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in mi mi mi ii iiiiii ii ii min in mi AlllDfc, 031L77

NBSIR 85-3134

The Buoyant Plume-Driven

Adiabatic Ceiling Temperature

Revisited

Leonard Y. Cooper and

Anne Woodhouse (Guest Worker, Univ. of MD)

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

April 1985

•QC

100

, U56

85-3134

1985

J.S. DEPARTMENT OF COMMERCE

IATIONAL BUREAU OF STANDARDS

Rational burea

Pf standards

library

NBSIR 85-3134

A • A THE BUOYANT PLUME-DRIVEN

ADIABATIC CEILING TEMPERATURE

REVISITED

Leonard Y. Cooper and

Anne Woodhouse (Guest Worker, Univ. of MD)

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

April 1985

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Emmt Ambl«r. Director

TABLE OF CONTENTS

Page

List of Figures iv

Nomenclature v

Abstract 1

1. INTRODUCTION 1

2. A RE-EVALUATION OF EXISTING DATA 4

3. A FORMULA FOR ATjd

(r/H) AND AN INDEPENDENT INDICATION OF THE

THE "TRUE” VALUE OF e 8

4. CONCLUSIONS 10

5. ACKNOWLEDGMENTS 11

6. REFERENCES 12

APPENDIX 13
LIST OF FIGURES

Page

Figure 1. Heat transfer to an unconfined ceiling 18

Figure 2. Plot of the h data of [5], 7.3(1 0^ ) < Re^ < 27(10^),

and the estimate of Eq. (6) - in the stagnation zone,

0 _< r/H < 0.2 19

Figure 3. Plot of the h data of [5], 7.3(10^) < Rejj < 27(10^),

and the estimate of Eq . (6) - outside the stagnation

zone, 0 .2 <_ r/H 20

* , Figure 4. Plots of the new AT , data for e = 0 (test 1: 0,

test 2: A, test 3: a X [2]) 21

* Figure 5. Plots of the new AT , data for e = 1 (test 1: 0,

test 2: A, test 3: a X [2]) 22

Figure 6. Plots of the normalized AT , data of [9] and the

estimate of Eq. (7) ..••••? 23

* Figure 7. Plot of the new AT , data for e = 0^1 (test 1: 0,

test 2: A, test 3? X [2]), the AT estimate of

Eqs. (7) and (8) ( ), and the correlation

(Eqs. (11) and (12) of [3]) for the measured, long-time,

near-ceiling gas temperatures of [2] ( ) 24

-iv-
NOliENCLATURE

A, B constants, Eq . (2)

b nozzle-to-wall separation distance

Cp specific heat at constant pressure

f, fj, functions, Eqs. (7) and (A2) , respectively

g acceleration of gravity

H plume source-to-ceiling distance

h heat transfer coefficient, Eq . (1)

h characteristic heat transfer coefficient, Eq . (4)

k kinematic momentum flux of jet

Pr Prandtl number

Q energy release rate of fire

Qj enthalpy flux of jet

Q* dimensionless Q, Eq . (4)

q rate of heat transfer, Eq . (1)

q c rate of in-plate radial heat transfer

q^ rate of heat transfer to insulation

Reg Reynold’s number, Eq . (4)

Re^ Reynold’s number defined in [3, 5]

r radial distance from impingement point

* 1/2 r jet where velocity is half of axial value

T

a g

surface temperature for adiabatic ceiling

Tamb ambient temperature

T

g ceiling surface temperature

t time

V velocity profile of wall/ceiling jet

V „ maximum of V

max

-v-
function, Eq . (2)

b divided by diameter of jet orifice

distance from wall

constant, Eq. (A4)

(largest) z where V/Vmax = 1/2

plate emissivity

rate of heat transfer retained in plate

dimensionless T

a(j, Eq . (5)

fraction of Q lost by radiation

kinematic viscosity

Stephan-Bolzmann constant
THE BUOYANT PLUME-DRIVEN ADIABATIC CEILING TEMPERATURE REVISITED

