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Combined radiant and convective heat transfer to laminar steam flow between gray parallel plates with uniform heat flux
J. Quanr.
Speclrosc. Radial.
Transfer. Vol. 15, pp. IMI-1081.
Pergamon Press 1975. Printed in Great Britam
COMBINED RADIANT AND CONVECTIVE HEAT TRANSFER TO LAMINAR STEAM FLOW BETWEEN GRAY PARALLEL PLATES WITH UNIFORM HEAT FLUX J. K. MARTIN and C. C. HWANG Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh Pa. 15261,U.S.A. (Received 29 October 1974) Abstract-The non-linear integro-differential equations describing combined radiation and convection heat transfer to non-gray non-isothermal steam in laminar flow between gray parallel plates are developed and solved numerically. The results are discussed and compared with a gray, linear, analytical solution and with available experimental data.
NOMENCLATURE Planck’s function, Btu-cm/ft’-hr specific heat, Btu/lbm-“F radiant intensity, Btu/ft*-hr-steradian thermal conducticity, Btu/ft-hr-“F pressure, atm heat flux, Btu/ft’-hr wall heat flux, Btu/ft*-hr wall radiosity temperature, “F local velocity and average velocity, respectively, ftlsec plate spacing, ft transverse coordinate, ft emissivity density, lbm/ft’ temperature variable, dimensionless absorption coefficient, ft-’ wavelength, ft radiant heat flux variable, dimensionless dimensionless distance from lower plate solid angle, steradian wave number, cm-’ Stefan-Boltzman constant
Subscript m mean
A evaluation at wavelength b evaluation at black body conditions x axial direction y transverse direction w evaluation at wave number
Superscript r radiant quantity
+ -
directed-in for&ard hemisphere directed in backward hemisphere
Conversion factors 1 Btu = 0.251996Kcal
1 ft = 0.3048m “F = 1g”C + 32 1atm = 1.013X 106dyne/cm2 1071 QSRT Vol. I5 No. 12-A
1072
J. K. MARTINand
C. C.
HWANC
1. STATEMENTOF THE PROBLEM THEPURPOSE of this study is to investigate the interaction of simultaneous convective and radiant heat transfer to superheated steam flowing laminarly between gray diffuse parallel plates. The boundary condition at the plate walls is that of uniform heat flux. The walls are to have equal emissivities. The steam is represented as non-isothermal and non-gray with five absorption-emission bands. Scattering is assumed to be negligible and the steam refractive index is assumed to be unity. The frequency and temperature dependence of the absorption in each band is represented. The flow and temperature profiles are assumed to be fully developed. The thermodynamic and transport properties of the steam are evaluated at mean conditions and assumed constant. Although the analysis is presented for superheated steam, it could be applied with small alteration to any non-scattering absorbing-emitting gas. 2. ANALYSIS 2.1 Energy equation For steady flow of an incompressible fluid with constant thermal conductivity, the energy equation can be written as
where the viscous dissipation term has been neglected. For the fully developed laminar velocity profile, the first term on the left is zero. The second term on the right, which represents conduction in the axial direction, may be neglected for a system with a Peclet number much greater than 1.“’ For such a system, conduction in the axial direction is much smaller than axial convection or transverse conduction. When radiation effects are not too strong, a criterion may also be developed for neglecting the axial component of the radiant heat flux. For this term to be negligible, it must be shown that
(2) The criterion for this condition to be met is developed in Ref. (1). The criterion can be expressed as
The present study treats the cases in which conduction and radiation in the axial direction can be neglected in comparison with the transverse direction. Employing this assumption and the constant heat flux boundary condition, we may write c?T 2Q -=dx @WC,’
(4)
The laminar flow profile is given as U, = 68$1-
y/W).
(5)
Substituting eqns (4) and (5) into eqn (1) and simplifying yields
,2&&y,W)=kfl-a9y’ W2
ayz
ay.
(6)
We may define *= 2YlW,
(7)
1073
Combined radiant and convective heat transfer
T, - T H$) = QW,k 9
With these definitions, eqn (6) may be non-dimensionalized as
The boundary conditions are 1
I0
Ou, de= 0,
(11)
(12) The first boundary condition represents the specified mean temperature of the system while the second states conditions of symmetry. Integrating eqn (10) from JI to 1 and then form zero to $ yields (13) where
e,= Tm-Tw ’
(14)
QWIk ’
and is to be evaluated from eqn (11). Equation (13) is the desired equation, in non-dimensional form, for the temperature distribution. The dimensionless radiant heat flux, 4, is, however, a complex non-linear integral function of the temperature distribution. Once 4 is expressed as a function of 4, eqn (13) is solvable. 2.2 Radiant heat flux equations Beginning with the quasi-steady state equation of transfer in spherical coordinates”’ with no scattering, assuming local thermodynamic equilibrium and defining k = cos 19,where 13is the polar angle, the following equation may be written:
where the optical path length TV,is defined as
r,& =
I
KA
dy.
(16)
Formal solution of eqn (15) give?’ -K7*-T*‘HWl
L+(T*,p) = IwA emcTA”‘)+ IA Ib,Aep-
dTA’,
(0 < CL< l),
(17)
1074
J. K.
MARTINand C. C. HWANC
where IA+is the positively directed intensity, I,- is the negatively directed intensity, and the wall intensity is used as a boundary condition. The positively and negatively directed radiant heat fluxes in the wavelength interval dh at A are defined, respectively, as
(20) where the superscipt r has been omitted for clarity. The following transformation is useful: dR = sin 0 de da,
(21)
da=-dkd&
(22)
or
where 4 is the azimuthal angle. Using this transformation on eqns (19) and (20) yields (23) and similar result for the negatively directed radiant heat flux. The wall intensity may be related to the wall radiosity and the black body intensity to Planck’s function through the following relations, the first of which assumes a diffuse wall I w.,,
=
R/T,
(24)
&,A= BAIT,
(25)
Equations (17), (23), (24) and (25) may be combined to give qh+(rh)= 2R, s’ e-(‘A’r)p dp + 2 ITA1’ BA e-‘(TA-ri)‘pldp d’r:.
