Post on 07-Mar-2020
VERIFICATION OF NUREG-1805 BY HAND CALCULATIONS Purpose: Verification of spreadsheets from NUREG-1805 (FDTS) using hand calculations. Design Input: Most inputs for calculations were from Benchmark Exercise #2 (“Experimental Study of the Localized Room Fires,” NFDC2 Test Series, VTT Research Notes 2104) and Benchmark Exercise #3 (Report of Experimental Results for the International Fire Model Benchmarking and Validation Exercise #3, NUREG/CR-6905, NIST Special Publication 1013-1). Assumptions: For one test (Heskestad's Flame Height Correlation), heptane was used as the fuel to determine the flame height. Documentation for Assumptions: The properties of heptane used for the hand calculations and FDTS spreadsheet are from the SFPE handbook. Procedure: A total of nine hand calculations were performed to verify their respective spreadsheets.
Spreadsheets used in Calculation Calculation Excel File Name Natural Ventilation: Method of McCaffrey, Quintiere, and Harkleroad (MQH)
02.1_Temperature_NV.xls (v. 1805.0)
Natural Ventilation: (Smoke Filling): The Non-Steady State Yamana and Tanaka Method
02.1_Temperature_NV.xls (v. 1805.0)
Forced Ventilation: Method of Foote, Pagni, and Alvares (FPA)
02.2_Temperature_FV.xls (v. 1805.0)
Forced Ventilation: Method of Deal and Beyler
02.2_Temperature_FV.xls (v. 1805.0)
Natural Ventilation (Compartment Closed): Method of Beyler
02.3_Temperature_CC.xls (v. 1805.1)
Heskestad's Flame Height Correlation 03_HRR_Flame_Height_Burning_Duration_Calculations.xls (v. 1805.0)
Point Source Radiation Model 05.1_Heat_Flux_Calculations_Wind_Free.xls (v. 1805.0)
Solid Flame Radiation Models (Above Ground)
05.1_Heat_Flux_Calculations_Wind_Free.xls (v. 1805.0)
Heskestad's Plume Temperature Correlation 09_Plume_Temperature_Calculations.xls (v. 1805.0)
Calculation: The calculation of each spreadsheet was performed with a methodical process. All of the inputs were listed first, followed by the applicable equations and description of variables. Lastly, the calculation was performed and the results for comparison are presented in a table. Summary: All of the spreadsheets used in the verification exercise were within 1% of the hand calculation, except the spreadsheet pertaining to Natural Ventilation: (Smoke Filling): The Non-Steady State Yamana and Tanaka Method, which had a large difference in output. However, this discrepancy is due to FDTs calculating the layer height only until the smoke starts to exit through the vent. In performing the hand calculations, an error was discovered for the Forced Ventilation: Method of Deal and Beyler. An errata and updated/revised excel spreadsheet to NUREG-1805 is in the process of completion. Conclusions: All of the spreadsheets tested have been verified by hand calculations. This supports NUREG-1824.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Hot Gas Layer Temperature Process: Natural Ventilation: Method of McCaffrey, Quintiere, and Harkleroad (MQH) Design Input:
air,T∞ = 300.8KC27.8 =o
air,ρ∞ = 1.2 kg/m3 g = 9.81 m/s2 Room size: Width (m) = 7.04 Length (m) = 21.7 Height (m) = 3.82 Wall properties: Interior lining thermal inertia [(kW/m2-K)2-sec] = 0.11 Interior lining thermal conductivity (kW/m-K) = 0.00012 Interior lining specific heat (kJ/kg-K) = 1.26 Interior lining density (kg/m3) = 737 Interior lining thickness (m) = 0.0254 Ventilation: Vent width (m) = 2 Vent height (m) = 2 Top of vent from floor (m) = 2 Heat Release Rate: 1190 kW Calculation: Natural Ventilation: Method of McCaffrey, Quintiere, and Harkleroad (MQH)
( )( )
31
kTvv
2
g hAhAQ6.85∆T ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
& (1)
Where: ∆Tg = upper layer gas temperature rise above ambient (Tg – Ta) (K) Q& = heat release rate of the fire (kW) Av = total area of ventilation opening(s) (m2) hv = height of ventilation opening (m) hk = heat transfer coefficient (kW/m2-K) AT = total area of compartment enclosing surfaces (m2), excluding area of vent opening(s).