Leonard Y. Cooper and Anne Woodhouse

Abstract

In previous works, convective heat transfer from buoyant plume-driven

ceiling jets to unconfined ceilings has been estimated using a formula for the

temperature distribution below an adiabatic ceiling, T

a(j, obtained from experimental data in the range 0 <_ r/H <0.7 (r is the radial distance from the

plume, H is the plume source-to-ceiling distance). The present study reevaluates this data, and develops an independent estimate for T

a

^. The

analysis takes account of the effect of ceiling surface re-radiation, and use

is made of the previously established similarity between plume/ceiling- and

jet/wall-driven heat transfer phenomena. The latter similarity is the basis

of a correlation of recently reported free jet-wall jet "recovery temperature"

data into a normalized T

a(j distribution. All of the analysis leads to new

formulae for estimating the convective heat transfer to ceilings during enclosure fires. These new results confirm previous formulae, and extend them into

the larger range 0 _< r/H _< 2.2.

1 . INTRODUCTION

The convective heat transfer to ceilings during enclosure fires can be

related to the heat transfer to unconfined ceiling surfaces from buoyant

plume-driven ceiling jets [1]. Results of unconfined ceiling experiments [2)

suggest a relatively simple means of estimating this unconfined ceiling heat

transfer rate, q, to an unconfined non-adiabatic ceiling, as

- 1 -
<1 - h ( Tad - V ( 1 )

Figure 1 depicts the phenomena in question near the fire plume. In Eq. (1),

T

ad is defined as the temperature distribution (a function of radial distance,

r, from the plume-ceiling impingement point) established at the surface of an

adiabatic ceiling (i.e., perfectly insulated and with zero absorbtivity and

emissivity) by the gas flow from the plume of a fire of strength, Q, and

effective, fire-to-ceiling distance, H, T is the instantaneous lower surface

ceiling temperature distribution, and h is the heat transfer coefficient based

on the T

a(j - T

g temperature difference. In the experiments [2] the transient

thermal response of unconfined 0.00159 m thick, cold-rolled steel plate

ceilings, insulated on the top side and initially at ambient temperature were

measured when heated by weakly radiating premixed propane fires of constant

strength between 1.17-1.53 kW (H's were between 0.58 and 0.81 m) . Analysis of

the data which led to the Eq. (1) formulation used long-time, "near"-ceiling

gas temperatures to estimate T (t + »), which was then used as a surrogate for

s the T

a(j distribution.

The original h results [2] did not distinguish between variations in the

characteristic Reynold's number, Re^, of the plume (which only varied in the

narrow range 1.5(1 0^ ) < Re^ < 2.0(10^) in the laboratory experiments). In a

subsequent study [3], it was noted that inertial forces compared to buoyancy

forces are generally large (i.e., characteristic Richardson numbers are small)

in plume-driven ceiling jets for the most interesting range of small to

moderate values of r/H. For this range it was conjectured that the nearceiling flow and heat transfer characteristics of interest could be directly

related to the analogous characteristics of wall or ceiling jets driven by
heated or unheated free turbulent jets (e.g., the same configuration of Figure

1, but with a turbulent jet replacing the fire and its buoyant plume). This

led to a proposed correlation for h which does depend on the impingement point

Reynold's number, Re^, namely

I

A Re"

1/2 Pr"

2/3

(1 + B r/H) , 0 < r/H < 0.2;

H.

( 2 )

C Re”

0,3 Pr“

2/3 w(r/H), 0.2 < r/H

n

where A and C are numerical constants, B = B(Re^) , h/h is a continuous

function of r/H and

lim w ~ (r/H)

^* 2

(3)

r/H -> ®

In the above

£ _ n r 1/2 „l/2 *1/3

h * pambC

p

8 H Q

R.h

- g

1/2 H3/2 Q* 1/3 /v (4)

Q* - (1 - X )Q/(p C T g

1/2 H5/2

) r amb p amb

where Xr is the fraction of Q which is lost by radiation from the fire's

combustion zone and from the plume itself, i.e., (1 - X r ) Q is the effective

portion of Q which actually drives the plume's upward convection. Also,

Pamb> Cp> ^amb» v > an^ ?r are the density, specific heat at constant pressur- ,

temperature, kinematic viscosity and Prandtl number of the ambient environment.

