Modeling the trajectory of window flames with regard to flow attachment to the adjacent wall

ARTICLE IN PRESS Fire Safety Journal 44 (2009) 250–258

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Modeling the trajectory of window flames with regard to flow attachment to the adjacent wall Keisuke Himoto a,, Tsuneto Tsuchihashi b, Yoshiaki Tanaka b, Takeyoshi Tanaka c a b c

Pioneering Research Unit for Next Generation, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan General Building Research Corporation of Japan, Fujishirodai 5-8-1, Suita, Osaka 565-0873, Japan Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan

a r t i c l e in f o

a b s t r a c t

Article history: Received 2 October 2006 Received in revised form 22 June 2008 Accepted 30 June 2008 Available online 9 August 2008

A model for predicting the trajectory of window flame ejected from a fire compartment was formulated incorporating the effect of wall above the opening. Based on the observation in the reduced scale experiments, window flames were divided into the following categories with regard to its trajectory configuration: the flow which ascends almost vertically up after ejection maintaining a certain separation from the wall; and the flow which ascends upward after ejection and gradually approaches to the wall in the downstream. In the model, trajectories of these flows were approximated by cubic polynomials whose coefficients were given as functions of a dimensionless parameter F*. The parameter F* was derived from the conservation equation of momentum which incorporates the effect of pressure gradient across the ascending flows. Critical condition for the occurrence of flow attachment was described as a proportion of the maximum separation from the wall versus the opening width. Trajectories predicted by the proposed model were then compared with the measurement data which indicated reasonable agreements. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Fire spread Window flame Trajectory Reduced-scale experiment Scale model

1. Introduction When the fire inside a compartment enlarges and reaches its fully developed stage, flame will be ejected from the opening after the failure of window glass. The heat flux transferred from the window flame causes ignition of combustibles stored inside the upper floors as well as those in the adjacent buildings. As it is one of the most important contributing factors for fire spread between rooms, window flame has received considerable attention and there are already several models available. In such models, the greatest effort has been devoted to the prediction of temperature rise, as it is closely related to the hazard evaluation of window flame [1–11]. The derived knowledge is widely applied to the practice of building design in which it provides quantitative measures for fire safety evaluation. However, most of the existing models predict temperature rise along trajectory, i.e., the curve sequentially connecting points of the largest temperature rise at each height. In other words, the hazard of a window flame cannot be quantified without knowing its trajectory configuration, as the rate of heat transfer from the window flame to adjacent combustible needs to be evaluated in terms of the separation between them.  Corresponding author. Tel.: +81774 38 4481; fax: +81774 38 4039.

E-mail address: [email protected] (K. Himoto). 0379-7112/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2008.06.007

In contrast to the substantial progress in the temperature rise prediction methodology, there is not much work done on the measurement of predicting trajectory configuration. One of the few examples is the model proposed by Yokoi [1] which is given by (   3=2 )2 ,  z 1 x þ x0 3=2 x0 x0 ¼  (1) H  Z N 9bT 1 H  ZN H  ZN H  ZN where x is the separation from the wall, z is the vertical height, x0 is the depth of the virtual origin, H is the height of the opening, ZN is the height of the neutral plane, and b is the thermal expansion coefficient. The model describes the trajectory configuration with the cubic function: the flow is initially ejected in the horizontal direction and gains vertical momentum due to buoyancy in the downstream. It shows reasonable agreement with the experimental data when no interaction is involved between the window flame and the wall above the opening [1]. However, this is not a common assumption for buildings in general. One of the typical consequences of the interaction is the attachment of the flow to the wall above the opening. This is often observed when the width of the opening is large. Under these conditions, the pressure in the space between the wall and the window flame decreases as the flame entrains air from this space, while the supply of air (to this space) is restricted due to the presence of the wall [1,11]. This yields the pressure gradient across the flow and, as a result, draws