Compartment interior surface area
( ) ( ) ( )[ ] vcccccct Alxh2wxh2lxw2A −++= ( ) ( ) ( ) 22
t 521.11mm4m21.7xm3.822m7.04xm3.822m21.7xm7.042A =−++= Thermal penetration time
2
p 2δ
kρct ⎟
⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛=
Where: ρ = density of interior lining (kg/m3) c = thermal capacity of interior lining (kJ/kg-K) k = thermal conductivity of the interior lining (kW/m-K) δ= thickness of the interior lining (m) Heat Transfer Coefficient
tckhkρ
= for ptt < or δkhk = for ptt >
hk = heat transfer coefficient (kW/m2-K)
ckρ = interior construction thermal inertia [(kW/m2-K)2-sec] δ= thickness of the interior lining (m) t = time after ignition in seconds (characteristic burning time) Calc:
sec14.12482
0.0254m
KmkW0.00012
KkgkJ1.26m
kg737t
23
p =⎟⎠⎞
⎜⎝⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅
⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
60 sec:
ptt < , where ( )
KmkW 0.0428
sec60
secKmkW0.11
h 2
2
2
k ⋅=
⋅⋅=
( )( )( )( ) K454.17K300.8
KmkW0.0428m521.11m2m4
kW11906.85T3
1
22
2
g =+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=
300 sec:
ptt < , where ( )
KmkW 0.01914
sec300
secKmkW0.11
h 2
2
2
k ⋅=
⋅⋅=
( )( )( )( ) K501.36K300.8
KmkW0.01914m521.11m2m4
kW11906.85T3
1
22
2
g =+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=
600 sec:
ptt < , where ( )
KmkW 0.01354
sec600
secKmkW0.11
h 2
2
2
k ⋅=
⋅⋅=
( )( )( )( ) K525.89K300.8
KmkW0.01354m521.11m2m4
kW11906.85T3
1
22
2
g =+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=
900 sec:
ptt < , where( )
KmkW 0.011055
sec900
secKmkW0.11
h 2
2
2
k ⋅=
⋅⋅=
( )( )( )( ) K541.62K300.8
KmkW0.011055m521.11m2m4
kW11906.85T3
1
22
2
g =+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=
1200 sec:
ptt < , where( )
KmkW 0.00957
sec1200
secKmkW0.11
h 2
2
2
k ⋅=
⋅⋅=
( )( )( )( ) K553.49K300.8
KmkW0.00957m521.11m2m4
kW11906.85T3
1
22
2
g =+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=
1500 sec:
ptt > , where KmkW0.00472
m0.0254Km
kW0.00012h 2k ⋅
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛⋅=
( )( )( )( ) K620.62K300.8
KmkW0.00472m521.11m2m4
kW11906.85T3
1
22
2
g =+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅
=
For sec1500t > , Tg will be constant at 620.62 K Results: Time (sec) FDT (K) Calculation (K) Temp. Difference (K)
60 454.15 454.17 0.02 300 501.33 501.36 0.03 600 525.89 525.89 0.00 900 541.62 541.62 0.00 1200 553.45 553.49 0.04 1500 620.52 620.62 0.10
Summary/Conclusions: Spreadsheet (02.1_Temperature_NV.xls) for Predicting Hot Gas Layer Temperature and Smoke Layer Height in a Room Fire With Natural Ventilation Compartment (v. 1805.0) is valid against hand calculations. Reference: 1)McCaffrey, B.J., J.G. Quintiere, and M.F. Harkleroad, “Estimating Room Temperature and Likelihood of Flashover Using Fire Test Data Correlation,” Fire Technology, Volume 17, No. 2, pp. 98-119, 1981.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Smoke Layer Temperature Process: Natural Ventilation (Smoke Filling): The Non-Steady State Yamana and Tanaka Method Design Input:
air,T∞ = 300.8KC27.8 =o
air,ρ∞ = 1.2 kg/m3 g = 9.81 m/s2 cp = 1 kJ/kg-K Room size: Width (m) = 7.04 Length (m) = 21.7 Height (m) = 3.82 Wall properties: Interior lining thermal inertia [(kW/m2-K)2-sec] = 0.11 Interior lining thermal conductivity (kW/m-K) = 0.00012 Interior lining specific heat (kJ/kg-K) = 1.26 Interior lining density (kg/m3) = 737 Interior lining thickness (m) = 0.0254 Ventilation: Vent width (m) = 2 Vent height (m) = 2 Top of vent from floor (m) = 2 Heat Release Rate: 1190 kW Calculation: Natural Ventilation (Smoke Filling): The Non-Steady State Yamana and Tanaka Method
23
32
cc
31
h1
A3tQk2z
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+=
& (1)
Where: z = height (m) of the smoke later interface above the floor Q&= heat release rate of the fire (kW) t = time after ignition (sec) Ac = compartment floor area (m2) hc = compartment height (m)
Compartment floor area ( )( ) 2
ccc m152.768m7.04m21.7wxlA === And: k = a constant given by the following equation
31
ap
2a
g Tcgρ
ρ0.21k ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
Where:
gρ = hot gas density (kg/m3)
aρ = ambient density (kg/m3) g = acceleration of gravity (m/s2) cp = specific heat of air (kJ/kg-K) Ta = ambient air temperature Where density of the hot gas (pg), layer is given by:
gg T
353ρ =
Where: Tg = hot gas layer temperature (k) calculated from Method of McCaffrey, Quintiere, and Harkleroad (MQH) Calc:
Time (sec) (Tg) Input from MQH method (K) 60 454.17 300 501.36
60 sec:
3777.017.454
353m
kgK
pg ==
( )( )
0975.08.3001
81.92.1
777.0
21.0
31
2
2
3
3
=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅
⎟⎠⎞⎜
⎝⎛
=KKkg
kJs
mm
kg
mkg
k
( )( ) ( )( ) ( )
mmm
kWz 78.182.3
1768.1523
sec6011900975.02 23
322
31
=⎟⎟⎠
⎞⎜⎜⎝
⎛+=
−
300 sec:
3704.036.501
353m
kgK
pg ==
( )( )
1076.08.3001
81.92.1
704.0
21.0
31
2
2
3
3
=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅
⎟⎠⎞⎜
⎝⎛
=KKkg
kJs
mm
kg
mkg
k
( )( ) ( )( ) ( )
mmm
kWz 38.082.3
1768.1523
sec30011901076.02 23
322
31
=⎟⎟⎠
⎞⎜⎜⎝
⎛+=
−
Results: Time (sec) FDT (m) Calculation (m) Height Difference (m)
60 2 1.78 0.22 300 2 0.38 1.62
The discrepancy in results is due to FDTs calculating the layer height only until the smoke starts to exit through the vent. Summary/Conclusions: Spreadsheet (02.1_Temperature_NV.xls) for Predicting Hot Gas Layer Temperature and Smoke Layer Height in a Room Fire With Natural Ventilation Compartment (v. 1805.0) is valid against hand calculations. Reference: 1) Yamana, T., and T. Tanaka, “Smoke Control in Large Spaces, Part 1: Analytical Theories for Simple Smoke Control Problems,” Fire Science and Technology, Volume 5, No. 1, 1985.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Hot Gas Layer Temperature Process: Forced Ventilation: Method of Foote, Pagni, and Alvares (FPA) Design Input:
air,T∞ = Co20 = 293 K
air,ρ∞ = 1.2 kg/m3 cp = 1 kJ/kg-K Room size: Width (m) = 27.0 Length (m) = 13.8 Height (m) = 15.8 Wall properties: Interior lining thermal inertia [(kW/m2-K)2-sec] = 0.015 Interior lining thermal conductivity (kW/m-K) = 0.0002 Interior lining specific heat (kJ/kg-K) = 0.15 Interior lining density (kg/m3) = 500 Interior lining thickness (m) = 0.05 Ventilation: 23,500 cfm = 11.0907 m3/sec Heat Release Rate: 3640 kW Calculation: Forced Ventilation: Method of Foote, Pagni, and Alvares (FPA)
0.36
p
Tk
0.72
apa
g
cmAh
TcmQ0.63
T∆T
−
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
&&
& (1)
Where: ∆Tg = hot gas layer temperature rise above ambient (Tg – Ta) (K) Ta = ambient air temperature (K) Q& = heat release rate of the fire (kW) m& = mass of the gas in the compartment (kg) cp = specific heat of air (kJ/kg-k) hk = heat transfer coefficient (kW/m2-K) AT = total area of compartment enclosing surfaces (m2)
nventilatioxpm air,∞=&
( ) skg13.309s
m11.0907mkg1.2m
33 =⎟⎠⎞⎜
⎝⎛=&
Compartment interior surface area ( ) ( ) ( )cccccct lxh2wxh2lxw2A ++= ( ) ( ) ( ) 2
t 2034.48mm13.8xm272m13.8xm15.82m27xm15.82A =++= Thermal penetration time
2
p 2δ
kρct ⎟
⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛=
Where: tp = thermal penetration time (sec) ρ = interior construction density (kg/m3) c = interior construction heat capacity (kJ/kg-K) k = interior construction thermal conductivity (kW/m-K) δ= interior construction thickness (m) Heat Transfer Coefficient
tckρ
=kh for ptt < or δkhk = for ptt >
hk = heat transfer coefficient (kW/m2-K)
ckρ = interior construction thermal inertia (kW/m2-K)2-sec t = time after ignition (sec) Calc:
sec37.2342
05.00002.0
15.0500 23
=⎟⎠⎞
⎜⎝⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅
⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=m
KmkW
KkgkJ
mkg
tp
60 sec:
ptt < , where ( )
KKm
kWhk ⋅
=⋅
⋅= 2
2
2
mkW 0.0158
sec60
sec015.0
( )
( )( )( )
KK
skgkJ
skJ
mKmkW
KskgkJ
skg
kWTg 2932931309.13
48.20340158.0
2931309.13
364063.0
36.02
2
72.0
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅
⋅
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
−
KTg 88.420=
120 sec:
ptt < , where ( )
KKm
kWhk ⋅
=⋅
⋅= 2
2
2
mkW 0.01118
sec120
sec015.0
( )
( )( )( )
KK
skgkJ
skJ
mKmkW
KskgkJ
skg
kWTg 2932931309.13
48.203401118.0
2931309.13
364063.0
36.02
2
72.0
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅
⋅
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
−
KTg 74.437=
180 sec:
ptt < , where ( )
KKm
kWhk ⋅
=⋅
⋅= 2
2
2
mkW 0.00913
sec180
sec015.0
( )
( )( )( )
KK
skgkJ
skJ
mKmkW
KskgkJ
skg
kWTg 2932931309.13
48.203400913.0
2931309.13
364063.0
36.02
2
72.0
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅
⋅
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
−
KTg 8.448=
240 sec:
ptt > , where KmKm
kWhk ⋅
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛⋅= 2m
kW 0.00405.0
0002.0
( )
( )( )( )
KK
skgkJ
skJ
mKmkW
KskgkJ
skg
kWTg 2932931309.13
48.2034004.0
2931309.13
364063.0
36.02
2
72.0
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅
⋅
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
−
KTg 7.502=
Any time above 240 seconds will be constant at 502.7 K
Results: Time (sec) FDT (K) Calculation (K) Temp. Difference (K)
60 420.66 420.88 0.22 120 437.63 437.74 0.11 180 448.58 448.8 0.22 240 502.39 502.7 0.31
Summary/Conclusions: Spreadsheet (02.2_Temperature_FV.xls) for Predicting Hot Gas Layer Temperature in a Room Fire With Forced Ventilation Compartment (v. 1805.0) is valid against hand calculations. Reference: 1) Foote, K.L., P.J. Pagni, and N.L. Alvares, “Temperatures Correlations for Forced-Ventilated Compartment Fires,” Fire Safety Science-Proceedings of the First International Symposium, International Association of Fire Safety Science (IAFSS), Grant and Pagni, Editors, Hemisphere Publishing Corporation, New York, pp. 139–148, 1985.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Hot Gas Layer Temperature Process: Forced Ventilation: Method of Deal and Beyler Design Input:
air,T∞ = Co20 = 293 K
air,ρ∞ = 1.2 kg/m3 cp = 1 kJ/kg-K Room size: Width (m) = 27.0 Length (m) = 13.8 Height (m) = 15.8 Wall properties: Interior lining thermal inertia [(kW/m2-K)2-sec] = 0.015 Interior lining thermal conductivity (kW/m-K) = 0.0002 Interior lining specific heat (kJ/kg-K) = 0.15 Interior lining density (kg/m3) = 500 Interior lining thickness (m) = 0.05 Ventilation: 23,500 cfm = 11.0907 m3/sec Heat Release Rate: 3640 kW Calculation: Forced Ventilation: Method of Deal and Beyler
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+=−= ⋅
⋅
tkp
agg
Ahcm
QTT∆T (1)
Where: ∆Tg = hot gas layer temperature rise above ambient (Tg – Ta) (K) Ta = ambient air temperature (K) Q& = heat release rate of the fire (kW) m& = mass of the gas in the compartment (kg) cp = specific heat of air (kJ/kg-k) hk = convective heat transfer coefficient (kW/m2-K) At = total area of compartment enclosing surfaces (m2)
nventilatioxρm air,∞=&
( ) skg13.309s
m11.0907mkg1.2m
33 =⎟⎠⎞⎜
⎝⎛=&
Compartment interior surface area ( ) ( ) ( )cccccct lxh2wxh2lxw2A ++= ( ) ( ) ( ) 2
t 2034.48mm13.8xm272m13.8xm15.82m27xm15.82A =++= Thermal penetration time
( )2p δkρct ⎟
⎠⎞
⎜⎝⎛=
Where: tp = thermal penetration time (sec) ρ = interior construction density (kg/m3) c = interior construction heat capacity (kJ/kg-K) k = interior construction thermal conductivity (kW/m-K) δ= interior construction thickness (m) Heat Transfer Coefficient
tck0.4hk
ρ= for ptt < or ⎟
⎠⎞
⎜⎝⎛=δk0.4hk for ptt >
Where: hk = heat transfer coefficient (kW/m2-K)
ckρ = interior construction thermal inertia (kW/m2-K)2-sec t = time after ignition (sec) Calc:
( ) sec5.93705.00002.0
15.05002
3
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅
⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
= mKm
kWKkg
kJm
kg
tp
60 sec:
ptt < , where ( )
KKm
kWhk ⋅
=⋅
⋅= 2
2
2
mkW 0.00632
sec60
sec015.04.0
( )( )KK
mKmkW
skgkJ
skg
kWTg 06.43229348.203400632.01309.13
36402
2
=+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅+⎟
⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
180 sec:
ptt < , where ( )
KKm
kWhk ⋅
=⋅
⋅= 2
2
2
mkW 0.00365
sec180
sec015.04.0
( )( )KK
mKmkW
skgkJ
skg
kWTg 52.46829348.203400365.01309.13
36402
2
=+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅+⎟
⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
1200 sec:
ptt > , where Km
KmkW
hk ⋅=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⋅= 2m
kW 0.001605.0
0002.04.0
( )( )KK
mKmkW
skgkJ
skg
kWTg 75.51229348.20340016.01309.13
36402
2
=+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅+⎟
⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
=
Any time above 1200 seconds, the temperature will be constant at 512.75 K Results: Time (sec) FDT (K) Calculation (K) Temp. Difference (K)
60 431.78 432.06 0.28 180 468.08 468.52 0.44 1200 512.05 512.75 0.70