- 3 -
The above correlation was based on measured heat transfer (to an

elevated-temperature wall) [5] and flow properties [10] of unheated free jetdriven wall jets (where T

a(j = Tam^) , and on the application of a criterion of

equivalence for free jets and buoyant plumes at their respective sections of

wall/ceiling impingement (Appendix of [3]).

With the data of [2], data from the similar "reduced-scale" fire-plumedriven heat transfer experiments of [4]

,

and unheated, free jet-driven wall

jet heat transfer measurements of [5], it was shown in [3] that with the

proposed h correlation of Eq. (2), the Eq. (1) formulation would provide

reasonable q estimates for Re^ at least in the range 0.73(104 )-27(10 4

)

.

(Full scale hazardous fires with fire-to-ceiling distances of 2-3 m and

convected energy fluxes of 200-2000 kW would have Re^'s in the range

20(104

) < ReH < 50(104

) [1]).

In [3] the h values from the [2] and [4] experiments were derived from

Eq. (1) with the use of measured values of q, Ts , and the long-time, "near"-

ceiling gas temperature measurements of [2] which were assumed to be representative of T a(j. The h values from the free-jet experiments of [5] were derived

in a similar way, but, as mentioned earlier, the values of T

fl(j in this case

required no special measurements as they are identically equal to Tam^.

2. A RE-EVALUATION OF EXISTING DATA

The present investigation involves a re-evaluation of the data of [2]

within the context of the Eq. (1) formulation. The goal is to obtain new

estimates for Ta<j and h which would 1) be consistent with an Eq. (2)-type of

-4-
ReH-dependent h; 2) would only use the initial transient data (t 1, 2, 3, 5

and 7 min) of [2], and not the 25-30 min, long-time, near surface gas temperatures which were previously used to represent the "true" T

a(j [2]; 3) would

include the previously neglected effect of exposed ceiling surface re-radiation to far-field surfaces; and which would 4) use recently reported data of

free jet-driven wall jet heat transfer experiments to extend the r/H range of

validity.

With regard to item 2) above, there are two reasons for avoiding the use

of the long-time measurements in a new analysis of the data of [2]. First,

there is indication from the transient data that in-plate radial conduction is

important even at steady-state for r/H at least up to 0.25. Also, a non-zero

plate emissivity could lead to non-negligible plate radiation to the farfield. Under either of these circumstances the steady-state plate response

would differ from that of an adiabatic ceiling. Second, all the reported

transient data are from thermocouples mounted directly on the plate itself,

whereas the long-time temperature data in reference [2] were measured in the

gas at a distance below the plate surface of 0.0016 m.

Consistent with the above remarks, another analysis of the tabulated data

of [2] was carried out. The analysis involved application of energy conservation of elemental volumes of the plate, at the seven locations of platemounted thermocouples (placed at .07 m radial intervals from the point of

plume-ceiling impingement), and at different times during the plate heating.

New, dimensionless T

a^ data were generated from the expression

-5 -
( 5 )

AT

(T - T

ad amb'

ad

Tamb Q

*27T

( T

s Tamb^

T wn

amb Q hT

ambQ

JJJ + q

c + AE + eo(T

4 - T , s amb )]

where o is the Stef an-Bolzmann constant and where q^ , q c and AE are the

respective instantaneous rates of heat transfer per unit area of plate

conducted and lost to the rear insulation, conducted radially outward, and

absorbed and retained in the plate itself. The right-hand term of Eq. (5)

represents the lower plate radiation to far-field ambient temperature surfaces

which are taken to be at the ambient temperature, and e is the effective plate

emissivity

.

