1-7 × 10 5
5.34 x 10-8
-5.4 x 10 11
3-9 x 10 14
1-14 x 10 5
1-56 x 10 7
-1.9 x 10-1°
2-7 x 10 13
K (W/m °C)
2.37 x 10 z
9.22x 10 5
-1-05 x 10 7
8-4 x 10 11
Cp (J/kg °C)
1.006
-4.9 × 10 4
8-46 × 10-6
-3.5 × 10 s
/a (kg/m s) v (m2/s)
a (From: y + Yl + y2Tp+ y3T 2 + y4T 3)
Numerical simulation on the cooling of the fire-induced airflow
269
between the air and water droplet:
Vr~,d
Re = ~
(9)
V
where Re = Reynolds n u m b e r of air, Vro~= relative velocity between the hot air and water droplets (m/s), d = diameter of water droplet (m), and v = kinematic viscosity (m2/s). The laminar Prandtl n u m b e r al is taken to be 0.7 and the turbulent one at is c o m p u t e d from k and e. Since air dragging forces will be experienced by the water droplets, the hot air layer itself may be pulled down by water spray. A sprinkler-induced air flow is obtained and the situation becomes more complicated when there is also a fire source. This is important in studying the complex fire suppression process. The air drag components Dx, Dy, Dz experienced by the water droplets are given by Dx -
D,= Oz =
p(CD)A.u~ 2
(10)
p(CD)Ayv2
(11)
2
p(CD)Azw~ 2
(12)
where p = air density (kg/m3), Co = drag coefficient which varies with different Reynolds number, Ax, Ay, Az = frontal area of water droplet (m2), ur, Vr, W~= relative velocity c o m p o n e n t s between air and water droplets in x, y and z directions (m/s). The value of CD is given by 3° Re < 1 1 < Re < 1 ×
Co = 24 (Red) 103
1 X 103 < Re < 3 x 105
CD = 4 CD = 0.47
Re>3x105 CD=0"20 Note that the air drag terms Dx, Dy and Dz appeared only in the source term of the x-, y- and z - m o m e n t u m equations in regions containing water. This is the main objective in the particle-source-in-cell method. Both experimental 12,13 and theoretical works 27 reported that the evaporative heat loss is very small in comparison with the convective ones. The a m o u n t of heat extracted by the sprinkler water spray Qcoo~ can be computed by considering the convection term only. The convective heat is expressed in terms of the Nusselt N u m b e r Nu (e.g. Whitaker 3°'37) for a hot gas stream flowing towards a water sphere of
270
W. K. Chow, N. K. Fong
Hot. Air : T© Veloeit. X _ ~
Hg. 1. Water droplet in a hot air flow field.
diameter dw (shown in Fig. 1) by /
.
x 1/4
Nu = 2 + [0-4(Red) 1/2 + 0.06(Red)~3]pr24{ ~1 }-
(13)
\~w /
where N u = N u s s e l t number, #1 =viscosity of the free air stream (kg/ms), #w = viscosity of water (kg/ms), and Red = Reynolds number of water droplets. Pr = Prandtl Number. The heat transfer coefficient h (W m 2 °C) for each water droplet can then be computed by (14) where k is the thermal conductivity of air (W/m °C). The convective heat loss of air q due to one water droplet is
(15)
q = f t A w ( T w - T~)
where T~ is the air temperature, and Aw is the surface area of droplets (me). Therefore, the total heat extracted by the water spray can be obtained by summing all the terms q in the enthalpy equation for those regions containing water droplets.
g.
E
$
Fig. 2. Control volume.
Numerical simulation on the cooling of the fire-induced air flow
271
To solve the above set of equations, the enclosure concerned is divided into a number of grid points surrounded by control volumes. Equations (1)-(6) are integrated in each control volume, using a staggered grid system. 31 The following form of finite-difference equations is achieved: apt~p : aEriE -I- awq~w + aN~bN+ as~s + ax+~x+ + ax-tPx- + c (16) where ~ are the variables on u, v, w, P, T, k and e. The subscripts E, S, W, N, X+, X-- label the node points surrounding any arbitrary point P, as shown in Fig. 2. The pressure-implicit with splitting of operators (PISO) 32 algorithm is used for solving the set of implicit coupled, time-dependent, discretized fluid flow equations. This method is a noniterative one which dispenses
First predictor f o r velocity
J Solve for U~,v~,w~ (implicit)
I
First predictor for pressure
Use u",v",w~,p n Solve for p " (implicit)
First corrector for velocity
Use U~,V",w",p * J Solve for u~,vWW,www (expl c t) I
I
I
First predictor for J Solve for scalar scalar variables variables h~,k~,e*
vn -V~'~
I
Wn = W ~ pn = p . . .
(implicit) First corrector for pressure
Second c o r r e c t o r for velocity
I
J Solve for p " " (implicit)
I
I
Solve for U~ , V"~, W~ (explicit)
Yes next time
Scheme 1.
Two stage PISO scheme.
272
W. K. Chow, N. K. Fong TABLE
2
Algorithm for PISO Scheme First predictor step for velocity (implicit) Substitute the following eqns into continuity eqn First predictor step for pressure (implicit) First corrector step for velocity (explicit) First predictor step for scalar variables (implicit) First corrector step for pressure (implicit) Second corrector step for velocity (explicit)
a~u* = ~ anbU*b "t- Ae(p~ - p~) + b'~ a,,v*= E a,,bv,,b* + An(pp-p~)-t-b'~" * n n afw'~ = E anbW,,b + Af(pp --PF) + b~ u¢ = ft~ + d~(p~, - p *~) v * * = f2,, + d , , ( p p - P N )
w~'* = g', + d,(pp - p~) app ~ - a¢p~* + awPw* + a,,p~ + a~p~* + a~p~' + ahp~. + b" *
*
n
aeu** = E a,,t,u,,b + Ae(pp - p ~ ) + be a,,v** = E a,,bV,,b + A . ( p p --pN) + bT, a,w~* = ~ a,,bW*b + A t ( p p --p~) + b'~ a p ( I ) p* --- E anbtYP*b+ b ~
where ~ , : h, k, e b": source term appp* = aep** + awp** + a,p** + asp** + afp~* + abp~* + b ~ **
**
n
aeu***= Eanbu** + Ac(pp --PE ) + b e a,,v*** = E a,~bV,~b**+ A,,(pp* - p~*) + b~ afw?** = E a,,bW,,b** + A t ( p ; * - p ~ * ) + b,'~
" T h e converged solution at the previous time step. * Solutions at the present time step at the predictor level. ** Solutions at the first corrector level. * * * Solutions at the second corrector level. ^ Pseudovelcity.
the iteration process on the coupled sets of equations for different variables. It is also applicable to both compressible and incompressible flows. The convection and diffusion terms of the partial differential equations given by eqns (1)-(7) are discretized, using the power law scheme. 31 A single predictor step for the scalar variables is used in the two-stage PISO scheme as illustrated by the flow chart in Scheme 1. A more detailed set of equations is shown in Table 2. The independent under-relaxation method, with an under-relaxation factor of 0-3, is used for solving the equations. From the analysis performed by I s s a , 33 the temporal errors introduced by the splitting procedure were shown to vanish with the time increment At. If more corrector stages were introduced, the order of accuracy would be increased by one for each of the additional stages. Therefore, the temporal errors introduced by a two-stage PISO scheme
Numerical simulation on the cooling of the fire-induced airflow
273
is of second order in the time increment At, i.e. 0(ATE). Because the discretized equations contained discretization errors of first order, the PISO algorithm should not affect the accuracy dominated by the difference scheme. A smaller time step, i.e. At = 2 s, is used in this paper. No stability problems were encountered even though buoyancy appeared in the y-momentum equation. The converging criteria at every interior node are chosen as (q~, _ q~,-1)/q~, < 0-02
(17)
where the superscripts n - 1 and n denote the local values of the variable ~ at the previous and present time step, respectively. Concerning the boundary conditions, solid walls are assumed to be adiabatic and impermeable. The x, y and z components of velocity are taken to be zero at the wall. The conventional wall function method 34 is used to describe near-wall effects. The boundary conditions for the opening (i.e. free boundaries) is Ou
Ox
Ov
0;
ax
Ow
0;
ax
0
(18)
3 NUMERICAL EXPERIMENTS The model is applied to predict the resultant air flow and temperature field due to sprinkler water sprays and fire. To the best of our present knowledge, no 'field' measurement on fire sprinkler flow had been reported in the literature. Therefore, it is an illustration of the capability of the model rather than a validation of the results. However, some data on the temperature in a sprinklered building fire have been recorded. 35 This building is taken as an example and comparison with the experimental and predicted values is attempted. Numerical experiments are performed on a building with the geometrical configuration shown in Fig. 3. The enclosure is similar to the full-scale physical model used at the Fire Research Station. 35 Here, sprinkler heads are installed as shown in Fig. 3. A wood crib fire source (rate of heat released shown in Fig. 4), with size 2 m x 1 m x 0.9 m, is located 3 m from the left wall. Since the rate of heat release was not reported, it was estimated by fitting the temperature-time curve at the monitor point measured in the experiment 35 before actuating the sprinkler. The method involved inserting different thermal powers as the source term of the enthalpy equation (i.e. eqn (7)). The temperature at the monitored point was computed and compared with the
274
W, K. Chow, N. K. Fong
Sprin~k|e ~
(i)
(ii)
5.5m
(iii)
(iv) Fig. 3.
