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Scaling the initial convective flow of power law fires
ARTICLE IN PRESS
Fire Safety Journal 42 (2007) 240–242 www.elsevier.com/locate/firesaf
Short communication
Scaling the initial convective flow of power law fires Gunnar Heskestad 10 Hilltop Road, Dover, MA 02030, USA Received 1 August 2006; received in revised form 5 December 2006; accepted 5 December 2006 Available online 30 January 2007
Abstract Scaling relations for the initial convective environment generated by power-law fires in geometrically similar spaces are derived, comprising functional relations for nondimensional temperature rise and gas velocity as functions of nondimensional time and location coordinates for arbitrary power-law behavior. Many fires initially can be modeled as t-squared fires, where heat release rate increases with the second power of time. r 2007 Elsevier Ltd. All rights reserved. Keywords: Fire scaling; Power law fires; t-squared fires; Fire detection
1. Introduction
2. The weak, turbulent fire plume
The author has on several occasions had a need to reference his paper of several years past on scaling of the initial fire environment [1], especially useful in fire detection problems. Unfortunately, this paper is not archival. Material therein relating to power-law fires is not covered elsewhere and, hence, is not easily accessible. We take the opportunity here to briefly redevelop this material for future reference, consistent with the original derivation [1]. The derivation depends on scaling relations for steady fire sources and these are reviewed first. The ambient air is assumed to be of uniform temperature and quiescent except for plume-induced air movements. However, effects of nonuniform temperature, e.g., in the form of vertical temperature stratifications [2,3], as well as ambient air currents, can be scaled with the aid of additional nondimensional groups based on variables defined in the derivation, but this extension is not within the scope of this paper. The motion of the front of so-called starting plumes is one significant aspect of the initial convective flow. Its scaling has been investigated previously for suddenly initiated steady fires [4] as well as t-squared fires [5], the latter with the aid of the scaling relationships redeveloped here.
A weak (axisymmetric) plume has small excess temperatures above ambient, as would be the case some distance above a fire source. Except for very small fire sources, the plume flow is turbulent. Past work, e.g. Rouse et al. [6], has established the manner in which plume width (b) and centerline values of velocity ðu0 Þ and excess temperature ðDT 0 Þ vary with height, z, expressed here as proportionalities:
Tel.: +1 508 785 1872.
E-mail address: [email protected]. 0379-7112/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2006.12.004
b/z
(1)
u0 /
cp T 1 r1 "
DT 0 /
1=3
g
1=3 Q1=3 , c z
2=3 # T1 5=3 . Q2=3 c z cp T 1 r1 g g
(2)
(3)
3. Steady fire under a ceiling Consider a steady fire under a ceiling. The fire plume impinges on the ceiling and forms a turbulent ceiling jet, subject to the continuity, momentum and energy equations. The fire plume is assumed to have reached a self-preserving state before impacting on the ceiling, hence can differ from
ARTICLE IN PRESS G. Heskestad / Fire Safety Journal 42 (2007) 240–242
Nomenclature defined in (6) (m4/s2 kJ) plume width (m) specific heat at constant pressure (kJ/kg K) acceleration of gravity (m/s2) ceiling height above virtual origin of fire source (m) thermal conductivity (kW/m K) positive exponent convective heat flux (kW) characteristic convective heat flux defined in (11) (kW)
A b cp g H k p Qc Qc0
one fire to the next only because of differences in plume width and centerline values of velocity and excess temperature. Suitable reference scales for these variables are established from (1)–(3) setting z ¼ H, where H is the ceiling height above the fire source (or more appropriately, above the virtual origin [7] of the fire source). When the governing equations are made dimensionless with the reference scales, taking H=u0 ðHÞ as a time scale, three parameters appear which govern the flows: Richardson number ðDT 0 gH=T 1 u20 Þ, Reynolds number ðru0 H=mÞ and Prandtl number ðcp m=kÞ. The Richardson number is constant for self-preserving turbulent plumes (from (1)— (3), setting z ¼ H) and the Prandtl number may also be considered constant. Effects of the Reynolds number can be considered small within turbulent convective flows with a free boundary. Although related friction and heat loss at solid boundaries like a ceiling may be of some consequence, such effects are also assumed small for the main flow. Accordingly, we establish the following functional relations in the flow field away from solid boundaries for nondimensional velocity and temperature rise in geometrically similar spaces: " # u t ¼ f 1 1=3 1=3 4=3 ; x=H , (4) 1=3 A1=3 Q1=3 A Qc H c H "
DT 5=3 A2=3 ðT 1 =gÞQ2=3 c H
¼ f2
t H 4=3 A1=3 Q1=3 c
# ; x=H .
(5)
Here, t is time from ignition or sudden initiation of a steady fire, x is the vectorial location of observation point, and A¼
g cp T 1 r1
.