A review of the data of [2] indicates that by seven minutes into the

experiments a value of e greater than a few tenths would have led to a significant contribution (on the order of 35 percent for e = 1) to the net value of

the bracketed term of Eq. (5). It would therefore appear that an Eq. (5)-type

of analysis of the data requires an estimate for the plate's e. Unfortunately, the literature has little to offer for e values of cold-rolled steel

plate. Indeed, while tables of the e's of solid surfaces typically provide

values of 0. 6-0.8 for rolled steel plate of various descriptions, the authors

found only one primary reference [6] for the e of cold-rolled steel. Such

an e is reported to have been measured at the remarkably low values of 0.075

(93°C) and 0.085 (260°C). (Reference [7] reports these results of [6]

incorrectly as 0.75-0.85.) To confirm this the authors had the e of available

samples of "old and uncleaned" cold-rolled steel measured at the National

Bureau of Standards. A room temperature value of 0.12 was obtained [8]. As

-6 -
will be seen below, the new analysis of the data of [2] suggests that the

"true" e of the plate of [2] was indeed in the range of these low values.

Except for e (assumed for now to be unknown) and h, all terms on the

right hand side of Eq. (5) are available from tabular data presented in [2].

To carry out the Eq. (5) analysis the following h distribution was used

8.82 Re^

1/2

Pr“

2/3

[l - (5.0 - 0.284 Re°*

2 )(r/H)], 0 < r/H < 0.2

h/h = ( 6 )

0.283 Re

-0.3 „_-2/3

H

-1.2 (r/H - 0.0771)

Pr (r/H) ’ (V/HTT27 9

r

’ °* 2 * r/H

This expression, which is in the form of the correlation of Eqs. (2)— (3)

,

was developed by using the free jet/buoyant plume equivalency relations

proposed in Eqs. (A9) and (A10) of [3] and by a curve fitting of the free-jetdriven wall jet h data of [5] (7.3(10^) < Re^ < 27(10^)). Plots of the h data

of [5] and the Eq. (6) correlation are presented in Figures 2 and 3 (residual

standard deviations, rsd, of .87 and .04, respectively) for r/H values within

(0 < r/H < 0.2) and outside the stagnation zone, respectively.^

^Note that the data of [5] were incorrectly plotted in Figures 6-8 of [3].

Instead of Re^*^ Pr^^ h/K and Re^'^ Pr^^ h/K as indicated on the

ordinates of Figures 6-7 and Figure 8, respectively, the actual plots are of

Re^’^ Pr^/^ h/K and Re^*^ Pr^^ h/K, where Re^ is defined in [3, 5] and

is approximately Re^ = 0.422 Re^. All of the data are correctly plotted lu-r.*

in Figures 2 and 3.

- 7 -
The new AT^ values were computed from Eqs. (5) and (6) and the data of

Tables IV-IX of [2] for different assumed values of e ranging from 0 to 1.

Figures 4 and 5 are plots of these data for the e = 0 and e 1 cases

,

respectively.

3. A FORMULA FOR AT* d (r/H) AND AN INDEPENDENT INDICATION OF

THE "TRUE" VALUE OF e

The correspondence between the flow and heat transfer characteristics of

plume-driven ceiling jets and heated or unheated free turbulent jet-driven

wall jets was mentioned earlier. Relative to this correspondence which was

conjectured and supported in [3], and which was introduced into the present

analysis via the Eq. (6) description for h, important new experimental results

have been reported recently in [9]. Using data from that work, which studied

heated jet-driven wall flows, the development of the following expression for

* the normalized AT

^ distribution was presented in the Appendix

AT

ad

(r/H)

= f(r/R)

1-1 . 10(r/H)°*

8 + 0.808(r/H)

1,6

(?)

AT* (r/H-0) 1-1. 10(r/H)

0 * 8

+ 2.20(r/H)

1,6 + Q.690(r/H)

2,4

ad

This result is plotted in Figure 6 together with the (equivalent) data from

which it was derived.

A To use an Eq. (7) correlation for the new AT^ values it is necessary to

determine the "best" value for the e, AT ,(r/H=0) pair. Toward this end the

ad

* following procedure was carried out: 1) compute new AT^ values for a given

e as discussed above; 2) obtain the corresponding AT

a(

j(r/H=0) which provides a

least squares, Eq. (7) fit to the new values.

