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Masters Theses Student Theses and Dissertations
1970
Heat transfer through a spherical gas shell Heat transfer through a spherical gas shell
Adel Nassif Saad
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HEAT TRANSFER THROUGH A SPHERICAL GAS SHELL
by
ADEL NASSIF SAAD, 1941-
A
THESIS
submitted to the faculty of
UNIVERSITY OF MISSOURI-ROLLA
in partial fulfillment of the requirements for the
Degree of
~mSTER OF SCIENCE IN MECHANICAL ENGINEERING
Rolla, Missouri
1971
Approved by
(advisor} ( 7
ii
ABSTRACT
The radiative transfer through an absorbing-emitting
gas shell contained within black concentric spheres at
uniform but different temperatures was investigated. A
close form exact solution and an approximate solution
were derived for the case of isothermal gas layer. The
two solutions appear to compare well specially at low
diameter ratios, and both agree with the radiative equili
brium solution in the thin limit.
An approximate method was developed for the radiative
equilibrium, non-isothermal gas layer, of the above problem.
The method is based on the assumption of a hyperbolic
emissive power distribution through the gas shell.
The jump boundary conditions were applied to calculate
the constant coefficients. The results of this approxi
mate method compare very well with the exact radiative
equilibrium solution.
The combined problem of radiative and convective
energy transfer between two concentric spheres was investi
gated experimentally. Helium was used as a non-partici
Pating gas to evaluate the natural convection contribution.
The results compare favorably with the predicted values
specially at higher pressures. Carbon dioxide was used
to evaluate the radiation contribution. The sum of the
predicted nutural convection and radiation was within
11 percent of the experimental results.
iii
ACKNOWLEDGEMEN'l'S
Deepest gratitude is expressed to my brother Dr.
Afif H. Saad and his wife Mrs. Linda C. Saad for their
financial support, encouragement and assistance in all
phases of my graduate program.
iv
The author is particularly grateful to his advisor,
Dr. Bassern F. Arrnaly, for his guidance, advice and always
ready assistance during this research.
The author wishes also to express his sincere appre
ciation to Dr. A.L. Crosbie for his assistance and valuable
suggestions throughout the preparation of this thesis, to
Dr. C.Y. Ho for his participation in the Oral Committee,
and to Mrs. Connie Hendrix for typing the manuscript.
This work is dedicated to my mother, Martha, for
her invaluable encouragement and endurance, shown during
the tenure of my entire education.
v
TABLE OF CONTENTS
Page
ABSTRACT . ii
ACKNOWLEDGEMENTS iv
LIST OF FIGURES vi
LIST OF TABLES .viii
NO!'-lENCLATURE . ix
I. INTRODUCTION 1
II. REVIEW OF LITERATURE 3
III. ISOTHERMAL GAS ANALYSIS 8
IV. NON-ISOTHERMAL GAS AND RADIATIVE EQUILIBRIUM 23
V. DESCRIPTION OF THE EXPERIMENTAL APPARATUS . 35
VI. EXPERIMENTAL PROCEDURE AND DATA REDUCTION . 43
VII. RESULTS AND DISCUSSION 54
VIII. CONCLUSIONS AND RECOMMENDATIONS . 57
IX. APPENDICES . 59
A. Isothermal and Non-isothermal Gas Analysis Relations and Integrations 60
B. Application of the Diffusion Approximation to Radiative Transfer Through a Spherical Shell of an Absorbing-Emitting Gray Medium with Jump Boundary Conditions . 67
c . Tab 1 e s . 7 3
D. Experimental Data and Results . 83
BIBLIOGRAPHY 86
VITA . 89
Figure
1.
2 .
3.
4 .
5 .
6 •
7.
8.
9 .
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
LIST OF FIGURES
Physical System, Concentric Spheres .
Concentric Spheres, Coordinate System . Isothermal Gas Flux Function for ( T 1 ; T
2 )= 0 .1 .
Isothermal Gas Flux Function for (Tl/T2)= 0.2857 . Isothermal Gas Flux Function for (-r 1 /T 2 )=0. 5.
Isothermal Gas Flux Function for (T 1 /-r 2 )=0.9.
Radiative Equilibrium Flux Function for (-r
1/-r
2)=0.1 . . .
Radiative Equilibrium Flux Function for (-r 1/-r 2 )=0.2857
Radiative Equilibrium Flux Function for (-r
1/-r
2)=0.5
Radiative Equilibrium Flux Function for (-r
1/-r 2 )=0.9 .
Experimental Set-Up .
Schematic of Experimental Set-Up
Concentric Spheres Assembly .
Cross Section of the Inner Sphere .
Instruments and Electrical Circuit Diagram.
Conduction Losses .
Inner Sphere Temperature, Vacuum Run
Emissivity Factor of the Concentric Spheres
Experimental Results for Helium and Carbon Dioxide Run .
vi
Page
9
15
17
18
19
20
30
31
32
33
36
37
38
39
42
45
46
48
49
Figure
20.
21.
22.
LIST OF FIGURES (continued)
Heat Transfer Contributions for Helium .
Heat Transfer Contributions for Carbon Dioxide .
Spherical Shell Coordinate System
vii
Page
51
53
68
Tables
C.l.
c. 2.
c. 3.
c. 4.
c. 5.
c. 6.
C.7.
c. 8.
c. 9.
D.l.
D. 2.
D. 3.
LIST OF TABLES
Isothermal Gas Flux Function for (-rl/-r2)=0.1.
Isothermal Gas Flux Function for (-rl/-r2)=0 .2857
Isothermal Gas Flux Function for (-rl/-r2)=0.5
Isothermal Gas Flux Function for (-rl/-r2)=0.9 •
Radiative Equilibrium Flux Function for (-rl/-r2)=0.1 .
Radiative Equilibrium Flux Function for (-rl/-r2)=0.2857
Radiative Equilibrium Flux Function for (-rl/-r2)=0.5 •
Radiative Equilibrium Flux Function for ( '"[ l/ '"C 2 ) =0 . 9 .
Comparison Between the Exact and Approximate Values of ¢(-r)
variation of the Power Input, Conduction Losses and the Emissivity Factor with the Inner Sphere Temperature
Experimental Data and Results for Helium Run . E xper imen ta 1 Data and Results for Carbon Dioxide Run .
viii
Page
73
74
75
76
77
78
79
80
81
83
84
85
Symbol
A
B
E g
F
F e
Gr
g
H
K
k
L. 1
L
Nu
Nu*
p
Q
q
s
T
NOMENCLJ\.'l'URE
Quantity
Area
Radiosity
Total emissive power of black body
Total emissive power of gas
Radiative flux leaving a surface
Emissivity factor of concentric spheres
Grashof number
Acceleration due to gravity
Irradiation
Coefficient of thermal conductivity
Hean absorption coefficient
length of link i
Gap distance
Nusselt number
Reference Nusselt number
Pressure
Rate of heat transfer
Heat flux
Coordinate distance given in Figure 1
Absolute temperature
ix
Units
2 Btu/hr.ft
2 Btu/hr.ft
2 Btu/hr.ft
2 Btu/hr.ft
2 ft/sec
2 Btu/hr.ft
Btu/hr.ft°F
atm-1-ft-l
ft
ft
Atmosphere
Btu/hr
2 Btu/hr.ft
Symbol
s
(J
T
p
E
1
2
in
loss
m
r
av
NOMENCLATURE (continued)
Quantity
Greek Symbols
Coefficient of thermal expansion
Stefan-Boltzman Constant
Solid angle
Optical distance
Viscosity
Density
Delta, pertains to a difference
Emissivity
Displacement angle
Subscripts
Refers to inner radius
Refers to outer radius
Refers to input power
Refers to conduction losses
Refers to mean temperature
Refers to radiative transfer
Refers to average quantity
X
Units
l/°F
Btu/hr.ft2 R4
Steradians
Lb /ft.hr m
Lb /ft 3 m
Radians
1
I. INTRODUCTION
The challenging field of radiative transfer through
bounded radiating media, has received increased atten
tion from engineers concerned with combustion, modern
high-temperature power plants and transport processes
involving radiative gases. The high-speed and space
technology has made considerable progress in the areas
of plasma and shock layers surrounding re-entry vehicles.
As a result, a better understanding has been achieved
regarding the basic mechanism of radiative transport
phenomenon.
The number of articles dealing with the problem of
heat transfer through a gas shell contained within two
concentric spheres has been increasing rapidly during the
past few years. Exact numerical solution of the problem
has been obtained and different methods of approximate
solutions are presented. Experimental investigations of
the above problem have been limited to only one dealing
with natural convection between two isothermal concentric
spheres.
The main objective of this investigation is to examine
experimentally the combined problem of natural convection
and radiation through a gas layer contained within two
concentric spheres. The experiment covers a wide range
of pressures starting from the region where the radiation
is the predominant mechanism to the region where the
natural convection is the predominant mechanism. The
possibility of analytically predicting the combined
effect by adding the natural convection contribution
to the radiation contribution is also examined.
A second objective is to consider the assumption
2
of isothermal gas as a means of simplifying the problem
of radiative transfer through an absorbing emitting gas
shell contained within isothermal black concentric
spheres at uniform but different temperatures, and to
check the validity of an approximate method similar to
the mean beam length approach as applied to this problem.
The radiative equilibrium of the same problem,a non
isothermal gas, was also treated by assuming a hyper
bolic emissive power distribution through the gas with
JUmp boundary conditions. All the methods are compared
with the existing exact numerical solution of the problem.
Sections III and IV deal with the theoretical and
analytical approach to the problem while the remaining
sections consider the experimental investigation.
3
II. REVIEW OF LITERATURE
In the past few years there has been considerable
interest in the subject of radiative heat transfer
through gas layers contained within non-planar geometries.
Certain aspects of such problems are discussed by
Chandrasekhar [1]* in connection with radiative transfer
in spherical atmosphere. The simple shape of the
spherical gas and the spherical layer between concentric
spheres have been treated by several authors. Heaslet
and Warming [2] analyzed the radiative transport through
a finite spherically symmetric and uniform generating
medium. Sparrow, Usiskin and Hubbard [3], using numeri-
cal techniques, solved the problem of radiative transfer
between two concentric black spheres maintained at the
same temperature and containing an absorbing-emitting
and heat-generating gray gas.
