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The integral form of the equation of transfer in finite, two-dimensional, cylindrical media
J. Quanr. Spcc~ros~~.Rudicrr. Transfer Vol. 42, No. 2. pp. Printed in Great Britain. All rights reserved
I 17-l
36, 1989
0022-4073189
Copyright
THE INTEGRAL FORM OF THE EQUATION TRANSFER IN FINITE, TWO-DIMENSIONAL, CYLINDRICAL MEDIA S.
Department
of Mechanical
T.
$3.00 + 0.00
(1 1989 Maxwell Pergamon Macmillan plc
OF
THYNELL
Engineering, The Pennsylvania PA 16802. U.S.A.
State University,
University
Park.
Abstract-The corresponding integral form of the equation of radiative transfer in absorbing, emitting, linear-anisotropic scattering media bounded by emitting, diffusely-reflecting walls is formulated. The formulation includes a development of expressions for the radiation intensity, the incident radiation and the forward and backward radiation heat fluxes in the axial and radial directions, respectively. The integral form is represented by a system of coupled Fredholm type integral equations, which can be solved accurately by numerical techniques. The two-dimensional formulation is subsequently used to develop the corresponding integral equations in one-dimensional, axi-symmetric media. In addition, it is also shown that the respectively, are developed expressions for the net radiant fluxes in the r- and :-directions. different from the equations presented recently in the literature.
I. INTRODUCTION Radiative transfer is an important mode of energy transfer in numerous areas. Examples of such areas include, among others, modeling of heat-transfer in liquid and solid-propellant propulsion systems; development of methods for improving the efficiency of pulverized coal, natural gas and oil fired furnaces; analysis of heat transfer through various fibrous and foam insulations; in the studies of nuclear energy generation and explosions; and in the prediction and examination of i.r. signatures from rocket plumes. Thus, radiative transfer has received a considerable amount of attention in the literature. To analyze the radiative heat transfer, numerous approaches have been proposed for solving the equation of transfer either approximately or rigorously. A brief discussion of several solution methods has been presented by Viskanta.’ Examination of these methods has revealed that the integral form of the equation of transfer is often employed to develop the solution.2-7 The use of the integral equation rather than the equation transfer yields, in most cases, highly accurate results for the quantities of engineering interest, which are primarily the incident radiation and the net radiation heat fluxes. The high accuracy is, in part, attributed to the smoothing that occurs when integrating the intensity over all solid angles. That is, the radiation intensity is often a highly singular function, whereas the incident radiation and the net radiation heat fluxes are usually much smoother functions. As shown by other workers8m’0 and also revealed in the following sections, however, it is quite a laborious task to transform the equation of transfer into an equivalent integral form for the incident radiation and the radiation heat fluxes. The transformation yields one or more coupled singular Fredholm type integral equations of the second kind, which involve integrals that in most cases must be evaluated numerically. The objective of this work is to present a development of the integral form of the equation of transfer in absorbing, emitting, linear-anisotropic scattering, two-dimensional cylindrical media, which is bounded by emitting and diffusely reflecting walls at known temperatures. The development in the following section, however, leads to expressions that are partially different from the corresponding ones presented recently by Lin,” who considered the effects of isotropically scattering media. In addition, the corresponding integral equations for the incident radiation and the net radiant heat flux for one-dimensional, axi-symmetric media are also formulated utilizing the two-dimensional analysis as the basis.
PSRTC--C
117
S.T.THYNELL
IIX
ANALYSIS
2.
-3.I. Equation
?f’ transfb
We consider radiative heat transfer in absorbing, emitting, linear-anisotropic scattering. twodimensional cylindrical media bounded by diffusely emitting and reflecting opaque walls. The radiative properties are assumed gray and spatially independent, but the analysis is also applicable to a specific wavelength. The medium contains spatially distributed and varying energy sources of intensity Z,[T(r. z)], where T(r, :) is the temperature. Z,(T) is the Planck function. which for the gray analysis becomes Z,(T) = n’c?T’:‘n. A schematic of the physical model and coordinates arc illustrated in Fig. I. The mathematical description to this problem is given by”
=(I
[G(r, z) + aq,. (r. -_)cos tl + aq,r(r, z)sin 8 cos 41,
- Q)Zh[T(r. :)I + 5
inO
OdO
conditions
OGd)<2n.
(1)
are given by
Z(r,-c.d,$)=c,Zh,(r)+cqj_
(r,-c)=J,(r),
77
Z(R.~.H.~)=(~Z~~(;)+~~~(R.-_)=J~(Z).
Z(r. c, 8, 4) = cJZbI(r) + :(I;
O
CR.
O
Izl
(r. c) = J3(r),
O
o,
n 5 < 0 < 71,
0 d r < R.
0 < (b < 27-t.
(2Ll)
(2b)
(2c)
Here, Z(r, 2, (b, 0) is the radiation intensity, r and ; are the optical variables in the radial and axial directions, respectively, 0 and 4 are the polar and azimuth angles, respectively c, = 1 - p,, i = I. 2, 3 the surface emissivities of the opaque bounding surfaces from the inside, and Z,[T(r, z)] is the spatially distributed energy source due to the temperature T(r, z) of the medium, and for simplicity we define at the wall Z,i = Z,(T,). The incident radiation G(r, z)? and the forward and backward are, respectively, defined by radiation heat fluxes in the r- and z-directions T I( r, z, 8, ~$)sin 0 db’ dq5, ’ l C$=o s o=o
(3a)
q$(r,z)=2
’ I( r, z,t3, &)sin2 l3 cos C$de d@. ” .i @=I1 1‘0=0
(3b)
q<:(r,-_)=2
’ r;7 Z( r, z, 8. 4)cos i o=o i lI=O
(3c)
G(r.z)=Z
q:,(r,z)=2
”
’
8 sin 0 d8 d$,
(3d)
Z(r,z,8,7r--$))sin’Qcos$dHd4,
(1I$=II l,=0 n s;; (r, z)
=
rr,2
Z(r, z, n - 8, $)cos
2
8 sin tI do d4.
(3ei
i‘ @=O s H=O The net radiation
flux in the r- and z-directions q+(r’. 2) = q;(r,
are, respectively,
specified
as
z) - qc; (r, z),
(4a) (4b)
4L:I(r1=)=q~(r,=)-ql~(r,Z). The anisotropic
scattering
has been represented
in an approximate
P(cos 0) = 1 + c1cos 0,
manner
as (5a)
Integral form of the equation of transfer
where P(cos 0) is the scattering phase function and the scattering incoming (e’, 4’) and outgoing (0, 4) d irections of the electromagnetic cos0
=sinesin9’cos(4
119
angle 0 is related waves by
to the
-~f)+c0sec0s8f,
(5b)
and - 1 d a d 1 is the expansion coefficient to the scattering phase function. Equation (l), subject to boundary conditions given by Eqs. (2) represents a singular integro-differential equation due to the possibility of different surface temperatures and emissivities, as well as the effects of the< corners of the enclosure. As such, Eq. (1) is difficult to solve accurately, but the integro-differential equation is transformable into a corresponding integral equation of the Fredholm type, in which all such singularities are smoothed out. In the present analysis, however, it is necessary to construct three coupled integral equations. The reasons for this coupling of the integral equations are: (1) the formulation includes the effects of diffusely reflecting bounding surfaces, and (2) the effects of the linear-anisotropic scattering. To construct the integral equations, we first present the expressions for the radiation intensities. 2.2. Formal solutions
to the intensity
By following the development of, e.g., ijzigik,‘* it is possible the equation of transfer of the form
to construct
a formal
solution
to
s0 Z(s = 0,0)
= exp( - S,)Z,(S,)
+
exp( - s’)S(s’, a’) ds’.
(6)
s0 Here, S, is the optical
distance
the rays have traveled
Fig. 1. Schematic of the physical model and coordmates.
from their origin
(at s = S,), I,(&)
is the
Fig. 2. Illustration of the intensity having a contribution from the bottom bounding surface at z = --c. The expression for the intensity and the limits of the angular variables (0 +, 4 *) are specified by Eq. (9).
S.T. THYNELL
IX
(diffuse) intensity of the rays leaving the wall, and S(s’. Q’) is the source function in direction R‘ at .Y’. To transform Eq. (6) into the (1..I, 8. 4) coordinates of cylindrical geometry. we consider the following. First. the intensity is symmetric with respect to a plane formed by the r- and --axes. permitting us to limit the bounds on the azimuth angle given by 0 d # d n. Second, we introduce a forward (&‘) and a backward (0 ) intensity in the axial direction. and a forward (4’) and a backward intensity (4 ) in the radial direction; these are defined as
The source
function
is then given by
S(l., :. 0’. 4s‘) = (I - to)Ih[T(r, +z
in
[G(i+,~)+fzq,~_(r.~)(
O
;)I tcosf3’)+uq,~(r,r)sinff’(rt
/-_/cc,O
O
cos$*)J.