ARTICLE IN PRESS K. Himoto et al. / Fire Safety Journal 44 (2009) 250–258

pN pN,W Dp Q_  Q0 r0 TN um u0 UE

Nomenclature A b B cp Dx Dy F* g H HA HE LE :

m0 pm

area of the opening (m2) half width of the flow velocity (m) horizontal width of the opening (m) heat capacity of the flow (kJ/(kgK)) length of the by-wall region in the x-direction (m) length of the by-wall region in the y-direction (m) dimensionless parameter defined in Eq. (28) () acceleration due to gravity (m/s2) height of the opening (m) height of the attachment point (m) height of the equilibrium point (m) separation between the attachment point and the wall (m) air supply rate (kg/s) pressure of the flame region (Pa)

251

pressure of the ambient region (Pa) pressure of the wall region (Pa) pressure gradient across the current (pNpN,W) (Pa) apparent heat release rate of the window flame (kW) dimensionless heat release rate (–) equivalent radius of the opening (m) ambient gas temperature (K) flow velocity along trajectory (m/s) maximum flow velocity at the opening (m/s) velocity of flow from the ambient region to the flame region (m/s) velocity of flow from the wall region to the flame region (m/s) velocity of flow from the ambient region to the wall region (m/s) height of the neutral plane (m) ambient gas density (kg/m3)

UE,W VE,W ZN

rN

space which may be considered as the ambient environment (ambient region). Moreover, the entrainment of ambient air into the window flame is assumed to be induced by the pressure difference between the flame region and the other regions. As the flame region entrains air from the other regions, the pressure inside the flame region pm should be the minimum among the three regions. However, as the entrainment of air from the by-wall region tends to be restricted due to the lack of space, while it is not from the ambient region, the following relation generally applies:

the window flame to the wall [1]. In order to characterize such behavior, a trajectory model which incorporates the effect of the wall–flame interaction was derived and verified with the data obtained in the previous experiments [11].

2. Trajectory of window flame 2.1. Difference in pressure between wall-side and open-side of the trajectory

pm op1;W pp1

The first parameter which needs to be modeled is the pressure difference Dp, which causes the attachment of the window flame to the wall. Fig. 1 shows the horizontal section of window flame in relation to the adjacent wall. For the convenience of description, a coordinate system is introduced: the origin is at the point of maximum temperature rise in the plane of the window; the x-axis is perpendicular to the window plane; the z-axis is vertically upward; and the y-axis perpendicular to the other axes. We now assume a thin layer element in the z-direction which is divided into three regions of different characteristics: (1) the buoyancy-driven region that constitutes the main body of the window flame (flame region); (2) the region between the flame region and the wall (the ‘‘by-wall region’’); and (3) the rest of the

(2)

where pN,W and pN are the characteristic pressure of the by-wall region and the ambient region, respectively. The flow attachment is caused by the pressure difference between the by-wall region and the ambient region Dp, which is given by

Dp ¼ p1  p1;W

(3)

When the ejected flow is incompressible and ascends almost vertically upward, mass transfer between the three regions is evaluated by Bernoulli’s theory. The pressure difference between the regions is equivalent to the flow momentum across the

Dy

By-wall region y

z 

p∞,W VE,W

x

VE,W

Dx

UE,W

um

Δp

pm

Ambient region x

UE p∞

Fig. 1. Pressure difference inducing flow around the window flame.

Flame region

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K. Himoto et al. / Fire Safety Journal 44 (2009) 250–258

boundary, i.e., p1  pm  p1  p1;W

1 1 r U 2 ; p1;W  pm  r1 U 2E;W , 2 1 E 2 1  r1 V 2E;W , 2

(4)

where UE is the characteristic flow velocity from the ambient region to the flame region, UE,W is that from the by-wall region to the flame region, VE,W is that from the ambient region to the bywall region, and rN is the gas density in the ambient region and the by-wall region. Eliminating the pressure terms from Eq. (4), we obtain U 2E  U 2E;W þ V 2E;W

(5)

Assuming that the effect of flow–wall interaction on the entraining flow velocity UE is minor, and the entrainment rate is proportional to the trajectory velocity um, which is a common assumption for the classical models of vertically ascending fire plumes [12], we obtain U E ¼ aum

(6)

Approximating the shapes of the flame region (abcd) and that of the by-wall region (a0 b0 c0 d0 ) by rectangles, and neglecting overall mass transfer in z-direction for the by-wall region, then the mass conservation equation for the by-wall region can be expressed as Dy U E;W ¼ 2Dx V E;W

aum U E;W  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ ðDy =2Dx Þ2

aum V E;W  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð2Dx =Dy Þ2

(8)