Ryhming [4] used the method of undetermined para
meters in solving the transfer of radiant energy between
two concentric black spheres kept at different uniform
temperatures and separated by an absorbing and emitting
gray gas. The temperatu~e distribution was determined
for only three ratios of inner to outer wall temperature
T1/T
2 = 2,5 and 25. For the two limiting cases of thin
* Numbers ln brackets designate references in Bibliography.
4
and thick optical thickness, a closed analytical expres
sion for the heat flux was deduced. Viskanta and Crosbie
[5] also treated the radiative transfer between two con
centric spheres, but in a more general way than Ryhming.
The method of successive approximation was used to obtain
a solution. The results were expressed in terms of a
dimensionless emissive power distribution and flux func
tions. Their study indicates that the validity of opti
cally thin and thick approximation is very limited and
the effect of curvature is appreciable even when the
radius ratio is close to 0.95.
Different approximate methods were also used to solve
the above problem. Deissler [6], using the diffusion
approximation and jump boundary conditions, obtained
generalized expressions for radiative diffusion in a non
gray gas layer. The diffusion approximation was improved
considerably by using the second order energy jump at the
wall. The range of validity for this approximation was
extended to lower values of optical thickness. Other
approximate methods were used by Olfe [7] and Chou and
Tien [8] to obtain an expression for the temperature dis
tribution and the radiative heat flux through a gray gas
enclosed between two concentric spheres. Olfe used a
modified differential approximation while the others used
a modified moment method. The boundary conditions for
use with the differential approximation was discussed by
5
Finkleman [9]. The method used is restricted neither
to gray nor to non-scattering gas and may be applied
to general geometries. Hunt [10] examined the modified
moment method for the case of spherical symmetry, and
included certain curvature terms which were neglected
in the original paper [8].
A procedure involving iteration of the differential
approximation was used by Lee and Olfe [11] to carry out
radiative transfer calculations for the problem of heat
generating gray medium contained between concentric
spheres. The first iteration yields an approximate solu
tion which compares favorably with the other approximate
solutions. Another differential approximation, based
upon half-range moments, was proposed by Denner and
Sibulkin [12]. A comparison with the exact solutions
illustrates the deficiencies of the differential approxi
mation for the spherically syn~etric case. Large error
exist in the optically thin limit for walls of different
temperatures. Smaller but appreciable errors are also
found at all optical depths for equal temperature walls
with heat generation in the gas. An improved differen
tial approximation was developed by Traugott [13]. The
method was used to construct a purely differential equa
tion that, with the associated boundary conditions,
describes radiative transfer with spherical symmetry. A
comparison with the exact solution indicates a superiority
6
of this method to other conventional moment approxima-
tion methods specially at the thick limit.
The radiative energy transfer from a small sphere
situated in a quiscent gas was considered by Emanual [14].
This represents the limiting case of concentric spheres
when the outer sphere is at infinity. The problem of
heat transfer by combined conduction and radiation
between concentric spheres separated by a radiating
medium was reported by Viskanta and Merriam [15]. An
iterative scheme was used to solve the governing equations
and the effect of the conduction parameter on the energy
transfer was investigated.
The natural convection between two isothermal con-
centric spheres was solved theoretically by Mack and
Hardee [ 16] . All the fluid properties were treated as
constants while the density was allowed to vary with
temperature. Bishop, Mack and Scanlan [17] conducted an
experimental investigation of the above problem with air
at various diameter ratios. Two Nusselt-Grashof numbers
correlations were presented for the measured heat flux
data for four different diameter ratios. The Grashof
4 6 number ranged from 2.0 x 10 to 3.6 x 10 based on gap
thickness and one correlation fits the data to within
15 . 5 per cent .
It is clear from the above literature review that
experimental investigations of heat transfer between two
7
concentric isothermal spheres, where the radiative
transfer is the predominant mechanism, does not exist.
The purpose of this research is to investigate this
problem experimentally.
8
III. ISOTHEH.HAL GAS ANALYSIS
The problem of radiative energy transfer between
two concentric black spheres separated by an absorbing-
emitting gray gas has been considered in detail in the
literature [4,5]. Approximate and exact numerical solu-
tions for the heat flux and the temperature distribution
exist as discussed in section II.
The objective of this investigation is to examine
the assumption of isothermal gas as a means of simplifying
the problem, to obtain a closed form expression for the
flux and to check the validity of an approximate method
similar to the mean beam length approach as applied to
this spherical geometry. There has not been any published
information on this problem in the open literature.
The exact solution for the radiative flux distribu-
tion in a spherical gas shell enclosed between two concen-
tric isothermal spheres with different surface temperatures,
Figure 1, is given in reference [5] as
'T2
c 2q(c) ~ 2[F1
h 1 (c)+F 2h 2 (c)+ J H(c,t)Eb(t)dt] (1)
'Tl
where F =E , F =Eb2
and represent the radiative fluxes 1 bl 2
leaving the inner and the outer surfaces respectively,
Eb=oT 4 is the black body emissive power, 'T=kpr is the
optical thickness for gray gas with constant absorption
9
FIGURE 1. PHYSICAL SYSTEM, CONCENTRIC SPHERES
10
coefficient k at a pressure p. The remaining expressions
are given by
h 1 (T) = TTlEJ(T-T 1 )-(T-Tl)E 4 (T-Tl)-E 5 (T-Tl)
2 2 1/2 2 2 1/2 2 2 1/2 + (T -T 1 ) E 4 [ (T -T 1 ) J+E
5 [ (T -T
1) J (2)
h 2 (T) = -TT 2E 3 (T 2 -T)+(T 2 -T)E 4(T
2-T)+ES(T
2-T)
2 2 1/2 2 2 1/2 2 2 1/2 2 2 1/2 -(T2-Tl) (T-Tl) E3[(T2-Tl) +(T-Tl) ]
2 2 l/2 2 2 l/ 2 2 2 l/ 2 2 2 J/2 - ( ( T 2 - T l ) + ( T - T l ) ] E 4 ( ( T 2 - T
1 ) + ( T - T l)
2 2 1/2 2 2 1/2. -E5 ( (T 2 -T 1 ) + (T -Tl) ] (3)
and
H(T,t) = {T sign(T-t)E 2 (jT-tj)+E3
(jT-tj)
2 2 1/2 2 2 1/2 2 2 1/2 - (T -Tl) E2 [ (t -Tl) + (T -Tl) ]
2 2 1/2 2 2 1/2 -E 3 ( (t -T 1
) + (T -T 1 ) ] }t (4)
where Tl and T2 are the optical thickness evaluated at the
inner and the outer surface, sign(T-t)=l for (T-t)>O,
sign(T-t)=-1 for (T-t)<O and E (T) is the exponential n
integral defined as 1
En(T) = JO ~n-2e-T/~d~ ( 5)
11
The solution for the flux in a radiative equilibrium
requires the simultaneous solution of equation (1) and
the energy equation to obtain the temperature distribu-
tion in the gas. Under such a condition, the actual
numerical solution presented in reference [5] is difficult
and lengthy.
The special case of isothermal gas was reduced from
the governing equation (1) and a closed form expression
for the heat flux was derived. For this special case,
equation (1) can be reduced to
2 T q ( T)
T2
= 2[Eblhl(T)+Eb2h 2 (-r)+Ebg (
JTl
where Ebg = Eb(t) =constant Ebl+Eb2
2
H (T ,t) dt) (6)
This constant
value was chosen based on an energy balance, energy
absorbed by the gas from the two bounding black surfaces
must equal to the energy emitted from the gas to these
surfaces.
A closed form solution was obtained for the above
equation by performing the integration. The details can
be found in Appendix A. An expression for the flux at
the inner and outer sphere was derived from equation (6)
by substituting -r=-r 1 and -r=-r 2 respectively.
expressions are given by
The final
12
and
q ( l 2)
( 8)
For the comparison with the exact solution the flux
can be expressed as
( 9)
and
(10)
13
where Q (T 2)
and - 2-- are nondimcns ional functions T2
similar to those obtained in reference [5].
A computer program using the Exponential Integral
subroutine was prepared and processed in UMR IBM 360
Model 50 Digital Computer to evaluate the different
values of the exponential integrals appearing in expres-
sions (7) and (8). The parameter (T 2-T
1) was varied
from 0.01 to 10, and the corresponding Q(T)/T 2 for the
inner and the outer sphere were evaluated for optical
radii ratios T 1 /T 2=0.l, 0.2857, 0.5 and 0.9.
of 0.2857 was the experimental ratio.
The value
The same physical problem was approached from a
different point of view. The approximate approach used
is similar in nature to the one used in the calculations
of radiant energy transfer through isothermal gas using
the mean beam length method [18]. The objective of
using this approach is to examine the validity of this
method by a comparison with the exact isothermal gas
solution obtained in this section.
Using this method the flux at the inner sphere can
be written in terms of the radiosity and irradiation in
the form:
( 11)
For a black surface, the radiosity is equal to the
black body emissive power. From the isothermal gas
14
analysis [18], the irradiation H1 is due to two contri
butions. One is the energy transmitted from the outer
surface and can be written as
Hl2~ f Eb2e-ks cosn~l dQ
Q
(12)
where ¢1
is the displaced angle from the normal to the
inner sphere element dA1
as viewed from a point on the
outer sphere and dQ is the solid angle subtended by the
outer sphere element dA2 at dA1 as shown in Figure 2.
For this geometry the solid angle is given by
( 13)
and the resulting expression for the irradiation becomes
(14)
The other contribution to the irradiation is the emission
of the gas expressed as
E (l-e-ks) bg
cos¢1 dQ
lT
For isothermal gas this expression becomes
(15)
(16)
15
OUTER SPHERE
INNER SPHERE
FIGURE 2. CONCENTRIC SPHERES, COORDINATE SYSTEM
16
The flux can be expressed by combining the two expressions
(14) and (16) and using equation (11) as
fTT/2-ks
q(T 1 )=Ebl-Ebg-2(Eb2-Ebg) e cos¢ 1sin¢ 1 d¢ 1
0
(17)
where
( 18)
The gas emissive power was taken as Ebg = (Eb 1+Eb 2 )/2
similar to the value used in the exact solution, and the
expression for the flux can be written as
(19)
A numerical integration scheme using Simpson's rule was
prepared to evaluate the integral in equation (17).