-8-
Carrying out the procedure led to the following results: for t in the

range 0 < e < 1, the values of AT*^(r/H=0) and the rsd increase monotonically

with e over the relatively narrow ranges from 8.59 and 0.56 to 9.75 and 0.78,

respectively. Thus, the present correlation is consistent with "very low"

values of plate emissivity for which the smallest rds’s are obtained.

The above procedure favors e = 0 as the "best" or "true" value. However,

in view of the above-mentioned results of [6] and [8]

,

AT* (r/H = 0) = 8.70 (8) ad

which corresponds to e = 0.1 and a rsd of 0.57 would appear to be a preferred

value for the normalizing factor of Eq. (7). Using Eqs. (7) and (8) leads to

* a new estimate for AT

^ which is plotted in Figure 7 along with the new,

*

e = 0.1, AT , data,

ad

Also plotted in Figure 7 is a correlation from [3] of the measured, latetime, near-ceiling gas temperatures of [2]. As can be seen, except for small

* r/H values the new estimate for AT^ is close to, and slightly above the

measurements. Near the stagnation point, however, the measured temperatures

generally suggest a plate temperature distribution with relatively large

radial gradients and with an r/H = 0 value which actually exceeds the new

* AT^. The large gradients can be explained by the non-adiabatic, in-plate,

radial conduction effect discussed earlier. However, near-ceiling gas temperatures would never have exceeded T

a(j, and the fact that they appear to do so

* would seem to disqualify the Eq. (7) estimate of AT^ near the stagnation

point.

- 9 -
In [2] it was noted that "appreciable error" in the estimates of in-plate

radial conduction (q^ of Eq. (5)) "has to be expected" near r/H - 0. This

fact together with the above discussion and previous remarks suggest 1) that

the long-time near-surface temperature measurements of [2] continue to be the

* basis of the AT^ estimate within the stagnation zone, albeit with some

uneasiness; 2) the latter estimate be continued at r/H = 0.2 by a new Eq. (7)-

* type AT ^ estimate which would be used outside the stagnation zone; and

3) that the selection of an optimum normalizing factor to use in Eq. (7) be

* made without consideration of those new AT

&^ values which correspond to the

two smallest radial positions, r = 0 and r = 0.07 m, of the experiments.

The above ideas were implemented, and they resulted in the following

* newly recommended estimate for AT ,

ad

10.22 - 14.9 r/H, 0 < r/H < 0.2;

8.39 f(r/H), 0.2 < r/H

(9)

where f(r/H) is defined in Eq. (7).

4. CONCLUSIONS

* The new AT^ estimate of Eq. (9) provides an experimentally based

extension of earlier results from a maximum r/H value of 0.7 to a r/H value of

2.2. The results of the present analysis provide independent confirmation

that, except for some uncertainty in the stagnation zone, the measured, longtime, near-ceiling gas temperatures of [2] accurately represent values of T

a(j,

- 10-
and that ceiling re-radiation in those experiments was negligible because of

very low effective values of surface emissivity. Also, the results of this

paper are consistent with and provide additional support for the notion,

introduced in [3], that up to moderate values of r/H there is an equivalence

between the flow dynamics and heat transfer characteristics of free jet-driven

wall flows and buoyant plume-driven ceiling jets.

A recommended procedure for estimating the convective heat transfer from

a fire plume to an unconfined ceiling, from the point of plume impingement to

moderate r/H was developed in [3]. This procedure is still recommended.

However, the new results for Tad and h, developed here and presented in Eqs.

(6), (7) and (9) should replace the corresponding Eq. (11), (12), (17), (18)

and (20) estimates of [3]. The new results should also be used to modify Eqs.

(5), (6), (16) and (17) of [1] where the unconfined ceiling equations are used

to estimate heat transfer to the confined ceilings of real compartment fire

scenarios.