(26)
With the definition of the familiar exponential integral function’4.5’ E,(~)
=
’ p“-* emx’&dp,
I0 eqn (26) may be rewritten as I&+
=
2&&(n)
+2
I
‘Ok
B, (T)&(T,
- 7:) dT:.
(28)
Th
The same procedure yields a similar result for the negatively directed radiant heat flux I*-
=
~RA&(TOA- 7~)+ 2
I
TOA
B,(T)Ez(TL-TA)~T:,
TA
(29)
where ToA is the optical path length between plates at wavelength A. The total radiant heat flux at a point is defined as 4r ‘z
Iom
(qh+- qh-)dh.
(30)
1075
Combined radiant and convective heat transfer
Thus, inserting eqns (28) and (29) into eqn (30) and recalling the definition of the dimensionless radiant heat flux function,
I
-
TO&
TA
Bn(T)E2(7:-
Th)dT: dh. I
(31)
It now remains to specify the wall radiosity, R,. The radiosity may be expressed as
E)&=o.
RA = eB* + (I-
(32)
Inserting eqn (29), evaluated at y = 0, into eqn (32) yields
I
Toon
Bh(T)&(T:)dT:
0
1 .
(33)
Equation (33) may be solved for the radiosity as follows, EBB+2(1-e) Rh =
TOA
Il
B*(T)Ez(T:)dT:
1 - 2(1”_ )&(%~)
’
(34)
Equations (31) and (34) now completely specify the radiant heat flux distribution for a given temperature distribution. 2.3 Absorption coeficient model EDWARDS et aI.‘clo’ recommend using the average line intensity correlation of the wide band exponential model for the absorption coefficient of an absorbing-emitting gas such as steam. This correlation gives the frequency and temperature dependence of the gas and takes the form K (3 T) ~ = &exp
[-lo - ool/C31,
P
(35)
where R is the steam gas constant, o. is the wave number at the band center, and C1 and Cs are different function of temperature for each band of the gas. These functions have been reported for the four vibration-rotation bands and one pure rotation band of water vapor.‘> The two important bands for radiant heat transfer calculations are the 1600 cm-’ band and the 3750 cm-’ band, respectively. 2.4 Gray gas-linear radiation analysis For the purpose of comparing with the exact numerical solution, as presented in the next section, an approximate analytical solution of eqns (10) and (31) was developed. In order to arrive at an analytical solution, it is necessary to make the rather severe approximation that the gas is gray, and to restrict the problem to one of black walls. In addition, the exponential kernal approximation is used, along with the assumption of small temperature differences in the gas. The latter assumption allows the radiation to be linearized and allows the temperature dependence of the absorption coefficient to be neglected. See Ref. (11) for details of the derivation. The solutions thus obtained for the radiant heat flux distribution and the temperature distribution are
4 (5)= fl sinh~5+ e(Z)= +osh
MZ’ 2(1
+
M)
+
f&
y - cash r[) + A( 1- 6’) + 16;1-+t;) + eo,
(37)
1076
J. K.
MARTIN
and C. C. HWANG
where, for convenience, a coordinate transformation to the midpoint between the plates has been performed. Thus ‘$=*-I.
(38)
The coefficients in eqns (36) and (37) are defined as f = ’
9MTO*(yZ~o - 37”- 3) 16y“(y cash y + 7asinh y)’
(39) (40)
f3=
970*(1 + M) + 32M 24~,2(1+M)* ’
(41)
where M
=
16uTIn3 3K,K
y= $1
(42)
’
+ M)“2.
(43)
Substituting eqn (37) into eqn (11) and performing the indicated integration yields for 2 ++[(+3)coshY+~]. & = 35(If M)
w
The Planck mean absorption coefficientC1”for steam, K~, was used in this analysis to represent the emission-absorption characteristic of the gas. As seen from the above equation, the heat transfer is characterized by the parameter M, which is the ratio of the radiation conductivity to the molecular conductivity, and the parameter G, which is the system optical thickness.
3. SOLUTION
Since the radiant heat flux function, c$($), is an integral function of the temperature distribution, we might expect it to be a smooth well behaved function of $. We would also expect, from physical considerations and from the boundary condition expressed by eqn (12), 4 to be an odd symmetric function about (I,= 1 and to have the value zero at $ = 1. From these considerations, then, we should be able to represent I$ as an odd polynomial of some order N. Thus
where the coefficients may be determined from a direct evaluation of 4 at (IV + 1)/2 points, assuming the temperature distribution is known. Substituting eqn (45) into eqn (13) and performing the integration yields O(l))
$ [$#
=&++;+-;
- ,)“+I-
l)] + eo.