This equation shows that the magnitudes of the boundary flow velocities UE,W and VE,W are dependent on the aspect ratio of the by-wall region. That is, when the lengths of the air entrainment boundaries a0 d0 and b0 c0 are short (Dx5Dy), the boundary flow velocities become UE,W5VE,W. Whereas, when the lengths of the air entraining boundaries a0 d0 and b0 c0 are large (DxbDy), the boundary flow velocities become UE,WbVE,W. As for the latter case, the entrainment rates become congruent regardless of their position, i.e., UEffiUE,W, as abundant air is supplied into the bywall region. This relation, UEffiUE,W, is commonly observed in the vertically ascending fire plumes independent of wall interference. Finally, by substituting the boundary flow velocity VE,W into Eqs. (3) and (4), then the pressure difference which causes the attachment of the window flame is obtained as follows: 1 2

"

1

1 þ ð2Dx =Dy Þ2

# (9)

Now we assume that the characteristic x-wise length of the bywall region Dx can be approximated by the distance of the trajectory from the wall surface, i.e., Dx  x

(10)

In addition to this, because the spreading width of window flame in the direction perpendicular to the trajectory is generally short [4,9,11], the characteristic y-wise length of the by-wall region is approximated by the opening width Dy  B

This equation shows that when the window flame becomes well separated from the wall surface (xbB), then the pressure difference Dp which draws the flame to the wall becomes negligible (Dp-0). 2.2. Modeling the trajectory of the window flame The approach we take here for the trajectory modeling is to approximate it with a curve and describe its relevant coefficients in terms of the pressure difference Dp in Eq. (12). According to the observation in the previous experiment [11], trajectories of window flames can be divided into two types: one in which the flow ascends almost vertically maintaining a certain separation from the wall; and the other in which the flow ascends upward after ejection and attaches to the wall downstream. Firstly, we approximate the latter by the following (n+2)-variable polynominal 8  nþ2  nþ1  n > z z z :0 ðHA pzÞ (13)

(7)

where Dx is the characteristic length of the by-wall region in the xdirection and Dy is that in the y-direction. By solving Eqs. (5)–(7) simultaneously, flow velocities at the boundaries UE,W and VE,W are obtained as follows:

Dp  r1 ðaum Þ2

By substituting Eqs. (10) and (11) into Eq. (9), then the pressure difference Dp can be rewritten as follows: " # 1 1 Dp  r1 ðaum Þ2 (12) 2 1 þ ð2x=BÞ2

(11)

where LE is the separation between the equilibrium point and the wall surface, HA is the vertical height of the attachment point, and c0, cn, cn+1, and cn+2 are the coefficients. The equilibrium point is the position at which the flow momentum in the horizontal direction becomes zero, and the attachment point is the position at which the trajectory touches the wall surface (Fig. 2). In Eq. (13), the window flame ascends vertically upward along the wall surface after the attachment. Note that the flow velocity becomes zero on the wall surface, and the trajectory does not attach to the wall in a strict sense. However, on the basis of the observed trajectory configurations in the experiments [11], this approximation is reasonable in a practical sense. Applying the following boundary conditions for the ejection point and the attachment point 9 z¼0: x¼0 ðejection pointÞ = qx (14) z ¼ HA : x ¼ 0; ¼ 0 ðattachment pointÞ ; qz into Eq. (13), the polynominal is transformed into 8  n   > z 2 :0 ðHA pzÞ

(15)

in which the coefficient cn+2 has been replaced by the symbol c. By differentiating Eq. (15) with respect to z, then the gradient of the trajectory within the range 0pzpHA is given as follows:  n1    qx LE z z n z  ¼ cðn þ 2Þ 1 (16) HA n þ 2 HA qz HA HA The left-hand side of the equation becomes zero, i.e., the trajectory becomes parallel to the wall surface, either when z ¼ n/ (n+2) HAHE or when z ¼ HA. These values of z represent the heights of the equilibrium point HE and that of the attachment point HA, respectively. Eq. (16) also shows that the height of the equilibrium point HE is proportional to both the height of the attachment point HA and the parameter n. The value of n is adjusted so that the model reproduces the experimental data well.