(8)
That is, the change in the sign on 0’ is related to the sign on +cos Q*, and similarly 4 7 is related to kcos 4 ‘. I*o construct the formal solution. we first consider directions of the intensity containing contributions from the bottom bounding surface at I = -- c. By considering Fig. 2, the transformation yields
exp[ - (2 - :‘)sec O+]S(r’, z’, N’, # *‘)sec If+ d-_‘,
O
t(r,R.(Pf) c+z '
(9)
where r’=[r’+(=
-z’f’tan?@+
-2rjz
--‘jtan8+(_tcos~‘)]“‘,
r’sin$“=rsin4=,
(lOa) (lob)
r(r, R, tp *) = r( f cos +4‘) + (R2 - rz sin’ # ‘)’ ‘. r, = r’j._.=_i,
(lOc) (10d)
Similarly, the intensity containing contributions from the bounding wall at Y = R are split up into axially forward (Fig. 3) and backward components. Such split up yields
I(f.:.W.#*)=J2(zZ)exp[-(z-q)secd+l exp[-(2
-:‘)secB+]S(r’,~‘,8+,$r’)sec8+d;’.
Integral
form of the equation
of transfer
and I(r, z, 8-, C#I ‘) = Jz(zz)exp[ - (z2 - z)sec 0-1 12 +
exp[-(z’-z)sec&]S(r’,z’,8~,~“)sec&dz’, s ..‘Z_
(12) where lz-~~I=t(r,R,~‘)cot8’. Finally, the radiation intensity containing is represented according to Fig. 4 as
(13)
a contribution
from the top bounding
surface at z = c
I(r,z,F,4+)=J3(r3)exp[-(c-z)sece-] exp[ - (z’ - z)sec &]S(r’,
+ o
t(r, R 4 ‘> c-z ’
z', 8 -, C#I *')sec 0 -
,,,*,q,
dz’,
(14)
where r3 = r’l,,=+,
(15)
The above formal solutions are used to construct integral expressions to the quantities of most interest in engineering applications, namely, the incident radiation and the radiation heat fluxes.
-r
Fig. 3. Illustration of the intensity having a contribution from the side bounding surface at r = R for z’ <.z. The expression for the intensity and the limits of the angular variables (O+, 4 *) are specified by Eq. (11).
Fig. 4. Illustration of the intensity having a contribution from the top bounding surface at z = c. The expression for the intensity and the limits of the angular variables (&, 4 *) are specified by Eq. (14).
S. T. THYNELI.
122
2.3. Incident radiation
G(r, 2)
To obtain an integral equation for the incident radiation, the formal solutions to the radiation intensities given by Eqs. (9). (11). (12) and (14) are substituted into the definition of G(r, z) given by Eq. (3a). To simplify the subsequent analysis. however, we represent the incident radiation as the sum of eight components: G(r. -) = i [G,+(r. -_) + G, (r. z)]. /= I
(16)
where
G!(r,z)=2
H+,d)*)sinH+dOi
d4*.
GT(r. r) = 2
(17c) tan,, -
G$(r.z)=2
(1%)
” s * - = 0 1 I, = (I
I,,.
R. .-@+ 1 < :
l(r,-_,H
,d*)sinC
dH
d$I.
(t7d)
The following analysis describes how the angular integrations are transformed into equivalent surface and volume integrals of the two-dimensional cylinder. However, we limit the details to those defined by G,‘(r, z), i = 1.2. since the expressions for these can be used to deduce the corresponding expressions for G,(r-, z), i = 3.4 due to symmetry in the integrals. 2.3.1. E.xpression for G :(r, :). First we introduce the Dirac-delta function 6(x) to write the definition of G:(r, 2) as
represent the angular variables of integration variable (r’. r), as shown in Fig. 5, the following
TO
The Jacobian
(8+, 4 +) in terms of a spatial transformation is introduced:
and an angular
(z - z’)tan H+ = [r’+ r” - 2rr’cos
cx]’ ‘.
(19a)
tan C#I+ = r’ sin cc/(r - r’cos
CC).
(I9b)
of the transformation
is given by (z - z’)r’
s(e+.$-) ;I(r’, J)
= d:(r, r’. r, z - z’) d,(r, r’, LX,0)’
(20)
where d,,(r. r’, 2, u) = [r’ + r” - 2rr’ cos tl + u~]~,~.
(21)
We also find that sinH’=d,(r.r’,a,O)jd,(r,r’,r,z-2’).
(22a)
cosfl+=(;-:‘),id,(r,r’.cr,z-2’).
(2%)
i = r’sin sin C#J
x/d,(r,
r’, c(, 0),
(22c)
cos C$T = (r - r’ cos ol)/d, (r, r’, do,0).
(22d)
cos g5’-’ = (r cos r - r’)/d,(r. r’, 4%.0).
(22e)
Integral
form of the equation
123
of transfer
(a)
TOP
VIEW
OF CYLINDER
r 4
(Z-Z’)
tone+
(b)
Fig.
5. Definition
of dummy variables and a.
of integration
r
Fig. 6. Illustration of the limits used on r’ and a in the various integrals. Here, it is assumed that z’< z with (a) applicable to r’ and z if z’ > z:. and (b) applicable to r’ and a: if z’
To specify the limits of integration, we use Figs. 6 and 7 to establish that the radiant energy propagating towards the point at (r, z) must be confined to within the circle of radius r:, which is centered at a radial distance rd+ from the z-axis, the line r’ = r set a and the r-axis. By substituting the above transformation into Eq. (18) and using Fig. 8 to express r: and ri in terms of known variables, we obtain Rcosa=r G T(r, 2) =
,S~CS
J, (r’)(c + z)K,(r,
2 s !l=O +2
j”
r’, a, c + z)r’
j”
J, (r’)(c
+ z)K,(r,
r’, a, c + z)r’
z’)K,(r,
ss G
+
dr’ da
s r'= 0
~,‘,I,
x=a+ r’= r&,
(-)r’
dr’ da
dr’ da
r’, r, z - z’)r’
dr’ da
1
dz’
(23)
The (s) symbol is introduced
to reduce the length of the expressions,
and it implies that the integrand
S. T. THYNELL
114
(a)
--
r
(b) Fig.
7. The slanted
r’-plane
Fig. 8. The radial distance r; to the center of the clrclc 01 using ratios of equ.tl radius r<’ and z: are determined triangles.
lines tllustrate the (r’. 1) domain in the that contributes to G,+(r. :). i = I. 2, which are components to the incident radiation.
is repeated.
Furthermore,
it is necessary
s*(l-. 1.‘. Iz. 1. z’) = (I - c0)Z,[T(r’. +
E
G(r’. z;) + ay,
to define a new source
function
as
:‘)I (z - z’) (r’. z’)
l/,(r. r’, 2. z -z
[
,+
)
q,.r(r’.
z’)
(1.cos x - 1.‘) d,(r. 1.‘. x, I - I’) 1 *
(74)
and
r A, = r;
cos
y
+
[r;’
rh = r(c * z’)/(c r’=Rlz
_ r$‘sin? ~(1’ ?,
+ z),
-z’j/(cf~).
tan c(5 = [ri+’ - (r - r,;)‘]’ The kernels
K,,(r, r’, x, u ) are introduced
‘jr.
as
K,,(r. 1.‘. x, II) = exp[ - d,(r, r’, x. u)]/d,,(r. r’, 2. 24). 23.2. Expression for G;(r, z). By substituting
G, (r. 2).
We continue the analysis by developing an expression Eq. (9) for 4 _ into Eq. (17a) and using the Dirac-delta function
t2h)
for in a
Integral
similar
manner
as in Eq. (18)
form of the equation
the resulting
ss
expression
of transfer
125
becomes
n?
G, (r,z)=2
[J,(r’)6(c+z’)+S(r’,z’,0+,&‘)sec0+]
$J =0 II+- 0
xexp[-(z-z’)sec0+]dz’sinB+dB+d&. In this case, the previous
transformation
(27)
given by Eqs. (19) can still be applied,
cos&‘=(r’-r
but we note that
cos c()/d, (r. Y’, c(, O),
(28)
which results in the same expression for the source function S*(r, r’, ~1,z, z’) as defined by Eq. (24). Using Figs. 6 and 7, the equivalent expression for the component of the incident radiation G ;(r, z) is Rcou=r
R
J, (r’)(c + z)K,(r, r’, 2, c + z)r’ dr’ da
G, (r,:)=2 s r = r \ec1
s I=0
2.3.3. Expression for G:(r, z). We continue by developing tution of Eq. (11) into Eq. (17b) applied to 4’ yields
+ To transform of integration
an expression
_ S(r’, z’ ,8’,$+‘)exp[-(z-z’)sec8+]sec0+dz’ __j:-_:
for G:(r, z). Substi-
sinO+dtI+d4+.
the surface integral involving Jz(zz) in Eq. (30) we treat zz = z’ as the dummy and introduce the transformation (z - z’)tan 0’ = [r’ + R’ - 2rR cos a]’ ‘,
The Jacobian
of this transformation
By utilizing Figs. 6 and 7 to deduce eliminate sin 0 +, we find ’ ss_‘
n
J?(z’)(R
(31b)
is
;(e+, 4’) d (z’, a)
G:(r,z)=2
variable
(314
= R sin r/(r - R cos 51).
tan$+
(30)
I
R(R -r
cosa) (32)
= d?(r, R, ‘A,z - z’)d,(r, R, CC,0)’ the limits
of integration
and by employing
Eq. (22a) to
- r cos a)K,(r, R, a, z - z’)R da dz’
RCOSZ-,
Rcosx=r
~SXJI
R (.)r’ dr’ dcc +
1
(.)r’ dr’ dcc dz’ s 2= 2”:
s r=o
S. T. THYNELL
IX
2.3.4.
Espression
for
in Eq. (I I ). Then,
Gz (r, z). To obtain
S(r’.