(46)
Odd
Substituting eqn (46) into eqn (11) and performing the integration yields for (47) Odd
1077
Combined radiant and convective heat transfer
Notice that if the radiant heat flux is assumed to be identically zero, eqns (46) and (47) reduce to the familiar convective result for constant heat flux. Equations (31), (34) and (46) may be solved through a directive iterative technique. The procedure is to assume a temperature distribution and, based on this distribution, calculate the radiant heat flux at (N + 1)/2 evenly spaced points using eqns (31) and (34). Equations (31) and (34) are evaluated at the desired points by numerically performing the required integrations over space and frequency. The values of the radiant heat flux coefficients, & are then determined through the simultaneous matrix solution of (N + 1)/2 linear algebraic equations similar to eqn (45). These coefficients, when inserted into eqns (46) and (47) give the new temperature distribution, which is used for the next iteration. Convergence is obtained when & (hence the radiant heat flux) ceases to change by a preset amount. The solution is basically oscillatory due to the inherent non-linearity of the equations. Thus, if the assumed initial temperature gradient is too severe, the calculated radiant heat fluxes are too large, which yields a temperature gradient which is too small for the next iteration. These oscillations may be damped out and convergence enhanced, through application of a “relaxation factor”. Thus, the radiant heat flux change between iterations is multiplied by this relaxation factor (usually approx. 0.5) before the new radiant heat flux distribution is used as input to calculate the temperature distribution for the next iteration. One of the more important parameters, both from the standpoint of accuracy and from the standpoint of computational time, is the order of expansion of the radiant flux distribution. Little improvement in the calculated Nusselt number was obtained in going from a fifth order expansion to a seventh order expansion, and even less improvement, approximately one fourth of one percent, was obtained in going from a seventh order to a ninth order expansion. Therefore, a seventh order expansion of the radiant heat flux distribution was used in the calculation. A Fortran program’“’ was written for the IBM 360 computer to do the above calculations and perform the iterations. 4. COMPARISON
WITH EXPERIMENTAL
DATA
Few experimental investigations have been performed to measure the effect of radiation on heat transfer to laminar steam flow. A literature survey, however, revealed an experimental evaluation of combined radiation and convection to steam flow in round tubes.‘13’Although the study was concerned primarily with turbulent and transitional flow, three data points were reported for laminar flow. Uniform heat flux, 310 stainless steel tubes with 0.498 in. inner diameter were used in the experiment. It should be pointed out here that the criteria for neglecting the axial conduction and radiation are met with the experimental data. The difficulty of comparing the analysis presented herein with the experimental data is obvious, due to the different geometries. It is postulated, however, that the geometric effects on the radiation can be approximately accounted for by using a flat plate geometry whose mean beam length is the same as that of the tubes used in the experiment. Since the mean beam length of plates, based on their spacing, is twice that of a tube,‘14’based on its diameter, the spacing of the plates used for the comparison was 0.249 in. Also, since the uniform flux, laminar flow Nusselt number for convection in tube is 4.36’14’while that for plate is 8.235, a comparison of the magnitudes of the experimental and calculated Nusselt numbers is not valid. A more meaningful quantity to compare is the fractional increase in Nusselt number brought about by the radiant effects. This, then, was the quantity used to compare the calculated results with the experiment. Three experimental runs were made with laminar flo~.“~’ Each run yielded only one valid data point for the fully developed Nusselt number since the fully developed condition was not reached except near the end of the tube. The tube wall emissivity was measured and is shown, along with the other pertinent experimental conditions, in Table 1. Also in Table 1 is the experimentally measured increase in Nusselt number due to radiation, and that calculated with the method presented herein. Considering the difficulty of comparing the geometry of the analysis and that of the experiment, surprisingly good agreement is found between the calculated radiation effects and those measured. As indicated in the table, the discrepancy between the measured effect and that calculated is approximately 8.3% for the second data point, while the calculated and measured effects for the other two points agree to within less than 1%. This close agreement indicates the
1078
J. K. MARTIN and C. C. HWANC Table 1. Comparison between the measured’*‘andcalculated effect of radiation on Nusselt number.
Experiment a”* Number
*“be I.D.
Plate spac1nq USed Analysis,
Q Tm
3.n
Pressure
Tube Emissivity
PCZrCc”t 1ncrcase
in
Pfrcfnt I”CTf2Sf In
_-%!A
Nusselt,
russelt,
ft *-hy
+!“:easure?
CelculaLeZ
In.
Atm.
CF
45
.438
,249
1.754
865
,848
2503
43.l.i
47
.498
,243
1.725
048
,852
2500
31.65
x3.35
49
.438
,243
5.155
lOC6
,855
2300
7U.69
73.25
Ill.
‘IO.XB
validity of the analysis, although it obviously does not fully substantiate it due to the limited number of data points available for comparison. 5. DISCUSSION
OF RESULS
Several representative situations of combined radiant and convective heat transfer to steam were analyzed by the method discussed in the preceding sections. 5.1 Radiation
effect on heat flux
Figure 1 shows the dimensionless radiant and conductive heat flux distribution between two sets of black parallel plates. The parameters for these two configurations are given on the figure. Since the heat flux distributions are symmetric, only the lower half of the distributions are shown in the figure. It is interesting to note that the radiant heat flux peaks at some distance into the fluid stream rather than at the wall. This effect was also noted by VISKANTA”” in his gray gas analysis of radiation and convection between plates of constant temperature. The reason for this is that, at the wall, the effect of the positive radiant heat flux from the wall is partially cancelled by the negative flux from the layers of hot gas next to the wall. At small distances into the stream, however, the positive flux from both the wall and the hot gas combine to give a maximum heat flux. Farther into the stream, the flux from the wall and hot gas is attenuated by the intervening cooler gas and, of course, partially cancelled, and at 4 = 1.0 exactly balanced, by flux from the opposite side of the channel. 5.2 The effect of radiation on the Nusselt number Combined radiant and convective heat transfer to a flowing participating gas may yield, depending on the system parameters, significantly higher heat transfer coefficients than would be expected if conduction were the only heat transfer mechanism considered. The total Nusselt number, as a function of plate spacing, is shown in Fig. 2 for black plates with the parameters indicated on the figure. Note that all the curves in Fig. 2 converge to the pure convective Nusselt number of 8.235 as the plate spacings approach zero. I
1
c I T,,= 600 g
I
’
I
8
W = 1.0 INCH Q = 200 BTU/FTZ
HR
PATM
2
1.0 -
if
_
i 1
-
3
-_-
,
l
1.0 0.5
.a_
Fig. 1.Dimensionless heat flux distributions.