(6)
Under an unconfined ceiling discharging the ceiling flow into free space, the nondimensional variables eventually become steady, no longer depending on nondimensional time, t=ðA1=3 Q1=3 H 4=3 Þ. Under a confined ceiling, fire c gases accumulate and the nondimensional variables never reach a steady state.
t T1 DT DT 0 u u0 x z ac m r r1
241
time from ignition (s) ambient temperature (K) excess temperature above ambient (K) DT on plume centerline (K) velocity (m/s) velocity on plume centerline (m/s) vectorial location of observation point (m) height above virtual origin of fire (m) convective fire growth coefficient defined in (8) (kW/s2) dynamic viscosity (Pa s) density (kg/m3) density of ambient air (kg/m3)
One may object to selecting the reference variables at the ceiling level, H, where the flow is not really established instantaneously upon ignition. However, the same results follow from assuming that a self-preserving turbulent plume is established somewhere above the fire site, at some fraction of H, within a time period which is small compared to time intervals of appreciable temporal variation (other than turbulent fluctuations) in the ensuing convection. 4. Power law fires For unsteady fires, viewing the gas supply to be at a fraction of the ceiling height where the flow practically tracks the instantaneous development of a fire, provides a handle for developing scaling relations for some of these fires. In the expressions for the reference scales which follow from putting z proportional to H in (1)–(3), the instantaneous convective heat flux,Qc , is replaced by the equivalent ðQc =Qc0 ÞQc0 , where Qc0 is a characteristic convective heat flux. Nondimensionalizing the governing equations for the convective flow, it is deduced that nondimensional forms of velocity and excess temperature for geometrically similar spaces are related to nondimensional time and space coordinates as in (4) and (5), except that Qc is replaced by Qc0 , and provided that a unique relation exists between Qc =Qc0 and nondimensional time: ! t Qc =Qc0 ¼ f 3 . (7) 1=3 A1=3 Qc0 H 4=3 This formulation may seem confusing and difficult to use, but if attention is limited to particular types of fire growth some readily usable and interpretable results can be obtained. Consider the large number of fire histories, which initially can be represented as power law growth Qc ¼ ac tp .
(8)
Here, ac is a convective fire growth coefficient and p is a positive exponent. In order to deduce the convective flow in
ARTICLE IN PRESS G. Heskestad / Fire Safety Journal 42 (2007) 240–242
242
a particular space from measurements in a geometrically similar space it is required that the ratios of instantaneous to characteristic convective heat fluxes in the two spaces be identical functions of nondimensional time, as formalized in (7). To examine whether this requirement can be satisfied, (8) is recast into a form consistent with (7): p 3" 2 #p 1=3 ac A1=3 Qc0 H 4=3 t 5 Qc =Qc0 ¼ 4 . 1=3 Qc0 A1=3 Qc0 H 4=3 (9) This equation shows that identical functions of nondimensional time are indeed possible, provided the first bracketed term is a constant and the exponent p is invariant. Choosing the first bracketed term equal to one, without loss of generality, the common function of nondimensional time is: " #p t Qc =Qc0 ¼ . (10) 1=3 A1=3 Qc0 H 4=3 However, the important outcome is that the value of the characteristic convective heat flux has been fixed by setting the first bracketed term equal to one: Qc0 ¼ Ap=ð3þpÞ ac3=ð3þpÞ H 4p=ð3þpÞ .
(11)
What has been found then is that power-law fires in geometrically similar spaces can be related by replacing Qc in (4) and (5) with Qc0 from (11). The exponent p must have the same value for all these fires. The final scaling relations for power-law fires then become: u 1=ð3þpÞ
A1=ð3þpÞ ac " ¼ f4
H ðp1Þ=ð3þpÞ
#
t 1=ð3þpÞ
A1=ð3þpÞ ac
H 4=ð3þpÞ
; x=H ,
ð12Þ
DT 2=ð3þpÞ
2=ð3þpÞ
T 1 =gÞac "
ðA
¼ f5
H ð5pÞ=ð3þpÞ
t 1=ð3þpÞ
A1=ð3þpÞ ac
H 4=ð3þpÞ
#
; x=H .