The isothermal gas solution has been obtained for a
wide range of physical parameters. The influence of radii
ratio T1/T 2 on the flux function Q(T 1 i/T~ is shown in
Figures 3 through 6, and Tables C.l through C.4 (Appendix
C), and correspond to ratios 0.1, 0.2857, 0.5 and 0.9,
respectively. In each figure the exact isothermal radia-
tive flux and the results of the mean beam length approach
are presented and compared with the exact radiative
equilibrium solution of the problem [5,19]. As expected,
the effect of curvature on the flux functions increases
1.0 ·--
0.8
0.6 N
t-' '-.. ..-.. t-'
........ 0
z 0.4 0 H 8 u z ::J ~
X ::J 0.2 H ~
0.01
·-.......~
·~
RADIATIVE EQUILIBRIUM ~ [REF. 5] '
' EXACT ISOTHEW.AL
------ APPROXU1ATE ISOTHERMAL (H. B. L)
' INNER SPHERE _]// """ ·~ ---~.-..--·-·
~· ........
. ---·~
0.1
/
OUTER SPHERE ---------....., .--· /.
/
/ /
/
./ /
1
/
OPTICAL THICKNESS (12
-11
)
FIGURE 3. ISOTHERMAL GAS FLUX FUNCTION FOR ( 11/1
2) = 0. 1
10
f-' --.1
N <-'
..........
<-' ....... 0
z 0 H E-t u 5 ~
X ::> ..:1 ~
1.0
0.8
0. 6
0.4
0.2
·--·'"=·-=:..:::-...:....-._. __ --.---:...- - - -- -........_ --- -- ...... ~. '~·::--..........
.......... . ......... ......... .
'· ' INNER SPHERE
., ' .
' " ' . ', ' ...
OUTER SPHERE ' .......... '"'· -"::..:. ~----.-.
RADIATIVE EQUILIBRIUM [REF. 19]
-·-·-·- EXACT ISOTHERMAL
-------APPROXIMATE ,....... ISOTHERMAL (MBLf.-' ·------·--·-· ·-·-
.,-/
/
/./
/./
--· -· .,.,. ......-· --,..
0.01 0.1 1
OPTICAL THICKNESS (12
-11
)
FIGURE 4. ISOTHERMAL GAS FLUX FUNCTION FOR (1 1/1 2) = 0.2857
10
I-" co
1.0 r=:·-· -- -==-==-. --- . --- -=------------- i --- ·--. 0.8
N .....
' ..... '(; 0.6
~ H E-1 u z ::J li-4 0.4 X ::J H li-4
0.2
0.01
--- ---· ......... -.... ......... ......... "-- .......
' ' ' ----RADIATIVE ' EQUILIBRIUM [REF. 5] "-,
-·-·-·-EXACT ISOTHERMAL
------APPROXIMATE ISOTHERMAL
' " ', "
INNER SPHERE
)' '· (H.B.L ' ' .............. .........
......... ....__ ""'\:-._ .---·- ... ..::a....-.).__ --. ~. ~ --
·---·--- .. --- .-.~ .--·
0.1
_... ./'
_,/ _.,... ·~ OUTER SPHERE --
1
OPTICAL THICKNESS (12
-1 1)
FIGURE 5. ISOTHE~~ GAS FLUX FUNCTION FOR (1 1/1 2 ) = 0.5
10
1-' 1.0
N !-'
.....__
-!-'
1.0
0.8
0 0. 6 z 0 H E:-i u ~ li-l
:X: ::J ..:I ~
0. 4
0.2
-......... ...................
"' ........ ·----~-·-·-·-·-·--.
" " ' ' ' ' .........
...............
...........
'·
RADIATIVE EQUILIBRIUM [REF. 5]
-·-·-·-· EXACT ISOTHERMAL
-------- APPROXIMATE ISOTHERMAL (M.B.L.)
0.01
INNER SPHERE
OUTER SPHERE
~ . ............ ~·-~-----
OPTICAL THICKNESS (1 2-1 1)
FIGURE 6. ISOTHERMAL GAS FLUX FUNCTION FOR (1 1/1 2) = 0.9
10
N 0
21
In examining the results presented in Figures 3
through 6, it appears that the exact isothermal gas
solution compares well with the mean beam length approxi-
mation at small radii ratio. However, as this ratio
increases the difference between the two solutions
increases. At the thick limit, the radiative flux function
approaches a constant value of 0.5 as predicted by the two
methods of solutions. For this limiting case all the
exponential integrals in equation (7) and the transmission
integral in equation (17) approach zero and the expression
for the flux at the inner sphere reduces to
This limit can be interpreted physically as the irradiation
at the inner sphere is due to the emission of the gas
only while the energy emitted from the outer sphere is
decayed completely by the absorption of the gas.
A comparison between the exact isothermal gas solu
tion with the exact radiative equilibrium solution indi
cates a reasonable agreement at thin limit. As (T 2-t 1 )
increases the deviation between the two solutions increases
with the largest error at small radii ratios. A wide range
of agreement exists for the cases of large radii ratios
where the geometry can be approximated by a planar medium
for which the exact solution at the thin limit [20] agrees
with the isothermal gas solution Eg=(Eb 1+Eb 2 )/2. This
22
behavior implies that the emissive power distribution
can be considered uniform, at the specified values, for
the case of thin shell at thin limit as shown in Figure
6 •
2 The flux function for the outer sphere Q(T 2 )/T 2 as
obtained from the exact isothermal solution also
approaches the value of 0.5 at the thick limit. The
explanation of this behavior is similar to the one given
above for the inner sphere. The value of this function
This at the thin limit increases as T1/T 2 increases.
behavior can be predicted from the fact that, at the
thin limit the isothermal gas solution agrees with the
exact radiative equilibrium solution which imposes the
relation q(T2
)=q(T1
) (r1/r 2 ) 2 . As the radii ratio in
creases the flux at the outer sphere increases proper-
tionally with the square of this ratio. This agrees
well with the results obtained from isothermal gas for
the flux function at the outer sphere as shown in Figures
3 through 6.
23
IV. NON-ISOTIIEHMAL GAS AND RADIATIVE EQUILIBRIUM
The exact numerical solution of radiative equili
brium through a gas shell contained between two concen
tric spheres at uniform but different temperatures was
considered in reference [5]. Due to the fact that this
solution is quite a formidable task and time consuming,
an approximate analysis is considered to predict the radia~
tive equilibrium flux. The method assumes an emissive
power distribution through the gas and thus uncouples
the energy equation from the expression for the flux (1) .
Different emissive power distributions were considered
and the one which compares best with the exact solution
was found to be a hyperbolic type of the form
( 2 0)
where a and b are constants depending on the temperature
of the gas next to the surface of each sphere. The
temperature in the gas next to the wall differs from the
wall temperature due to what is known as radiative jump
condition. This jump condition can be explained physi
cally as follows. The radiative flux passing through a
plane next to the wall is made up of flux coming from the
wall and from gas which, on the average, is a radiation
mean free path away from the wall. Thus, the average
24
temperature of the radiation passing through the plane
next to the wall will lie between the wall temperature
and the temperature of the mean free path away from the
wall.
In order to relate the emissive power of the gas
surface to the emissive power of the boundary surface,
the diffusion approximation method with jump boundary
conditions was used. The general method, valid only
in the thick limit, was outlined by Deissler [6] and
detail derivations for the case of concentric black
spheres separated by a gray gas are given in Appendix B.
The final results are
E
E
where
to the
= gl
= g2
E gl
3 rl l 3 rl 2 l -1" ( 1--) + -2 +-- + (-) (-4 l r 2 8-r 1 r 2 2
l (Ebl -Eb2) (2 +
3 aT)
3 a:r> 2
Ebl-3 rl .!. + 3 r 2
(____!) (l -1" (1--) + 4 l r 2 2 8-r 1 + r 2 2
(Ebl-Eb2) rl 2 l 3 (-) (- - S""T) r 2 2
Eb2+ 2
3 rl r 2 l + 3 (____!) (.!. -T (1--)+
8-r 1 +
4 l r 2 2 r2 2
and E g2 are the emissive power of the
inner and the outer sphere respectively,
( 21)
( 2 2) 3 g:r)
2
( 2 3) 3 g:r)
2
gas close
Ebl and
25
Eb 2 are the emissive power of the inner and the outer
sphere respectively.
The diffusion approximation and the associated jump
boundary conditions equations (22) and (23) are applicable
only in the thick limit. The exact region where this
approximation fails is not well defined. However, it can
be seen that for the case of T 2 <0.75 the temperature of
the gas, predicted using these equations, reaches a lower
value than the temperature of the boundaries which is
physically impossible. As a result, these jump conditions
could not be used in equation (20) in the thin limit.
The analysis used to predict the flux, equation (21)
and Appendix B, will be referred to as Deissler analysis.
For any fixed ratio of T 1 /L 2 , this flux function has a
maximum at
= :2 [1- ( ~) 3 J
1 r1 2 ( 1--)
r2
1/2
( 2 4)
In the region where L 2 >L 2p the flux compares well with
the exact solution and when L 2 <L 2 p a large deviation
exist between the two solutions (Figures 7 through 10).
As a result, this or a proportional value of L 2p can be
used to specify the region where the jump boundary con-
ditions, equations (22), (23) are applicable and accurate.
26
In the region where T2 <2T 2 p the jump boundary con
dition is taken from the exact solution [5] for the case
of thin limit and is specified by
These jump boundary conditions are exact only if T2
approaches zero. However in this approximate solution
( 2 5)
(26)
they are used through the whole range where T2
<2T2p.
There is no exact closed form expression specifying the
jump boundary condition through the whole range of T2
•
A comparison between the approximate jump boundary con-
ditions, obtained using equations (22), (23) when
T2 >2T 2 p and equations (25), (26) when T2 <2T 2p' and the
exact values [5] are shown in Table C.9. The comparison
is expressed in terms of a dimensionless emissive power
function
(2 7)
The approximate values appear to agree favorably with the
available exact values.
The appropriate approximate values for Egl and E92
are then used as boundary conditions to evaluate the
coefficients a and b in the assumed gas emissive power
distribution equation (2~ as
27
( 2 8)
and
( 2 9)
Therefore,
and b = (30)
where now Egl and Eg 2 are known functions of Ebl and Eb 2
which are known quantities.