Geometrical configuration of the building.
~3m
Numerical simulation on the cooling of the fire-induced air flow
275
MW
2
/
"d 0
0
I
100
I
I
1
I
3O0
I
1
600
Time (see) Fig. 4.
Heat source release rate.
experimental data. By trial and error, the heat released rate given by Fig. 4 can be estimated. The r o o m is divided into 8721 grid points as shown and the developed computer package is known as BSESPRI. The computer core m e m o r y storage required is 3.4 megabytes in a Vax 8500. Three-dimensional simulation is performed and the total C P U time required is about 20 C P U h. No comparison has been made for coaser grids. However, divergent results are to be expected if a much larger grid size is used. The steady-state fire envrionment before actuating the sprinkler is shown in Fig. 5. A stratified hot air layer is formed because heat is continuously injected into the enclosure. Now, the effect of sprinkler water spray is studied. In the experiment, 35 a sprinkler head was actuated when the ambient temperature raised to 120 °C, i.e. 2 min 35 s after turning on the plume. The total water flow rate was 20 dm3/min. A second sprinkler head was then actuated at 2 min 45 s with the water flow rate kept at 20 dm3/min. The third sprinkler was operated 15 s later. The total water flow rate was 128 dm3/min and then further increased to 217 dm3/min at 3 min 15 s. The water flow rate was reduced to 159 dm3/min at 4 min and finally operated at 119 dm3/min. The above conditions are simulated closely in our numerical experiments. The sprinkler water droplets are assigned a mean diameter of 0.6 m m and are distributed uniformly inside the spray. The area of coverage for each water envelope is 14.6 m 2. The spray patterns are parabolic hollow cones with a radius of 2.156 m, as shown in Fig. 3. With those input parameters, the resultant 'sprinkler-plume' flow and temperature fields can be predicted by solving the equations numerically. Here, a first sprinkler head was actuated when the ambient
276
W. K. Chow, N. K. Fong
~
(c)
O)
a.
1
(ii)
(iii)
(iv) Fig. 5 Steady state results of the fire environment: (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
"MO!AUeld (!!!A) '.0 'q 'e saueld Joj sJ,Insoa pa)~!posd (!!^) './(~,.t,~lOA oq~, Su.~oqs saueld (!A) '.ao~vns sno]uoo X OBt' (A)
"plUO~--$ "$.~l
(:::^)
m ~
, , , o
(::^)
2-22:::::::
.......
,
~:~s'~"
(:^)
(^)
x
LLZ
0oi,
=
£
~,o~/~.w pa~npuz-aJ~/aye,to ~m.lOO~ ~!~ uo uo.,almul.s l~Z.aatunN
278
W. K. Chow, N. K. Fong
dm 3/min
300~Wa t e r flow r a t e for s i m u l a t i o n
~ 200 ID
Wa t e r flow r a t e for FRS e x p e r i m e n t
i00
o
0
5
10
Time (rain)
Fig. 6. Water flow rate curve.
temperature was raised to 120 °C, i.e. 155 s after turning on the plume. The second sprinkler head was actuated 10 s later with a water flow rate equal to 20 dm3/min. The third sprinkler was operated at 180 s. The total water flow rate was 128 dm3/min and then further increased to 217 dm3/min at 3 min 15 s. The water flow rate was then maintained at 217 dm3/min for the rest of the calculation. The water flow conditions are shown in Fig. 6. The horizontal velocity component of the water droplets is determined, using the distance travelled by the droplets. All the input conditions are very similar to the experimental ones. The flow and temperature fields predicted are shown in Figs 7-9. Figure 7 shows the air flow and temperature field when the first sprinkler is actuated. The results after actuating the second sprinkler are shown in Fig. 8. Figure 9 shows the results when all the sprinklers are turned on. These diagrams illustrate how the fire-induced flow and temperature field would be influenced by sprinkler water sprays. Obviously, the temperature inside the compartment has been reduced. Heat is not spreading outside when all the sprinklers are turned on. The water sprays are effective in confining the fire. However, smoke-logging will be found in positions close to the sprinkler, as indicated by the downward air movement in those regions. Natural venting should not be designed adjacent to a sprinkler head! The transient variation of the temperature predicted at the monitor position (in Fig. 3) is shown in Fig. 10, together with the experimentally determined values. Because only the cooling effect on the smoke layer is simulated, the temperature does not decay to the ambient values. This would only be predicted when fire suppression is considered.
Numerical simulation on the cooling of the fire-induced air flow
279
However, the water spray can stop the spreading of hot air out of the burning compartment. A steady-state flow pattern is achieved because the fire source is continuously injecting heat (see the curve shown in Fig. 4) to the enclosure. A temperature drop of about 100°C is predicted. An important factor, known as drag-to-buoyancy, i.e. D / B ratio, 4'5 which accounts for the stability of the smoke layer, can also be found. This is achieved by summing all the total downward air drag experienced by the water droplets and the bouyancy of air in every control volume containing water. The drag D and buoyancy B are given by 2
D = ~ CoAfVrelNop
(all cells containing water)
(19)
2 B = ~ 9-81(p - p0)Az
(all cells containing water)
(20)
where Co = drag coefficient, Af = frontal area of droplet, Vre~= relative velocity in y direction, No = number of droplets, p = density, P0 = initial density, and Air = control volume. The transient results are plotted in Fig. 11. It has been proposed that if D / B is greater than one, the stratified smoke layer will be disturbed. 'Smoke-logging' therefore occurs and so the D / B ratio would indicate the stability of the hot air layer. In our simulation, the values for D / B are greater than one after the first sprinkler is operated. Disturbance of the smoke layer occurs as illustrated also by the downward air flow patterns in Figs 7 and 9. Therefore, if natural venting is designed, smoke is unable to be extracted. Further, another macroscopic quantity, the total heat removed by the sprinkler Qcool can be calculated by summing all the cooling terms for the droplets (i.e. eqn (15)): Qcool = E hAw(T,~ - T~)
(all cells containing water)
(21)
w h e r e / i = convective heat transfer coefficient (W/m 2 °C), A,, = droplet surface area (m2), Tw= droplet temperature (K), T~--gas temperature (K). The transient curve of Qcoo~is shown in Fig. 12 and the rate of heat extraction is up to 600 kW. This is quite close to the value computed from the empirical equation of Morgan: 5 Qcoo~ = 440Ms 6 A TAs
(22)
where Ms is the sprinkler water flow rate (about 1.1 kg/s/m), 6 is the smoke layer thickness (about 0-5 m), AT is the increased temperature (about 200 °C), and As is the average area covered by the sprinkler.
280
W. K. Chow, N. K. Fong
(i) a.
1
(ii) b.
(iii)
C.
(iv) Fig. 7. Fire field predicted when first sprinkler head is actuated, i.e. t = 163 s. (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
Numerical simulation on the cooling of the fire-induced air flow T =
480
281
K
(v)
(vi)
(vii)
144
. . . . ;;'"~":ZZZZZZZZZZ~ZZZZZL2ZZZZZZZZ, I~.