ð13Þ
Note that, for steady fires (p ¼ 0 and ac ¼ Qc ), these equations revert to the forms of (4) and (5). Since their derivation [1], these scaling relations have been successfully tested against data on smoke arrival times measured by O’Dogherty [8] under a horizontal ceiling from linearly growing fires (p ¼ 1) [9] as well as temperature and velocity data under horizontal ceilings from t-squared fires (p ¼ 2) [10,11]. 5. Conclusion The scaling relations derived for power-law fire growth, away from immediately next to solid boundaries, are given in Eqs. (12) and (13). The relations have been successfully tested against data from linear and t-squared fire growth. References [1] Heskestad G. Similarity relations for the initial convective flow generated by fire. American Society of Mechanical Engineers, Paper No. 72-WA/HT-17, November 1972. [2] Morton BR, Taylor GI, Turner JS. Turbulent gravitational convection from maintained and instantaneous sources. Proc R Soc 1956;A234:1–23. [3] Heskestad G. Fire plume behavior in temperature stratified ambients. Combust Sci Technol 1995;106:207–28. [4] Tanaka T, Fujita T, Yamaguchi J. Investigation into rise time of buoyant fire plume fronts. Int J Eng Performance-Based Fire Codes 2000;2:14–25. [5] Heskestad G. Rise of plume front from starting fires. Fire Saf J 2001;36:201–4. [6] Rouse H, Yih CS, Humphries HW. Gravitational convection from a boundary source. Tellus 1952;4:201–10. [7] Heskestad G. The SFPE handbook of fire protection engineering. 3rd ed. National Fire Protection Association, 2002, p. 2-1. [8] O’Dogherty MJ. The detection of fires by smoke, Part 2: slowly developing wood crib fires, F.R. Note No. 793, November 1969, Fire Research Station, Boreham Wood, Herts. [9] Heskestad G. Physical modeling of fire. J Fire Flammability 1975;6:253–73. [10] Heskestad G, Delichatsios MA. The initial convective flow in fire, Seventeenth symposium (international) on combustion. Pittsburgh, PA, The Combustion Institute, 1979, 1113-23. [11] Heskestad G. Heat of combustion in spreading wood crib fires with application to ceiling jets. Fire Saf J 2006;41:343–8.
Fire Safety Journal 42 (2007) 240–242 www.elsevier.com/locate/firesaf
Short communication
Scaling the initial convective flow of power law fires Gunnar Heskestad 10 Hilltop Road, Dover, MA 02030, USA Received 1 August 2006; received in revised form 5 December 2006; accepted 5 December 2006 Available online 30 January 2007
Abstract Scaling relations for the initial convective environment generated by power-law fires in geometrically similar spaces are derived, comprising functional relations for nondimensional temperature rise and gas velocity as functions of nondimensional time and location coordinates for arbitrary power-law behavior. Many fires initially can be modeled as t-squared fires, where heat release rate increases with the second power of time. r 2007 Elsevier Ltd. All rights reserved. Keywords: Fire scaling; Power law fires; t-squared fires; Fire detection
1. Introduction
2. The weak, turbulent fire plume
The author has on several occasions had a need to reference his paper of several years past on scaling of the initial fire environment [1], especially useful in fire detection problems. Unfortunately, this paper is not archival. Material therein relating to power-law fires is not covered elsewhere and, hence, is not easily accessible. We take the opportunity here to briefly redevelop this material for future reference, consistent with the original derivation [1]. The derivation depends on scaling relations for steady fire sources and these are reviewed first. The ambient air is assumed to be of uniform temperature and quiescent except for plume-induced air movements. However, effects of nonuniform temperature, e.g., in the form of vertical temperature stratifications [2,3], as well as ambient air currents, can be scaled with the aid of additional nondimensional groups based on variables defined in the derivation, but this extension is not within the scope of this paper. The motion of the front of so-called starting plumes is one significant aspect of the initial convective flow. Its scaling has been investigated previously for suddenly initiated steady fires [4] as well as t-squared fires [5], the latter with the aid of the scaling relationships redeveloped here.
A weak (axisymmetric) plume has small excess temperatures above ambient, as would be the case some distance above a fire source. Except for very small fire sources, the plume flow is turbulent. Past work, e.g. Rouse et al. [6], has established the manner in which plume width (b) and centerline values of velocity ðu0 Þ and excess temperature ðDT 0 Þ vary with height, z, expressed here as proportionalities:
Tel.: +1 508 785 1872.