Combined radiant and convective heat transfer
1079
Fig. 2. Nusselt number as a function of plate spacing.
As would be expected, the higher pressures yield higher Nusselt numbers. This is due to the increased optical path length of the system, which enhances the radiant heat transfer. Figure 2 indicates, however, that the effect of increasing Nusselt number with increasing pressure is less pronounced at the higher pressures shown. This is explained by the fact that, at some pressure, the system becomes saturated (i.e. the absorptivity approaches 1-O) and further increase in pressure has less effect on the radiation heat transfer. Figure 2 also illustrates the fact that an increase system mean temperature significantly increases the effect of radiation, thereby yielding higher Nusselt numbers. It will be noted from Fig. 2 that a relatively small plate spacings, with intermediate pressures (i.e. greater than 1.0 atm) and temperatures, radiation is a significant heat transfer mechanism. With plate spacing as small as 0.20 in. at the higher pressures and temperatures, radiation may be the dominant mode of heat transfer. Figure 3 illustrates the effect of wall emissivity on the total Nusselt number. The curves in Fig. 3 were generated with the parameters shown. Figure 3 shows that, due to the reflection of the radiant heat flux incident on the wall, a reduced wall emissivity results in a reduced Nusselt number.
1
I
I
I
T,= 600 “F W = I.0
INCH
P=I
ATM
-
IO -
0
I .2
I .4 PLATE
I .6
I .S
1.0
EMISSIVITY
Fig. 3. Nusselt number as a function of plate emissivity.
1080
J. K. MARTIN and C. C.
.=
1.0
Q = 200 -
HWANG
STU/FT’-HR
GRAY
/
APPROX.
---EXACT
1
1
.2
_I PLATE
I
I
.3
.4
.5
SPACING,
INCH
I
Fig. 4. Nusselt number as a function of plate spacing.
5.3 Comparison
between the gray gas-linear
radiation approximation
and the exact solution
Several cases have been analyzed using both the gray gas-linear radiation approximation, as developed in Ref. (11) and presented in Section 2.4 and the exact solution. The purpose is to investigate the accuracy of the approximate solution and to see if it demonstrates the expected trends. As expected, the approximation overestimates the radiant heat flux. This is not surprising since a participating gas is actually transparent over large regions of the spectrum and may be practically opaque in other regions. Thus any attempt to represent the entire spectrum with a mean absorption coefficient will lead to considerable overestimate of the radiant heat transfer in the transparent and opaque regions. No mean absorption coefficient has yet been found which will, in the general case, yield accurate results for radiant heat transfer in the non-gray media. Figure 4 shows the variation of the Nusselt number with plate spacing, as calculated with both the exact solution and the approximate one. Curves for two sets of system parameters, which are listed on the figure, are shown. Figure 4 shows that the gray approximation produces the proper parametric trends but yields a high estimate of the Nusselt number, as would be expected. The magnitude of the error resulting from using the gray approximation is a sensitive function of the system being analyzed. A small plate spacing, low temperature, or pressure yields less radiant effect on the Nusselt number and therefore less error is introduced by using the gray gas-linear radiation approximation. From these data, then, it appears that the gray gas-liner radiation approximation consistently provides a high estimate of the Nusselt number for a combined convection and radiation heat transfer system. The error may be quite severe for a system in which radiation is the dominant heat transfer mechanism. The approximation is useful, however, in providing insight into the parametric trends to be expected in the system heat transfer characteristics and in estimating the significance of radiation. REFERENCES 1. R. D. CESSand E. M. SPARROW, Radiation Heat Transfer. Brooks-Cole, Belmont, California (1966). 2. B. T. CHAO(Ed.). Advanced Heat Transfer, DD.75-156. Universitv of Illinois Press, Urbana, Illinois (1%9). 3. T. F. IRWIN, h. ‘Ad J. P. HARTNE~T (Ed:.), ;idvances in Heat Tr&er. Academic Press, New York (1966). 4. H. C. H~TTELand A. F. SAROL~M, Radiative Transfer. McGraw-Hill, New York (1967). 5. T. ROCKWELL, Reactor Shielding Design Manual. Van Nostrand, Princeton, New Jersey (1956). 6. S. DE&TO and D. K. EDWARDS, Proc. of 1%5 Heat Transfer and Fluid Mechs. Inst. pp. 358-372.Stanford University Press (1965). 7. D. K. EDWARDS, B. J. FLORNES, L. K. GLASSEN and W. SUN,Appl. Opt. 4, 715 (1965). 8. D. K. EDWARDS, L. K. GLASSEN, W. C. HAUSERand J. S. TLJCHSCHER, J. Heat Trans. 89, 219 (1967). 9. D. K. EDWARDS and W. A. MENARD,Appl. Opt. 3, 621 (1964). 10. D. K. EDWARDS and W. A. MENARD,Appl. Opt. 3, 847 (1964). 11. J. K. MARTIN, MS. Thesis, Mechanical Engineering, University of Pittsburgh (1971).
Combined radiant and convective heat transfer
1081
12. M. M. ABU-ROMIA and C. L. RF& J. Heat Trans. 89, 321 (1967). 13. P. S. LARSEN, H. A. LORDand R. F. FARMEN, 4th Int. Heat Trans. Conf. Paris-Versailles 1970,Vol. III, P.R2.6, Elsevier, Amsterdam (1970). 14. W. M. ROHSENOW and H. CHOI,Heat, Mass and Momentum Transfer. Prentice Hall, Englewood Cliffs, New Jersey (1971). 15. R. VISKANTA, Appl. Scientific Res. A13, 291 (1964).