E E E Z Z 2 2 2
[
;
[
[
(viii)
Fig. 7 contd.
(v) 4 8 0 K contour surface; (vi) planes showing the velocity; (vii) predicted results for planes a, b, c; (viii) plan view.
W. K. Chow, N. K. Fong
282
Icl
(i) a.
\ @
/
1
(ii) b.
\ 1
S (iii) C.
(iv) Fig. 8. Fire field predicted when second sprinkler head is actuated, i.e. t = 175 s. (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
Numerical simulation on the cooling of the ]ire-induced air flow T=
283
480K
(v)
(vi)
(vii)
(viii) Fig. 8--contd.
(v) 480K contour surface; (vi) planes showing the velocity; (vii) predicted results for planes a, b, c; (viii) plan view.
284
W. K. Chow, N. K. Fong
~__T~
la)
(i) a. t
/ 1
(ii)
b.
(iii) C°
~1-..)
i
(iv) Fig. 9. Fire field predicted when third sprinkler head is actuated, i.e. t = 217 s. (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
Numerical simulation on the cooling of the fire-induced air flow
285
T = 480 K
(v)
(vi)
(vii)
I ~ I Illlt"
l. , I' LbZ/ / , ', ' f~WL' ' b ' ~
-
~..~-.2..~2: :
L
(viii)
Fig. ~----contd. (v) 480K contour surface; (vi) planes showing the velocity; (vii) predicted results for planes a, b, c; (viii) plan view.
286
Fong
W. K. Chow, N. K. *C
o 2°° l "E
/%
I
~l o o ~ - ~ o~
\
/// J
w
Curve from
nu=~eal
~
///
~\
// 0
~ 0
'
~perlment
~.-._ , 1 5
'
Time
Fig. 10.
,
1
,_~_
l 10
(min)
Temperature-time ratio.
In comparison with others, our BSESPRI model predicts the fire environment in a building of size 2 7 . 8 m × 6 - 8 m x 3 m , being discretized into 8721 grid points. The CPU time required is 20 h in a Vax 8500. There is no numerical work reported for this special compartment. 34 However, comparison can be made with one of our models, known as BSEFM125 for fire simulation. There, a compartment 19 of size 9 m x 6 m x 3 m had been considered. This was divided into 5491 grid points and the numerical scheme used was the SIMPLER. 31 The total CPU time required there was about 24 h in a
m
Limit 2or
Lair
120
I
I
180
240
3O0
rime (=ec) Fig. 11.
D/B ratio plotted against time.
Numerical simulation on the cooling of the fire-induced air flow
287
800
t~ t~ 200 t~
120
180
I 240
300
Time (sec)
Fig. 12. Heat extraction plotted against time.
Convex C1 (i.e. about 72 h in a Vax 8500). These two simulations are very similar, but the BSESPRI is much faster than BSEFMI.
4 CONCLUSIONS The interaction of a fire-induced hot air layer with sprinkler water sprays, i.e. 'sprinkler-plume' interaction, is illustrated in a 'microscopic' manner. This is achieved by imposing water sprays into the fire-induced air flow. The air drag effect and convective heat transfer between the hot air and water droplets are included only in the source term of the momentum and enthalpy equations. The predicted resultant air flow and temperature field indicates how the hot air layer is affected by sprinkler. This information is very useful in designing sprinkler layout, selecting sprinkler heads water flow rate, justifying the smoke extraction systems, etc. Further, it is possible to see whether the water spray is capable of confining the fire. This model, BSESPRI, is very useful for fire engineers as it can be executed in a minicomputer. However, the model is not yet perfect. Since fire suppression of the fuel has not been simulated, the predicted temperature-time curve does not decrease as in the experimental one. 34 The evaporation effect of water droplets has not been included, and so a smaller prediction value on the heat extraction rate is expected. However, whether two-phase analysis '8 is necessary or not needs to be investigated as the evaporation heat has been shown to be much smaller than the convective one. 12,'3,27"28 Also, the shape of the spray pattern is imposed in this simulation without computing the droplet trajectories. In this way, the water penetration ratio ~4'~5cannot be determined. Putting all these into
288
w. K. Chow, N. K. Fong
a field model, however, is not easy, as illustrated by Boysan et al. 36 in studying fuel spray in a furnace. Investigation is in progress and further results will be reported later. Finally, this is just an illustration of the capability of a fire field model. No experimental work on measuring the fire field in a sprinklered building is reported. Validation by carrying out a full-scale burning test is therefore needed to criticize the model. Measurement on the heat release rate of the fire, air flow and temperature fields, smoke concentration, sprinkler water droplet sizes and velocities are urgently needed. ACKNOWLEDGEMENTS The project is supported by the Hong Kong Polytechnic Research Committee, and the authors wish to thank Drs H. S. Ward and A. M. Marsden for their encouraging support.
REFERENCES 1. Nash, P. & Young, R. A., Sprinklers in High-rise Buildings. Building Research Establishment Current Paper CP29/78, Borehamwood, UK, 1978. 2. Watson, B., Sprinkler-early suppression and fast response. Fire Prevention, 17 (1987) 174-82. 3. Rasbash, D. J., The extinction of fire with plain water--a review. Proc. of the First Int. Symp. on Fire Safety Science, 1985, pp. 1145-62. 4. Bullen, M. L., The Effect of a Sprinkler on the Stability of a Smoke Layer Beneath a Ceiling. Fire Research Note 1016, Fire Research Station, London, 1974. 5. Morgan, H. P., Heat transfer from a buoyant smoke layer beneath a ceiling to a sprinkler spray. Fire and Materials, 3 (1979) 27-33. 6. Heselden, A. J. M., The Interaction of Sprinkler and Roof Venting in Industrial Buildings: the Current Knowledge. Building Research Establishment, Borehamwood, UK 1984. 7. Hinkley, P. L., The effect of smoke/fire venting on the opening of sprinklers. Fire Safety J., U (1986) 211. 8. Hinkley, P. L., The effect of smoke venting on the operation of sprinklers subsequent to the first. Fire Safety J., 14 (1989) 221. 9. Chow, W. K., Sprinklers in atrium building. The Hong Kong Engineer, (Sept.) (1988) 41-3. 10. Chow, W. K., More on sprinklers in atrium buildings. The Hong Kong Engineer, (Feb.) (1989) 7-8. 11. Fire Offices' Committee, Rules for Automatic Sprinkler Installation, Installation Regulations, 1986.