Returning to the exact expression for the flux at
the inner sphere, equation (1), and considering black
surfaces, the following equations are obtained using the
assumed hyperbolic emissive power distribution, equation
( 2 0) •
where the expressions h 1 (T 1 ), h 2 (T 1 ) and
given in Appendix A and
( 31}
are
28
rl2 + j E 3 (jc 1 -tj)dt-
Ll
A detail and closed form integration for the first and the
second integrals in equation (32) can be found in Appendix
A.
The final expression for the flux at the inner sphere
takes the following form
Note that this expression is algebraic in terms of the
exponential integrals except the last integral term which
29
requires numerical integration. For the comparison
with the exact solution, the flux can be expressed as
(34)
A computer program was used to determine the con-
stants a and b, to evaluate the different values of the
exponential integrals and to execute the last integral
in equation (33). The program uses Simpsons Rule and
the Exponential Integral subroutine. The parameter
(T 2 -T 1 ) was varied from 0.01 to 10 and the corresponding
2 Q(T 1 )/T 1 was evaluated.
The approximate radiative equilibrium solution has
been obtained for a wide range of physical parameters.
The influence of the radii ratio T 1/T 2 on the flux func
tion Q(T 1 )/Ti is shown in Figures 7 through 10 and
Tables C.5 through C.B, and corresponds to ratios 0.1,
0.2857, 0.5 and 0.9, respectively. In each Figure the
flux function Q(T 1 )/T 12 from the approximate analysis,
equations (33,34), and Deissler analysis, equation
(21,34), is presented and compared with the exact radia-
tive equilibrium solution [5,19].
A comparison between the approximate solution and
the exact radiative equilibrium solution indicates an
excellent agreement through the whole range of optical
thickness. The maximum deviation is 6 percent lower than
Nr-i l-'
......... .........
r-i l-' ...... 0
z 0 H E-1 u 5 ~
X ::> H ~
-·=·===·-·-· -·-·-·---1 0
·--
• r
--· 0.9 EXACT SOLUTION --0.8 -· -·-·- APPROXIMATE SOLUTION
0.7 ------ DEISSLER SOLUTION
0.6
0.5
0.4
0.3 / /
/ 0.2 /
/
/wl'
//
0.1 .......... / ___ ,.., 0.01 0.1 1
OPTICAL THICKNESS (t 2-t 1)
............ ........_
"""·,
/ I
/
,I I
1/ II
I
,.,.---/ ....
/ '
10
FIGURE 7. RADIATIVE EQUILIBRIUM FLUX FUNCTION FOR (t 1/t 2)=0.1
w 0
Nr-i ~
...........
r-i ~
0
z 0 H 8 u 5 ~
~ ::J H ~
0.9
----·-·-·-·-·---1.01 ' ·-· ·--.... ·-.......
EXACT SOLUTION
-·-·-·- APPROXIMATE SOLUTION
------ DEISSLER SOLUTION
-----//
...,/'
0.1
/
/ /
/ /
/
, I
I I
I
I I
I I
I
1
OPTICAL THICKNESS (1 2-1 1)
/ /
FIGURE 8. RADIATIVE EQUILIBRIUM FLUX FUNCTION FOR (1 1/1 2)=0.2857
I
w ......
Nr-i 1-' '-.. .........
r-i 1-'
........ 0
z 0 H 8 u '"7 1'--1
:::::> 1::4
:X: :::::> H ~
1. 0 r· · ..... ·--- ·-·---.....
EXACT SOLUTION
0.8 APPROXIMATE SOLUTION
------- DEISSLER SOLUTION
0.6
0.4
0.2
_____ _,
0.01
..,"" ..........
/ /
/ /
/ /
0.1
//
I I
/
I I
I I
I I
/
//
1
OPTICAL THICKNESS (t 2-t 1 )
FIGURE 9. RADIATIVE EQUILIBRIUM FLUX FUNCTION FOR (t 1/t2) = 0.5
10
w N
1.0 --.
// .,.,..--
,/
Nr-i0.8 // <-' "'-.........
r-i <-' ..._,
0
z 0 H 8 u z ::::> ~
X ::::> H ~
// /
I /
0.6 I I
I
0. 4 EXACT SOLUTION
APPROXIMATE SOLUTION
0.2 ----- DEISSLER SOLUTION
0.01 0.1 1
OPTICAL THICKNESS (t 2-t 1)
FIGURE 10. RADIATIVE EQUILIBRIUM FLUX FUNCTION FOR (t 1/t 2)=0.9
10
w w
34
the exact solution for radius ratio of T1 /T 2=0.l. As
a result of this good agreement between the two solutions,
it is reasonable to assume that the actual emissive power
distribution is close to the assumed hyperbolic distri-
bution. The major error in the results is due to the
discrepancy between the imposed and the exact jump
boundary conditions.
The results obtained by Deissler analysis, equation
(21), are compared also with the approximate analysis.
It 1s clear from Figures 7 through 10, that the validity
of the first is limited to the thick limit T2 >2T 2p.
Deissler analysis was used to predict the jump conditions
for the approximate solution when T2 was larger than 2T 2p
and equations (25), (26) were used to predict this jump
when T2 was smaller than 2T 2p. The proposed approximate
analysis improves appreciably the prediction of the flux
at the thin limit. The main advantage of the proposed
method over other existing approximate methods is its
simplicity and clarity. The flux in the thick limit,
when T2 >2T 2p could be predicted accurately by the closed
form expression appearing in equation (21) .
35
V. DESCRIPTION OF THE EXPERIMENTAL APPARATUS
The apparatus consists of a spherical assembly,
where the inner and the outer sphere were used as a
heater and a sink respectively, a cooling system consis
ting of a dewar and a circulator, a vacuum pump, a D.C.
power supply, an ammeter, a voltmeter and a potentio
meter (shown in Figures 11 and 12).
The spherical assembly is a structure supporting
two concentric spheres as shown in Figure 13. The inner
sphere used as a heater, is a hollow copper sphere with
a diameter of 2 inches and a 1/4 inch wall thickness.
It was made of two halves which were assembled together
by a thread as shown in Figure 14. This copper sphere
houses the heating element consisting of a lava core and
a nichrome wire. The Lava core is Alsimag Grade A Lava
and was machined to a spherical shape 1.5 inches diameter
which fitted into the cavity of the copper sphere. The
outer surface of the lava sphere has a helical groove
with a pitch of twelve threads per inch. After machining,
the lava core was cured by heating it in an oven to 600°F
for a period of 8 hours and allowed to cool slowly. A
26 gauge nichrome wire, approximately 2 feet long, was
wrapped around the lava core inside the peripheral grooves.
36
FIGURE 11. EXPERIMENTAL SET-UP
VACUUM
PUiviP
SYSTEM VACUUM GAUGE
SYSTEM PRESSURE GAUGE
VACUUM VALVE PRESSURE VALVE
t j E-i E-i ril ril H H E-i z ;:J H 0
CIRCULA
TOR
THERMOCOUPLE
T ----bf: I \m SPHERES
-- / ASSEHBLY
~lt~{~ DISTILLED _-_:_-:_·_ WATER -
DEWAR
FIGURE 12. SCHEHATIC OF EXPERIMENTAL SET-UP
VllliVE
I GAS
I TANK
TAl~K
GAUGE
w -...]
VACUUH
38
INSTRUMENT FEED THROUGH
PRESSURE LINE
FIGURE 13. CONCENTRIC SPHERES ASSEMBLY
LIGHT PERFORATED STEM
COPPER
39
POWER SUPPLY LEADS
INNER SPHERE THERI-10COUPLE
INSULATING CEMENT
HEATER WIRE
LAVA CORE
FIGURE 14. CROSS SECTION OF THE INNER SPIIERE
40
Sauereisen Cement No. 7, a high grade silica cement, was
used to coat the wire and cement it to the lava core.
This assembly was then fitted well inside the cavity of
the copper sphere and used as a heater. The nichrome
wire was well insulated to prevent any electrical con
tact with the copper sphere and was connected to the
electrical power supply. Electrical power, supplied to
the nichrome wire, was then dissipated by conduction
through the wall of the copper sphere.
The outer sphere, made of steel, is 1/16 inch thick
and 7 inches in diameter. To allow the insertion of the
smaller sphere an opening was made in the outer sphere
dividing it into two sections. A flange was welded on
the edge of each section, so that when the flanges were
brought together, the spherical shape and the dimensions
of the outer sphere were not altered.
A 22 gauge 3/4" inch stainless steel tube was welded
to the upper half of the outer sphere and was used as a
supporting column and as a path for evacuating the spheri
cal assembly. The heater, inner copper sphere, was
supported from the upper half of the outer sphere by a very
thin perforated 1/4 inch stainless steel tubing in such a
manner that when the two halves of the outer sphere were
brought and held together v1a the flanges, the two
spheres were concentric. All the surfaces internal to the
external sphere were blackened by an acetylene torch
before assembly.
Thermocouples were attached to the outer surface
41
of the inner sphere (heater) and to the inner and outer
surfaces of the outer sphere. The thermocouples,
chromel-alumel, were connected to a selector switch and
a potentiometer. A D.C. power supply was used to supply
power to the inner sphere. This power was measured by
an ammeter and a voltmeter. The instruments and electri-
cal connections are shown in Figure 15.
The concentric spheres assembly can be evacuated
through a piping system and a vacuum pump. The piping
assembly is fitted with a valving system to permit
pressurizing or evacuating the system independently and
it is equipped with both vacuum and pressure gauges.
A cooling system consisting of a dewar and a circula
tor was used to control the external temperature of the
concentric spheres assembly. The circulator maintained the
liquid temperature in the dewar at a prescribed constant
value for a given run.
u.s. c. H I D. C. Digita
VOLTf!.ETER I 1 I
T. C. 3
DJNER SPHERE
I WESTING-HOUSE
I I D 0 c 0 I
AMMETER I
INNER SPHERE T. C.
~--------------------~~
TIIERHOCOUPLE
SELECTOR SWITCH
I HEvJLETT PACKARD
I D.C. POWER I SUPPLY
OUTER SPHERE T.C. 's 2&3
T.C. = THE~~OCOUPLE
POTENTIOMETER
OUTER SPHERE
FIGURE 15. I:~STRUMENTS AND ELECTRICAL CIRCUIT DIAGRAH
,!:>. N
43
VI. EXPERIMENTAL PROCEDURE 1\J.'JD DATA REDUCTION
To determine experimentally the effect of gas pres
sure on the radiant energy transferred through an absor
bing emitting media the experiment was divided into three
parts, a vacuum run, a helium run and a carbon dioxide
run.