+
z’ ,o+,&‘)exp[-(=-=‘)sec6,+]secH+d;’
I=::
Using
G, (r, z), we first consider the backward direction of this expression over the appropriate solid angles yields
an integration
the preceding
sinf!I’dH+d$
4
(34)
i
development
G’(r.
involving
z), we deduce
immediately
that
Hco\1= , G,(r,;)=Z
J2(z’)(R I---
Hu,,r ( +
- r cos r)K,(r.
R. 2,;
- r’)R
da dz’
L
K (*)r’ dr’ da
i 1 7.’
i , = rwc1
(35)
dz’. 1
23.5. Expression for G(r, z). A close examination of the integrals presented in Eqs. (23) (29), (33) and (35) for G’(r, z). i = 1.2, reveals that the sum of these four equations add up to one volume integral and two surface integrals. The limits of these integrals are confined to within a cylinder whose top surface is located at z. It is thus obvious that the sum of the four expressions for G ,?(r. : L i = 3.4. adds UP to an equivalent volume integral and two surface integrals, whose limits are within a cylinder located above the point at z. By using symmetry in the integrals. WC deduce that the integral equation for the incident radiation is given by I? GO,. z) = 3 I-1e (I I , =,,
J,(r’)(c
+ z)K7(r,
J,(z’)(R
+2 i-
- r cos x)K,(r,
R, x. 2 - z’)R
da dz’
IL J?(r’)(c
+2
r’. ;I, c’ + :)r’dr’dr
- z)K,(r,
r’. IX,c - z)r’ dr’ dr
I-x=I, !^i -0 I
+q_
R
,; 1= 0 j r =o
S*(r.
r’, x. 2. z’)K?(r,
r’. Y(.: - z’)r’dr’dr
dz’.
(36)
where the source function S*(r, r’, 2. z. z ‘) is defined by Eq. (24). Since the source function contains the net radiant heat flux in the radial and axial directions, it is also necessary to develop expressions for these quantities in order to complete the formulation. The formulation is also necessary in the case of nonblack surfaces which enclose a purely absorbing/emitting medium. It should be noted that Eq. (36), applied to the special case of isotropic scattering (a = 0), is readily reduced to the of Eq. (36) integral equation for the source function as derived by Lin. ” In addition, evaluation at z = -c’ results in the contribution 271J,(r) from the first double integral; similarly, evaluation of Eq. (36) at z = c results in the contribution 2nJ,(r) from the third double integral. 2.4. Radial
radiant
heat ,fiuxes
q$ (r, z)
The forward and backward radiant heat fluxes, qg (r. z) and q, (r, z), respectively, angular coordinates (0 *, C#J ‘), are defined by r:? n? de+ q‘t (r. :) = 2 I(r, z, 8+. 4’)sin’Q’ s $$t=0 [S i)+= 0 +
a’ Z(r, z,8-,~*)sin28-d& s o- =0
1
based on the
cos4’d@‘.
(37)
Integral
Using the specific definitions we find
form of the equation
of the intensities
127
of transfer
Z(r, z, 9 +, I#J*) given by Eqs. (9)
(11),(12)
and (14)
J2(z2)exp[ - (z - z,)sec 0 ‘1
S(r', z', 13+, C#I*‘)exp[
+2
-
(Z - z')sec 8 +]sec 0
+dz’
sin* 8 +dO +cos 4 f d$ ?r
{;I=0 JIro_ i.l,(z2)exp[ - (z2- z)sece-1 _,cr;R.f*l
s. =z
+
,s(r’,z’,e-,@*‘)exp[-(z’-z)sec&]sec&dz Z,= _
+S(r’,
sin20~dO~cos~*dQ,*
z’, 8-,~“)sec8-]exp[-(z’-z)sece-]dz’sin2e--de~cos~’d~~.
(38)
To transform these integrals from the angular domain of (Ok, C#J ‘) to the (r’, CX)domain, we consider first the forward radial heat flux, qi (r, z). For the integrals in Eq. (38), we introduce for the surface integrals involving the contributions from the bounding intensities J,(r) and J3(r) and the volume integrals involving the source function, respectively, the transformation given by Eqs. (19) resulting in
(~-r’COS~)IZ--‘lr,dr’dr d4(r,r,,C(,Z_z,)
sin’O’cos~+d~*d~+=
,
(39)
where cOse*=Iz-z’l/d,(Y,r’,CI,Z-~‘).
For the two surface integrals involving the boundary transformation given by Eqs. (31), yielding
intensity
Jz(z?) in Eq. (38), we employ
(r-Rcoscr)(R-rcosa)
sin*8*cos#+dO*d~+=
d4(r, R, a, z - z’)
To establish the corresponding limits of integration, expression for the forward radial heat flux: Rcora=r
(40)
R da dz’.
we use Fig. 9, resulting
the
(41) in the following
rseca
q; (r, z) = 2
[~,(r’)(~+z)K~(r,r’,tl,c+z)+J~(r’)(c-z)K~(r,r’,~,c-z)l s 0=0
s I'= 0 11
x (Y - Y' cos
R
cc)r’ dr’ dcr + 2
(.)r’ dr’ dcr s Rcosa=r s r'=O
J,(z’)(r-Rcosa)(R-rcosa)K,(r,R,u,z-z’)Rdudz’
S*(r,r’,cr,z,z’)(r
(.)r’dr’dcr
1
dz’.
-r’cosc1)K3(r,r’,cI,z
-z’)r’dr’dcr
(42)
1%
S.T.THYNELL
The (e) symbol implies that the integrand is repeated and introduced to reduce the length of the expressions. In the case of the backward radial heat flux, it is now readily shown that the corresponding expressions to Eqs. (39) and (41’ are, respectively, given by (43) and sin-‘@* cosd,’
d@* d4”
-_
(Rcosa-r)(R-rcosff) d43(r,R, a, z - 2’)
--- R da dz’.
(44)
By using Eqs. (43) and (44) in Eq. (38) for 4 - and by deducing the limits from Fig. 9. we obtain Rcosd=r y<;:
(1.,
z)
2
=
s f =0
R
V,(r’)(c’ + z)&(r. Y’,~1,c + z) s ‘i;iTSCCZ
-t-Jl,(r’)(C -z)iY,(r,r’,x,c
-z)](r’cosa
-r)Y’dr’da
x K3(r, r’, a, ; I z’)r’ dr’ da dz’.
(45)
The resulting expression for the net radiant heat flux, which is defined by Eq. (4a), takes the form y,:,(v,zf=2
R [J, (Y’)(C + z)K 4(r,y’,a,c n s x=0 s I =n x(r +zJ
+z)+J3(r’)(c
-z)iY,(r,r’,ff,c
-z)]
- P cos a)K,(r,
R, a, z - z’)R dor dz’
-r’cosa)t.‘dr’da ii .J?(z’)(r - R cos a)(R s f z --( I 1= 0 S*(r,t”,a,z,z’)(r
-r’co~a)K~(r,r’,a,z
-z’)r’dr’dcl
dz’.
(46)
It should be noted that the above expression for the net radiai heat flux, reduced to the special case of isotropic scattering, does not completely agree with the one presented by Lin.” The difference between his (on p. 595) and Eq. (46) is found in the integrand of the surface integral involving the bounding intensity J,(z); Lin’s formula contains the term (I - R cos a)‘, whereas Eq. (46) contains the term (r - R cos a) (R - r cos a). The expression presented by Lin yields numerical results of the net radial heat flux that appears to be in error and physically unsound. For example, if we suppose that the radiation transfer originates from the bounding surface at Y = R. that is the bounding intensities J1 (r) = 0 and &(P) = 0 (cold and purely absorbing boundary surfaces at z = kc). and that the source function is negligibly small (cold and nonscattering medium); then the net radial heat flux is always positive according to Lin’s expression. This result does not seem to be possible. Thus, it appears that there is either a typographicai error, or that the application of Lin’s general formulas have resulted in incorrect integral expressions in this particular case. 2.5 Axial radiant heat fluxes 4: (r, z) The development of expressions for the forward and backward radiant heat fluxes in the axial direction, q$ (r, z) and qe: (r, z), respectively, proceeds in a similar manner as for the development
129
Integral form of the equation of transfer
of the heat fluxes in the radial direction. The axial heat fluxes are defined by
s; (r, z>= 2
ss x/2
4
,$+=o e*=o
Z(r,z,8*,~+)cos8*sin8’de~d~+
ss d-2
+2
n/2
Z(r,z,8’,~-)cos8’sin8’dB’d~~. ,#-=lJ 0+=0
(47)
Using the definitions of the intensities Z(r, z. 0 *, 4 ‘) given by Eqs. (9), (1 l), (12) and (14), the axial heat fluxes are q~~~,z~=2~~~=o~~~~~=~~~=_~~~,~~~~~~~+I.)+S(~..i.;R+,~+.)rers'j
+
s
+
s
’ S(r’, z’, 0 +, 4 +‘)exp[ - (Z - z')sec0 +]sec8 + dz’ sin 8 + cos 8 + de + d4 + i’=i2
’ S(r’, z’, 8 +, 4 -‘)exp[ - (Z - z')sec8 +]sec8 + dz’ sin 8 + cos 8 + de + d4 -, 2’=22 I (48)
and [J,(r')d(c -z') + S(r',~',e-,4+')sece-1
xexp[-(z’-z)secO-]dz’sin&cosO-de-d$+ [J,(r’)d(c -z’)+
S(r',z',e-, +-')sec e-1
xexp[-(z’-z)secO-]dz’sinfJ-cos8-d&d& +2C.oJ~O_
,,,,,,+,~*(z2)exp[-(z2-z)sece-l c--L
22 +
s I’=z
S(r’,z’,&,4+‘)exp[-(z’-z)secO-]sec&dz
sine-cm&de-d4+
tbw-,{ +2J;:_oJ::e
4 (z2)exp[ - (z2 - z)sec 8 -1
+
s
c--z
z2S(’r ,z’,O-,4-‘)exp[-(z’-z)secO-]sec&dz
I’- z
sin&cos8-de-d4-. (49)
S.T. THYNELL
Fig. 9. The slanted lines illustrate the (r’. 2) domain in the -‘-plane that contributes to the forward and backward radiation heat fluxes in the radial direction, y;t (r. z) and 4,; (r. z ). respectively.