Numerical simulation on the cooling of the fire-induced air flow
289
12. Kung, H. C., Cooling of room fires by sprinkler spray. A S M E Trans.--J. Heat Transfer, 99 (1977) 353-9. 13. You, H. Z., Kung, H. C. & Han, Z., Spray Cooling in Room Fires. NBS-GCR-86-515, National Bureau of Standards, Washington, DC, 1986. 14. Alpert, R. L., Calculated interactions of sprays with large-scale buoyant flows. A S M E J. Heat Transfer, 106 (1984) 310-17. 15. Alpert, R. L., Calculated Spray Water-droplet Flows in a Fire Environment. FMRC-RC86-BT-6, Factory Mutual Research Corp. Norwood, MA, USA, 1986. 16. Crowe, C. T., Sharma, M. P. & Stock, D. E., The particle source-in cell model for gas-droplet flows. A S M E J. Fluids Engng, 99 (1977) 325-32. 17. Chow, W. K. & Fong, N. K., Cooling effect of a fire-induced stratified hot air layer by sprinkler water spray. Fire J. (Aust. Fire Prot. Assoc.), (June) (1988) 20-7. 18. Hoffmann, N., Galea, E. R. & Markatos, N. C., Mathematical modelling of fire sprinkler systems. Appl. Math Modelling, 13 (1989) 298-306. 19. Markatos, N. C., Malin, M. R. & Cox, G., Mathematical modelling of buoyant-induced smoke flow in enclosures. Int. J. Heat Mass Transfer, 25 (1982) 63-75. 20. Yang, K. T., Lloyd, J. R., Kanury A. M. & Satoh, K., Modelling of turbulent buoyant flows in aircraft cabin. Combust. Sci. Technol., 39 (1984) 107-18. 21. Galea, E. R. & Markatos, N. C., A review of mathematical modelling of aircraft cabin fires. Appl. Math Modelling, 11 (1987) 162-76. 22. Cox, G. & Kumar, S., Field modelling of fire in forced ventilation enclosures. Combust. Sci. Technol., 52 (1987) 7-23. 23. Chow, W. K. & Leung, W. M., Application of field model to tunnel fire services design. B H R A 6th Int. Symp. on the Aerodynamics & Ventilation of Vehicle Tunnel, Durham, UK, Paper No. H1, Sept. 27-29 1988, pp. 495-510. 24. van de Leur, P. H. E., Kleijn, C. R. & Hoogendoorn, C. J., Numerical study of the stratified smoke flow in a corridor-fuU scale calculations. Fire Safety J., 14 (1989) 287. 25. Chow, W. K. & Leung, W. M., Fire-induced convective flow inside an enclosure before flashover: numerical experiments. Building Services Engng Res. Technol., 10 (1989) 51-9. 26. Markatos, N. C., Pericleous, K. A. & Cox, G., A novel approach to the field modelling of fires. Physiochem. Hydrodyn., 7 (1986) 125-43. 27. Beyler, C. L., The interaction of Fire and Sprinklers. NBS-GCR-78-121, National Bureau of Standards, Washington, DC, 1978. 28. Chow, W. K., On the evaporation effect of a sprinkler water spray. Fire Technol, Nov., (1989) 364-73. 29. Markatos, N. C., The mathematical modelling of turbulent flows. Appl. Math Modelling, 10 (1986) 190. 30. Holman, J. P., Heat Transfer. 6th edn, McGraw-Hill, New York, 1986, p. 294. 31. Patankar, S. V., Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York, 1980. 32. Issa, R. T., Gosman, A. D. & Watkins, A. P., The computation of
290
33. 34. 35. 36. 37.
W. K. Chow, N. K. Fong compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comp. Phys., 62 (1986) 66-82. Issa, R. I., Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Computational Phys. 62 (1985) 40-65. Launder, B. E. & Spalding, D. B., The numerical computation of turbulent flows. Computer Meth. in Appl. Mech. Eng., 3 (1974) 269. Rogers, S. P. & Young, R. A., The Performance of an Extra Light Hazard Sprinkler Installation. Fire Research Note 1065, Fire Research Station, London, 1977. Boysan, F., Ayres, W. H., Swithenbank, J. & Pan, Z., Three-dimensional model of spray combustion in gas turbine combustor. J. Energy, 6 (1982) 368. Whitaker, S., Forced convection heat-transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and flow in packed bids and tube bundles. AIChe J., 18 (1972) 361.
Numerical Simulation on Cooling of the Fire-induced Air Flow by Sprinkler Water Sprays W. K. C h o w & N. K. F o n g Department of BuildingServicesEngineering, Hong Kong Polytechnic,Hung Hom, Hong Kong (Received 29 May 1989; accepted 10 September 1990)
ABSTRACT By the air drag and cooling effect of water droplets, the cooling of the fire-induced hot air flow by sprinkler water sprays is studied by using a three-dimensional field modelling technique. This gives a 'microscopic' description of visualizing how the sprinkler interacts with a smoke layer. Macroscopic parameters, such as drag-to-buoyancy ratio, convective cooling rate, etc., can be computed accordingly. With such a model, it is possible for the fire-safety engineers to design proper location, type and performance of sprinkler systems. Since experimental data for validating the model are absent, it is concluded that measurement from full-scale burning tests in a sprinklered compartment is needed.
NOTATION Coefficient of the linear equation Buoyancy of smoke B Downward air drag D Air drag components (kg/ms 2) Ox, Oy, Oz Gravity (m/s 2) g Enthalpy of air (kJ/kg) h Turbulent kinetic energy (m2/s 3) k Nu Nusselt number of water drops Pressure of air (N/m 2) P 263 Fire Safety Journal 0379-7112/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland ai
264
W. K. Chow, N. K. Fong
U, /3, W
Air velocity components (m/s)
E
Turbulent energy dissipation Effective viscosity (kg/ms) Turbulent viscosity of air (kg/ms) Laminar viscosity of air (kg/ms) Density of air (kg/m 3) Variables (u, v, w, P, h, k or e)
~etI ~,~i
#l P
1 INTRODUCTION Past statistical records on fires illustrate that sprinklered buildings have a much lower risk of having uncontrolled fires. 1,2 This leads to a reduction of the insurance premium for buildings installed with automatic sprinkler systems. However, the fire extinguishing mechanism of water sprays is not yet clearly understood) Out of the many possible outcomes, interaction of a water spray with the smoke layer is an important issue for study. 4-8 This is because the fire-induced buoyant smoke layer might be cooled down significantly by the water spray. Therefore, it could lose buoyancy and fall to a lower level. The occurrence of 'smoke logging' reduces the efficiency of smoke extraction systems, causing hazards to building occupants. The situation becomes very significant for atrium buildings. 9,1° Secondly, it is interesting to know whether the sprinkler water spray obtained under certain operating conditions is able to confine a fire. With such knowledge, the minimum amount of water can be discharged from an appropriate sprinkler head in order not to 'overflood' the compartment and cause water damage. A good understanding of the interaction between the fire-induced flow and sprinkler water spray enables engineers to improve the present design for sprinkler layout, H by applying a reasonable amount of water, and giving guidance for modifying the physical features of the sprinkler head itself. The sprinkler system can then be designed to perform more efficiently in controlling a fire. Solving this problem is very difficult, however, because many physical phenomena such as direct cooling of the smoke, air entrainment into the spray and water evaporation ~2'~3 have all to be considered concurrently. To the best of our present knowledge, there is still no good theory which is capable of explaining how the sprinkler water spray interacts with a smoke layer realistically. Papers which have appeared in the literature include those on the stability analysis of the smoke layer, by investigating the drag-to-buoyancy ratio, by e.g. Bullen4 and Morgan, 5 sprinkler-induced air flow by Alpert, 14"1~study of the interac-
Numerical simulation on the cooling of the ]ire-induced airflow
265
tion between a ceiling jet and a sprinkler water spray, using the particle-source-in-cell method, 16 interaction of smoke venting by using a simple zone model with ceiling jet by Hinkley, 7'8 the simple heat extraction model by Chow and Fong, 17 and the more recent works with a two-phase field model by Hoffmann et al. TM None of these have satisfactorily indicated a clear picture of the interaction of a smoke layer with the sprinkler water spray inside an enclosure. Previously, the resultant air convective flow and temperature fields induced by a fire plume can be simulated successfully from the existing fire field 19-25 models with minicomputers (such as a V A X 8500 system). Alpert ~4"~5 and the current authors have investigated the effect of a sprinkler water curtain on the fire-induced flow field by using twodimensional simulation (Chow, W. K & Fong, N. K., 1988, unpublished). It is the objective of the present article to investigate how sprinkler water sprays will interact with a fire-induced thermal statified layer by using a similar technique, but extended to a three-dimensional picture. The predicted result is useful as the aerodynamics would indicate the smoke movement pattern. Other macroscopic parameters, such as convective heat extraction by the water spray and drag-tobuoyancy ratio of the hot air layer, 4'5 can all be computed under different sprinkler operating conditions. Section 2 gives the background theory; Section 3 describes the numerical experiments performed; Section 4 shows the results achieved; and the paper ends in Section 5 with conclusions. Although there are no satisfactory experimental data available for validating the results, the capability of applying the fire field model to solving this problem is illustrated.