During the vacuum run the spherical cavity was
evacuated to approximately 25~(microns). The temperature
of the outer sphere was maintained constant at 113°F while
the electrical power input to the inner sphere was varied
over the range of 10-71 watts. A water circulator was
used as a heater to maintain the outer sphere temperature
at a constant value. For a given power input to the inner
sphere, in the range of 10-71 watts, a steady state con
dition was reached and the temperatures were recorded.
The time required to reach steady state varied from 6 to
12 hours depending on the magnitude of the flux (higher
flux corresponded to a longer time period). The steady
power input and the temperature were used to determine the
effective emissivity factor for this spherical geometry.
Part of the energy was conducted through the supporting
link and the instrument wires while the other part was
transferred by radiation to the outer sphere. The con-
duction losses can be approximated by Fourier's conduction
law.
llT = L: I<. A. l l L.
l l
44
(35)
where K., A. and L. are the thermal conductivity, area l l l
and length of the ith link respectively and 6T is the
difference between the inner and the outer sphere tempera-
tures. These conduction links are four current and voltage
wires, two thermocouple leads and the supporting stem.
This value represents only 11% or less of the total energy
input. A plot of the conduction losses versus the inner
sphere temperature is shown in Figure 16. The radiative
transfer can be calculated by taking the difference between
the input energy and the conduction losses as
(36)
where Q and Q. are the radiative transfer and the energy r ln
input respectively. The variation of both Q. and Q ln r
versus the inner sphere temperature is shown in Figure 17.
In order to determine the emissivity factor, the radiative
transfer between the spheres can be written as
where F = e
= F e
1
(37)
is the emissivity factor,
S8SSO~ NOiiliJOGNOJ
0 0 r--
0 0 \0
0 0 1..{)
0 0 '<;f'
0 0 C"'l
~ 0
I
~ ::J E-i
~ ~
~ ~ E-i
~ ~ ::c ~ ())
~ rLl z z H
45
())
~ ()) ())
0 ....:1
z 0 H E-i u ::J ~ z 0 u
\0 r-1
~ ::J t.'J H ~
p::; ...... .......
" ::_) 8 c:Q
I
p::; li1 8::: 0 P-1
250
200
150
100
50
-----40 POWER INPUT
----¢-----POWER RADIATED (INPUT MINUS
;) /
%.,/" ..-
LOSSES)
~/ /
o/'/ //
/ y
0~~----~------~----~------~----~------~----~----~~----~----~
300 400 500 600 700
TEMPERATURE AT THE INNER SPHEP~ - F0
FIGURE 17. INNER SPHERE TEMPERATURE, VACUUM RUN .:::>-0'1
47
El and E2 are the 1nner and the outer sphere emissivities
respectively, and r 1 and r2
are the inner and outer sphere
radii respectively. A plot of the emissivity factor
versus the inner sphere temperature is shown in Figure 18.
During the helium run, helium gas at various pressures
(in the range 0-3 atmospheres) was maintained in the
spherical gap between the two spheres. The power input to
the inner sphere and the outer sphere temperature were
maintained constants at 52.5 watts and 113°F respectively.
The gap pressure was the only controlled variable which in
effect dictated the steady state temperature of the inner
sphere. The results of this run, steady state energy input
and the inner sphere temperature, with the final results
of the vacuum run were used to determine the natural con-
vection contribution to the energy transfer.
The radiative component for helium, a non-participating
gas, is evaluated in a similar method as the vacuum case by
using equation (37). The steady state inner sphere tern-
perature, which is a function of the gas pressure shown in
Figure 19 was used to select the emissivity factor F from e
Figure 18. The conduction losses corresponding to the
inner sphere temperatures are obtained from Figure 16. The
difference between the input energy and the sum of the last
two components is the natural convection component. This
can also be predicted by using the results of reference [12]
as
p::; 0 E:-i u f:!! ~
~ E:-i H :> H U) U)
H ::8 ~
1.0~---------------------------------------------------------
0.8
0.6
0.4
0. 2
0
D
D
0
0 Q
D
400 600
INNER SPHERE TEMPERATURE - °F
FIGURE 18. EHISSIVITY FACTOR OF THE CONCENTRIC SPHERES
D
800
.;::. co
1oo 1 CARBON DIOXIDE INPUT POWER - 179 BTU/HR
OUTER SPHERE TE!vlP. - 113°F
-<1 HELIUM
l\t ~
0
I
~ ;:J 8
~ ~ 500 ~ ~ 8
~ r:::::l ...,... >"'-<
P-1 lf.l
~ r:4 z z H
300 0 1 2 3
PRESSURE - ATMOSPHERES
FIGURE 19. EXPERIMENTAL RESULTS FOR HELIUM AND CARBON DIOXIDE
~
\.0
50
(38)
where Qc is the heat transfer by natural convection, K is
the gas thermal conductivity evaluated at a volume-
weighted mean temperature
T m (39)
where r av
= L * and Nu = (1 + - )Nu, Nu is Nusselt rl
* number, L = r 2 - r 1 is the gap distance and Nu is a refer-
ence Nusselt number given by
N~ = 0.106 ~ 0 · 276 (40)
2 3
and Gr = gSp L (T 1-T 2 )
2 is the Grashof number evaluated
]l
at T , g is the acceleration of the gravity, S is the m
coefficient of thermal expansion and Jl is the gas vis-
cosity, all evaluated at T . m The predicted and experimen-
tal natural convection contribution for the case of helium
are shown in Figure 20.
The carbon dioxide run was made in similar fashion
to the helium run with the exception that carbon dioxide
was used in place of helium. The results of vacuum and
helium run were used with the results of this run to
180
160
140
120 ~ ::r: .......... ;::::l ~ 100 r:Q
~ li:l 8: 0 (1;
80
60
40
20
I I
0
LEVEL OF INPUT POI'IER
··---··--------·---··---·· ---··--·· ---·· --·· --·· --·· ----·· ---·· ---- .. ---·· -~o EXPERIMENTAL NATURAL CONVECTION
---<0>---
/ /
I
/
/ /
PREDICTED NATURAL CONVECTION EQN. (38)
RADIATION
CONDUCTION LOSSES ................ ---- --- .-·
---------------..--------
1 2
PRESSURE ATMOSPHEP-ES
FIGURE 20. HEAT TRANSFER CONTRIBUTIONS FOR HELIUM RUN
3
(.;,
f-'
52
determinG the effects of this participating gas on the
radiant energy transfer. The conduction losses and the
natural convection contribution were subtracted from the
energy input to the inner sphere to isolate the radiant
energy transferred. The natural convection contribution
was calculated using equation (38) while the conduction
losses were obtained from Figure 16. The results of
these calculations are shown in Figure 21.
To evaluate the experimental results, the radiative
component is predicted by using the results of reference
[5] as
( 41)
where T1
= kpr1
is the optical radius at the inner sphere,
k = 12 ft- 1 . atrn-l is the Planck mean absorption coef-
ficient for carbon dioxide and p is the pressure in atrnos-
pheres. The function Q(T 1 )/Ti corresponding to the opti-
cal radii difference (T 2 - T1 ) and the radii ratio
Tl/T2
= 0.2857 can be obtained from reference [19], El
and E2
are evaluated by making use of Figure 18 and the
emissivity factor F . e The predicted and experimental
values are compared in Figure 21.
180
160-
140
~ 120 ::r::
" ::::> ~ 100
~ 80 ~ 3: 0 P-4
60
40
20
\ -,
0
\
" ",
--0-- EXPERIMENTAL RADIATION
PREDICTED RADIATION EQ. (41)
NATURAL CONVECTION EQ. (38)
CONDUCTION LOSSES
LEVEL OF INPUT POI'JER
"-, -· '-....... -............ ....._ ---·
~·-·-·-·-·-·-·-
/
/ /.
/'.,/"'
.......... ~. -- -----. -----~ -------------_.....
1 2
PRESSURE ATMOSPHERES
FIGURE 21. HEAT TRfu~SFER CONTRIBUTIONS FOR CARBON DIOXIDE RUN
3
\.n w
54
VII. RESULTS AND DISCUSSION
The results of the vacuum run are presented in
Figures 16, 17 and 18. It is evident from the Figures
that the energy loss by conduction through the supporting
stern and the instrument wires is less than 11% of the
energy input. The emissivity of the inner sphere surface
decreases with increase in temperature. This behavior
is common to carbon and lampblack surfaces. The measured
emissivity compares well with the published [21] values
for lampblack (0.78 - 0.84) in the experimental range.
The experimental results for the helium and carbon
dioxide runs are presented in Figure 19, 20 and 21. The
steady state temperature of the inner sphere for the
carbon dioxide run is always higher than that for helium
at the same pressure and energy input. This behavior is
due to the lower thermal conductivity (0.0125 btu/hr ft°F)
and the higher absorption coefficient (12 ft-l atm-1 )[20] of
carbon dioxide as compared to the higher thermal conducti
vity (0.097 Btu/hr ft°F) and zero absorption coefficient
of helium at 200°F. The lower thermal conductivity re-
duces the natural convection contribution from the
inner sphere. The higher absorption coefficient causes
the temperature of the carbon dioxide to be higher than
that of helium and thus decreasing the net radiant energy
removed from the inner sphere. The above two effects are
55
responsible for the higher inner sphere temperature for
the carbon dioxide run.
A comparison between the predicted and the experi-
mental natural convection contribution for helium is shown
in Figure 20. A reasonable agreement exists between the
predicted and the experimental values specially at higher
pressures where the natural convection is the predominant
energy transfer mechanism. The expression used to predict
this contribution is in effect limited to a Grashof number
4 6 range of 2 x 10 to 3.6 x 10 , to a gap radius ratio of
0.25 < (r 2-r1 )/r1 ~ 1.5 and to a1r as the convective
medium. The present experimental gap ratio is 2.5 and
the Grashof number range covered was 25 < Gr < 38 x 10 4
for the helium run and 2.7 x 10 2 < Gr < 74 x 10 6 for the
carbon dioxide run. The maximum error in the natural con-
vection contribution is in the low pressures or low
Grashof number range which is outside the applicability
of reference [12].