Fig. 10. The slanted lines illustrate the (T’%z) domain in the =,-plane that contributes to the forward and backward radiation heat fluxes in the axial direction. (I,~ (r. -_) and y,,; (r. z). respectively
To transform these integrals from the angular domain of (0 +, C#I ‘) to the domain of (r’, x ), we employ the previous transformation given by Eqs. (19). For the surface integrals involving J,(r) and J3(r), and the volume integral involving the source function. we introduce the substitutions, respectively, given by (z
sin0’cosQ’dCI’d+‘=
_
=‘)Z
dd( 1. r’, rl, z - z’)
r’dr’da,
(50)
and sintI*
de* d4’
=
/z - Z’I
r’ dr’ da.
d-1(r, r’, a, z - z’)
For the surface integral involving the boundary the transformation given by Eqs. (31), yielding
intensity
J,(zl)
(51)
in Eqs. (48) and (49). we employ
(52) The limits of the integration for the z’ variable are readily established from Fig. 10, and we must have 0 d r ,< TCand 0 s r’ G R. The expressions for the forward and backward axial heat fluxes, respectively, are constructed as: n
R
J, (r’)(c
ql(r.z)=2
+ z)‘K,(r,
r’, CI,c + z)r’ dr’ dcr
s X=0 s ,'= 0 ’
+2 s.
s
J>(z’)(z
- z’)(R
- r cos x)K,(r,
R, CC,z - z’)R dcc dz’
z=O
')(z -
z’)K3(r,
r’, a, z - z’)r’ dr’ do! dz’,
(53a)
131
Integral form of the equation of transfer
and R
qP;(r,z)=2
n
J3(r’)(c
-
z)‘K,(r,
r’,
a,
c
z)r’dr’ dcc
-
s ;r=O s r'= 0 ‘
+2
n
SJ
J2(z’)(z’ - z)(R - r cos a)K,(r, R, a, z - z’)R dcr dz’
3-0
c
n
R
a=0
s r'=O
S*(r, r’, a, z, z’)(z’ - z)K,(r, r’, a, z - z’)r’dr’
+2 LS
dcc dz’.
(53b)
The resulting expression for the net radiant heat flux, which is defined by Eq. (4b), takes the form qt,(r, z) = 2
s’ sR
[J,(r’)(c + z12K(r, r’, a, c + z) - J3(r’)(c
Z=O
4
- z)‘K,(r,
r’, cx,c - z)]
r'= 0
c
x r’ dr’ dg + 2
l-S
J,(z’)(z - z’)(R - r cos a)K,(r, R, a, z - z’)R dcr dz’
0=0
S*(r, r’, LY,z, z’)(z - z’)K,(r, r’, LX,z - z’)r’ dr’da dz’. +~s:.-c~~~os.lo
n
(54)
It should be noted that there is also a discrepancy between the above Eq. (54), applied to the special case of isotropic scattering, and the corresponding one developed by Lin.” The difference between the two expressions is observed in the integrand of the integral involving the bounding intensity J,(z). In Lin’s work, the integrand contains the term (r - R cos a), whereas in this work it contains the term (R - r cos cc). The effect of Lin’s result is that it appears to produce physically unrealistic results; that is, for c( < cos~‘(r/R), a negative contribution to the net flux is predicted. However, such negative contribution to the net axial heat flux is possible only for z’ > z.
3. RADIATION
TRANSFER AXI-SYMMETRIC
IN
ONE-DIMENSIONAL, MEDIA
In numerous applications, the radiative energy transfer takes place in axi-symmetric media. In such media, the radiation transfer is independent of rotation about the z-axis as well as translation along the z-axis. In previous works, the integral form of the equation of radiative transfer in absorbing, emitting, isotropically scattering media has been formulated’0 and solvedI using the appropriate expansion functions. The objective of this section is to reduce the formulae developed in Sec. 2 for the incident radiation and the net radiant heat flux in the radial direction to the corresponding ones in the one-dimensional case. 3.1. Incident radiation G(r) We consider an infinitely long cylinder, in which the radiation transfer depends only on the radial variable r. Using Eq. (36) as the basis for the subsequent analysis, the integral equation for the incident radiation is written as 00 n G(r) = 45,
s &-‘=a s x=0
(R-rcosa)K,(r,R,u,z’)Rdccdz n:
n
R
r’, a, z’)r’ dr’ du dz’.
(55)
Here, we arbitrarily set z = 0 and note that the integrals are symmetric in the z/-direction. source function S&(r, r’, a, z’) in the one-dimensional case is defined by
The
+4
S?,(r,
.?‘=!I r-0 sss
r’, a, z’) = (1 - o)Z,[T(r’)]
Sy,(r, r’, a, z’)K,(r,
r’=O
+ z
G(r’) + aq+(r’)
(rcosa-r’) d, (r, r ‘, a, z’)
1’
(56)
S. T.
131
THYNELL
It is now desirable to reduce the number of integrals such reduction, we consider first the integral defined
By introducing
in Eq. (55) from three to two. To perform by
the transformation (58)
d, (r, r’, a, z’) = [I, (r. r’. 2. O)sec 1~. we find dz’ = o’,(r, r ‘, z, 0)sec’ 1~d/c Substitution
of this transformation
(5%
into Eq. (57) yields
(r cos x - r’)_ G(r’)Ki, [d, (Y, r’, a, 0)] + aqCr(r’)Ki2[d, (r. r’, CC.O)] --~---~ n, cr. r’. L-i.01
F,(r)=4
x -~--~~~ ri, (r. r’. 1. 0) where Q(x)
is the Bickley
function14.‘” defined
(60)
by
nz! K,(s)
exp( - .Y set P)COS’ ’ p dp.
=
(hl)
s0 By using its relation
to the modified
Bessel function Ki, (s) =
and the addition
theorem”
applied
to r’ < r given by
F,(r)
that the function
(03,
K,(u) du,
I’
K,[d, (r, r’, x, O)lexp(il$) = it may be shown
K,,(x), namely.
ci
,I:
K,+,Cr)ln(r’)exp(inu),
can be written
(63)
as
I F,(r) = 47~
T ,=n u=l SS[
G(r’)K,,(ru)Z,(r’u)
+ aqGT(r’) k K,,(ru)l,(r’u)
ss[ R
+ 4n
du r’dr’
1
I
T=T ti=I
G(r’)Z,(ru)&(r’u)
Here, i = v; - 1 and the angle $ is located opposite the surface integral in Eq. (55) and define %
- aqGr(r’) i &(ru)K,(r’u)
du r’ dr’. (64) 1 to r’ for r’ < r and vice versa. Next we consider
*
F?(r) = 4J,
(R - r cos a )K,(r, R. r, z’)R dcc dz’.
(65)
1: -01 z=O Using
the transformation
given by Eq. (58), Eq. (65) becomes
F,(r) = 4J, The use of the addition
’ (Rs z=o
theorem
r cos r)K&[d, (r, R, x, O)]
yields the desirable
R dcc d2 (r? R, u, 0)’
(66)
result given by
1 F:(r) = 4nRJ, By introducing
the kernel
L,,,(r,
% s
r’) defined
L,, (r, r ‘) = 47t
L K, (Ru)Z,(ru) du. s [,=I u
u=I
by
K,(ru)Z,(r’u)
du ~ t?1+?7’ r’ < r, u
%
L,.,(r,
r’) = 4n( - l),+’
(67)
i u= I
I,(rU)K,,(r’u)
du z, u
(6W
r’ > r,
(68b)
Integral
the integral
equation
form of the equation
for the incident
G(r) = RJ&,,,(R
radiation
133
of transfer
is represented
by
r)
s[i
1
R
+
(1 -w)lb[T(r’)]
r’=O
3.2. Net radiative heat flux
r’) r’dr’.
L,,,(r, r’)+azqgT(r’)L,,,(r,
+zG(r’)
(69)
q;,(r)
The development of the integral equation for the net radiative heat flux follows the procedure used above for the incident radiation. Using Eq. (46) as the basis for the analysis, we express the net radiative heat flux in the form q2, (r) = 4J,
n (r-R x s I=0 s a=0
cos
T. f4 By defining
n
a)(R - r
F,(r)
function
a)K4(r, R, Z, z’)R da dz’
R
:‘=0 z=o /=O sss
a working
cos
S*(r, r’, E, z’)(r - r’ cos cc)K,(r, r’, 2, z’)r’ dr’ dcr dz’. as
x (r -r’cosa)K,(r,r’,cc,z’)dz’dccr’dr’, and using the transformation
given by Eq. (58)
G(r ‘)I& [d, (r. r ‘. a,
&:7(r) = 4
this function
WI+ q,rWG
is expressed
F3(r) = 471
ss “’
x
r=O
u=l
(rcostl
Finally,
we introduce
3
7
, = r’ u=l
of the modified
dz(r, r’, LX,0)’
(72)
Bessel function
and
1
cr’dr’
1
-G(r’)Z,(ru)K,(r’u)+uq,~(r’)~Z,(ru)K,(rIU)
of the surface
’
(r-R
s a=0
cr’dr’.