2 THEORY The fire field generated by a thermal plume inside an enclosure can be simulated by field models. 19-25 The transient behaviour of air flow and temperature field can be predicted successfully by visualizing the plume as a heat source. Since the combustion process during a fire is very complicated, realistic treatment of combustion in a fire field model has not yet been achieved. 22 At the moment, fire can only be treated as a simple chemical reacting system. Extension by a two-fluid model 26 is possible and has been developed in studying the two-dimensional diffusion flame. Application of a laminar flamelet model (Moss, J. B. et al., 1989, pers. comm.) to fire simulation is in progress and results are to be investigated. In this paper, the smoke movement field model treats the flow as primarily a natural convection problem. However,
266
W. K. Chow, N. K. Fong
this is good enough to define the fire environment within a building at the preflashover stage, zs It is particularly useful for studying the interaction between fire-induced air flow and a water spray as the sprinkler head is expected to be actuated during the initial stage of a fire. The effects of thermal radiation have been ignored. The fire field model employed is the one developed by Chow and Leung, z3,z5 which is able to predict the transient three-dimensional air flow, temperature and pressure fields induced by a volumetric heat source. It is based on finite-difference calculations of a set of conservation equations for mass, m o m e n t u m and energy. The model is now applied in studying how a sprinkler water spray interacts with the fire induced hot air layer. The particle-source-in-cell m e t h o d 16 is used to describe the interaction between the water droplets and hot air. The flow is still visualized as 'single-phase' with those air drags and convective cooling terms appearing only in the source term in the set of m o m e n t u m and enthalpy equations. In this way, only ' m o m e n t u m ' and 'heat' coupling effects between hot air and water droplets are considered. This is reasonable, as evaporation heat lost for a water droplet is much smaller than the convective one (e.g. Beyler, z7 Kung, 12 and Chow28). The problem now becomes one of studying how an axisymmetric sprinkler flow would interact with a transient, horizontally flowing hot air stream. A three-dimensional simulation picture of the air flow is necessary, and the turbulent effect is described by a k - e model, z9 The set of equations becomes
-
ap+ a(ou)+ a(ov)+ a(ow) - -
at
ax
9y
az
0
(continuity equation)
(1)
where p is the air density, and u, v, w are the velocity components in a Cartesian coordinate system. The x - m o m e n t u m for the horizontal velocity c o m p o n e n t u is
a(pu) +
a
a--T- Tx
(puu) +
a
Yy ( o u r )
+
(puw)
a( au) a( a~) a( a.) a(a.)
= - ~ap + 7x
#°"-~x
+ 7y
~'°"
+ 7z
+O---z /~t-~-x
~'°"~
+ Tx #'--~x
ay ~U'oxx - D x
(2)
267
Numerical simulation on the cooling of the fire-induced air flow The y - m o m e n t u m for the vertical velocity c o m p o n e n t v is
a(pv) v
~
(OUr)+
at
_
a(po)
g(p
ay
s (pvv) + o (pvw)
~y
~z
p,a) +
_
+--~x
-~x
,ue. ~
"'
"eft
ay
'~-
"JV--~Z "t
as "t
"eli
--By
(3)
The z - m o m e n t u m for the velocity component w is
a(pw) ~
~
at
=-a~
o (pvw) + o (pww)
(ouw) +
-~y
7z
,9
~ "°"
ap + a (a_~xx) + ~a
(
aw + a
" e .aw ~
)
av
(ow)
+_a_a az ""~zzz - D =
(4)
Here, Dx, Dr, D= are the air-drag components experienced by the water droplets and appear only in regions containing water droplets. The equation for the turbulence kinetic energy k is
a,< a,< Ty a,< w ~a,<- -~,{a[,.~]+_~ r,.,~,
--+
u
p
+
v
+
--
kay/
\azlJ
Lay
r ao
axJ
aw]h
+ [~+~J
1 ap
.[- ~ +"'~p2 ay o)
The equation for the turbulence energy dissipation e is
a,~ --+u
at
ae cae ae 1{ a [ / . , a s ] a [.tone -] + +w~=-+ -~x v--~y az p -~x Lo, ax J ~ La~ ay J +~z[~t~z])+
p
kt
L\ax/ + \ O r / + \ O z / J
[~u ~1~ [~u ~wl~ [~o ~wl~ ~ ~
+ ~+axJ + ~ + a z J
+ ~+ayJJ
(6)
where C1 = 1.44, C 2 = 1.92, ok = 1.0, a, = 1-3, #en = #, + #1, #, = Copk2/e, CD = 0-09, and #t = 1.82 x 10 -5 kg -1 s -1.
W. K. Chow, N. K. Fong
268
The equation for the enthalpy h is
+-~x (pUhl +--~y(pvh) +-~z (pWh)=-~x e\o,
Ot /
+ --Oyt\olO [(~-2 +/~t]at/0hyh] + -~z0 [(/~_~+ la~)~zOh] + q "
(7)
4 " is a heat source expressed by the heat released rate per unit volume (W m -3) of burning fuel in regions containing the fire source. H o w e v e r , it becomes a sink term in regions containing water droplets and has to be computed from the rate of convective cooling. A description on the fire source z3,z5 has b e e n reported elsewhere and will not be r e p e a t e d here. W h e n the sprinkler head is actuated, a water spray will be discharged. In this model, the water envelope is taken to be a constant hollow cone (Wraight, H. G. & Morgan, H. P., 1986, pers. c o m m . ) with uniform distribution of water droplets. The shape is parabolic and the size depends on the physical features of the sprinkler head, water flow rate, pressure, etc. No attempt is m a d e to c o m p u t e the droplet trajectories and the water spray is, therefore, treated as an imposed water envelope. Hot air properties such as /~, v, K, Cp are calculated, using the following t e m p e r a t u r e d e p e n d e n t equations: y = y, + y2Tp + y3T~ + •4 T3
(8)
where Yi, i = 1 , . . . , 4 are empirical coefficients; y can be the values of either /~, v, K or Cp, and a list is shown in Table 1. The Reynolds n u m b e r Re of air can be calculated from the relative velocity Vre,
TABLE 1 Temperature Dependent Quantities a ~1
~2
~3
~4
1-7 × 10 5
5.34 x 10-8
-5.4 x 10 11
3-9 x 10 14
1-14 x 10 5
1-56 x 10 7
-1.9 x 10-1°
2-7 x 10 13
K (W/m °C)
2.37 x 10 z
9.22x 10 5
-1-05 x 10 7
8-4 x 10 11
Cp (J/kg °C)
1.006
-4.9 × 10 4
8-46 × 10-6
-3.5 × 10 s
/a (kg/m s) v (m2/s)
a (From: y + Yl + y2Tp+ y3T 2 + y4T 3)
Numerical simulation on the cooling of the fire-induced airflow
269
between the air and water droplet:
Vr~,d
Re = ~
(9)
V
where Re = Reynolds n u m b e r of air, Vro~= relative velocity between the hot air and water droplets (m/s), d = diameter of water droplet (m), and v = kinematic viscosity (m2/s). The laminar Prandtl n u m b e r al is taken to be 0.7 and the turbulent one at is c o m p u t e d from k and e. Since air dragging forces will be experienced by the water droplets, the hot air layer itself may be pulled down by water spray. A sprinkler-induced air flow is obtained and the situation becomes more complicated when there is also a fire source. This is important in studying the complex fire suppression process. The air drag components Dx, Dy, Dz experienced by the water droplets are given by Dx -
D,= Oz =
p(CD)A.u~ 2
(10)
p(CD)Ayv2
(11)
2
p(CD)Azw~ 2
(12)
where p = air density (kg/m3), Co = drag coefficient which varies with different Reynolds number, Ax, Ay, Az = frontal area of water droplet (m2), ur, Vr, W~= relative velocity c o m p o n e n t s between air and water droplets in x, y and z directions (m/s). The value of CD is given by 3° Re < 1 1 < Re < 1 ×
Co = 24 (Red) 103
1 X 103 < Re < 3 x 105
CD = 4 CD = 0.47
Re>3x105 CD=0"20 Note that the air drag terms Dx, Dy and Dz appeared only in the source term of the x-, y- and z - m o m e n t u m equations in regions containing water. This is the main objective in the particle-source-in-cell method. Both experimental 12,13 and theoretical works 27 reported that the evaporative heat loss is very small in comparison with the convective ones. The a m o u n t of heat extracted by the sprinkler water spray Qcoo~ can be computed by considering the convection term only. The convective heat is expressed in terms of the Nusselt N u m b e r Nu (e.g. Whitaker 3°'37) for a hot gas stream flowing towards a water sphere of
270
W. K. Chow, N. K. Fong
Hot. Air : T© Veloeit. X _ ~
Hg. 1. Water droplet in a hot air flow field.
diameter dw (shown in Fig. 1) by /
.
x 1/4
Nu = 2 + [0-4(Red) 1/2 + 0.06(Red)~3]pr24{ ~1 }-
(13)
\~w /
where N u = N u s s e l t number, #1 =viscosity of the free air stream (kg/ms), #w = viscosity of water (kg/ms), and Red = Reynolds number of water droplets. Pr = Prandtl Number. The heat transfer coefficient h (W m 2 °C) for each water droplet can then be computed by (14) where k is the thermal conductivity of air (W/m °C). The convective heat loss of air q due to one water droplet is
(15)
q = f t A w ( T w - T~)
where T~ is the air temperature, and Aw is the surface area of droplets (me). Therefore, the total heat extracted by the water spray can be obtained by summing all the terms q in the enthalpy equation for those regions containing water droplets.
g.