ARTICLE IN PRESS K. Himoto et al. / Fire Safety Journal 44 (2009) 250–258

z

253

z

Attachment point

HA

HE

HE

Venting point

Equilibrium point

Equilibrium point

x LE

Venting point

x LE

Fig. 2. Trajectory model for the window flame: (A) attaching window flame and (B) non-attaching window flame.

Height of the attachment point HA (m)

1.5 n = 1/2

As a result of these discussions, we obtain an expression for the attaching flame trajectory, 8    > z 2 < 27 z ð0pzpHA Þ 1 x 4 HA HA Attaching flame : ¼ (19) LE > :0 ðHA pzÞ

n=1

1.25

1

0.75

As for the trajectory of the non-attaching flame, we assume that the curved configuration of the trajectory from the venting point to the equilibrium point (x ¼ LE, z ¼ HE) is determined by the same mechanism as that of the attaching flame. As a result, the trajectory of the non-attaching flame takes the identical configuration to that of Eq. (19) before it reaches the equilibrium point. In the subsequent phase, the trajectory is drawn towards the wall due to Dp for the attaching flame, while the effect of Dp can be neglected for the non-attaching flame. As for the nonattaching flame, flow momentum in the horizontal direction is balanced out with the work done by Dp before it reaches the equilibrium point. Thus, the flame rises vertically upward in the downstream as 8    > z 2 < 27 z ð0pzpHE Þ 1 x 4 HA HA Non-attaching flame : ¼ LE > :1 ðHE pzÞ

n=2

0.5

0.25

0 0

0.25 Height of the equilibrium point HE (m)

0.5

Fig. 3. The equilibrium point HE and the attachment point HA.

Out of the 20 test cases carried out, HA and HE were distinguishable in 7 cases [11]. The relationship between HA and HE is shown in Fig. 3 with lines at different values of n. Although there is a certain variation between the plots, n ¼ 1 is adopted referring to these results. In this case, the trajectory is approximated by a cubic function, and the height of the equilibrium point is given by HE ¼

HA ðn ¼ 1Þ 3

(17)

Additionally, as the trajectory passes through the equilibrium point (x ¼ LE when z ¼ HA/3), the value of the remaining coefficient c is obtained from Eq. (15) c¼

27 4

(18)

(20) 2.3. The equivalent point and the attachment point Configuration of the trajectory can be estimated using Eqs. (19) and (20). However, the expressions have no physical background at this moment except for the boundary conditions. Thus, the separation of the equilibrium point LE and the height of the attachment point HA which are involved in these equations, are modeled in order to fulfill the momentum balance in the horizontal (x) direction. For simplicity, we assume that the momentum in the x-direction decreases mainly due to the pressure difference d Dp ðbu2m Þ   dx r1

(21)

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K. Himoto et al. / Fire Safety Journal 44 (2009) 250–258

where b is the characteristic width of the window flame when the velocity in the x direction takes a Gaussian profile. Based upon the experimental observation [11], we assume that the virtual heat source is just below the upper edge of the window, and has an elongated shape in the y-direction. In other words, the line heat source assumption is invoked, and profiles of temperature and velocity in the y-direction are neglected. Substituting the pressure difference Dp into Eq. (21), we obtain " # d ðaum Þ2 1 ðbu2m Þ   (22) dx 2 1 þ ð2x=BÞ2 As defined previously, the equilibrium point (x ¼ LE, z ¼ HE) is the position at which the velocity in the x-direction becomes zero. In other words, it is the point at which the momentum in the horizontal direction (pu20(HZN)) balances out with the work done by the pressure. Thus, by integrating Eq. (22) with regard to x " # Z LE ðaum Þ2 1 2 f1 u0 ðH  Z N Þ   dx (23) 2 1 þ ð2x=BÞ2 0 where f1 is the coefficient, and u0 is the maximum venting velocity at the window. Among the terms in Eq. (23), the trajectory velocity um can be broken down into fundamental parameters by incorporating the results of the dimensional analysis [11]. Although the window flame can be divided into either the flaming regime or the plume regime, we adopt the results for the plume regime in order to derive an uninterrupted formulation. Thus, the velocity along the trajectory is expressed as [11] um ¼ f2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1=3 gðH  Z N ÞQ