(74)
given by Eq. (58) into Eq. (74) to obtain R da
cos cr)(R - r cos a)Ki3[d, (r, R, a, 0)]
(75) 4
Expressing the Bickley function in terms of repeated using the addition theorem, we find I;,(r) = -4nRJ2 ‘Then, based on the developed
(73)
in Eq. (70) as
cos cc)(R - r cos a)K,(r, R, u, z’)R du dz’.
’ (r-R m s :‘=0 s ol=lJ
the transformation
integral
1
dcr r’ dr’
m
a definition
F4(r) = 4J2
integrals
3
ss[
47c
F4(r) = 45, Next, we substitute
-r’)
k4 (r, r’, @,O)ld,cr r, c1oj
G(r’)K,(ru)Zo(r’u)+oq,T(r’)~K,(ru)Z,(r’u) R
+
in terms of repeated Eq. (63), we find
(71)
as
x (r -r’coscr)
Expressing the Bickley function applying the addition theorem,
(70)
expressions
integrals
(r,
of the modified
R,
a,
0)’
Bessel function
CC 1 1 K, (Ru)Z, (ru) du. s u=l n for F,(r)
and F4(r), we express
and
(76) the net radiative
heat
S. T. THYNELL
134
flux as q,r(r) =
-RJ2L,,(R, r)
sci R
+
“’= 0
(1 - w)I~[T(Y’)] + 2
L,,o(r, r’) + a z
G(r)
qpr(r’)L,,,
(r, r’)
r’ dr’.
(77)
I
To eliminate the boundary intensity J2 in Eqs. (69) and (77), we use Eqs. (2b) and (4a) applied to the one-dimensional case for r = R. This yields 1 -tz
nJz = T& + ___(z
Evaluation
q,;,(R).
(78)
of Eq. (77) at r = R and the use of Eq. (78) to eliminate
q;,(R)
yield
~27143 RJ2=(zn
+p2
+
P2
RL,., (R,R) (1 -co)1,[7’(r’)]+~G(r’)
I
L:,(R,r’)+a~~q,~(r’)L:,(R,r’)
r’dr’,
(79)
where
L&CR, r’) = Substitution
RL.,(R r’) (271 + P~RL,.,(R
of Eq. (79) into Eqs. (69) and (77) produces
(80)
RI’
the desirable
result given by
(1 -o)Ih[T(ri)]+zG(r’)
G(r) = c27db2LTn(R,r) +
x [Lo,o(r,r’) + P~L~,~(R, r)L?, UC r')l +~~g,~(r’)~Lo,~~r.r’)-~2~l,o(R,r~L~l(R,r’)l
r’dr’
(81)
1 and
s[i R
qB,(r)= - ~2~~b2L~l(R r>+
r =0
(1 - wN,[T(r’)]
G(r’)
+ z
x LL,(r, r’) - PAL,.,CRrWiYoCRr')l
+~~q,,(~‘)[LI.,(r,r’)-~2LI,,(R,r)L~l(R,r’)l
r’dr’.
(82)
1
To obtain an accurate solution to the coupled integral equations given by Eqs. (81) and (82), it is used to construct an initial estimate to G(r) and qi,(r). is suggested that the P,-approximation” A significantly improved value is expected to both of these quantities if the P, solution is substituted into the right-hand side of Eqs. (81) and (82). The integrals over r’ can be performed analytically.” thereby reducing the number of required numerical integrations in Eqs. (55) and (70) from three to one. Such reduction in the computational effort represents a significant achievement, and it is of particular interest to those workers who analyze the interaction of radiation with other modes of energy transfer. 4.
SUMMARY
A rigorous approach has been implemented to develop expressions for the radiation intensities, incident radiation and the forward and backward radiation heat fluxes in the radial and axial directions, respectively, in absorbing, emitting, linear-anisotropic scattering, two-dimensional
Integral form of the equation of transfer media bounded for the incident
135
by diffusely emitting and diffusely reflecting surfaces. Subsequently, the expressions radiation and the net radiant heat flux in the radial direction have been reduced
to integral equations in axi-symmetric, one-dimensional media. A solution to the formulated integral equations may be constructed by various numerical and semi-analytical means. ~C~~~wZe~ge~e~r-The author is grateful to acknowledge the partial support from the Office of Naval Research under contract No. ~~~4-86-K-~8. REFERENCES I. R. Viskanta. Fortschr. Yer- Tech. 22, 51 (1984). N. D. Sze, JQSRT 16, 763 (1976). S. A. Elwakil, J. Appl. Phys. D: Appl. Phys. 13, 339 (1980). G. Spiga, F. Santarelli, and C. Stramigioli, Int. J. Hent Muss Tran~/kr 23, 841 (1980). M. N. &isik and Y. Yener, J. Heat Transfer 104, 351 (1982). M. M. R. Williams, lM,4 J. Appl. cash. 31, 37 (1983). 7. S. T. Thynell and M. N. &igik, Appl. Phys. 60, 541 (1986). 8. M. G. Smith, Proc. Camb. Phil. Sot. 60, 909 (1964). 9. G. C. Pomraning and C. E. Siewert, JQSRT 28, 503 (1982). 10. S. T. Thynell and M. N. Ozisik, JQSRT 36, 492 (1986). 11. J. D. Lin, JQSRT 37, 591 (1987). 12. M. N. &isik, Radiative Transfer, Wiley, New York, NY (1973). 13. S. T. Thynell and M. N. t)zigik, JQSRT 38, 413 (1987). 14. W. G. Bickley and J. Nayior, Phil. Mug., 7th Ser., No. 20, 343 (1935). 15. W. W. Yuen and L. W. Wong, JQSRT 29, 145 (1983). 16. I. S. Gradshteyn and I. M. Ryzhik, Table oflntegrafs, Series, and Products, p. 979, Academic Press, FL (1980). 2. 3. 4. 5. 6.
NOMENCLATURE a = Linear anisotropic
scattering coefficient c = Half the optical axial length of the enclosure F;(r) = Functions defined in Sec. 3 G(r, z) = Incident radiation G:(r, z) = Components to the incident radiation, Eqs. (17) f(r, z, 0,4) = Radiation intensity &(T) = n2BT4/lr, the Planck function & = Emission from the ith wall I,(X) = Modified Bessel function of the first kind J, (r) = Diffuse intensity of bounding surface at z = -c Jr(z) = Diffuse intensity of bounding surface at r = R J3(r) = Diffuse intensity of bounding surface at z = c K,,(r, r’, CI,u) = Kernels of integral equations, Eq. (26) Kn(x) = Modified Bessel function of the second kind Xi,(x) = Bickley function .L,,,(r, P’) = Kernei for one-dimensional case, Eqs. (68) L,&(R, I’) = Kernel for one-dimensional case, Eq. (80) n = Index of refraction q#r(r, z) = Net radiation heat flux in r-direction q$ (r, t) = Radiation heat flux in forward ( + ) and backward ( - ) r-direction qez(r, z) = Net radiation heat flux in z-direction qt (r, z) = Radiation heat flux in forward (+) and backward ( -) z-direction P(0) = Scattering phase function r = Optical radial variable r,i = Radius of circle, Eq. (25f) t-2 = Radial distance, Eq. (25e) r&, = Radial distance, Eq. (2%) rfmax= Radial distance, Eq. (25d) rl = Radial distance, Eq. (IOd) r, = Radial distance, Eq. (15). R = Optical radius S(r, z, 8, #) = Source function, Eq. (8) S*(r, r’, GI,z, 2’) = Source function, Eq. (24) %p(r, r’, tl, 2’) = Source function, Eq. (56) f (r, R, rb *) = Distance defined by Eq. (10~) T(r, z) = Temperature
S. T.
THYNEL.~.
T, = Wall temperature : = Optical axial variable Z? = Axial distance, Ey. (I 3) z& = Axial distance, Eq. (2Sa) Greek letters 2,: = Angle, Eq. (Xb), illustrated in Figs. 6 zi = Angle. Eq. (25g). illustrated in Figs. 6 6(s) = Dirac-delta function C,= Emissivity of bounding surface 0 = Polar angle (I= = Forward and backward polar angles. Eqs. (7a) and (7b) Q = Scattering angle # = Azimuth angle 4’ = Forward and backward azimuth angles. Eqs. (7~) and (7d) 6 = Stefan-Boltzmann constant \jJ = Diffuse reflectivity of bounding surface $ = Angle defined in the addition theorem, Eq. (63) (1)-1 Single scattering afbedo
I 17-l
36, 1989
0022-4073189
Copyright
THE INTEGRAL FORM OF THE EQUATION TRANSFER IN FINITE, TWO-DIMENSIONAL, CYLINDRICAL MEDIA S.
Department
of Mechanical
T.
$3.00 + 0.00
(1 1989 Maxwell Pergamon Macmillan plc
OF
THYNELL
Engineering, The Pennsylvania PA 16802. U.S.A.