Speclrosc. Radial.
Transfer. Vol. 15, pp. IMI-1081.
Pergamon Press 1975. Printed in Great Britam
COMBINED RADIANT AND CONVECTIVE HEAT TRANSFER TO LAMINAR STEAM FLOW BETWEEN GRAY PARALLEL PLATES WITH UNIFORM HEAT FLUX J. K. MARTIN and C. C. HWANG Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh Pa. 15261,U.S.A. (Received 29 October 1974) Abstract-The non-linear integro-differential equations describing combined radiation and convection heat transfer to non-gray non-isothermal steam in laminar flow between gray parallel plates are developed and solved numerically. The results are discussed and compared with a gray, linear, analytical solution and with available experimental data.
NOMENCLATURE Planck’s function, Btu-cm/ft’-hr specific heat, Btu/lbm-“F radiant intensity, Btu/ft*-hr-steradian thermal conducticity, Btu/ft-hr-“F pressure, atm heat flux, Btu/ft’-hr wall heat flux, Btu/ft*-hr wall radiosity temperature, “F local velocity and average velocity, respectively, ftlsec plate spacing, ft transverse coordinate, ft emissivity density, lbm/ft’ temperature variable, dimensionless absorption coefficient, ft-’ wavelength, ft radiant heat flux variable, dimensionless dimensionless distance from lower plate solid angle, steradian wave number, cm-’ Stefan-Boltzman constant
Subscript m mean
A evaluation at wavelength b evaluation at black body conditions x axial direction y transverse direction w evaluation at wave number
Superscript r radiant quantity
+ -
directed-in for&ard hemisphere directed in backward hemisphere
Conversion factors 1 Btu = 0.251996Kcal
1 ft = 0.3048m “F = 1g”C + 32 1atm = 1.013X 106dyne/cm2 1071 QSRT Vol. I5 No. 12-A
1072
J. K. MARTINand
C. C.
HWANC
1. STATEMENTOF THE PROBLEM THEPURPOSE of this study is to investigate the interaction of simultaneous convective and radiant heat transfer to superheated steam flowing laminarly between gray diffuse parallel plates. The boundary condition at the plate walls is that of uniform heat flux. The walls are to have equal emissivities. The steam is represented as non-isothermal and non-gray with five absorption-emission bands. Scattering is assumed to be negligible and the steam refractive index is assumed to be unity. The frequency and temperature dependence of the absorption in each band is represented. The flow and temperature profiles are assumed to be fully developed. The thermodynamic and transport properties of the steam are evaluated at mean conditions and assumed constant. Although the analysis is presented for superheated steam, it could be applied with small alteration to any non-scattering absorbing-emitting gas. 2. ANALYSIS 2.1 Energy equation For steady flow of an incompressible fluid with constant thermal conductivity, the energy equation can be written as
where the viscous dissipation term has been neglected. For the fully developed laminar velocity profile, the first term on the left is zero. The second term on the right, which represents conduction in the axial direction, may be neglected for a system with a Peclet number much greater than 1.“’ For such a system, conduction in the axial direction is much smaller than axial convection or transverse conduction. When radiation effects are not too strong, a criterion may also be developed for neglecting the axial component of the radiant heat flux. For this term to be negligible, it must be shown that
(2) The criterion for this condition to be met is developed in Ref. (1). The criterion can be expressed as
The present study treats the cases in which conduction and radiation in the axial direction can be neglected in comparison with the transverse direction. Employing this assumption and the constant heat flux boundary condition, we may write c?T 2Q -=dx @WC,’
(4)
The laminar flow profile is given as U, = 68$1-
y/W).
(5)
Substituting eqns (4) and (5) into eqn (1) and simplifying yields
,2&&y,W)=kfl-a9y’ W2
ayz
ay.
(6)
We may define *= 2YlW,
(7)
1073
Combined radiant and convective heat transfer
T, - T H$) = QW,k 9
With these definitions, eqn (6) may be non-dimensionalized as
The boundary conditions are 1
I0
Ou, de= 0,
(11)
(12) The first boundary condition represents the specified mean temperature of the system while the second states conditions of symmetry. Integrating eqn (10) from JI to 1 and then form zero to $ yields (13) where
e,= Tm-Tw ’
(14)
QWIk ’
and is to be evaluated from eqn (11). Equation (13) is the desired equation, in non-dimensional form, for the temperature distribution. The dimensionless radiant heat flux, 4, is, however, a complex non-linear integral function of the temperature distribution. Once 4 is expressed as a function of 4, eqn (13) is solvable. 2.2 Radiant heat flux equations Beginning with the quasi-steady state equation of transfer in spherical coordinates”’ with no scattering, assuming local thermodynamic equilibrium and defining k = cos 19,where 13is the polar angle, the following equation may be written:
where the optical path length TV,is defined as
r,& =
I
KA
dy.
(16)
Formal solution of eqn (15) give?’ -K7*-T*‘HWl
L+(T*,p) = IwA emcTA”‘)+ IA Ib,Aep-
dTA’,
(0 < CL< l),
(17)
1074
J. K.
MARTINand C. C. HWANC
where IA+is the positively directed intensity, I,- is the negatively directed intensity, and the wall intensity is used as a boundary condition. The positively and negatively directed radiant heat fluxes in the wavelength interval dh at A are defined, respectively, as
(20) where the superscipt r has been omitted for clarity. The following transformation is useful: dR = sin 0 de da,
(21)
da=-dkd&
(22)
or
where 4 is the azimuthal angle. Using this transformation on eqns (19) and (20) yields (23) and similar result for the negatively directed radiant heat flux. The wall intensity may be related to the wall radiosity and the black body intensity to Planck’s function through the following relations, the first of which assumes a diffuse wall I w.,,
=
R/T,
(24)
&,A= BAIT,
(25)
Equations (17), (23), (24) and (25) may be combined to give qh+(rh)= 2R, s’ e-(‘A’r)p dp + 2 ITA1’ BA e-‘(TA-ri)‘pldp d’r:.