E-mail address: [email protected]. 0379-7112/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2006.12.004
b/z
(1)
u0 /
cp T 1 r1 "
DT 0 /
1=3
g
1=3 Q1=3 , c z
2=3 # T1 5=3 . Q2=3 c z cp T 1 r1 g g
(2)
(3)
3. Steady fire under a ceiling Consider a steady fire under a ceiling. The fire plume impinges on the ceiling and forms a turbulent ceiling jet, subject to the continuity, momentum and energy equations. The fire plume is assumed to have reached a self-preserving state before impacting on the ceiling, hence can differ from
ARTICLE IN PRESS G. Heskestad / Fire Safety Journal 42 (2007) 240–242
Nomenclature defined in (6) (m4/s2 kJ) plume width (m) specific heat at constant pressure (kJ/kg K) acceleration of gravity (m/s2) ceiling height above virtual origin of fire source (m) thermal conductivity (kW/m K) positive exponent convective heat flux (kW) characteristic convective heat flux defined in (11) (kW)
A b cp g H k p Qc Qc0
one fire to the next only because of differences in plume width and centerline values of velocity and excess temperature. Suitable reference scales for these variables are established from (1)–(3) setting z ¼ H, where H is the ceiling height above the fire source (or more appropriately, above the virtual origin [7] of the fire source). When the governing equations are made dimensionless with the reference scales, taking H=u0 ðHÞ as a time scale, three parameters appear which govern the flows: Richardson number ðDT 0 gH=T 1 u20 Þ, Reynolds number ðru0 H=mÞ and Prandtl number ðcp m=kÞ. The Richardson number is constant for self-preserving turbulent plumes (from (1)— (3), setting z ¼ H) and the Prandtl number may also be considered constant. Effects of the Reynolds number can be considered small within turbulent convective flows with a free boundary. Although related friction and heat loss at solid boundaries like a ceiling may be of some consequence, such effects are also assumed small for the main flow. Accordingly, we establish the following functional relations in the flow field away from solid boundaries for nondimensional velocity and temperature rise in geometrically similar spaces: " # u t ¼ f 1 1=3 1=3 4=3 ; x=H , (4) 1=3 A1=3 Q1=3 A Qc H c H "
DT 5=3 A2=3 ðT 1 =gÞQ2=3 c H
¼ f2
t H 4=3 A1=3 Q1=3 c
# ; x=H .
(5)
Here, t is time from ignition or sudden initiation of a steady fire, x is the vectorial location of observation point, and A¼
g cp T 1 r1
.
(6)
Under an unconfined ceiling discharging the ceiling flow into free space, the nondimensional variables eventually become steady, no longer depending on nondimensional time, t=ðA1=3 Q1=3 H 4=3 Þ. Under a confined ceiling, fire c gases accumulate and the nondimensional variables never reach a steady state.
t T1 DT DT 0 u u0 x z ac m r r1
241
time from ignition (s) ambient temperature (K) excess temperature above ambient (K) DT on plume centerline (K) velocity (m/s) velocity on plume centerline (m/s) vectorial location of observation point (m) height above virtual origin of fire (m) convective fire growth coefficient defined in (8) (kW/s2) dynamic viscosity (Pa s) density (kg/m3) density of ambient air (kg/m3)
One may object to selecting the reference variables at the ceiling level, H, where the flow is not really established instantaneously upon ignition. However, the same results follow from assuming that a self-preserving turbulent plume is established somewhere above the fire site, at some fraction of H, within a time period which is small compared to time intervals of appreciable temporal variation (other than turbulent fluctuations) in the ensuing convection. 4. Power law fires For unsteady fires, viewing the gas supply to be at a fraction of the ceiling height where the flow practically tracks the instantaneous development of a fire, provides a handle for developing scaling relations for some of these fires. In the expressions for the reference scales which follow from putting z proportional to H in (1)–(3), the instantaneous convective heat flux,Qc , is replaced by the equivalent ðQc =Qc0 ÞQc0 , where Qc0 is a characteristic convective heat flux. Nondimensionalizing the governing equations for the convective flow, it is deduced that nondimensional forms of velocity and excess temperature for geometrically similar spaces are related to nondimensional time and space coordinates as in (4) and (5), except that Qc is replaced by Qc0 , and provided that a unique relation exists between Qc =Qc0 and nondimensional time: ! t Qc =Qc0 ¼ f 3 . (7) 1=3 A1=3 Qc0 H 4=3 This formulation may seem confusing and difficult to use, but if attention is limited to particular types of fire growth some readily usable and interpretable results can be obtained. Consider the large number of fire histories, which initially can be represented as power law growth Qc ¼ ac tp .