State University,
University
Park.
Abstract-The corresponding integral form of the equation of radiative transfer in absorbing, emitting, linear-anisotropic scattering media bounded by emitting, diffusely-reflecting walls is formulated. The formulation includes a development of expressions for the radiation intensity, the incident radiation and the forward and backward radiation heat fluxes in the axial and radial directions, respectively. The integral form is represented by a system of coupled Fredholm type integral equations, which can be solved accurately by numerical techniques. The two-dimensional formulation is subsequently used to develop the corresponding integral equations in one-dimensional, axi-symmetric media. In addition, it is also shown that the respectively, are developed expressions for the net radiant fluxes in the r- and :-directions. different from the equations presented recently in the literature.
I. INTRODUCTION Radiative transfer is an important mode of energy transfer in numerous areas. Examples of such areas include, among others, modeling of heat-transfer in liquid and solid-propellant propulsion systems; development of methods for improving the efficiency of pulverized coal, natural gas and oil fired furnaces; analysis of heat transfer through various fibrous and foam insulations; in the studies of nuclear energy generation and explosions; and in the prediction and examination of i.r. signatures from rocket plumes. Thus, radiative transfer has received a considerable amount of attention in the literature. To analyze the radiative heat transfer, numerous approaches have been proposed for solving the equation of transfer either approximately or rigorously. A brief discussion of several solution methods has been presented by Viskanta.’ Examination of these methods has revealed that the integral form of the equation of transfer is often employed to develop the solution.2-7 The use of the integral equation rather than the equation transfer yields, in most cases, highly accurate results for the quantities of engineering interest, which are primarily the incident radiation and the net radiation heat fluxes. The high accuracy is, in part, attributed to the smoothing that occurs when integrating the intensity over all solid angles. That is, the radiation intensity is often a highly singular function, whereas the incident radiation and the net radiation heat fluxes are usually much smoother functions. As shown by other workers8m’0 and also revealed in the following sections, however, it is quite a laborious task to transform the equation of transfer into an equivalent integral form for the incident radiation and the radiation heat fluxes. The transformation yields one or more coupled singular Fredholm type integral equations of the second kind, which involve integrals that in most cases must be evaluated numerically. The objective of this work is to present a development of the integral form of the equation of transfer in absorbing, emitting, linear-anisotropic scattering, two-dimensional cylindrical media, which is bounded by emitting and diffusely reflecting walls at known temperatures. The development in the following section, however, leads to expressions that are partially different from the corresponding ones presented recently by Lin,” who considered the effects of isotropically scattering media. In addition, the corresponding integral equations for the incident radiation and the net radiant heat flux for one-dimensional, axi-symmetric media are also formulated utilizing the two-dimensional analysis as the basis.
PSRTC--C
117
S.T.THYNELL
IIX
ANALYSIS
2.
-3.I. Equation
?f’ transfb
We consider radiative heat transfer in absorbing, emitting, linear-anisotropic scattering. twodimensional cylindrical media bounded by diffusely emitting and reflecting opaque walls. The radiative properties are assumed gray and spatially independent, but the analysis is also applicable to a specific wavelength. The medium contains spatially distributed and varying energy sources of intensity Z,[T(r. z)], where T(r, :) is the temperature. Z,(T) is the Planck function. which for the gray analysis becomes Z,(T) = n’c?T’:‘n. A schematic of the physical model and coordinates arc illustrated in Fig. I. The mathematical description to this problem is given by”
=(I
[G(r, z) + aq,. (r. -_)cos tl + aq,r(r, z)sin 8 cos 41,
- Q)Zh[T(r. :)I + 5
inO
OdO
conditions
OGd)<2n.
(1)
are given by
Z(r,-c.d,$)=c,Zh,(r)+cqj_
(r,-c)=J,(r),
77
Z(R.~.H.~)=(~Z~~(;)+~~~(R.-_)=J~(Z).
Z(r. c, 8, 4) = cJZbI(r) + :(I;
O
CR.
O
Izl
(r. c) = J3(r),
O
o,
n 5 < 0 < 71,
0 d r < R.
0 < (b < 27-t.
(2Ll)
(2b)
(2c)
Here, Z(r, 2, (b, 0) is the radiation intensity, r and ; are the optical variables in the radial and axial directions, respectively, 0 and 4 are the polar and azimuth angles, respectively c, = 1 - p,, i = I. 2, 3 the surface emissivities of the opaque bounding surfaces from the inside, and Z,[T(r, z)] is the spatially distributed energy source due to the temperature T(r, z) of the medium, and for simplicity we define at the wall Z,i = Z,(T,). The incident radiation G(r, z)? and the forward and backward are, respectively, defined by radiation heat fluxes in the r- and z-directions T I( r, z, 8, ~$)sin 0 db’ dq5, ’ l C$=o s o=o
(3a)
q$(r,z)=2
’ I( r, z,t3, &)sin2 l3 cos C$de d@. ” .i @=I1 1‘0=0
(3b)
q<:(r,-_)=2
’ r;7 Z( r, z, 8. 4)cos i o=o i lI=O
(3c)
G(r.z)=Z
q:,(r,z)=2
”
’
8 sin 0 d8 d$,
(3d)
Z(r,z,8,7r--$))sin’Qcos$dHd4,
(1I$=II l,=0 n s;; (r, z)
=
rr,2
Z(r, z, n - 8, $)cos
2
8 sin tI do d4.
(3ei
i‘ @=O s H=O The net radiation
flux in the r- and z-directions q+(r’. 2) = q;(r,
are, respectively,
specified
as
z) - qc; (r, z),
(4a) (4b)
4L:I(r1=)=q~(r,=)-ql~(r,Z). The anisotropic
scattering
has been represented
in an approximate
P(cos 0) = 1 + c1cos 0,
manner
as (5a)
Integral form of the equation of transfer
where P(cos 0) is the scattering phase function and the scattering incoming (e’, 4’) and outgoing (0, 4) d irections of the electromagnetic cos0
=sinesin9’cos(4
119
angle 0 is related waves by
to the
-~f)+c0sec0s8f,
(5b)
and - 1 d a d 1 is the expansion coefficient to the scattering phase function. Equation (l), subject to boundary conditions given by Eqs. (2) represents a singular integro-differential equation due to the possibility of different surface temperatures and emissivities, as well as the effects of the< corners of the enclosure. As such, Eq. (1) is difficult to solve accurately, but the integro-differential equation is transformable into a corresponding integral equation of the Fredholm type, in which all such singularities are smoothed out. In the present analysis, however, it is necessary to construct three coupled integral equations. The reasons for this coupling of the integral equations are: (1) the formulation includes the effects of diffusely reflecting bounding surfaces, and (2) the effects of the linear-anisotropic scattering. To construct the integral equations, we first present the expressions for the radiation intensities. 2.2. Formal solutions
to the intensity
By following the development of, e.g., ijzigik,‘* it is possible the equation of transfer of the form
to construct
a formal
solution
to
s0 Z(s = 0,0)
= exp( - S,)Z,(S,)
+
exp( - s’)S(s’, a’) ds’.
(6)
s0 Here, S, is the optical
distance
the rays have traveled
Fig. 1. Schematic of the physical model and coordmates.
from their origin
(at s = S,), I,(&)
is the
Fig. 2. Illustration of the intensity having a contribution from the bottom bounding surface at z = --c. The expression for the intensity and the limits of the angular variables (0 +, 4 *) are specified by Eq. (9).
S.T. THYNELL
IX
(diffuse) intensity of the rays leaving the wall, and S(s’. Q’) is the source function in direction R‘ at .Y’. To transform Eq. (6) into the (1..I, 8. 4) coordinates of cylindrical geometry. we consider the following. First. the intensity is symmetric with respect to a plane formed by the r- and --axes. permitting us to limit the bounds on the azimuth angle given by 0 d # d n. Second, we introduce a forward (&‘) and a backward (0 ) intensity in the axial direction. and a forward (4’) and a backward intensity (4 ) in the radial direction; these are defined as
The source
function
is then given by
S(l., :. 0’. 4s‘) = (I - to)Ih[T(r, +z
in
[G(i+,~)+fzq,~_(r.~)(
O
;)I tcosf3’)+uq,~(r,r)sinff’(rt
/-_/cc,O
O
cos$*)J.
(8)
That is, the change in the sign on 0’ is related to the sign on +cos Q*, and similarly 4 7 is related to kcos 4 ‘. I*o construct the formal solution. we first consider directions of the intensity containing contributions from the bottom bounding surface at I = -- c. By considering Fig. 2, the transformation yields
exp[ - (2 - :‘)sec O+]S(r’, z’, N’, # *‘)sec If+ d-_‘,
O
t(r,R.(Pf) c+z '
(9)
where r’=[r’+(=
-z’f’tan?@+
-2rjz
--‘jtan8+(_tcos~‘)]“‘,
r’sin$“=rsin4=,
(lOa) (lob)
r(r, R, tp *) = r( f cos +4‘) + (R2 - rz sin’ # ‘)’ ‘. r, = r’j._.=_i,
(lOc) (10d)
Similarly, the intensity containing contributions from the bounding wall at Y = R are split up into axially forward (Fig. 3) and backward components. Such split up yields
I(f.:.W.#*)=J2(zZ)exp[-(z-q)secd+l exp[-(2
-:‘)secB+]S(r’,~‘,8+,$r’)sec8+d;’.