The experimental radiation contribution in the case
of carbon dioxide, was compared with the predicted values
from reference [19,5]. The difference between the two
could be due to inaccuracies in predicting the natural con-
vection and thus effecting the experimental radiation con-
tribution, or due to the assumption of gray gas in the
analysis and thus effecting the predicted radiation con-
tribution. The use of Planck mean absorption coefficient
56
in the analysis is also questionable. This mean coef
ficient has been shown to give reasonable results only
in the thin radiation limit. The actual carbon dioxide
absorption mechanism is due to its principal infrared
absorption bands at l5w, 4.3w and 2.7w appearing in the
experimental temperature range.
Another significant point to bring out when comparing
the experimental and predicted results is the nonlinearity
of the combined radiation and natural convection problem.
The simple sum of each contribution independent of the
other does not correspond to the physical problem. How
ever the results of this experiment indicate that such a
simple sum is a good approximation. The error between
the actual and the predicted input is less than eleven
percent.
57
VIII. CONCLUSIONS AND RECOMMENDATIONS
The results of the analytical analysis indicate that
the exact isothermal gas solution compares well with the
mean beam length approximation at small radii ratios. A
reasonable agreement between the exact isothermal gas
solution and the radiative equilibrium solution exists
only at the thin limit. As the optical thickness increases
the deviation between the two solutions lncreases with the
largest error at small radii ratios.
A comparison between the approximate,non-isothermal
gas solution, and the exact radiative equilibrium solu
tion indicat~that the emissive power distribution through
the gas is close to a hyperbolic with the proper jump
boundary conditions. The approximate solution improves
the prediction of flux function as compared with Deissler
analysis at the thin limit.
The results obtained from the experimental investi
gations indicate a reasonable agreement between the pre
dicted and the experimental natural convection contribution
for helium at higher pressures where natural convection is
the predominant mechanism. The experimental radiation con
tribution for the ~arbon dioxide compare favorably with the
predicted values. Some of the deviation could be due to
inaccuracies in predicting the natural convection contri
bution. The assumption of gray gas and the simple sum of
58
the natural convection and radiation without interaction
could also contribute to some of the deviation between
the experimental and the predicted values. The error
between the measured and predicted total flux is less
than 11 percent.
The experimental apparatus should be modified to
measure the temperature distribution in the gas layer
and to determine the effect of changing the surface
emissivities and diameter ratio on the energy transfer.
59
IX. APPENDICES
APPENDIX A
ISOTHERMAL AND NON-ISOTHERMAL GAS ANALYSIS RELATIONS AND INTEGRATIONS
A. Isothermal Gas
60
The exact expression for the radiative flux distri-
bution through isothermal gas shell contained between two
black isothermal concentric spheres is given by T2
c2q(T) = 2(Eblhl(T)+Eb2h 2 (c)+Ebg J H(c,t)dt]
Tl
where
(A. 1)
2 2 1/2 2 2 1/2 2 2 1/2 2 2 V2 -[(-r
2--r
1) +(T -T 1 ) )E
4[(T 2 --r 1 ) +(T --r 1 ) )
2 2 1/2 2 2 1/2 -E
5[(-r 2 --r 1 ) +(T --r 1 ) ] (A. 3)
and
H(T,t) = {T sign(T-t)E 2 (1-r-ti)+E 3 (1-r-tl)
2 2 1/2 2 2 1/2 2 2 1/2 -(T-Tl) E2[(t -Tl) +(T-Tl) ]
61
(A. 4)
where
sign(T-t)=+l for (T-t) > 0
sign(T-t)=-1 for (T-t) < 0
Using equation (A.4), the integral in (A.l) is divided
into four integrals as follows:
where
fT
2
H(T,t)dt = I 1 (T)+I2
(T)+I3
(T)+I4
(T)
Tl
= fT2T Il(T) sign(T-t)E 2 (jT-tj)t dt
Tl
fT2
=T E 2 (T-t)t
Tl
dt -T dt
These types of integrals are easy to integrate and can
be obtained from reference [22] as follows
(A. 5)
T2 T2
I l ( T) = T[t E ( T-t) - E ( T- t) ) - T [ - tE ( t-T ) - E ( t-T ) ] 3 4 3 4
Tl T
62
(A. 6)
dt+
(A. 7)
T2
I 2 2 1/2 2 2 1/2 2 2 1/2
=- (T-Tl) E2[(t --rl) +(T --rl) ]t dt
Tl
2 2 1/2 = -(T -T ) 1
2 2 1/2 = -(T -T ) 1
2 T2
f 2 2 l/2 2 2 l/2 1
E [(t -T) +(T --r ) )-2 1 1 2
2 Tl
fT~-Ti
2 2 l/2 2 2 l/2 1 2 2 E [(t -T) +(T -T ) ]-d6;:-T)
2 1 1 2 1
0
by letting x and c = constant
then,
I J ( T)
63
(A. 8)
and
Using the same transformation as I 3 (T), this integral
reduces to 2 2 1/2
I4
(T) ~- J:• 2 -T 1~ 3 (x+c)x dx
At the inner sphere T = Tl then the abo~! relations
reduce to
64
hl(l:l) 2
= 1:1/2
h2(1:1) = -'(1'(2 E3(1:2-1:1)+(1:2-1:l)E4(1:2-1:1)
2 2 1/2 2 2) 1/2
+E ( 1: -1: ) - ( 1: -1: ) E4(1: 2 -'(
5 2 1 2 1 1
2 2 1/2 -E ( 1: -1: )
5 2 1 2
1 1(1:1) ll ll
+ 1:1E4(1:2-1:1)+1:11:2E3(1:2-1:1) = 2 3
I 2
At the outer sphere 1: = 1: 2 and the above relations
reduce to
2 2 1/2 -E ( T -T )
5 2 1
B. Non-Isothermal Gas
65
For the case of radiative equilibrium dealing with
approximate analysis the following integrals are used to
evaluate the radiative flux at the lnner sphere.
-'[ 1
T2
= --rl[-E3(t--rl)] Tl
l
f E 3 (T 1-t)dt ~ ll
66
(A. 10)
(A.ll)
67
APPENDIX B
APPLICATION OF THE DIFFUSION APPROXIMATION 'I'O RADIATIVE TRANSFER TIIROUGII A SPHERICAL SHELL OF AN
ABSORBING-EHIT'l'ING GRAY MEDIUM WITH JUMP BOUNDARY CONDITIONS
The generalized equations for radiative diffusion in
a non gray gas and for energy jump at a wall are given in
reference [6]. For the case of a gray gas these equations
take the form
(B. 1)
(B. 2)
for a wall below the gas, and
(B. 3)
for a wall above the gas, where q is the radiant heat z
transfer per unit area in z direction, Eg and Eb is the
total black body emissive power of the gas and the surface
respectively. k is the mean absorption coefficient in ft-l
and the subscripts 1 and 2 are defined in Figure 22 and
68
(XI Y 1 Z)
FIGURE 22. SPHERICAL SHELL, COORDINA'rE SYSTEM
refer to the boundaries enclosing the gas.
For the case of concentric spheres and radiative
equilibrium the heat transfer per unit area q is r
inversily proportional to the square of the radius
Substituting in equation (B.l) and considering the
spherical symmetry
4~ - 3k dr
this equation can be integrated to give
3 rl Egl-Eg = -4 q kr (1---) rl :_1 r
at the outer sphere this expression takes the form
E 1-E 2 g g
qrl
The energy jump at the inner and outer walls are
69
(B. 4)
(B. 5)
(B. 6)
(B. 7)
obtained from equations (B.2) and (B.3). The derivatives 1/2
in those equations are obtained by setting r = (x2
+y2
+z2
)
~n equation (B.l), differentiating, and setting x = 0 and
y = 0. Thus,
aE __51
az (B. 8)
where
and
The results are
similarly,
a2E ( g)
d 2 Y x=O,y=O,z=r1
and
70
(B. 9)
2 2 2 312 2 2 2 2 11 2 (x +y +z ) -3z (x +y +z )
(x2+y2+z2)3
(B.lO)
(B.ll)
(B. 12)
a2E g 3k = () - q ax2 - -4r1 rl
x=O,y=O,z=r1 (B.l3)
qrl (B.l4)
Substituting the values of the derivatives (B.ll)
through (B.l4) in equation (B.2) and (B.3), the expressions
for the energy jump at the black spheres take the
form
and
1 + 3 2 8Tl
71
(B.lS)
(B. 16)
where Tl = kr 1 and T 2 = kr2 are the optical radii at the
inner and the outer sphere respectively.
The addition and rearranging of equations (B.7),
(B.lS) and (B.l6) gives expressions relating q(T1), Egl
and E92
to the emissive power at the boundaries as
E = Ebl -gl
and
E = Eb2 + g2
(Ebl-Eb2) ( l+
3 rl + 1 + -T (1--) 4 1 r 2 2
r 2
3 s:r>
1
3 8Tl
+
3 s:r>
2
rl 2 1 (-) (- -r 2 2
1 3 (Ebl-Eb2) (__.!_} (- - s:r> r2 2 2
r r 2 3 ___!_) + !+ _3_ + (_!} (! -T (1-4 1 r2 2 8T l r 2 2
(B.l7)
(B.l8)
3 1f[)
2
(B. 19) 3
s:-r-> 2
72
Based on physical argument the jump boundary condi-
tions Egl and Eg 2 must lie between Ebl and Eb2. Expressing
this limitation in dimensionless form implies that
1 2+
r 2-r (1- __!)+ 4 1 r 2
3 81"1
(B.20)
at the inner sphere and
3 r 1 3 _l_) +
E g2-Ebl -T (1- 2 +
81"1 4 1 r2 ¢(1"2) =
Eb2-Ebl r r 2 3 !+ 3 (1 _l_) + ( _l_) 3 -T (1- + - g:r) 4 1 r2 2 BTl r2 2 2
(B.2l)
at the outer sphere.
The second criterion, equation (B.2l),can be satis-
fied if -r2 ~ 0.75. This restriction on the magnitude of
-r 2 also satisfy the criterion is equation (B.20).
Based on the above discussion the jump boundary con-
ditions as presented in equations (B.l8) and (B.l9) are
applicable only in the thick limit where -r 2 ~ 0.75. To
evaluate the jump conditions in the thin limit -r 2 < 0.75
other methods should be used as discussed in section IV.