5. ACKNOWLEDGMENTS

The authors acknowledge C. Marks of the University of Maryland for his

valuable advice, and D.W. Stroup, S. Thorne and D. Walton who designed and

carried out preliminary experiments and data analysis which led to the ideas

presented here. The work was supported by the U.S. Department of Health and

Human Services, the Bureau of Mines and the National Park Service of the U.S.

Department of Interior, and the Federal Aviation Administration of the U.S.

Department of Transportation.

- 11 -
6. REFERENCES

[1] Cooper, L.Y. , Convective Heat Transfer to Ceilings Above Enclosure

Fires, 19th Symp. (Inter.) on Combustion, p. 933 (1982).

[2] Veldman, C.C., Kubota, T. and Zukoski, E.E. , An Experimental

Investigation of the Heat Transfer from a Buoyant Gas Plume to a Horizontal Ceiling - Part 1. Unobstructed Ceiling, Cal. Inst. Tech.

,

Nat. Bur. Stand. (U.S.) NBS-GCR-77-97 (1975).

[3] Cooper, L.Y., Heat Transfer from A Buoyant Plume to an Unconfined

Ceiling, J. Heat Transfer, 104, p. 446 (1982).

[4] You, H-Z., and Faeth, G.M. , Ceiling Heat Transfer During Fire Plume and

Fire Impingement, Fire and Materials, 103, p. 140 (1979).

[5] Donaldson, C. DuP. , Snedeker, R.S., and Margolis, D.P. , A Study of Free

Jet Impingement. Part 2. Free Jet Turbulent Structure and Impingement

Heat Transfer, J. Fluid Mech. , 45, Part 3, p. 477 (1971).

[6] McDermott, P.F., An Apparatus for Measurement for the Total Normal

Thermal Emissivity of Sheet Materials in the Temperature-Range

140-500°F, Rev. of Scientific Instruments, 8, p. 185 (1937).

[7] Gubareff, G.G., Janssen, J.E. and Torborg, R.H. , Thermal Radiation

Properties Survey, Honeywell Research Center, Minneapolis, p. 98 (1960).

[8] Roberts, W. , National Bureau of Standards, private communication.

[9] Hollworth, B.R., and Wilson, S.I., Entrainment Effects on Impingement

Heat Transfer. Part I: Measurements of Heated Jet Velocity and

Temperature Distributions and Recovery Temperatures on a Target Surface,

J. Heat Transfer, 106, p. 797 (1984).

[10]

Poreh, M. , Tsuei, Y.G., and Cermak, J.E., Investigation of a Turbulent

Radial Wall Jet, ASME Journal of Applied Mechanics, June 1967,

pp. 477-512.

- 12-
APPENDIX

T

a(j for the Wall Jet Away from the Stagnation Zone

Consider a wall jet driven by a heated turbulent jet impinging on a plane

adiabatic surface. Let be the constant enthalpy flux of the jet relative

to T

am^. Assume that jet temperatures near the impingement point are close

enough to Tam^ so that, for the purpose of applying conservation of energy to

the wall jet, the density field and Cp can be approximated by their ambient

values. Then

T - T

Q. = 2irrp ,C V (T

j amb p max'' ad

- T

amb ) /

o

amb

ad

- T

dz (Al)

amb

,

max

where the integration is over the thickness of the wall jet at the radial

position of interest. In the above, V is the distribution of the radial

component of velocity and Vmav. is its maximum value. T

a(^ is the temperature

distribution at the wall surface.

Let 6 be the distance from the wall, at the outer portion of the wall

jet, where V/VTnav = 1/2. Then the experimental study of [10] for unheated

jets indicates that V/Vmax

profiles at different r stations are similar in the

sense that they can be closely approximated by

V/V = f.(z/6) for r/b > 0.75

max 1 (A2)

- 13-
where b is the effective jet nozzle-to-wall separation distance. We assume

that this latter result is applicable even for heated jets since momentum

forces will typically dominate buoyancy forces in the zero to moderate r/b

values of interest. Furthermore, it is reasonable to assume that ( T-T

amb^/

< Tad-Tamb^ can a-*- so t> e approximated as a function of z/6 in the r/b > 0.75

range. Using the latter assumptions in Eq. (1) leads to

T - T = a Q./(p ,C V r6) , r/b > 0.75 ad amb j amb p max

] f-d(f) o \ ad amb / max

Besides Eq. (A2), other results in [10] of interest here are

vmax/(^/b) = 1.32 (r/b)" 1,1

; 5/b = 0.098 (r/b) 0,9

(A5)

where Pam^K is the (constant) momentum flux along axial sections of the

unheated jet prior to impingement.