(24) 0

where f2 is the coefficient, and Q is the dimensionless heat release rate appropriate for the line heat source flame Q_



Q0 

r1 cP T 1

g 1=2 BðH

 ZN Þ

(25)

3=2

function is well below 1, Eq. (27) can be approximated by ! LE 1 4f1  ¼ fF   F 2 a2 f2 B

(29)

2

where f is a substitute for the other coefficients. Although the range of application of this relationship is nominally limited, we assume that it is applicable to the cases in general. Regardless of this approximation, the dimensional relationship between parameters in Eq. (29) corresponds with those of the original relationship in Eq. (27). Given that the ascending velocity of the flow um is constant in the plume regime (Eq. (24)), the height of the attachment point HE is approximately proportional to the separation in the x-direction LE. In addition to this, as the height of the attachment point HA is proportional to that of the equilibrium point HE (Eq. (17)), then HA also becomes proportional to LE HA LE /  fF  B B

(30)

3. Verification of the model 3.1. The equilibrium point and the attachment point The observed trajectory data in the experiment were correlated with Eqs. (29) and (30). The results are shown in Fig. 4, where the data of the attaching flames are indicated with solid symbols and the rest with open symbols. Regression lines for both the separation between the equilibrium point and the wall surface LE, and the height of the attachment point HA were obtained by the least-squares method, respectively as follows: LE ¼ 0:21F  B HA ¼ 1:33F  B

ðr ¼ 0:96Þ

(31)

ðr ¼ 0:90Þ

(32)

:

where Q is the virtual heat release rate of the window flame, cp is the specific heat of the gas, TN is the ambient temperature, and g is the acceleration due to gravity. Now substituting Eqs. (24) into (23), and excluding parameters independent of x from the integral, we obtain

a2 f22

gQ 0

2=3

Z

LE

dx

1 þ ð2x=BÞ2   2 2 a f2 0 2=3 B 1 2LE tan gQ ¼ 2 2 B 2

HA/B = 1.33F*

0

(r = 0.90)

(26)

After further manipulation, the separation of the equilibrium point from the wall surface LE is given by ! LE 1 4f1   tan F (27) 2 B a2 f2 2

4 LE/B = 0.21F*

LE/B & HA/B (-)

f1 u20  

6

(r = 0.96)

2

where F* is another dimensionless parameter which is defined as F 

!2 !2=3 u0 Q_ pffiffiffiffiffi ffi gB r1 cP T 1 g 1=2 BðH  Z N Þ3=2

(28)

Eq. (27) describes LE as a tangent function of the parameter F*. However, there is a restriction with this expression in that the parameter of the tangent function must be within the range of p/2p/2. This restriction is attributed to the assumptions adopted in the derivation process and needs to be excluded for use in practice. Note that when the parameter of the tangent

LE/B < 0.7 (attachment condition) 0 0

5

10

15

F* (-) Fig. 4. Separations of the equilibrium point from the wall LE and height of the attachment points HA. Solid symbols designate the data for the attaching flows.

ARTICLE IN PRESS K. Himoto et al. / Fire Safety Journal 44 (2009) 250–258

Substituting these equations into Eqs. (19) and (20), expressions for the trajectory configuration are derived as ( Attaching flame : X  ¼

0:603Z  ð1:33  Z  Þ2

ð0pZ  p1:33Þ

0

ð1:33pZ  Þ

( Non-attaching flame : X  ¼

LE

Equilibrium point

B HA

Attachment point

yes

B

≡

x 1 B F*

LE B

=

0.21 F*

Eq.(31)

= 1.33 F*

≡

Z*

0:21

0:44pZ 

x 1 ; B F

Z 

z 1 B F

(35)

Prior to selecting the expression for the trajectory configuration either from Eqs. (33) or (34) the critical condition for flow attachment needs to be determined. Note that from Eq. (12), the magnitude of the pressure difference Dp which draws the plume towards the wall, depends on the ratio of the flame–wall separation to the opening width x/B. However, as the numerator x is a variable number, we adopt LE/B as the alternative to x/B. From Fig. 4, we can roughly deduce the critical condition for the flow attachment, as follows:

Eq.(28)

∞cPT∞g1/2B (H−ZN)3/2

gB

X*

−2/3

. Q

2

ð0pZ  p0:44Þ

where X* and Z* are dimensionless parameters which are defined as follows: X 

u0

0:603Z  ð1:33  Z  Þ2

(34)

(33)

F* ≡

255

Eq.(32)

LE ¼ 0:7 B

z 1

Eq.(35)

B F*

(36)

below which the window flame attaches to the wall surface. In order to organize the derived equations, a flowchart which shows the calculation process of the trajectory configuration is shown in Fig. 5.

no 7.0?

3.2. Comparison of the trajectory

Attaching flame X* =

(0

0.603Z* (1.33−Z*)2

Z*

(1.33

0

1.33) Z*)

As for the model verification, two different ventilation conditions (with and without the mechanical ventilation), at five different opening configurations were selected (Figs. 6–10). Mechanical ventilation was applied to the compartment in order to simulate pressure elevation inside the fire compartment which may take place under the effect of external wind. Series (A) shows : the results for the no mechanical ventilation (m0 ¼ 0 kg/s), and those of series (B) show the results for the mechanically ventilated : conditions (mo 4 ¼ 0 kg/s). The figures also show the results of the model proposed by Yokoi (Eq. (1)), which are shown with dotted lines. In fact, neither the present model nor that of Yokoi involves the effect of ventilation explicitly. However, it is implicitly

Eq.(33)

Non-attaching flame 0.603Z* (1.33−Z*)2

(0

0.21

(144

Z*

1.44) Z*)

Eq.(34)

Fig. 5. Flowchart of the trajectory calculation.

1.2

1.2

0.9

0.9

○ Experiment … Yokoi

0.6

○ Experiment

z (m)

z (m)

X* =

― Present

… Yokoi

0.6

― Present

m0 = 0.0528 (kg/s)

m0= 0.0 (kg/s)

0.3

F*= 4.49 (-)

0.3

* F = 9.56 (-)

0

0 0

0.3 x (m)

0.6

0

0.3 x (m)

0.6

_ o ¼ 0.0 kg/s and (B) m _ o ¼ 0.0528 kg/s. Fig. 6. Trajectory of the window flame (B ¼ 0.2 m, H ¼ 0.5 m): (A) m

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K. Himoto et al. / Fire Safety Journal 44 (2009) 250–258

1.2

1.2

0.9

0.9

○ Experiment

z (m)

z (m)

○ Experiment … Yokoi

0.6

― Present

… Yokoi

0.6

― Present

m0 = 0.0458 (kg/s)

m0= 0.0 (kg/s)

0.3

0.3

F*= 2.74(-)

F* = 4.77 (-)

0

0 0

0.3

0.6

0

x (m)

0.3 x (m)

0.6

_ o ¼ 0.0 kg/s and (B) m _ o ¼ 0.0458 kg/s. Fig. 7. Trajectory of the window flame (B ¼ 0.3 m, H ¼ 0.5 m): (A) m

1.2

1.2

0.9

0.9

… Yokoi

0.6

○ Experiment

z (m)

z (m)

○ Experiment

― Present

… Yokoi

0.6

― Present

m0 = 0.0548 (kg/s)

m0= 0.0 (kg/s)

0.3

F*= 1.34 (-)

0.3

F* = 1.76 (-)

0

0 0

0.3 x (m)

0.6

0

0.3

0.6

x (m)

_ o ¼ 0.0 kg/s and (B) m _ o ¼ 0.0548 kg/s. Fig. 8. Trajectory of the window flame (B ¼ 0.5 m, H ¼ 0.5 m): (A) m

reflected within the equations as the height of neutral plane ZN changes along with the compartment pressurization. The results of the present model show reasonable agreements with the experimental data, in which the values of Lx and Hz were predicted with fair accuracy. In most of the cases, the present model also predicts the occurrence of the attachment adequately, whereas Yokoi’s model disregards the attachment. However, the present model showed erroneous prediction in certain conditions. In particular, in Figs. 7(A) and 8(B) the discrepancy with the experimental data was significant, as the trajectory configuration was calculated on the basis of the wrong assumption that the flow

attachment occurs. With the relationship in Eq. (31), the critical condition for the occurrence of flow attachment (Eq. (36)) can be translated into the dimensionless number F* F  ¼ 3:33