(26)
With the definition of the familiar exponential integral function’4.5’ E,(~)
=
’ p“-* emx’&dp,
I0 eqn (26) may be rewritten as I&+
=
2&&(n)
+2
I
‘Ok
B, (T)&(T,
- 7:) dT:.
(28)
Th
The same procedure yields a similar result for the negatively directed radiant heat flux I*-
=
~RA&(TOA- 7~)+ 2
I
TOA
B,(T)Ez(TL-TA)~T:,
TA
(29)
where ToA is the optical path length between plates at wavelength A. The total radiant heat flux at a point is defined as 4r ‘z
Iom
(qh+- qh-)dh.
(30)
1075
Combined radiant and convective heat transfer
Thus, inserting eqns (28) and (29) into eqn (30) and recalling the definition of the dimensionless radiant heat flux function,
I
-
TO&
TA
Bn(T)E2(7:-
Th)dT: dh. I
(31)
It now remains to specify the wall radiosity, R,. The radiosity may be expressed as
E)&=o.
RA = eB* + (I-
(32)
Inserting eqn (29), evaluated at y = 0, into eqn (32) yields
I
Toon
Bh(T)&(T:)dT:
0
1 .
(33)
Equation (33) may be solved for the radiosity as follows, EBB+2(1-e) Rh =
TOA
Il
B*(T)Ez(T:)dT:
1 - 2(1”_ )&(%~)
’
(34)
Equations (31) and (34) now completely specify the radiant heat flux distribution for a given temperature distribution. 2.3 Absorption coeficient model EDWARDS et aI.‘clo’ recommend using the average line intensity correlation of the wide band exponential model for the absorption coefficient of an absorbing-emitting gas such as steam. This correlation gives the frequency and temperature dependence of the gas and takes the form K (3 T) ~ = &exp
[-lo - ool/C31,
P
(35)
where R is the steam gas constant, o. is the wave number at the band center, and C1 and Cs are different function of temperature for each band of the gas. These functions have been reported for the four vibration-rotation bands and one pure rotation band of water vapor.‘> The two important bands for radiant heat transfer calculations are the 1600 cm-’ band and the 3750 cm-’ band, respectively. 2.4 Gray gas-linear radiation analysis For the purpose of comparing with the exact numerical solution, as presented in the next section, an approximate analytical solution of eqns (10) and (31) was developed. In order to arrive at an analytical solution, it is necessary to make the rather severe approximation that the gas is gray, and to restrict the problem to one of black walls. In addition, the exponential kernal approximation is used, along with the assumption of small temperature differences in the gas. The latter assumption allows the radiation to be linearized and allows the temperature dependence of the absorption coefficient to be neglected. See Ref. (11) for details of the derivation. The solutions thus obtained for the radiant heat flux distribution and the temperature distribution are
4 (5)= fl sinh~5+ e(Z)= +osh
MZ’ 2(1
+
M)
+
f&
y - cash r[) + A( 1- 6’) + 16;1-+t;) + eo,
(37)
1076
J. K.
MARTIN
and C. C. HWANG
where, for convenience, a coordinate transformation to the midpoint between the plates has been performed. Thus ‘$=*-I.
(38)
The coefficients in eqns (36) and (37) are defined as f = ’
9MTO*(yZ~o - 37”- 3) 16y“(y cash y + 7asinh y)’
(39) (40)
f3=
970*(1 + M) + 32M 24~,2(1+M)* ’
(41)
where M
=
16uTIn3 3K,K
y= $1
(42)
’
+ M)“2.
(43)
Substituting eqn (37) into eqn (11) and performing the indicated integration yields for 2 ++[(+3)coshY+~]. & = 35(If M)
w
The Planck mean absorption coefficientC1”for steam, K~, was used in this analysis to represent the emission-absorption characteristic of the gas. As seen from the above equation, the heat transfer is characterized by the parameter M, which is the ratio of the radiation conductivity to the molecular conductivity, and the parameter G, which is the system optical thickness.
3. SOLUTION
Since the radiant heat flux function, c$($), is an integral function of the temperature distribution, we might expect it to be a smooth well behaved function of $. We would also expect, from physical considerations and from the boundary condition expressed by eqn (12), 4 to be an odd symmetric function about (I,= 1 and to have the value zero at $ = 1. From these considerations, then, we should be able to represent I$ as an odd polynomial of some order N. Thus
where the coefficients may be determined from a direct evaluation of 4 at (IV + 1)/2 points, assuming the temperature distribution is known. Substituting eqn (45) into eqn (13) and performing the integration yields O(l))
$ [$#
=&++;+-;
- ,)“+I-
l)] + eo.