For the case of the heated jet it is reasonable to expect that the

unheated jet results of Eqs. (A5) would still be valid provided K and b are

redefined. This could be done, for example, by drawing an equivalence at the

jet/wall impingement point between a heated jet of interest and the unheated

jets for which Eqs. (A5) are valid. This notion of equivalence was used,

e.g., in [3]. There, a buoyant plume (i.e., the far field of a limiting,

small-K or low-momentum jet) was taken to be equivalent to an unheated jet at

respective points of wall/ceiling impingement if the total mass and momentum

flux of the plume and jet were identical. Applying these criteria for equiva-

(A3)

(A4)

- 14-
lence led to the results in [3] that the flow of a buoyant plume-driven

ceiling jet, at least up to moderate values of r/H, would simulate the flow of

an unheated, free jet-driven wall jet if b and K in the results of [10] were

replaced by point-heat-source-driven plume parameters as follows

K

1/2

= 0.636 g

1/2 H3/2 Q* 1/3

; b = 1.17 H (A6)

Following the above, it is assumed that the idea of equivalence at

impingement between the K, b pair of an unheated jet and related parameters of

a heated jet has some generality. Accordingly, Eqs. (A5) are used in Eq. (A3)

with the result — . , 7 . 73 a ( r /b)

-0 * 8

; r/b < 0.75 (A7)

V(p-bV /K)

For the case of a buoyant plume, the equivalence relationships of Eq. (A6)

together with the substitution of (l - Xr )Q for Qj are used in Eq. (A7) to

yield

11.8 a(r/H)" 0,8

; r/H > 0.88

* A Normalized AT , Distribution

ad

(A8)

T

a(j data for heated jet-driven wall jets have been recently acquired and

plotted in Figure 9 of [9]. This plot is in the form

(r) - T

(0) - T

amb

amb

(A9)

- 15-
where * 1/2 is defined as the radial position of the impinging jet, at the

plane of impingement, where the jet velocity, in the absence of the wall,

would have dropped to 1/2 of its value on the jet axis. Plotted in this way,

the data (r/r^2 ^ 20.6) correlate well for all test cases where the ratio of

jet orif ice-to-wall-distance to jet orifice diameter, Z, exceeds 1 (i.e., for

jets of different Qj and with Z’s of 5, 10, and 15). Interpreting Eq. (A7) as

an asymptotic result for large r/rj^* the following curve fit to the Z > 1

data of [9] has been obtained with a residual standard deviation of 0.02

In the experiments of [9], buoyancy forces compared to inertial forces in

the impinging downward-directed jet were reported to be small enough so that

within jet axis distances of interest, centerline velocities varied on the

order of a few percent at most when the jet was turned upward. Accordingly it

is reasonable to assume that the velocity distributions (e.g. , the values of

r

- 1 / 2

) of the jets at impingement would be closely approximated by the corresponding velocity distributions of similarly configured but unheated free

jets. Equation (A9) of [3] suggests the following equivalence between the

r ^/2 °f an unheated jet and the H (distance from the source) of a buoyant

plume

l-0.199(r/r

1/2

)°* 8

+ 0.0258(r/r

1/2 ) 1,6

l-Q.199(r/r

1/2 ) 0 * 8

+ Q.0640(r/r

1/2 ) K6

+ 0.00443 0/r

1/2 ) 2 * 4

(A10)

r 1/2 = 0.109 H (All)

- 16 -
Using this in Eq. (A10) suggests the normalized AT , distribution for plumead

driven ceiling jets which is presented in Eq. (7). This together with the

data of [6] (made equivalent with the use of Eq. (All)) is plotted in Figure

6 .