E
$
Fig. 2. Control volume.
Numerical simulation on the cooling of the fire-induced air flow
271
To solve the above set of equations, the enclosure concerned is divided into a number of grid points surrounded by control volumes. Equations (1)-(6) are integrated in each control volume, using a staggered grid system. 31 The following form of finite-difference equations is achieved: apt~p : aEriE -I- awq~w + aN~bN+ as~s + ax+~x+ + ax-tPx- + c (16) where ~ are the variables on u, v, w, P, T, k and e. The subscripts E, S, W, N, X+, X-- label the node points surrounding any arbitrary point P, as shown in Fig. 2. The pressure-implicit with splitting of operators (PISO) 32 algorithm is used for solving the set of implicit coupled, time-dependent, discretized fluid flow equations. This method is a noniterative one which dispenses
First predictor f o r velocity
J Solve for U~,v~,w~ (implicit)
I
First predictor for pressure
Use u",v",w~,p n Solve for p " (implicit)
First corrector for velocity
Use U~,V",w",p * J Solve for u~,vWW,www (expl c t) I
I
I
First predictor for J Solve for scalar scalar variables variables h~,k~,e*
vn -V~'~
I
Wn = W ~ pn = p . . .
(implicit) First corrector for pressure
Second c o r r e c t o r for velocity
I
J Solve for p " " (implicit)
I
I
Solve for U~ , V"~, W~ (explicit)
Yes next time
Scheme 1.
Two stage PISO scheme.
272
W. K. Chow, N. K. Fong TABLE
2
Algorithm for PISO Scheme First predictor step for velocity (implicit) Substitute the following eqns into continuity eqn First predictor step for pressure (implicit) First corrector step for velocity (explicit) First predictor step for scalar variables (implicit) First corrector step for pressure (implicit) Second corrector step for velocity (explicit)
a~u* = ~ anbU*b "t- Ae(p~ - p~) + b'~ a,,v*= E a,,bv,,b* + An(pp-p~)-t-b'~" * n n afw'~ = E anbW,,b + Af(pp --PF) + b~ u¢ = ft~ + d~(p~, - p *~) v * * = f2,, + d , , ( p p - P N )
w~'* = g', + d,(pp - p~) app ~ - a¢p~* + awPw* + a,,p~ + a~p~* + a~p~' + ahp~. + b" *
*
n
aeu** = E a,,t,u,,b + Ae(pp - p ~ ) + be a,,v** = E a,,bV,,b + A . ( p p --pN) + bT, a,w~* = ~ a,,bW*b + A t ( p p --p~) + b'~ a p ( I ) p* --- E anbtYP*b+ b ~
where ~ , : h, k, e b": source term appp* = aep** + awp** + a,p** + asp** + afp~* + abp~* + b ~ **
**
n
aeu***= Eanbu** + Ac(pp --PE ) + b e a,,v*** = E a,~bV,~b**+ A,,(pp* - p~*) + b~ afw?** = E a,,bW,,b** + A t ( p ; * - p ~ * ) + b,'~
" T h e converged solution at the previous time step. * Solutions at the present time step at the predictor level. ** Solutions at the first corrector level. * * * Solutions at the second corrector level. ^ Pseudovelcity.
the iteration process on the coupled sets of equations for different variables. It is also applicable to both compressible and incompressible flows. The convection and diffusion terms of the partial differential equations given by eqns (1)-(7) are discretized, using the power law scheme. 31 A single predictor step for the scalar variables is used in the two-stage PISO scheme as illustrated by the flow chart in Scheme 1. A more detailed set of equations is shown in Table 2. The independent under-relaxation method, with an under-relaxation factor of 0-3, is used for solving the equations. From the analysis performed by I s s a , 33 the temporal errors introduced by the splitting procedure were shown to vanish with the time increment At. If more corrector stages were introduced, the order of accuracy would be increased by one for each of the additional stages. Therefore, the temporal errors introduced by a two-stage PISO scheme
Numerical simulation on the cooling of the fire-induced airflow
273
is of second order in the time increment At, i.e. 0(ATE). Because the discretized equations contained discretization errors of first order, the PISO algorithm should not affect the accuracy dominated by the difference scheme. A smaller time step, i.e. At = 2 s, is used in this paper. No stability problems were encountered even though buoyancy appeared in the y-momentum equation. The converging criteria at every interior node are chosen as (q~, _ q~,-1)/q~, < 0-02
(17)
where the superscripts n - 1 and n denote the local values of the variable ~ at the previous and present time step, respectively. Concerning the boundary conditions, solid walls are assumed to be adiabatic and impermeable. The x, y and z components of velocity are taken to be zero at the wall. The conventional wall function method 34 is used to describe near-wall effects. The boundary conditions for the opening (i.e. free boundaries) is Ou
Ox
Ov
0;
ax
Ow
0;
ax
0
(18)
3 NUMERICAL EXPERIMENTS The model is applied to predict the resultant air flow and temperature field due to sprinkler water sprays and fire. To the best of our present knowledge, no 'field' measurement on fire sprinkler flow had been reported in the literature. Therefore, it is an illustration of the capability of the model rather than a validation of the results. However, some data on the temperature in a sprinklered building fire have been recorded. 35 This building is taken as an example and comparison with the experimental and predicted values is attempted. Numerical experiments are performed on a building with the geometrical configuration shown in Fig. 3. The enclosure is similar to the full-scale physical model used at the Fire Research Station. 35 Here, sprinkler heads are installed as shown in Fig. 3. A wood crib fire source (rate of heat released shown in Fig. 4), with size 2 m x 1 m x 0.9 m, is located 3 m from the left wall. Since the rate of heat release was not reported, it was estimated by fitting the temperature-time curve at the monitor point measured in the experiment 35 before actuating the sprinkler. The method involved inserting different thermal powers as the source term of the enthalpy equation (i.e. eqn (7)). The temperature at the monitored point was computed and compared with the
274
W, K. Chow, N. K. Fong
Sprin~k|e ~
(i)
(ii)
5.5m
(iii)
(iv) Fig. 3.
Geometrical configuration of the building.
~3m
Numerical simulation on the cooling of the fire-induced air flow
275
MW
2
/
"d 0
0
I
100
I
I
1
I
3O0
I
1
600
Time (see) Fig. 4.
Heat source release rate.
experimental data. By trial and error, the heat released rate given by Fig. 4 can be estimated. The r o o m is divided into 8721 grid points as shown and the developed computer package is known as BSESPRI. The computer core m e m o r y storage required is 3.4 megabytes in a Vax 8500. Three-dimensional simulation is performed and the total C P U time required is about 20 C P U h. No comparison has been made for coaser grids. However, divergent results are to be expected if a much larger grid size is used. The steady-state fire envrionment before actuating the sprinkler is shown in Fig. 5. A stratified hot air layer is formed because heat is continuously injected into the enclosure. Now, the effect of sprinkler water spray is studied. In the experiment, 35 a sprinkler head was actuated when the ambient temperature raised to 120 °C, i.e. 2 min 35 s after turning on the plume. The total water flow rate was 20 dm3/min. A second sprinkler head was then actuated at 2 min 45 s with the water flow rate kept at 20 dm3/min. The third sprinkler was operated 15 s later. The total water flow rate was 128 dm3/min and then further increased to 217 dm3/min at 3 min 15 s. The water flow rate was reduced to 159 dm3/min at 4 min and finally operated at 119 dm3/min. The above conditions are simulated closely in our numerical experiments. The sprinkler water droplets are assigned a mean diameter of 0.6 m m and are distributed uniformly inside the spray. The area of coverage for each water envelope is 14.6 m 2. The spray patterns are parabolic hollow cones with a radius of 2.156 m, as shown in Fig. 3. With those input parameters, the resultant 'sprinkler-plume' flow and temperature fields can be predicted by solving the equations numerically. Here, a first sprinkler head was actuated when the ambient
276
W. K. Chow, N. K. Fong
~
(c)
O)
a.