(46)
Odd
Substituting eqn (46) into eqn (11) and performing the integration yields for (47) Odd
1077
Combined radiant and convective heat transfer
Notice that if the radiant heat flux is assumed to be identically zero, eqns (46) and (47) reduce to the familiar convective result for constant heat flux. Equations (31), (34) and (46) may be solved through a directive iterative technique. The procedure is to assume a temperature distribution and, based on this distribution, calculate the radiant heat flux at (N + 1)/2 evenly spaced points using eqns (31) and (34). Equations (31) and (34) are evaluated at the desired points by numerically performing the required integrations over space and frequency. The values of the radiant heat flux coefficients, & are then determined through the simultaneous matrix solution of (N + 1)/2 linear algebraic equations similar to eqn (45). These coefficients, when inserted into eqns (46) and (47) give the new temperature distribution, which is used for the next iteration. Convergence is obtained when & (hence the radiant heat flux) ceases to change by a preset amount. The solution is basically oscillatory due to the inherent non-linearity of the equations. Thus, if the assumed initial temperature gradient is too severe, the calculated radiant heat fluxes are too large, which yields a temperature gradient which is too small for the next iteration. These oscillations may be damped out and convergence enhanced, through application of a “relaxation factor”. Thus, the radiant heat flux change between iterations is multiplied by this relaxation factor (usually approx. 0.5) before the new radiant heat flux distribution is used as input to calculate the temperature distribution for the next iteration. One of the more important parameters, both from the standpoint of accuracy and from the standpoint of computational time, is the order of expansion of the radiant flux distribution. Little improvement in the calculated Nusselt number was obtained in going from a fifth order expansion to a seventh order expansion, and even less improvement, approximately one fourth of one percent, was obtained in going from a seventh order to a ninth order expansion. Therefore, a seventh order expansion of the radiant heat flux distribution was used in the calculation. A Fortran program’“’ was written for the IBM 360 computer to do the above calculations and perform the iterations. 4. COMPARISON
WITH EXPERIMENTAL
DATA
Few experimental investigations have been performed to measure the effect of radiation on heat transfer to laminar steam flow. A literature survey, however, revealed an experimental evaluation of combined radiation and convection to steam flow in round tubes.‘13’Although the study was concerned primarily with turbulent and transitional flow, three data points were reported for laminar flow. Uniform heat flux, 310 stainless steel tubes with 0.498 in. inner diameter were used in the experiment. It should be pointed out here that the criteria for neglecting the axial conduction and radiation are met with the experimental data. The difficulty of comparing the analysis presented herein with the experimental data is obvious, due to the different geometries. It is postulated, however, that the geometric effects on the radiation can be approximately accounted for by using a flat plate geometry whose mean beam length is the same as that of the tubes used in the experiment. Since the mean beam length of plates, based on their spacing, is twice that of a tube,‘14’based on its diameter, the spacing of the plates used for the comparison was 0.249 in. Also, since the uniform flux, laminar flow Nusselt number for convection in tube is 4.36’14’while that for plate is 8.235, a comparison of the magnitudes of the experimental and calculated Nusselt numbers is not valid. A more meaningful quantity to compare is the fractional increase in Nusselt number brought about by the radiant effects. This, then, was the quantity used to compare the calculated results with the experiment. Three experimental runs were made with laminar flo~.“~’ Each run yielded only one valid data point for the fully developed Nusselt number since the fully developed condition was not reached except near the end of the tube. The tube wall emissivity was measured and is shown, along with the other pertinent experimental conditions, in Table 1. Also in Table 1 is the experimentally measured increase in Nusselt number due to radiation, and that calculated with the method presented herein. Considering the difficulty of comparing the geometry of the analysis and that of the experiment, surprisingly good agreement is found between the calculated radiation effects and those measured. As indicated in the table, the discrepancy between the measured effect and that calculated is approximately 8.3% for the second data point, while the calculated and measured effects for the other two points agree to within less than 1%. This close agreement indicates the
1078
J. K. MARTIN and C. C. HWANC Table 1. Comparison between the measured’*‘andcalculated effect of radiation on Nusselt number.
Experiment a”* Number
*“be I.D.
Plate spac1nq USed Analysis,
Q Tm
3.n
Pressure
Tube Emissivity
PCZrCc”t 1ncrcase
in
Pfrcfnt I”CTf2Sf In
_-%!A
Nusselt,
russelt,
ft *-hy
+!“:easure?
CelculaLeZ
In.
Atm.
CF
45
.438
,249
1.754
865
,848
2503
43.l.i
47
.498
,243
1.725
048
,852
2500
31.65
x3.35
49
.438
,243
5.155
lOC6
,855
2300
7U.69
73.25
Ill.
‘IO.XB
validity of the analysis, although it obviously does not fully substantiate it due to the limited number of data points available for comparison. 5. DISCUSSION
OF RESULS
Several representative situations of combined radiant and convective heat transfer to steam were analyzed by the method discussed in the preceding sections. 5.1 Radiation
effect on heat flux
Figure 1 shows the dimensionless radiant and conductive heat flux distribution between two sets of black parallel plates. The parameters for these two configurations are given on the figure. Since the heat flux distributions are symmetric, only the lower half of the distributions are shown in the figure. It is interesting to note that the radiant heat flux peaks at some distance into the fluid stream rather than at the wall. This effect was also noted by VISKANTA”” in his gray gas analysis of radiation and convection between plates of constant temperature. The reason for this is that, at the wall, the effect of the positive radiant heat flux from the wall is partially cancelled by the negative flux from the layers of hot gas next to the wall. At small distances into the stream, however, the positive flux from both the wall and the hot gas combine to give a maximum heat flux. Farther into the stream, the flux from the wall and hot gas is attenuated by the intervening cooler gas and, of course, partially cancelled, and at 4 = 1.0 exactly balanced, by flux from the opposite side of the channel. 5.2 The effect of radiation on the Nusselt number Combined radiant and convective heat transfer to a flowing participating gas may yield, depending on the system parameters, significantly higher heat transfer coefficients than would be expected if conduction were the only heat transfer mechanism considered. The total Nusselt number, as a function of plate spacing, is shown in Fig. 2 for black plates with the parameters indicated on the figure. Note that all the curves in Fig. 2 converge to the pure convective Nusselt number of 8.235 as the plate spacings approach zero. I
1
c I T,,= 600 g
I
’
I
8
W = 1.0 INCH Q = 200 BTU/FTZ
HR
PATM
2
1.0 -
if
_
i 1
-
3
-_-
,
l
1.0 0.5
.a_
Fig. 1.Dimensionless heat flux distributions.