Summary/Conclusions: In performing the hand calculations, an error was discovered for the Forced Ventilation: Method of Deal and Beyler. An errata and updated/revised excel spreadsheet to NUREG-1805 is in the process of completion. Based on revision, spreadsheet (02.2_Temperature_FV.xls) for Predicting Hot Gas Layer Temperature in a Room Fire With Forced Ventilation Compartment (v. 1805.0) is valid against hand calculations. Reference: 1) Deal, S., and C.L. Beyler, “Correlating Preflashover Room Fire Temperatures,” SFPE Journal of Fire Protection Engineering, Volume 2, No. 2, pp. 33–48, 1990. Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Hot Gas Layer Temperature
Process: Natural Ventilation (Compartment Closed): Method of Beyler Design Input:
air,T∞ = Co20 = 293 K
air,ρ∞ = 1.2 kg/m3 cp = 1 kJ/kg-K Room size: Width (m) = 27.0 Length (m) = 13.8 Height (m) = 15.9 Wall properties: Interior lining thermal inertia [(kW/m2-K)2-sec] = 0.015 Interior lining thermal conductivity (kW/m-K) = 0.0002 Interior lining specific heat (kJ/kg-K) = 0.15 Interior lining density (kg/m3) = 500 Interior lining thickness (m) = 0.05 Heat Release Rate Time (sec) HRR (kW) 0 0 13.2 1251 90 1706 288 1858 327 1782 409.2 1365 438 0 Calculation: Natural Ventilation (Compartment Closed): Method of Beyler
( )tK12
1
2agg
1e1tKK2KTT∆T −+−=−= (1)
Where:
( )p
T1 mc
Ack0.42K
ρ=
p2 mc
QK&
=
And: ∆Tg = upper layer gas temperature rise above ambient (Tg – Ta) (K) AT = total area of internal compartment boundaries (m2) k = thermal conductivity of the interior lining (kW/m-K) ρ = density of the interior lining (kg/m3) c = thermal capacity of the interior lining (kJ/kg-K)
Q& = heat release rate of the fire (kW) m = mass of the gas in the compartment (kg) cp = specific heat of air (kJ/kg-k) t = exposure time (sec)
HWLV ⋅⋅= 35,924.34mm)(15.9m)(27m)(13.8V ==
Vρm air, ⋅= ∞
kg7,109.208)m)(5,924.34mkg(1.2m 3
3 ==
Compartment interior surface area
( ) ( ) ( )cccccct lxh2wxh2lxw2A ++= ( ) ( ) ( ) 2
t 2042.64mm13.8xm15.92m27xm15.92m13.8xm272A =++= Calc:
( ) ( )
( ) ⎟⎠⎞⎜
⎝⎛
⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞⎜
⎝⎛
⋅⎟⎠⎞⎜
⎝⎛
⋅=
KkgkJkg
mKkgkJ
mkg
KmkW
K1208.7109
64.204215.05000002.04.02 23
1 = 0.02815 s-1/2
90 sec:
( ) ⎟⎠⎞⎜
⎝⎛
⋅
=
KkgkJkg
kWK1208.7109
17062 = 0.23997 K/s
( )
Kesss
sK
Tss
g 293190 0.02815 0.02815
23997.02 90 0.028152
1
22
1
21
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟
⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
=⎟⎠⎞
⎜⎝⎛−−
−
−
= 312.80 K
288 sec:
( ) ⎟⎠⎞⎜
⎝⎛
⋅
=
KkgkJkg
kWK1208.7109
18582 = 0.26135 K/s
( )
Kesss
sK
Tss
g 2931288 0.02815 0.02815
26135.02 288 0.028152
1
22
1
21
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟
⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
=⎟⎠⎞
⎜⎝⎛−−
−
−
=
=gT 357.58 K 327 sec:
( ) ⎟⎠⎞⎜
⎝⎛
⋅
=
KkgkJkg
kWK1208.7109
17822 = 0.25066 K/s
( )
Kesss
sK
Tss
g 2931327 0.02815 0.02815
25066.02 327 0.028152
1
22
1
21
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟
⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
=⎟⎠⎞
⎜⎝⎛−−
−
−
=
=gT 362.66 K 409.2 sec:
( ) ⎟⎠⎞⎜
⎝⎛
⋅
=
KkgkJkg
kWK1208.7109
13652 = 0.192 K/s
( )
Kesss
sK
Tss
g 29312.409 0.02815 0.02815
192.02 2.409 0.028152
1
22
1
21
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟
⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
=⎟⎠⎞
⎜⎝⎛−−
−
−
=
=gT 358.56 K Results: Time (sec) FDT (K) Calculation (K) Temp. Difference (K)
90 312.72 312.8 0.08 288 357.37 357.58 0.21 327 362.43 362.66 0.23
409.2 358.34 358.56 0.22 Summary/Conclusions: Spreadsheet (02.3_Temperature_CC.xls) for Predicting Hot Gas Layer Temperature in a Fire Room With Door Closed (v. 1805.1) is valid against hand calculations. Reference: 1) Beyler, C.L., “Analysis of Compartment Fires with Overhead Forced Ventilation,” Fire Safety Science, Proceeding of the 3 International Symposium, International Association of Fire Safety Science (IAFSS), Cox and Langford, Editors, Elsevier Applied Science, New York, pp. 291-300, 1991.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Plume Temperature. Process: Heskestad’s Flame Height Correlation Design Input:
air,T∞ = Co20 = 293 K
air,ρ∞ = 1.2 kg/m3 cp = 1 kJ/kg-K g = 9.81 m/s2
Assumptions: Fuel = Heptane Afuel = 0.5 m2
m ′′& = 0.101 kg/m2-sec ∆Hc,eff = 44,600 kJ/kg βk = 1.1 m-1