1
(ii)
(iii)
(iv) Fig. 5 Steady state results of the fire environment: (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
"MO!AUeld (!!!A) '.0 'q 'e saueld Joj sJ,Insoa pa)~!posd (!!^) './(~,.t,~lOA oq~, Su.~oqs saueld (!A) '.ao~vns sno]uoo X OBt' (A)
"plUO~--$ "$.~l
(:::^)
m ~
, , , o
(::^)
2-22:::::::
.......
,
~:~s'~"
(:^)
(^)
x
LLZ
0oi,
=
£
~,o~/~.w pa~npuz-aJ~/aye,to ~m.lOO~ ~!~ uo uo.,almul.s l~Z.aatunN
278
W. K. Chow, N. K. Fong
dm 3/min
300~Wa t e r flow r a t e for s i m u l a t i o n
~ 200 ID
Wa t e r flow r a t e for FRS e x p e r i m e n t
i00
o
0
5
10
Time (rain)
Fig. 6. Water flow rate curve.
temperature was raised to 120 °C, i.e. 155 s after turning on the plume. The second sprinkler head was actuated 10 s later with a water flow rate equal to 20 dm3/min. The third sprinkler was operated at 180 s. The total water flow rate was 128 dm3/min and then further increased to 217 dm3/min at 3 min 15 s. The water flow rate was then maintained at 217 dm3/min for the rest of the calculation. The water flow conditions are shown in Fig. 6. The horizontal velocity component of the water droplets is determined, using the distance travelled by the droplets. All the input conditions are very similar to the experimental ones. The flow and temperature fields predicted are shown in Figs 7-9. Figure 7 shows the air flow and temperature field when the first sprinkler is actuated. The results after actuating the second sprinkler are shown in Fig. 8. Figure 9 shows the results when all the sprinklers are turned on. These diagrams illustrate how the fire-induced flow and temperature field would be influenced by sprinkler water sprays. Obviously, the temperature inside the compartment has been reduced. Heat is not spreading outside when all the sprinklers are turned on. The water sprays are effective in confining the fire. However, smoke-logging will be found in positions close to the sprinkler, as indicated by the downward air movement in those regions. Natural venting should not be designed adjacent to a sprinkler head! The transient variation of the temperature predicted at the monitor position (in Fig. 3) is shown in Fig. 10, together with the experimentally determined values. Because only the cooling effect on the smoke layer is simulated, the temperature does not decay to the ambient values. This would only be predicted when fire suppression is considered.
Numerical simulation on the cooling of the fire-induced air flow
279
However, the water spray can stop the spreading of hot air out of the burning compartment. A steady-state flow pattern is achieved because the fire source is continuously injecting heat (see the curve shown in Fig. 4) to the enclosure. A temperature drop of about 100°C is predicted. An important factor, known as drag-to-buoyancy, i.e. D / B ratio, 4'5 which accounts for the stability of the smoke layer, can also be found. This is achieved by summing all the total downward air drag experienced by the water droplets and the bouyancy of air in every control volume containing water. The drag D and buoyancy B are given by 2
D = ~ CoAfVrelNop
(all cells containing water)
(19)
2 B = ~ 9-81(p - p0)Az
(all cells containing water)
(20)
where Co = drag coefficient, Af = frontal area of droplet, Vre~= relative velocity in y direction, No = number of droplets, p = density, P0 = initial density, and Air = control volume. The transient results are plotted in Fig. 11. It has been proposed that if D / B is greater than one, the stratified smoke layer will be disturbed. 'Smoke-logging' therefore occurs and so the D / B ratio would indicate the stability of the hot air layer. In our simulation, the values for D / B are greater than one after the first sprinkler is operated. Disturbance of the smoke layer occurs as illustrated also by the downward air flow patterns in Figs 7 and 9. Therefore, if natural venting is designed, smoke is unable to be extracted. Further, another macroscopic quantity, the total heat removed by the sprinkler Qcool can be calculated by summing all the cooling terms for the droplets (i.e. eqn (15)): Qcool = E hAw(T,~ - T~)
(all cells containing water)
(21)
w h e r e / i = convective heat transfer coefficient (W/m 2 °C), A,, = droplet surface area (m2), Tw= droplet temperature (K), T~--gas temperature (K). The transient curve of Qcoo~is shown in Fig. 12 and the rate of heat extraction is up to 600 kW. This is quite close to the value computed from the empirical equation of Morgan: 5 Qcoo~ = 440Ms 6 A TAs
(22)
where Ms is the sprinkler water flow rate (about 1.1 kg/s/m), 6 is the smoke layer thickness (about 0-5 m), AT is the increased temperature (about 200 °C), and As is the average area covered by the sprinkler.
280
W. K. Chow, N. K. Fong
(i) a.
1
(ii) b.
(iii)
C.
(iv) Fig. 7. Fire field predicted when first sprinkler head is actuated, i.e. t = 163 s. (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
Numerical simulation on the cooling of the fire-induced air flow T =
480
281
K
(v)
(vi)
(vii)
144
. . . . ;;'"~":ZZZZZZZZZZ~ZZZZZL2ZZZZZZZZ, I~.
E E E Z Z 2 2 2
[
;
[
[
(viii)
Fig. 7 contd.
(v) 4 8 0 K contour surface; (vi) planes showing the velocity; (vii) predicted results for planes a, b, c; (viii) plan view.
W. K. Chow, N. K. Fong
282
Icl
(i) a.
\ @
/
1
(ii) b.
\ 1
S (iii) C.
(iv) Fig. 8. Fire field predicted when second sprinkler head is actuated, i.e. t = 175 s. (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
Numerical simulation on the cooling of the ]ire-induced air flow T=
283
480K
(v)
(vi)
(vii)
(viii) Fig. 8--contd.
(v) 480K contour surface; (vi) planes showing the velocity; (vii) predicted results for planes a, b, c; (viii) plan view.
284
W. K. Chow, N. K. Fong
~__T~
la)
(i) a. t
/ 1
(ii)
b.
(iii) C°
~1-..)
i
(iv) Fig. 9. Fire field predicted when third sprinkler head is actuated, i.e. t = 217 s. (i) planes showing the temperature; (ii) temperature contour at plane a; (iii) temperature contour at plane b; (iv) temperature contour at plane c.
Numerical simulation on the cooling of the fire-induced air flow
285
T = 480 K
(v)
(vi)
(vii)
I ~ I Illlt"
l. , I' LbZ/ / , ', ' f~WL' ' b ' ~
-
~..~-.2..~2: :
L
(viii)
Fig. ~----contd. (v) 480K contour surface; (vi) planes showing the velocity; (vii) predicted results for planes a, b, c; (viii) plan view.
286
Fong
W. K. Chow, N. K. *C
o 2°° l "E
/%
I
~l o o ~ - ~ o~
\
/// J
w
Curve from
nu=~eal
~
///
~\
// 0
~ 0
'
~perlment
~.-._ , 1 5
'
Time
Fig. 10.
,
1
,_~_
l 10
(min)
Temperature-time ratio.
In comparison with others, our BSESPRI model predicts the fire environment in a building of size 2 7 . 8 m × 6 - 8 m x 3 m , being discretized into 8721 grid points. The CPU time required is 20 h in a Vax 8500. There is no numerical work reported for this special compartment. 34 However, comparison can be made with one of our models, known as BSEFM125 for fire simulation. There, a compartment 19 of size 9 m x 6 m x 3 m had been considered. This was divided into 5491 grid points and the numerical scheme used was the SIMPLER. 31 The total CPU time required there was about 24 h in a
m
Limit 2or
Lair
120
I
I
180
240
3O0
rime (=ec) Fig. 11.
D/B ratio plotted against time.