Combined radiant and convective heat transfer
1079
Fig. 2. Nusselt number as a function of plate spacing.
As would be expected, the higher pressures yield higher Nusselt numbers. This is due to the increased optical path length of the system, which enhances the radiant heat transfer. Figure 2 indicates, however, that the effect of increasing Nusselt number with increasing pressure is less pronounced at the higher pressures shown. This is explained by the fact that, at some pressure, the system becomes saturated (i.e. the absorptivity approaches 1-O) and further increase in pressure has less effect on the radiation heat transfer. Figure 2 also illustrates the fact that an increase system mean temperature significantly increases the effect of radiation, thereby yielding higher Nusselt numbers. It will be noted from Fig. 2 that a relatively small plate spacings, with intermediate pressures (i.e. greater than 1.0 atm) and temperatures, radiation is a significant heat transfer mechanism. With plate spacing as small as 0.20 in. at the higher pressures and temperatures, radiation may be the dominant mode of heat transfer. Figure 3 illustrates the effect of wall emissivity on the total Nusselt number. The curves in Fig. 3 were generated with the parameters shown. Figure 3 shows that, due to the reflection of the radiant heat flux incident on the wall, a reduced wall emissivity results in a reduced Nusselt number.
1
I
I
I
T,= 600 “F W = I.0
INCH
P=I
ATM
-
IO -
0
I .2
I .4 PLATE
I .6
I .S
1.0
EMISSIVITY
Fig. 3. Nusselt number as a function of plate emissivity.
1080
J. K. MARTIN and C. C.
.=
1.0
Q = 200 -
HWANG
STU/FT’-HR
GRAY
/
APPROX.
---EXACT
1
1
.2
_I PLATE
I
I
.3
.4
.5
SPACING,
INCH
I
Fig. 4. Nusselt number as a function of plate spacing.
5.3 Comparison
between the gray gas-linear
radiation approximation
and the exact solution
Several cases have been analyzed using both the gray gas-linear radiation approximation, as developed in Ref. (11) and presented in Section 2.4 and the exact solution. The purpose is to investigate the accuracy of the approximate solution and to see if it demonstrates the expected trends. As expected, the approximation overestimates the radiant heat flux. This is not surprising since a participating gas is actually transparent over large regions of the spectrum and may be practically opaque in other regions. Thus any attempt to represent the entire spectrum with a mean absorption coefficient will lead to considerable overestimate of the radiant heat transfer in the transparent and opaque regions. No mean absorption coefficient has yet been found which will, in the general case, yield accurate results for radiant heat transfer in the non-gray media. Figure 4 shows the variation of the Nusselt number with plate spacing, as calculated with both the exact solution and the approximate one. Curves for two sets of system parameters, which are listed on the figure, are shown. Figure 4 shows that the gray approximation produces the proper parametric trends but yields a high estimate of the Nusselt number, as would be expected. The magnitude of the error resulting from using the gray approximation is a sensitive function of the system being analyzed. A small plate spacing, low temperature, or pressure yields less radiant effect on the Nusselt number and therefore less error is introduced by using the gray gas-linear radiation approximation. From these data, then, it appears that the gray gas-liner radiation approximation consistently provides a high estimate of the Nusselt number for a combined convection and radiation heat transfer system. The error may be quite severe for a system in which radiation is the dominant heat transfer mechanism. The approximation is useful, however, in providing insight into the parametric trends to be expected in the system heat transfer characteristics and in estimating the significance of radiation. REFERENCES 1. R. D. CESSand E. M. SPARROW, Radiation Heat Transfer. Brooks-Cole, Belmont, California (1966). 2. B. T. CHAO(Ed.). Advanced Heat Transfer, DD.75-156. Universitv of Illinois Press, Urbana, Illinois (1%9). 3. T. F. IRWIN, h. ‘Ad J. P. HARTNE~T (Ed:.), ;idvances in Heat Tr&er. Academic Press, New York (1966). 4. H. C. H~TTELand A. F. SAROL~M, Radiative Transfer. McGraw-Hill, New York (1967). 5. T. ROCKWELL, Reactor Shielding Design Manual. Van Nostrand, Princeton, New Jersey (1956). 6. S. DE&TO and D. K. EDWARDS, Proc. of 1%5 Heat Transfer and Fluid Mechs. Inst. pp. 358-372.Stanford University Press (1965). 7. D. K. EDWARDS, B. J. FLORNES, L. K. GLASSEN and W. SUN,Appl. Opt. 4, 715 (1965). 8. D. K. EDWARDS, L. K. GLASSEN, W. C. HAUSERand J. S. TLJCHSCHER, J. Heat Trans. 89, 219 (1967). 9. D. K. EDWARDS and W. A. MENARD,Appl. Opt. 3, 621 (1964). 10. D. K. EDWARDS and W. A. MENARD,Appl. Opt. 3, 847 (1964). 11. J. K. MARTIN, MS. Thesis, Mechanical Engineering, University of Pittsburgh (1971).
Combined radiant and convective heat transfer
1081
12. M. M. ABU-ROMIA and C. L. RF& J. Heat Trans. 89, 321 (1967). 13. P. S. LARSEN, H. A. LORDand R. F. FARMEN, 4th Int. Heat Trans. Conf. Paris-Versailles 1970,Vol. III, P.R2.6, Elsevier, Amsterdam (1970). 14. W. M. ROHSENOW and H. CHOI,Heat, Mass and Momentum Transfer. Prentice Hall, Englewood Cliffs, New Jersey (1971). 15. R. VISKANTA, Appl. Scientific Res. A13, 291 (1964).