- 17 -
-18

Figure

1.

Heat

transfer

to

an

unconfined

ceiling
10.00 - 19-

Figure

2.

Plot

of

the h

data

of

15),

7.3(1(T) <

ReH<

27(10 4

),

and

the

estimate

of

Eq.

(6)-

In

the

stagnation

zone, Q<

r/H <

0.2
20

Figure

3.

Plot

of

the h

data

of

[5],

7.3(10 )<

Re^ <

27(10

),

and

the

estimate

of

Eq.

(6)-

outside

the

stagnation
i i i i

i ri n i

i i i i ii i ii i

i i i i 00

<3X1

<W<H]

CO

o

>®x

<M

o

CM

O

<]<fl

®*ID

Lj i i i-i i i i i i i i i i l o

If) CM

O o

LO

o

o

o

If)

if) CM

O

d

o

o

pe

*1V

- 21 -

Figure

4.

Plots

of

the

new

AT

,

data

for

e

“0

(test

test

2:

A,

test

3:^C

12]

)
12.50

i i i i

|

i i i i

| r~ i i i

i

i i i i

i

i i

1

1

-

<3 <3 <3tE

— < <3 <33

- XXSSCDX

<3 <3 <333

]p<m

/ xym&

<33 <3<33

acos

-

-» ^ob&«f 1 1

1

i i i 1 i i i i 1 i i i i.

00

o

CO

a o

o

CM

o

o

o

o

o

o

in

o

o

LO

o

in

CM

o

o

pe

* IV

-22-

Figure

5.

Plots

of

the

new

AT

,

data

for

e

1

(test

test

2:

A,

test

3:

aaX

[2])
o

Q QD <D CM O

-r-

. O

. O

• O...O O

DB pB

(0 = H/J)

# 1\7/(H/J) # 1V

- 23 -

Figure

6.

Plots

of

the

normalized

AT^

data

of

[9]

and

the

estimate

of

Eq

.

(7)
12.50

24 -

Figure

7.

Plot

of

the

new

AT,

data

fore=

0^1

(test

1:

0,

test

2:

A,

test

3fX

[2]),

the

AT

estimate

of

Eqs.

(7)

and

(8) (

),

and

the

correlation

(Eqs.

(11)

and

(12)

of

13])

for

the

measured,

long-time,

near-ceiling

gas

temperatures

of

[2] ()
NBS-) 14A (REV. 2 -80) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

4. TITLE AND SUBTITLE

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

NBSIR 85-3134 April iy85

The Buoyant Plume-Driven Adiabatic Ceiling Temperature Revisited

5. AUTHOR(S)

Leonard Y. Cooper and Anne Woodhouse

6. PERFORMING ORGANIZATION (If joint or other than N BS, see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS _

DEPARTMENT OF COMMERCE S WASHINGTON, D.C. 20234

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

. Type of Report & Period Covered

10.

SUPPLEMENTARY NOTES

QJ Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

In previous works, the convective heat transfer from buoyant plume-driven ceiling

jets to unconfined ceilings has been estimated using a formula for the temperature

distribution below an adiabatic ceiling, Tad’ obtained from experimental data in

the range 0 < r/H < 0.7 (r is the radical distance from the plume, H is the plume

source-to-ceiling distance). The present study re-evaluates this data, and develops

an independent estimate for Tacj. The analysis takes account of the effect of ceiling

surface re-radiation, and use is made of the previously established similarity

between plume/ceiling- and jet /wall-driven heat transfer phenomena. The latter

similarity is the basis of a correlation of recently reported free jet-wall jet

"recovery temperature" data into a normalized Ta(j distribution. All of the analysis

leads to new formulae for estimating the convective heat transfer to ceilings

during enclosure fires. These new results confirm previous formulae, and extend

them into the larger range 0 < r/H < 2.2.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semco’on \ Ceilings; compartment fires; convective heat transfer; enclosure fires; fire

plumes; fire modeling; heat transfer; room fires; walls

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