Numerical simulation on the cooling of the fire-induced air flow
287
800
t~ t~ 200 t~
120
180
I 240
300
Time (sec)
Fig. 12. Heat extraction plotted against time.
Convex C1 (i.e. about 72 h in a Vax 8500). These two simulations are very similar, but the BSESPRI is much faster than BSEFMI.
4 CONCLUSIONS The interaction of a fire-induced hot air layer with sprinkler water sprays, i.e. 'sprinkler-plume' interaction, is illustrated in a 'microscopic' manner. This is achieved by imposing water sprays into the fire-induced air flow. The air drag effect and convective heat transfer between the hot air and water droplets are included only in the source term of the momentum and enthalpy equations. The predicted resultant air flow and temperature field indicates how the hot air layer is affected by sprinkler. This information is very useful in designing sprinkler layout, selecting sprinkler heads water flow rate, justifying the smoke extraction systems, etc. Further, it is possible to see whether the water spray is capable of confining the fire. This model, BSESPRI, is very useful for fire engineers as it can be executed in a minicomputer. However, the model is not yet perfect. Since fire suppression of the fuel has not been simulated, the predicted temperature-time curve does not decrease as in the experimental one. 34 The evaporation effect of water droplets has not been included, and so a smaller prediction value on the heat extraction rate is expected. However, whether two-phase analysis '8 is necessary or not needs to be investigated as the evaporation heat has been shown to be much smaller than the convective one. 12,'3,27"28 Also, the shape of the spray pattern is imposed in this simulation without computing the droplet trajectories. In this way, the water penetration ratio ~4'~5cannot be determined. Putting all these into
288
w. K. Chow, N. K. Fong
a field model, however, is not easy, as illustrated by Boysan et al. 36 in studying fuel spray in a furnace. Investigation is in progress and further results will be reported later. Finally, this is just an illustration of the capability of a fire field model. No experimental work on measuring the fire field in a sprinklered building is reported. Validation by carrying out a full-scale burning test is therefore needed to criticize the model. Measurement on the heat release rate of the fire, air flow and temperature fields, smoke concentration, sprinkler water droplet sizes and velocities are urgently needed. ACKNOWLEDGEMENTS The project is supported by the Hong Kong Polytechnic Research Committee, and the authors wish to thank Drs H. S. Ward and A. M. Marsden for their encouraging support.
REFERENCES 1. Nash, P. & Young, R. A., Sprinklers in High-rise Buildings. Building Research Establishment Current Paper CP29/78, Borehamwood, UK, 1978. 2. Watson, B., Sprinkler-early suppression and fast response. Fire Prevention, 17 (1987) 174-82. 3. Rasbash, D. J., The extinction of fire with plain water--a review. Proc. of the First Int. Symp. on Fire Safety Science, 1985, pp. 1145-62. 4. Bullen, M. L., The Effect of a Sprinkler on the Stability of a Smoke Layer Beneath a Ceiling. Fire Research Note 1016, Fire Research Station, London, 1974. 5. Morgan, H. P., Heat transfer from a buoyant smoke layer beneath a ceiling to a sprinkler spray. Fire and Materials, 3 (1979) 27-33. 6. Heselden, A. J. M., The Interaction of Sprinkler and Roof Venting in Industrial Buildings: the Current Knowledge. Building Research Establishment, Borehamwood, UK 1984. 7. Hinkley, P. L., The effect of smoke/fire venting on the opening of sprinklers. Fire Safety J., U (1986) 211. 8. Hinkley, P. L., The effect of smoke venting on the operation of sprinklers subsequent to the first. Fire Safety J., 14 (1989) 221. 9. Chow, W. K., Sprinklers in atrium building. The Hong Kong Engineer, (Sept.) (1988) 41-3. 10. Chow, W. K., More on sprinklers in atrium buildings. The Hong Kong Engineer, (Feb.) (1989) 7-8. 11. Fire Offices' Committee, Rules for Automatic Sprinkler Installation, Installation Regulations, 1986.
Numerical simulation on the cooling of the fire-induced air flow
289
12. Kung, H. C., Cooling of room fires by sprinkler spray. A S M E Trans.--J. Heat Transfer, 99 (1977) 353-9. 13. You, H. Z., Kung, H. C. & Han, Z., Spray Cooling in Room Fires. NBS-GCR-86-515, National Bureau of Standards, Washington, DC, 1986. 14. Alpert, R. L., Calculated interactions of sprays with large-scale buoyant flows. A S M E J. Heat Transfer, 106 (1984) 310-17. 15. Alpert, R. L., Calculated Spray Water-droplet Flows in a Fire Environment. FMRC-RC86-BT-6, Factory Mutual Research Corp. Norwood, MA, USA, 1986. 16. Crowe, C. T., Sharma, M. P. & Stock, D. E., The particle source-in cell model for gas-droplet flows. A S M E J. Fluids Engng, 99 (1977) 325-32. 17. Chow, W. K. & Fong, N. K., Cooling effect of a fire-induced stratified hot air layer by sprinkler water spray. Fire J. (Aust. Fire Prot. Assoc.), (June) (1988) 20-7. 18. Hoffmann, N., Galea, E. R. & Markatos, N. C., Mathematical modelling of fire sprinkler systems. Appl. Math Modelling, 13 (1989) 298-306. 19. Markatos, N. C., Malin, M. R. & Cox, G., Mathematical modelling of buoyant-induced smoke flow in enclosures. Int. J. Heat Mass Transfer, 25 (1982) 63-75. 20. Yang, K. T., Lloyd, J. R., Kanury A. M. & Satoh, K., Modelling of turbulent buoyant flows in aircraft cabin. Combust. Sci. Technol., 39 (1984) 107-18. 21. Galea, E. R. & Markatos, N. C., A review of mathematical modelling of aircraft cabin fires. Appl. Math Modelling, 11 (1987) 162-76. 22. Cox, G. & Kumar, S., Field modelling of fire in forced ventilation enclosures. Combust. Sci. Technol., 52 (1987) 7-23. 23. Chow, W. K. & Leung, W. M., Application of field model to tunnel fire services design. B H R A 6th Int. Symp. on the Aerodynamics & Ventilation of Vehicle Tunnel, Durham, UK, Paper No. H1, Sept. 27-29 1988, pp. 495-510. 24. van de Leur, P. H. E., Kleijn, C. R. & Hoogendoorn, C. J., Numerical study of the stratified smoke flow in a corridor-fuU scale calculations. Fire Safety J., 14 (1989) 287. 25. Chow, W. K. & Leung, W. M., Fire-induced convective flow inside an enclosure before flashover: numerical experiments. Building Services Engng Res. Technol., 10 (1989) 51-9. 26. Markatos, N. C., Pericleous, K. A. & Cox, G., A novel approach to the field modelling of fires. Physiochem. Hydrodyn., 7 (1986) 125-43. 27. Beyler, C. L., The interaction of Fire and Sprinklers. NBS-GCR-78-121, National Bureau of Standards, Washington, DC, 1978. 28. Chow, W. K., On the evaporation effect of a sprinkler water spray. Fire Technol, Nov., (1989) 364-73. 29. Markatos, N. C., The mathematical modelling of turbulent flows. Appl. Math Modelling, 10 (1986) 190. 30. Holman, J. P., Heat Transfer. 6th edn, McGraw-Hill, New York, 1986, p. 294. 31. Patankar, S. V., Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York, 1980. 32. Issa, R. T., Gosman, A. D. & Watkins, A. P., The computation of
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33. 34. 35. 36. 37.
W. K. Chow, N. K. Fong compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comp. Phys., 62 (1986) 66-82. Issa, R. I., Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Computational Phys. 62 (1985) 40-65. Launder, B. E. & Spalding, D. B., The numerical computation of turbulent flows. Computer Meth. in Appl. Mech. Eng., 3 (1974) 269. Rogers, S. P. & Young, R. A., The Performance of an Extra Light Hazard Sprinkler Installation. Fire Research Note 1065, Fire Research Station, London, 1977. Boysan, F., Ayres, W. H., Swithenbank, J. & Pan, Z., Three-dimensional model of spray combustion in gas turbine combustor. J. Energy, 6 (1982) 368. Whitaker, S., Forced convection heat-transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and flow in packed bids and tube bundles. AIChe J., 18 (1972) 361.