(8)
Here, ac is a convective fire growth coefficient and p is a positive exponent. In order to deduce the convective flow in
ARTICLE IN PRESS G. Heskestad / Fire Safety Journal 42 (2007) 240–242
242
a particular space from measurements in a geometrically similar space it is required that the ratios of instantaneous to characteristic convective heat fluxes in the two spaces be identical functions of nondimensional time, as formalized in (7). To examine whether this requirement can be satisfied, (8) is recast into a form consistent with (7): p 3" 2 #p 1=3 ac A1=3 Qc0 H 4=3 t 5 Qc =Qc0 ¼ 4 . 1=3 Qc0 A1=3 Qc0 H 4=3 (9) This equation shows that identical functions of nondimensional time are indeed possible, provided the first bracketed term is a constant and the exponent p is invariant. Choosing the first bracketed term equal to one, without loss of generality, the common function of nondimensional time is: " #p t Qc =Qc0 ¼ . (10) 1=3 A1=3 Qc0 H 4=3 However, the important outcome is that the value of the characteristic convective heat flux has been fixed by setting the first bracketed term equal to one: Qc0 ¼ Ap=ð3þpÞ ac3=ð3þpÞ H 4p=ð3þpÞ .
(11)
What has been found then is that power-law fires in geometrically similar spaces can be related by replacing Qc in (4) and (5) with Qc0 from (11). The exponent p must have the same value for all these fires. The final scaling relations for power-law fires then become: u 1=ð3þpÞ
A1=ð3þpÞ ac " ¼ f4
H ðp1Þ=ð3þpÞ
#
t 1=ð3þpÞ
A1=ð3þpÞ ac
H 4=ð3þpÞ
; x=H ,
ð12Þ
DT 2=ð3þpÞ
2=ð3þpÞ
T 1 =gÞac "
ðA
¼ f5
H ð5pÞ=ð3þpÞ
t 1=ð3þpÞ
A1=ð3þpÞ ac
H 4=ð3þpÞ
#
; x=H .
ð13Þ
Note that, for steady fires (p ¼ 0 and ac ¼ Qc ), these equations revert to the forms of (4) and (5). Since their derivation [1], these scaling relations have been successfully tested against data on smoke arrival times measured by O’Dogherty [8] under a horizontal ceiling from linearly growing fires (p ¼ 1) [9] as well as temperature and velocity data under horizontal ceilings from t-squared fires (p ¼ 2) [10,11]. 5. Conclusion The scaling relations derived for power-law fire growth, away from immediately next to solid boundaries, are given in Eqs. (12) and (13). The relations have been successfully tested against data from linear and t-squared fire growth. References [1] Heskestad G. Similarity relations for the initial convective flow generated by fire. American Society of Mechanical Engineers, Paper No. 72-WA/HT-17, November 1972. [2] Morton BR, Taylor GI, Turner JS. Turbulent gravitational convection from maintained and instantaneous sources. Proc R Soc 1956;A234:1–23. [3] Heskestad G. Fire plume behavior in temperature stratified ambients. Combust Sci Technol 1995;106:207–28. [4] Tanaka T, Fujita T, Yamaguchi J. Investigation into rise time of buoyant fire plume fronts. Int J Eng Performance-Based Fire Codes 2000;2:14–25. [5] Heskestad G. Rise of plume front from starting fires. Fire Saf J 2001;36:201–4. [6] Rouse H, Yih CS, Humphries HW. Gravitational convection from a boundary source. Tellus 1952;4:201–10. [7] Heskestad G. The SFPE handbook of fire protection engineering. 3rd ed. National Fire Protection Association, 2002, p. 2-1. [8] O’Dogherty MJ. The detection of fires by smoke, Part 2: slowly developing wood crib fires, F.R. Note No. 793, November 1969, Fire Research Station, Boreham Wood, Herts. [9] Heskestad G. Physical modeling of fire. J Fire Flammability 1975;6:253–73. [10] Heskestad G, Delichatsios MA. The initial convective flow in fire, Seventeenth symposium (international) on combustion. Pittsburgh, PA, The Combustion Institute, 1979, 1113-23. [11] Heskestad G. Heat of combustion in spreading wood crib fires with application to ceiling jets. Fire Saf J 2006;41:343–8.