Documentation for Assumptions: Fuel properties well known from Babrauskas, which is documented in SFPE Handbook of Fire Protection Engineering (Ref. 1). Calculation: Heskestad’s Flame Height Correlation The HRR of the fire can be determined by laboratory or field testing. In the absence of experimental data, the maximum HRR for the fire is given by the following equation:
( )kββfeffc, e1A∆HmQ −−′′= && (1)
Where: &Q = heat release rate of the fire (kW)
m ′′& = burning or mass loss rate per unit area per unit time (kg/m2-sec) ∆Hc,eff = effective heat of combustion (kJ/kg) Af = horizontal burning area of the fuel (m2) βk = empirical constant (m-1)
D = diameter of burning area (m) For non-circular pools, the effective diameter is defined as the diameter of a circular pool with an area equal to the actual area given by the following equation:
π4AD f=
Where: D = diameter of the fire (m)
Af = fuel spill area or curb area (m2)
1.02DQ0.235H5
2
f −=⋅
(2) Where: Hf = flame height (m) &Q = heat release rate of the fire (kW)
D = diameter of the fire (m) The above correlation can also be used to determine the length of the flame extension along the ceiling and to estimate radiative heat transfer to objects in the enclosure. Calc:
( )( ) kW1316.02e10.5mkgkJ44,600sm
kg0.101Q )(0.798m)(1.1m22
1
=−⎟⎠⎞⎜
⎝⎛⎟⎠⎞⎜
⎝⎛
⋅=
−−&
( ) m0.798π
m0.54D2
==
( ) ( ) 3.34mm0.7981.02kW1316.020.235H 5
2f =−=
Results:
Fuel Flame Height
FDT (m) Flame Height
Calculation (m) Height
Difference (m) Heptane 3.34 3.34 0.00
Summary/Conclusions: Spreadsheet (03_HRR_Flame_Height_Burning_Duration_Calculations.xls) for Estimating Burning Characteristics of Liquid Pool Fire, Heat Release Rate, Burning Duration, and Flame Height (Flame Height Only) (v. 1805.0) is valid against hand calculations. Reference: 1) Babrauskas, V., “Burning Rates,” Section 3, Chapter 3-1, SFPE Handbook of Fire Protection Engineering, 3rd Edition, P.J. DiNenno, Editor-in-Chief, National Fire Protection Association, Quincy, Massachusetts, 2002. 2) Heskestad, G., “Fire Plumes,” Section 2, Chapter 2-2, SFPE Handbook of Fire Protection Engineering, 2nd Edition, P.J. DiNenno, Editor-in-Chief, National Fire Protection Association, Quincy, Massachusetts, 1995.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Point Source Radiation Model Process: Point Source Radiation Model Design Input:
air,T∞ = 296.9KC23.9 =o
air,ρ∞ = 1.19 kg/m3 cp = 1 kJ/kg-K g = 9.81 m/s2
44.0=rχ Af = 0.79 m2 D = 1m Heat Release Rate: 1190 kW Distance from fire to target (m): varied for multiple calculations Calculation: Point Source Radiation Model
2r
4πQχqR
&& =′′ (1)
Where: q ′′& = radiant heat flux (kW/m2) Q& = heat release rate of the fire (kW) R = radial distance from the center of the flame to the edge of the target (m) χr = fraction of total energy radiated
Distance from Center of the Fire to Edge of the Target Calculation
2DLR +=
Where: R = distance from center of the pool fire to edge of the target (m) L = distance between pool fire and target (m) D = pool fire distance (m)
Distance from fire to radiant heat flux gauge: 4.88 m
( ) mmD 179.04 2
==π
mmmR 38.52
188.4 =+=
( )( )( ) 22 44.1
38.54119044.0
mkW
mkWq ==′′
⋅
π
Distance from fire to radiant heat flux gauge: 4.24 m
( ) mmD 179.04 2
==π
mmmR 74.42
124.4 =+=
( )( )( ) 22 85.1
74.44119044.0
mkW
mkWq ==′′
⋅
π
Distance from fire to radiant heat flux gauge: 3.80 m
( ) mmD 179.04 2
==π
mmmR 30.42
180.3 =+=
( )( )( ) 22 25.2
30.44119044.0
mkW
mkWq ==′′
⋅
π
Distance from fire to radiant heat flux gauge: 1.81 m
( ) mmD 179.04 2
==π
mmmR 31.22
181.1 =+=
( )( )( ) 22 80.7
31.24119044.0
mkW
mkWq ==′′
⋅
π
Results:
Distance Heat Flux
FDT (kW/m2) Heat Flux
Calculation (kW/m2) Heat Flux Difference
4.88 1.44 1.44 0.00
4.24 1.85 1.85 0.00
3.8 2.25 2.25 0.00
1.81 7.80 7.80 0.00 Summary/Conclusions: Spreadsheet (05.1_Heat_Flux_Calculations_Wind_Free.xls) for Estimating Radiant Heat Flux From Fire to a Target Fuel at Ground Level Under Wind-Free Condition (Point Source Radiation) (v.1805.0) is valid against hand calculations. Reference: 1) Drysdale, D.D., An Introduction to Fire Dynamics, Chapter 4, “Diffusion Flames and Fire Plumes,” 2nd Edition, John Wiley and Sons, New York, pp. 109-158, 1998. 2) Babrauskas, V., “Burning Rates,” Section 3, Chapter 3-1, SFPE Handbook of Fire Protection Engineering, 2nd Edition, P.J. DiNenno, Editor-in-Chief, National Fire Protection Association, Quincy, Massachusetts, 1995.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Solid Flame Radiation Model Process: Solid Flame Radiation Model (Above Ground) Design Input:
air,T∞ = KC 9.2969.23 =o
air,ρ∞ = 1.19 kg/m3 cp = 1 kJ/kg-K g = 9.81 m/s2 D = 1m Heat Release Rate: 1400 kW Distance and height of target to fire:
Distance (m) Height (m) 1.5 2.3
Calculation: Solid Flame Radiation Model
21EFq →=′′& (1) Where: &′′q = incident radiative heat flux (kW/m2)
E = average emissive power at flame surface (kW/m2) F1→2 = configuration factor
( )0.00823D1058E −= (2)
Where:
E = flame emissive power (kW/m2) D = diameter of pool fire (m)
The Heskestad correlation is widely used to determine the flame height of pool fires
1.02DQ0.235H5
2
f −=⋅
(3) Where: Hf = flame height (m) &Q = heat release rate of the fire (kW)
D = diameter of the fire (m)
Distance from center of the fire to edge of the target calculation
2DRL +=
Where: L = distance from center of the pool fire to edge of the target (m) R = distance between pool fire and target (m) D = pool fire distance (m)
The following expressions are used to estimate the configuration factor (or view factor) under wind-free conditions for targets above ground level: (4)
( )( )
( )( )( )( ) ⎟⎟
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
+−−+
−
++−
−⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
−
−−
→
1S1A1S1Atan
1AπS
hA
1S1Stan
πSh
1Sh.tan
πS1
F
1
11
21
11
112
11
V2,1 1
Where:
2S1Sh
A
D
2Hh
D2L
S
221
1
f1
1
++=
=
=
( )( )
( )( )( )( ) ⎟⎟
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
+−−+
−
++−
−⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
−
−−
→
1S1A1S1Atan
1AπS
hA
1S1Stan
πSh
1Sh.tan
πS1
F
2
21
22
22
122
21
V2,1 2
Where:
2S1ShA
D2H
h
D2LS
222
2
f2
2
++=
=
=
And:
L = the distance between the center of the cylinder (flame) to the target (m) Hf = the height of the cylinder (flame) (m) D = the cylinder (flame) diameter (m)
The total configuration factor or (view factor) at a point is given by the sum of two configuration factor as follows:
V22,1V12,1wind)V(no2,1 FFF →→−→ +=
Calc:
H (1m) = 3.24 mf = −0 235 1400 10225. ( ) .kW
mmmL 22
150.1 =+=
( ) 2
)0.00823(1m 91.561058E mkW== −
Vertical distance of target from ground (Hf1) = 2.3m
41m
m)2(2S ==
6.41m
2(2.3m)h1 ==
77.42(4)
1(4))6.4(A22
1 =++
=
( )( )
( )( )( )( )
114.0
14177.414177.4tan
1)77.4(π(4)
)(4.77)(4.6
1414tan
π(4)4.6
1(4)
4.6.tanπ(4)
1
F1
2
1
2
1
V2,1 1=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
+−−+
−
++−
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
−
−−
→
Hf2 = Hf – Hf1 = 3.24m – 2.30m = 0.94m
41m
m)2(2S ==
88.11m
2(0.94m)h 2 ==
57.22(4)
1(4))88.1(A22
2 =++
=
( )( )
( )( )( )( )
077.0
14157.214157.2tan
1)57.2(π(4)
8)(2.57)(1.8
1414tan
π(4)1.88
1)4(
1.88.tanπ(4)
1
F1
2
1
2
1
V2,1 2=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
+−−+
−
++−
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
−
−−
→
191.0077.0114.0F wind)V(no2,1 =+=−→
( )( )′′ = =& . . .q kWm
kWm56 91 0191 10872 2
Results:
Distance Height Heat Flux FDT
(kW/m2) Heat Flux Calculation
(kW/m2) Heat Flux Difference1.50 2.30 10.93 10.87 0.05
Summary/Conclusions: Spreadsheet (05.1_Heat_Flux_Calculations_Wind_Free.xls) for Estimating Radiant Heat Flux From Fire to a Target Fuel at Ground Level Under Wind-Free Condition (Solid Flame 2 Models) (v.1805.0) is valid against hand calculations. Reference: 1) Beyler, C.L., “Fire Hazard Calculations for Large Open Hydrogen Fires,” Section 3, Chapter 1, SFPE Handbook of Fire Protection Engineering, 3rd Edition, P.J. DiNenno, Editor-in-Chief, National Fire Protection Association, Quincy, Massachusetts, 2002. 2) Shokri, M., and C.L. Beyler, “Radiation from Large Pool Fires,” SFPE Journal of Fire Protection Engineering, Volume 1, No. 4, pp.141–150, 1989. 3) Heskestad, G., “Fire Plumes,” Section 2, Chapter 2-2, SFPE Handbook of Fire Protection Engineering, 2nd Edition, P.J. DiNenno, Editor-in-Chief, National Fire Protection Association, Quincy, Massachusetts, 1995. 4) Beyler, C.L., “Fire Hazard Calculations for Large Open Hydrogen Fires,” Section 3, Chapter 1, SFPE Handbook of Fire Protection Engineering, 3rd Edition, P.J. DiNenno, Editor-in-Chief, National Fire Protection Association, Quincy, Massachusetts, 2002.