Integral
form of the equation
of transfer
and I(r, z, 8-, C#I ‘) = Jz(zz)exp[ - (z2 - z)sec 0-1 12 +
exp[-(z’-z)sec&]S(r’,z’,8~,~“)sec&dz’, s ..‘Z_
(12) where lz-~~I=t(r,R,~‘)cot8’. Finally, the radiation intensity containing is represented according to Fig. 4 as
(13)
a contribution
from the top bounding
surface at z = c
I(r,z,F,4+)=J3(r3)exp[-(c-z)sece-] exp[ - (z’ - z)sec &]S(r’,
+ o
t(r, R 4 ‘> c-z ’
z', 8 -, C#I *')sec 0 -
,,,*,q,
dz’,
(14)
where r3 = r’l,,=+,
(15)
The above formal solutions are used to construct integral expressions to the quantities of most interest in engineering applications, namely, the incident radiation and the radiation heat fluxes.
-r
Fig. 3. Illustration of the intensity having a contribution from the side bounding surface at r = R for z’ <.z. The expression for the intensity and the limits of the angular variables (O+, 4 *) are specified by Eq. (11).
Fig. 4. Illustration of the intensity having a contribution from the top bounding surface at z = c. The expression for the intensity and the limits of the angular variables (&, 4 *) are specified by Eq. (14).
S. T. THYNELI.
122
2.3. Incident radiation
G(r, 2)
To obtain an integral equation for the incident radiation, the formal solutions to the radiation intensities given by Eqs. (9). (11). (12) and (14) are substituted into the definition of G(r, z) given by Eq. (3a). To simplify the subsequent analysis. however, we represent the incident radiation as the sum of eight components: G(r. -) = i [G,+(r. -_) + G, (r. z)]. /= I
(16)
where
G!(r,z)=2
H+,d)*)sinH+dOi
d4*.
GT(r. r) = 2
(17c) tan,, -
G$(r.z)=2
(1%)
” s * - = 0 1 I, = (I
I,,.
R. .-@+ 1 < :
l(r,-_,H
,d*)sinC
dH
d$I.
(t7d)
The following analysis describes how the angular integrations are transformed into equivalent surface and volume integrals of the two-dimensional cylinder. However, we limit the details to those defined by G,‘(r, z), i = 1.2. since the expressions for these can be used to deduce the corresponding expressions for G,(r-, z), i = 3.4 due to symmetry in the integrals. 2.3.1. E.xpression for G :(r, :). First we introduce the Dirac-delta function 6(x) to write the definition of G:(r, 2) as
represent the angular variables of integration variable (r’. r), as shown in Fig. 5, the following
TO
The Jacobian
(8+, 4 +) in terms of a spatial transformation is introduced:
and an angular
(z - z’)tan H+ = [r’+ r” - 2rr’cos
cx]’ ‘.
(19a)
tan C#I+ = r’ sin cc/(r - r’cos
CC).
(I9b)
of the transformation
is given by (z - z’)r’
s(e+.$-) ;I(r’, J)
= d:(r, r’. r, z - z’) d,(r, r’, LX,0)’
(20)
where d,,(r. r’, 2, u) = [r’ + r” - 2rr’ cos tl + u~]~,~.
(21)
We also find that sinH’=d,(r.r’,a,O)jd,(r,r’,r,z-2’).
(22a)
cosfl+=(;-:‘),id,(r,r’.cr,z-2’).
(2%)
i = r’sin sin C#J
x/d,(r,
r’, c(, 0),
(22c)
cos C$T = (r - r’ cos ol)/d, (r, r’, do,0).
(22d)
cos g5’-’ = (r cos r - r’)/d,(r. r’, 4%.0).
(22e)
Integral
form of the equation
123
of transfer
(a)
TOP
VIEW
OF CYLINDER
r 4
(Z-Z’)
tone+
(b)
Fig.
5. Definition
of dummy variables and a.
of integration
r
Fig. 6. Illustration of the limits used on r’ and a in the various integrals. Here, it is assumed that z’< z with (a) applicable to r’ and z if z’ > z:. and (b) applicable to r’ and a: if z’
To specify the limits of integration, we use Figs. 6 and 7 to establish that the radiant energy propagating towards the point at (r, z) must be confined to within the circle of radius r:, which is centered at a radial distance rd+ from the z-axis, the line r’ = r set a and the r-axis. By substituting the above transformation into Eq. (18) and using Fig. 8 to express r: and ri in terms of known variables, we obtain Rcosa=r G T(r, 2) =
,S~CS
J, (r’)(c + z)K,(r,
2 s !l=O +2
j”
r’, a, c + z)r’
j”
J, (r’)(c
+ z)K,(r,
r’, a, c + z)r’
z’)K,(r,
ss G
+
dr’ da
s r'= 0
~,‘,I,
x=a+ r’= r&,
(-)r’
dr’ da
dr’ da
r’, r, z - z’)r’
dr’ da
1
dz’
(23)
The (s) symbol is introduced
to reduce the length of the expressions,
and it implies that the integrand
S. T. THYNELL
114
(a)
--
r
(b) Fig.
7. The slanted
r’-plane
Fig. 8. The radial distance r; to the center of the clrclc 01 using ratios of equ.tl radius r<’ and z: are determined triangles.
lines tllustrate the (r’. 1) domain in the that contributes to G,+(r. :). i = I. 2, which are components to the incident radiation.
is repeated.
Furthermore,
it is necessary
s*(l-. 1.‘. Iz. 1. z’) = (I - c0)Z,[T(r’. +
E
G(r’. z;) + ay,
to define a new source
function
as
:‘)I (z - z’) (r’. z’)
l/,(r. r’, 2. z -z
[
,+
)
q,.r(r’.
z’)
(1.cos x - 1.‘) d,(r. 1.‘. x, I - I’) 1 *
(74)
and
r A, = r;
cos
y
+
[r;’
rh = r(c * z’)/(c r’=Rlz
_ r$‘sin? ~(1’ ?,
+ z),
-z’j/(cf~).
tan c(5 = [ri+’ - (r - r,;)‘]’ The kernels
K,,(r, r’, x, u ) are introduced
‘jr.
as
K,,(r. 1.‘. x, II) = exp[ - d,(r, r’, x. u)]/d,,(r. r’, 2. 24). 23.2. Expression for G;(r, z). By substituting
G, (r. 2).
We continue the analysis by developing an expression Eq. (9) for 4 _ into Eq. (17a) and using the Dirac-delta function
t2h)
for in a
Integral
similar
manner
as in Eq. (18)
form of the equation
the resulting
ss
expression
of transfer
125
becomes
n?
G, (r,z)=2
[J,(r’)6(c+z’)+S(r’,z’,0+,&‘)sec0+]
$J =0 II+- 0
xexp[-(z-z’)sec0+]dz’sinB+dB+d&. In this case, the previous
transformation
(27)
given by Eqs. (19) can still be applied,
cos&‘=(r’-r
but we note that
cos c()/d, (r. Y’, c(, O),
(28)
which results in the same expression for the source function S*(r, r’, ~1,z, z’) as defined by Eq. (24). Using Figs. 6 and 7, the equivalent expression for the component of the incident radiation G ;(r, z) is Rcou=r
R
J, (r’)(c + z)K,(r, r’, 2, c + z)r’ dr’ da
G, (r,:)=2 s r = r \ec1
s I=0
2.3.3. Expression for G:(r, z). We continue by developing tution of Eq. (11) into Eq. (17b) applied to 4’ yields
+ To transform of integration
an expression
_ S(r’, z’ ,8’,$+‘)exp[-(z-z’)sec8+]sec0+dz’ __j:-_:
for G:(r, z). Substi-
sinO+dtI+d4+.
the surface integral involving Jz(zz) in Eq. (30) we treat zz = z’ as the dummy and introduce the transformation (z - z’)tan 0’ = [r’ + R’ - 2rR cos a]’ ‘,
The Jacobian
of this transformation
By utilizing Figs. 6 and 7 to deduce eliminate sin 0 +, we find ’ ss_‘
n
J?(z’)(R
(31b)
is
;(e+, 4’) d (z’, a)
G:(r,z)=2
variable
(314
= R sin r/(r - R cos 51).
tan$+
(30)
I
R(R -r
cosa) (32)
= d?(r, R, ‘A,z - z’)d,(r, R, CC,0)’ the limits
of integration
and by employing
Eq. (22a) to
- r cos a)K,(r, R, a, z - z’)R da dz’
RCOSZ-,
Rcosx=r
~SXJI
R (.)r’ dr’ dcc +
1
(.)r’ dr’ dcc dz’ s 2= 2”:
s r=o
S. T. THYNELL
IX
2.3.4.
Espression
for
in Eq. (I I ). Then,
Gz (r, z). To obtain
S(r’.
+
z’ ,o+,&‘)exp[-(=-=‘)sec6,+]secH+d;’
I=::
Using
G, (r, z), we first consider the backward direction of this expression over the appropriate solid angles yields
an integration
the preceding
sinf!I’dH+d$
4
(34)
i
development
G’(r.
involving
z), we deduce
immediately
that
Hco\1= , G,(r,;)=Z
J2(z’)(R I---
Hu,,r ( +
- r cos r)K,(r.