< 1
< 1
APPENDIX C
TABLES
TABLE C.l
ISOTHERMAL GAS N~ALYSIS
73
VARIATION OF THE FLUX FUNCTION Q(T)/T 2 WITH (T2
-T1
)
FOR Tl/T 2 = 0.1
Exact Solution Approximate soln.
T2-Tl (Mean beam length)
Inner Outer Inner sph. Eqs (17-Sphere Eqs. Sphere Eqs. 19) ( 7 19) (8,10)
0.01 1. 0 415 0.0188 0.9947
0.02 0.9985 0.0246 0. 9 89 5
0.04 0.9817 0.0381 0.9792
0.06 0.9718 0.0514 0.9691
0.08 0.9607 0.0643 0.9592
0.10 0.9513 0.0768 0.9495
0.20 0.9067 0.1334 0.9041
0.40 0.8306 0.2226 0.8267
0.60 0.7689 0.2875 0.7640
0.80 0.7186 0.3352 0.7134
1. 0 0 0.6778 0.3705 0.6725
2.00 0.5633 0.4533 0.5595
4.00 0.5080 0.4874 0.5071
6. 0 0 0.5010 0.4944 0.5008
8.00 0.5001 0.4968 0.5001
10.00 0.5000 0. 49 80 0.5000
TABLE C.2
ISOTI-IER!'-1AL GAS AN! r,YSIS
74
VARIATION OF THE FLUX FUNCTION Q(-r)/-r 2 with (-r2
--r1
)
FOR -r 1J-r 2 = 0.2857
Exact Solution Approximate soln.
Inner Sphere Outer Sphere (Mean beam length) T2-Tl Eqs. ( 7 , 9 ) Eqs. ( 8 , 10) Inner sph. Eqs. ,
(17,19)
0.01 0.9956 0. 0 89 8 0.9931
0.02 0.9907 0. 09 72 0.9863
0.04 0.9788 0.1115 0.9730
0.06 0. 9 6 80 0.1255 0.9601
0.08 0.9578 0. 13 89 0.9475
0.10 0.9478 0.1518 0.9353
0.20 0. 9 010 0.2090 0.8790
0. 40 0. 8216 0.2941 0.7873
0.60 0. 7 5 80 0. 3516 0.7179
0.80 0.7071 0. 3909 0.6653
1.0 0.6662 0.4181 0.6254
2.0 0.5556 0. 4 7 36 0.5316
4.0 0.5063 0.4925 0.5020
6.0 0.5007 0. 49 6 5 0.5001
8.0 0.5001 0. 49 80 0.5000
10. 0.5000 0.4987 0.5000
TABLE C. 3
ISOTHERMAL GAS ANALYSIS
75
VARIATION OF THE FLUX FUI·.JCTION Q(T)/T 2 WITH (T2
-T1
)
FOR Tl/T 2 = 0o5
Exact Solution Approximate solno
Inner Sphere Outer Sphere (Mean beam length) T2-Tl Eqso (719) Eqso ( 8 I lO) Inner sph o 1 Eqso
( 17 1 19)
0 0 01 0.9940 0.2573 Oo9888
Oo02 Oo9884 0.2641 Oo9778
0. 0 4 0.9766 0.2771 Oo9566
0.06 0.9652 0.2894 Oo9364
0.08 0.9542 0 . 3011 0.9171
0 .10 0.9434 0.3122 0.8987
0.20 0. 89 33 0.3588 0.8183
0.40 0. 80 9 7 0.4199 0. 70 3 7
0.60 0.7441 0.4543 0.6311
0. 80 0.6926 0.4735 0.5848
1.0 0.6521 0.4843 0.5552
2.0 0.5473 0. 49 6 3 0.5069
4o0 Oo5048 0 0 49 7 3 Oo5001
6o0 Oo5005 Oo4984 0.5000
8.0 0.5001 0.4990 0.5000
10.0 0.5000 0.4994 0.5000
TABLE C.4
ISOTHERMl\L GAS l\NALYSIS
76
VARIATION OF TilE FLUX FUNCTION Q ( T) /T 2 WITH ( T 2
-T l)
FOR Tl/T 2 = 0.9
Exact Solution Approximate so1n.
T2-Tl Inner Sphere Outer Sphere (Mean beam length)
Eqs. ( 7, 9) Eqs. (8,10) Inner sph. Eqs. (17,19)
0.01 0.9923 0. 80 91 0.9294
0.02 0.9848 0. 80 80 0.8697
0.04 0.9701 0.8052 0.7761
0.06 0.9559 0. 8017 0.7084
0.08 0.9422 0.7977 0.6590
0. 10 0 . 9 2 89 0. 79 31 0.6225
0.20 0 . 86 91 0.7664 0.5389
0. 40 0.7754 0.7096 0.5070
0.60 0.7073 0.6612 0. 5019
0. 80 0.6573 0.6235 0.5006
1.00 0.6201 0.5948 0.5002
2.00 0.5333 0.5263 0.5000
4.00 0.5031 0.5023 0.5000
6.00 0. 5010 0.5007 0.5000
8.00 0.5000 0.5000 0.5000
10.00 0.5000 0.5000 0.5000
77
TABLE C.5
NON-ISOTHERMAL GAS ANALYSIS AND RADIATIVE EQUILIBRIUM
VARIATION OF THE FLUX FUNCTION Q(Tl)/Tf
WITH (T 2 -T 1 ) FOR Tl/T 2 = 0.1
Exact Analysis Approximate Deissler
Ref. [ 5] Analysis Analysis T2-Tl Eqs. (33,34) Eq. (21,23)
0. 01 0.9967 0.0030
0.02 0.9960 0.0059
0.04 0.9946 0.0118
0.06 0. 9 916 0.0176
0.09 0.9970 0. 9 89 7 0.0263
0. 10 0.9887 0.0292
0.20 0.9780 0.0575
0.45 0.9850 0.9570 0.1240
0.70 0. 9 3 89 0.1861
0.90 0.9680 0.9255 0.2315
1.00 0.9188 0.2531
2.00 0.8635 0.4272
4.50 0.8300 0.7800 0.6251
7.00 0.7279 0. 6 615
9.00 0.6839 0.6998 0.6432
10.00 0.6882 0.6281
TABLE C.6
NON-ISOTHERMAL GAS Al'>JALYSIS AND RADIATIVE EQUILIBRIUM
78
2 VARIATION OF THE FLUX FUNCTION Q(Tl)/Tl WITH (T 2 -T 1 )
FOR Tl/T 2 = 0.2857
Exact Analysis Approximate Deissler Ref. [19] Analysis Analysis
T2-T1 Eqs. (33,34) Eqs. (21,34)
0. 01 0.9965 0. 010 9
0.02 0 . 9 9 49 0.0216
0.04 0.9917 0.0427
0.07 0.992 0.9858 0.0733
0.08 0.9844 0.0833
0.10 0.9806 0.1029
0.25 0.9735 0.9570 0.2335
0. 40 0.9302 0.3430
0.60 0. 9017 0.4555
0.80 0.8762 0.5386
1.00 0. 89 2 0.8537 0.5985
2.50 0.74 0. 7 40 0 0.6925
4.00 0.6521 0.6147
6.00 0.5185 0.5053
8.00 0.4277 0.4220
10.00 0. 36 32 0.3603
TABLE C.7
NON-ISOTHERMAL GAS ANALYSIS AND RADIATIVE EQUILIBRIUM
79
2 VARIATION OF THE FLUX FUNCTION Q(Tl)/Tl WITH (T
2-T
1)
FOR Tl/T 2 = 0.5
Exact Analysis Approximate Deissler Ref. [ 5] Analysis Analysis
T2-Tl Eqs . ( 3 3, 3 4) Eqs. (21,34)
0.01 0.9968 0.0299
0.02 0.9937 0.0587
0.04 0.9879 0.1131
0.05 0.9900 0.9852 0.1388
0.08 0. 9 76 7 0. 210 2
0.10 0.9711 0.2536
0.25 0.9480 0.9440 0.4866
0.50 0. 89 80 0.8784 0. 6 80 9
0.60 0.8597 0.7159
0. 80 0.8263 0.7490
1. 0 0 0. 80 0 6 0.7976 0.7529
2.50 0.5800 0.6096 0.5937
5.00 0.3835 0 . 3915 0. 3 89 8
6.00 0.3423 0.3413
8.00 0.2731 0.2728
10.00 0.2250 0.2270 0.2269
TABLE C.8
NON-ISOTHERMAL GAS ANALYSIS A.t."JD RADIATIVE
EQUILIBRIUM
80
2 VARIATION OF THE FLUX FUNCTION Q(Tl)/Tl WITH (T 2 -T 1 )
FOR Tl/T 2 = 0.9
Exact Analysis Approximate Deissler
T2-T 1 Ref. [ 5] Analysis Analysis
Eqs. (33,34) Eqs. (21,34)
0.01 0.994 0. 99 39 0.4900
0.02 0.9880 0.6743
0.04 0.9764 0. 8235
0.05 0.9730 0. 9 70 8 0.8587
0.08 0.9543 0.9090
0 .10 0.9460 0.9438 0.9213
0.20 0. 89 50 0.8957 0. 9120
0.50 0.7625 0.7783 0.7905
0.60 0.7400 0.7525
0. 80 0.6741 0. 6 85 3
1.00 0.607 0.6191 0.6284
2.00 0.431 0.4396 0.4424
4.00 0.2769 0.2772
6.00 0.2017 0.2017
8. 0 0 0.1586 0.1586
10.00 0.1306 0.1306
TADLE C.9
COMPARISON BETWEEN THE EXACT VALUES OF THE FUNCTION cjJ ( T) AT TIIE BOUl\!DARIES AND TilE CORRT..::SPONDING APPROXIMATE VALUES FOR DIFFERENT VALUES OF
INNER TO OUTER OPTICAL RATIO T1
/T2
cjJ(T1) cjJ(T2)
81
T2-T1 Exact Ref. Approximate Exact Ref. Approximate [ 5] [ 5]
a. T1/T2 = 0.1
0.09 0.4983 0.5 0.9974 0.9975
0.45 0.4913 0. 5 0.9973 0.9975
0.90 0.4824 0.5 0.9971 0.9975
4.50 0.4097 0.5 0.9968 0.9975
9.00 0.3329 0.5 0.9972 0.9975
b. T1/T2 = 0.5
0.05 0.4941 0.5000 0.9325 0.9330
0.25 0.4707 0.5000 0.9311 0. 9 3 30
0.50 0.4420 0.5000 0. 9 30 6 0. 9 330
1.00 0.3890 0.5000 0.9325 0.9330
2.5 0.2737 0.3838 0.9449 0.9373
5 0.1763 0.2241 0.9613 0.9550
10 0.1011 0.1219 0.9767 0.9727
82
TABLE C.9 (continued)
¢ ( T 1) ¢(T2)
T2-T1 Exact Ref. Approximate Exact Ref ·I Approximate [5] [ 5]
c. T1/T2 = 0.9
0. 01 0.4963 0.5000 0.7180 0. 7179
0.05 0.4818 0.5000 0.7187 0.7179
0.10 0.4645 0.5000 0.7208 0. 7179
0.20 0.4324 0.5000 0.7275 0.7179
0.50 0.3563 0.4611 0.7554 0.7279
1. 0 0 0.2755 0.3404 0.7985 0.7646
2.00 0.1912 0.2304 0.8537 0.8276
83
APPENDIX D
EXPERIMENTAL DATA AND RESULTS
TABLE D.l
VARIATION OF THE POWER INPUT, CONDUCTION LOSSES AND THE EMISSIVITY FACTOR WITH THE INNER SPHERE
TEMPERATURE*
0 in Tl 0 1oss Qr F e
BTU/HR OF BTU/HR BTU/HR
34.13 300.5 3.82 30.31 0. 89 4
68.26 422. 6.38 61.88 0.832