(37)

While the values of the governing dimensionless number F* were 2.74 for Fig. 7(A) and 1.76 for Fig. 8(B). These were not close to the critical value 3.33, and it may not be assumed that they were within the margin of error. Possible causes for the lack of accuracy may be attributed to some inadequacies in the model assumptions. Specifically, these are: (1) incompressibility of the

ARTICLE IN PRESS K. Himoto et al. / Fire Safety Journal 44 (2009) 250–258

1.2

1.2

0.9

0.9

… Yokoi

0.6

○ Experiment

z (m)

z (m)

○ Experiment

― Present

… Yokoi

0.6

― Present

m0 = 0.0525 (kg/s)

m0= 0.0 (kg/s)

0.3

257

0.3

F*= 1.85 (-)

F* = 3.23 (-)

0

0 0

0.3

0

0.6

0.3

0.6

x (m)

x (m)

_ o ¼ 0.0 kg/s and (B) m _ o ¼ 0.0525 kg/s. Fig. 9. Trajectory of the window flame (B ¼ 0.3 m, H ¼ 0.3 m): (A) m

1.2

1.2

0.9

0.9

… Yokoi

0.6

○ Experiment

z (m)

z (m)

○ Experiment

― Present

… Yokoi

0.6

― Present

m0 = 0.0554 (kg/s)

m0= 0.0 (kg/s)

0.3

F*= 0.72 (-)

0

0.3

F* = 1.96 (-)

0 0

0.3 x (m)

0.6

0

0.3

0.6

x (m)

_ o ¼ 0.0 kg/s and (B) m _ o ¼ 0.0554 kg/s. Fig. 10. Trajectory of the window flame (B ¼ 0.5 m, H ¼ 0.3 m): (A) m

flow; (2) symmetry of temperature and velocity profiles along the trajectory and (3) division of the horizontal plane into three regions of different characteristics. Assumption (1) is often invoked in the classical plume modeling and its legitimacy is widely approved. However, as drop of the pressure inside the flame region due to volume expansion gives an increase in the rate of air entrainment, it seems fair to expect that this intrinsic flow characteristic influences the attachment behavior. Regarding Assumption (2), it is not always adequate to assume a symmetrical profile for window flames, especially for the attaching

flames. When the profiles are asymmetrical, the rate of air entrainment also becomes asymmetrical. This may bring some inconsistency with the homogeneity assumption on combustion and the pressure formation inside the segmented regions. In Assumption (3), we assumed that the attachment of the window flame is induced by the pressure difference between three regions. However, the adequacy of this assumption is not fully validated with quantitative experimental data. In order to verify, measurement of three dimensional profiles of the flow velocity as well as the static pressure will be needed.

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4. Conclusions

References

In this paper, a trajectory of window flame was modeled with special emphasis on the flow attachment to the wall. In the model, the flow attachment was assumed to be induced by the pressure difference between the by-wall region and the ambient region. The pressure difference Dp was modeled by considering the exchange of mass and momentum between the segmented regions around the window flame. The configuration of the flame trajectory was approximated by a cubic polynominal, which showed the best agreement to the experimental data. The parameters involved in the polynomial were determined in order to comply with the conservation equation of momentum, in which the modeled pressure difference Dp was incorporated. The results of the model prediction were reasonably comparable to the experimental data. However, the model gave erroneous predictions for the occurrence of flow attachment in certain conditions. As a result, trajectory configurations for these conditions were not even close to the observed ones. The reason for this discrepancy is considered due to the lack of accuracy in the governing dimensionless parameter F*. For further model refinement, we need to discuss the following issues which are invoked as assumptions in the model: (1) incompressibility of the flow; (2) symmetry of temperature and velocity profiles along the trajectory and (3) division of horizontal plane into three distinct regions.

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