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.