102.39 517 8.41 93.98 0.782
135.9 596 10.2 125.7 0.74
170.65 664.5 11.69 158.96 0.712
200.64 715.5 12.80 187.84 0.697
243.5 786.6 14.44 228.06 0.66
*Outer sphere temperature = 113°F (for all runs)
*Vacuum run
TABLE D.2
EXPERIMENTAL DATA AND RESULTS FOR HELIUM RUN*
PRESSURE T1 0 1oss Qr
Qc Qc ATMOSPHERE PREDICTED
OF BTU/HR BTU/HR ~~lJ)~~ CXPER2J1ENTAI BTJ.I lR
0.022 60 3. 10.35 131.18 18.24 37.47 0.049 567. 9.6 114.99 26.24 54.41 0.119 525.66 8.65 98.19 37.95 72.16 0.188 511. 8.32 92.33 46.96 78.35 0.236 503. 8.19 89.81 52.06 81.0 0.482 462.33 7.28 75.19 67.72 98.53 0.758 438.5 6.75 67.30 79.81 104.95 0. 9 83 427. 6.5 63.86 88.97 108.64 1.509 401.66 5.92 55.92 102.25 117.16 2.004 386.5 5.6 51.85 112.61 121.52 2.499 374. 5.35 48.20 121.13 125.45 3.001 363.5 5.1 45.40 127.61 128.5
~--'-- -··~ .. ~
*Power Input = 179 BRU/HR *Outer Sphere Temperature = 113°F
**ERROR= [(Q. -TOTAL PREDICTED VALUES)/Q. ]100 ln ln
ERROR
10.74
15.72
19.13
17.5
16.17
16.1
14.02
11.0
9.95
4. 9 8
2.42
0.5
**
I
co ~
TABLE D.3
EXPERIMENTAL DATA AND RESULTS FOR CARBON DIOXIDE RUN*
PRESSURE T1 01oss Qc PRE-j
ATMOSPHERE OF DICTED EQ. BT.U/HR (3~ BTU/HR
0.006 652.33 11.45 5.36
0.022 637.33 11.12 11.0 8
0.032 618.66 10.73 13.13
0.049 609.33 10.50 16.26
0.119 590.00 10.0 8 25.06
0.188 578.50 09.83 31.32
0.236 573.00 09.71 35.03
0.482 552.66 09.25 48.95
0.758 535.66 0 8. 90 59.74
0.997 527.00 08.68 67.64
1.049 521.66 0 8. 55 68.30
1. 509 504.50 0 8. 20 79.08
2.004 491.00 07.90 8 8. 53
2.506 483.00 07.72 97.39
3.022 473.00 07.50 104.31
*Power Input = 179 BRU/HR 0 *Outer Sphere Temperature = 113 F
Qr EXPERI.VillNTAL
BTU/HR
162.19
156.8
155.14
152.24
143.86
137.85
134.26
120.8
110.36
102.68
102.12
91.72
82.57
73.89
67.19
**ERROR= [(Q. -TOTAL PREDICTED VALUES)/Q. ]100 ln ln
Qr PREDICTED EQ.
ERROR** (41) BTU/HR
154.36 4.37
146.76 5.76
137.42 9 . 9
132.73 10.9
12 3. 0 7 11.6
116.32 12.02
111.75 12.58
98.63 12.38
84.31 14.56
76.57 14.58
73.80 15.82
61.68 16.78
52.02 17.06
45.44 15.9
3 8. 9 4 15.8
co vl
I3Il3LIOGRAPHY
1. Chandrasekhar, S. "Radiative Trunsfer," Dover Publications, New York, 1960, p. 287.
86
2. Heaslet, M.A. and vlarming, R.F. "1\pplication of Invariance Principles to a Radiative Transfer Problem in a Homogenous Spherical Medium," J. Quant. Spectrosc. Radiat. Transfer, Vol. 5, No. 5, Sept./Oct. 1965, p. 669.
3. Sparrow, E.M., Usiskin, C.M. and Hubbard, H.A. "Radiation Heat Transfer in a Spherical Enclosure Containing a Participating Heat-Generating Gas," Trans. ASME, Journal of Heat Transfer, Vol. 83, Series C. No. 2, May 1961, p. 199.
4. Rhyming, I.L. "Radiative Transfer Between Two Concentric Spheres Separated by an Absorbing and Emitting Gas," Int. J. Heat Hass Transfer, Vol. 9, No. 4, April 1966, p. 315.
5. Viskanta, R. and Crosbie, A.L. "Radiative Transfer Through a Spherical Shell of an AbsorbingEmitting Gray Medium", ~uant. Spectrosc. Radiat. Transfer, Vol.7, No. 6, Nov./Dec. 1967, p. 871.
6. Deissler, R.G. "Diffusion Approximation for Thermal Radiation in Gases with Jump Boundary Conditions," Trans. ASME, Journal of Heat Transfer, Vol. 86, Series C, No. 2, May 1964, p. 240.
7. Olfe, D.I3. "Application of a Modified Differential Approximation to Radiative Transfer in a Gray Medium Between Concentric Spheres and Cylinders," J. Quant. Spectrosc. Radiat. Transfer, Vol. 8, No. 3, March 1968, p. 899.
8. Tien, C.L. and Chou, Y.S. "A Modified Moment Method for Radiative Transfer in Non-Planer Systems," J. Quant. Spectrosc. Radiat. Transfer," Vol. 8, No. 3, March 1968, p. 919.
9. Finkleman, D. "A Note on Boundary Conditions for Use with the Differential Approximation to Radiative Transfer," Int. J. Heat Mass Transfer, Vol. 12, No. 5, May 1969, p. 653.
10. Hunt, B.L. "An Examination of the Method of Regional Averaging for Radiative Transfer Between Concentric Spheres," Int. J. Heat Mass Transfer, Vol. 11, No. 6,-June 1968, p. 1071.
11. Lee, R.L. and Olfe, D.B. "An Iterative Method
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for Non-Planar Radiative Transfer Problems," J. Quant. Spectrosc. Radiat. Transfer, Vol. 9, No. 2, Feb. 1969, p. 297.
12. Dennar, E. and Sibulkin, H. "An Evaluation of the Differential Approximation for Spherically Symmetric Radiative Transfer," Trans. ASME, Journal of Heat Transfer, Vol. 91, Series c, No. 1, February 1969, p. 73.
13. Traugott, S.C. "An Improved Differential Approximation for Radiative Transfer with Spherical Symmetry," AIAA Journal , Vol. 7, No. 10, Oct. 1969, p. 1825.
14. Emanuel, G. "Radiative Energy Transfer from a Small Sphere," Int. J. Heat t·1ass Transfer, Vol. 12, No. 10, Oct. 1969, p. 1327.
15. Viskanta, R. and Merriam, R.L. "Heat Transfer by Combined Conduction and Radiation Between Concentric Spheres Separated by Radiating Medium," Trans. ASME, Journal of Heat Transfer, Vol. 90, Series C., Mo. 2, May 1968, p. 248.
16. Mack, L.R. and Hardee, H.C. "Natural Convection Between Concentric Spheres at Low Rayleigh Numbers," Int. J. Heat Mass Transfer, Vol. 11, No. 3, March 1968, p. 387.
17. Bishop, E.M., Mack,L.R., and Scanlan, J.A. "Heat Transfer by Natural Convection Between Concentric Spheres," Int. J. Heat Mass Transfer, Vol. 9, No. 7, July 1966, p. 649.
18. McAdams, W.H., "Heat Transmission," McGraw-Hill, New York, 1954, p. 82.
19. Crosbie, A.L2 , Numerical Values of the Flux Function Q(; 1 )/Tl for t~e R~tio (T 1 /T 2 ) = 0.2857 Pr1vate Commun1cat1on.
20. Sparrow, E.M. and Cess, R.D. "Radiation Heat Transfer, Drooks/Cole Publishing Company, Belmont, Calif., 1966.
21. Holman, J.P. "Heat Transfer," McGraw-Hill, New York, 1968, p. 385.
22. Abramowitz, M. and Stegurn, I.A. (editors), "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," Dover publications, New York, 1965, p. 227.
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80
VITA
Adel Nassif Saad was born on January 15, 1941 in
Deirmawas, Egypt (U.A.R.). He received his primary
and secondary education in Deirmawas, Mallawi ~d cairo,
Egypt. In October, 1960 he enrolled in the Faculty of
Engineering Cairo University and in July, 1965 he re
ceived a Bachelor of Science Degree in Aeronautical
Engineering.
From October, 1965 to April, 1969 the author was
employed with the Aircraft Inspection Directorate -
Cairo International Airport. In May, 1966 he was granted
the International Civil Aviation Scholarship for Advanced
Training Program in Beirut, Lebanon.
In June, 1969 he has been enrolled in the Graduate
School of the University of Missouri-Rolla as a candidate
for the Master of Science Degree in Mechanical Engineering.