Purpose: Demonstrate compliance of FDT’s spreadsheet to that of hand calculations for Plume Temperature Process: Heskestad’s Plume Temperature Correlation Design Input:
air,T∞ = Co20 = 293 K
air,ρ∞ = 1.2 kg/m3 cp = 1 kJ/kg-K g = 9.81 m/s2 Afuel = 0.5 m2
65.0=cχ z = 7 m Heat Release Rate: varied for multiple calculations Calculation: Heskestad’s Plume Temperature Correlation
( ) 35
o
32
c
31
2a
2p
a
ane)p(centerlizz
Qρgc
T9.1
TT−
⎟⎟⎠
⎞⎜⎜⎝
⎛
=−
&
(1)
Where: Tp(centerline) = plume centerline temperature (K) Ta = ambient air temperature (K)
cQ& = convective HRR (kW) g = acceleration of gravity (m/sec2) cp = specific heat of air (kJ/kg-k)
aρ = ambient air density (kg/m3) z = elevation above the fire source (m) zo = hypothetical virtual origin of the fire (m) The virtual origin zo, depends on the diameter of the fire source and the total energy released, as follows:
DQ0.0831.02
Dz
52
o
⋅
+−= (1)
Where: zo = virtual origin (m) D = diameter of fire source (m)
Q& = total HRR (kW) For non-circular pools, the effective diameter is defined as the diameter of a circular pool with an area equal to the actual area given by the following equation:
π4AD f=
Where: D = diameter of the fire (m) Af = fuel spill area or curb area (m2) HRR: 1251 kW
( ) mmD 7978.05.04 2
==π
( ) ( ) mkWmzo 6264.01251083.07978.002.1 5
2=+−=
kWkWQQ cc 15.813)65.0)(1251( ===⋅⋅
χ
( )( )
( )KK
mm
kW
mkg
KkgkJ
sm
K
T centerlinep 42.3922936264.07
15.8132.1181.9
2931.9
35
32
31
2
3
2
2
)( =+
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛⎟
⎠⎞⎜
⎝⎛
⋅=
HRR: 1706 kW
( ) mmD 7978.05.04 2
==π
( ) ( ) mkWmzo 8151.01706083.07978.002.1 5
2=+−=
kWkWQQ cc 9.1108)65.0)(1706( ===⋅⋅
χ
( )( )
( )KK
mm
kW
mkg
KkgkJ
sm
K
T centerlinep 16.4312938716.07
7.12072.1181.9
2931.9
35
32
31
2
3
2
2
)( =+
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛⎟
⎠⎞⎜
⎝⎛
⋅=
HRR: 1858 kW
( ) mmD 7978.05.04 2
==π
( ) ( ) mkWmzo 8716.01858083.07978.002.1 5
2=+−=
kWkWQQ cc 7.1207)65.0)(1858( ===⋅⋅
χ
( )( )
( )KK
mm
kW
mkg
KkgkJ
sm
K
T centerlinep 16.4312938716.07
7.12072.1181.9
2931.9
35
32
31
2
3
2
2
)( =+
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞⎜
⎝⎛⎟
⎠⎞⎜
⎝⎛
⋅=
Results: HRR (kW) FDT (K) Calculation (K) Temp. Difference (K)
1251 392.11 392.42 0.31 1706 421.19 421.54 0.35 1858 430.79 431.16 0.37
Summary/Conclusions: Spreadsheet (09_Plume_Temperature_Calculations.xls) for Estimating Centerline Temperature of a Buoyant Fire Plume (v. 1805.0) is valid against hand calculations.
Reference: 1) Heskestad, G., “Fire Plumes,” Section 2, Chapter 2, SFPE Handbook of Fire Protection Engineering, 2 Edition, P.J. DiNenno, Editor-in-Chief, National Fire Protection Association, Quincy, Massachusetts, 1995.