R. 2,;
- r’)R
da dz’
L
K (*)r’ dr’ da
i 1 7.’
i , = rwc1
(35)
dz’. 1
23.5. Expression for G(r, z). A close examination of the integrals presented in Eqs. (23) (29), (33) and (35) for G’(r, z). i = 1.2, reveals that the sum of these four equations add up to one volume integral and two surface integrals. The limits of these integrals are confined to within a cylinder whose top surface is located at z. It is thus obvious that the sum of the four expressions for G ,?(r. : L i = 3.4. adds UP to an equivalent volume integral and two surface integrals, whose limits are within a cylinder located above the point at z. By using symmetry in the integrals. WC deduce that the integral equation for the incident radiation is given by I? GO,. z) = 3 I-1e (I I , =,,
J,(r’)(c
+ z)K7(r,
J,(z’)(R
+2 i-
- r cos x)K,(r,
R, x. 2 - z’)R
da dz’
IL J?(r’)(c
+2
r’. ;I, c’ + :)r’dr’dr
- z)K,(r,
r’. IX,c - z)r’ dr’ dr
I-x=I, !^i -0 I
+q_
R
,; 1= 0 j r =o
S*(r.
r’, x. 2. z’)K?(r,
r’. Y(.: - z’)r’dr’dr
dz’.
(36)
where the source function S*(r, r’, 2. z. z ‘) is defined by Eq. (24). Since the source function contains the net radiant heat flux in the radial and axial directions, it is also necessary to develop expressions for these quantities in order to complete the formulation. The formulation is also necessary in the case of nonblack surfaces which enclose a purely absorbing/emitting medium. It should be noted that Eq. (36), applied to the special case of isotropic scattering (a = 0), is readily reduced to the of Eq. (36) integral equation for the source function as derived by Lin. ” In addition, evaluation at z = -c’ results in the contribution 271J,(r) from the first double integral; similarly, evaluation of Eq. (36) at z = c results in the contribution 2nJ,(r) from the third double integral. 2.4. Radial
radiant
heat ,fiuxes
q$ (r, z)
The forward and backward radiant heat fluxes, qg (r. z) and q, (r, z), respectively, angular coordinates (0 *, C#J ‘), are defined by r:? n? de+ q‘t (r. :) = 2 I(r, z, 8+. 4’)sin’Q’ s $$t=0 [S i)+= 0 +
a’ Z(r, z,8-,~*)sin28-d& s o- =0
1
based on the
cos4’d@‘.
(37)
Integral
Using the specific definitions we find
form of the equation
of the intensities
127
of transfer
Z(r, z, 9 +, I#J*) given by Eqs. (9)
(11),(12)
and (14)
J2(z2)exp[ - (z - z,)sec 0 ‘1
S(r', z', 13+, C#I*‘)exp[
+2
-
(Z - z')sec 8 +]sec 0
+dz’
sin* 8 +dO +cos 4 f d$ ?r
{;I=0 JIro_ i.l,(z2)exp[ - (z2- z)sece-1 _,cr;R.f*l
s. =z
+
,s(r’,z’,e-,@*‘)exp[-(z’-z)sec&]sec&dz Z,= _
+S(r’,
sin20~dO~cos~*dQ,*
z’, 8-,~“)sec8-]exp[-(z’-z)sece-]dz’sin2e--de~cos~’d~~.
(38)
To transform these integrals from the angular domain of (Ok, C#J ‘) to the (r’, CX)domain, we consider first the forward radial heat flux, qi (r, z). For the integrals in Eq. (38), we introduce for the surface integrals involving the contributions from the bounding intensities J,(r) and J3(r) and the volume integrals involving the source function, respectively, the transformation given by Eqs. (19) resulting in
(~-r’COS~)IZ--‘lr,dr’dr d4(r,r,,C(,Z_z,)
sin’O’cos~+d~*d~+=
,
(39)
where cOse*=Iz-z’l/d,(Y,r’,CI,Z-~‘).
For the two surface integrals involving the boundary transformation given by Eqs. (31), yielding
intensity
Jz(z?) in Eq. (38), we employ
(r-Rcoscr)(R-rcosa)
sin*8*cos#+dO*d~+=
d4(r, R, a, z - z’)
To establish the corresponding limits of integration, expression for the forward radial heat flux: Rcora=r
(40)
R da dz’.
we use Fig. 9, resulting
the
(41) in the following
rseca
q; (r, z) = 2
[~,(r’)(~+z)K~(r,r’,tl,c+z)+J~(r’)(c-z)K~(r,r’,~,c-z)l s 0=0
s I'= 0 11
x (Y - Y' cos
R
cc)r’ dr’ dcr + 2
(.)r’ dr’ dcr s Rcosa=r s r'=O
J,(z’)(r-Rcosa)(R-rcosa)K,(r,R,u,z-z’)Rdudz’
S*(r,r’,cr,z,z’)(r
(.)r’dr’dcr
1
dz’.
-r’cosc1)K3(r,r’,cI,z
-z’)r’dr’dcr
(42)
1%
S.T.THYNELL
The (e) symbol implies that the integrand is repeated and introduced to reduce the length of the expressions. In the case of the backward radial heat flux, it is now readily shown that the corresponding expressions to Eqs. (39) and (41’ are, respectively, given by (43) and sin-‘@* cosd,’
d@* d4”
-_
(Rcosa-r)(R-rcosff) d43(r,R, a, z - 2’)
--- R da dz’.
(44)
By using Eqs. (43) and (44) in Eq. (38) for 4 - and by deducing the limits from Fig. 9. we obtain Rcosd=r y<;:
(1.,
z)
2
=
s f =0
R
V,(r’)(c’ + z)&(r. Y’,~1,c + z) s ‘i;iTSCCZ
-t-Jl,(r’)(C -z)iY,(r,r’,x,c
-z)](r’cosa
-r)Y’dr’da
x K3(r, r’, a, ; I z’)r’ dr’ da dz’.
(45)
The resulting expression for the net radiant heat flux, which is defined by Eq. (4a), takes the form y,:,(v,zf=2
R [J, (Y’)(C + z)K 4(r,y’,a,c n s x=0 s I =n x(r +zJ
+z)+J3(r’)(c
-z)iY,(r,r’,ff,c
-z)]
- P cos a)K,(r,
R, a, z - z’)R dor dz’
-r’cosa)t.‘dr’da ii .J?(z’)(r - R cos a)(R s f z --( I 1= 0 S*(r,t”,a,z,z’)(r
-r’co~a)K~(r,r’,a,z
-z’)r’dr’dcl
dz’.
(46)
It should be noted that the above expression for the net radiai heat flux, reduced to the special case of isotropic scattering, does not completely agree with the one presented by Lin.” The difference between his (on p. 595) and Eq. (46) is found in the integrand of the surface integral involving the bounding intensity J,(z); Lin’s formula contains the term (I - R cos a)‘, whereas Eq. (46) contains the term (r - R cos a) (R - r cos a). The expression presented by Lin yields numerical results of the net radial heat flux that appears to be in error and physically unsound. For example, if we suppose that the radiation transfer originates from the bounding surface at Y = R. that is the bounding intensities J1 (r) = 0 and &(P) = 0 (cold and purely absorbing boundary surfaces at z = kc). and that the source function is negligibly small (cold and nonscattering medium); then the net radial heat flux is always positive according to Lin’s expression. This result does not seem to be possible. Thus, it appears that there is either a typographicai error, or that the application of Lin’s general formulas have resulted in incorrect integral expressions in this particular case. 2.5 Axial radiant heat fluxes 4: (r, z) The development of expressions for the forward and backward radiant heat fluxes in the axial direction, q$ (r, z) and qe: (r, z), respectively, proceeds in a similar manner as for the development
129
Integral form of the equation of transfer
of the heat fluxes in the radial direction. The axial heat fluxes are defined by
s; (r, z>= 2
ss x/2
4
,$+=o e*=o
Z(r,z,8*,~+)cos8*sin8’de~d~+
ss d-2
+2
n/2
Z(r,z,8’,~-)cos8’sin8’dB’d~~. ,#-=lJ 0+=0
(47)
Using the definitions of the intensities Z(r, z. 0 *, 4 ‘) given by Eqs. (9), (1 l), (12) and (14), the axial heat fluxes are q~~~,z~=2~~~=o~~~~~=~~~=_~~~,~~~~~~~+I.)+S(~..i.;R+,~+.)rers'j
+
s
+
s
’ S(r’, z’, 0 +, 4 +‘)exp[ - (Z - z')sec0 +]sec8 + dz’ sin 8 + cos 8 + de + d4 + i’=i2
’ S(r’, z’, 8 +, 4 -‘)exp[ - (Z - z')sec8 +]sec8 + dz’ sin 8 + cos 8 + de + d4 -, 2’=22 I (48)
and [J,(r')d(c -z') + S(r',~',e-,4+')sece-1
xexp[-(z’-z)secO-]dz’sin&cosO-de-d$+ [J,(r’)d(c -z’)+
S(r',z',e-, +-')sec e-1
xexp[-(z’-z)secO-]dz’sinfJ-cos8-d&d& +2C.oJ~O_
,,,,,,+,~*(z2)exp[-(z2-z)sece-l c--L
22 +
s I’=z
S(r’,z’,&,4+‘)exp[-(z’-z)secO-]sec&dz
sine-cm&de-d4+
tbw-,{ +2J;:_oJ::e
4 (z2)exp[ - (z2 - z)sec 8 -1
+
s
c--z
z2S(’r ,z’,O-,4-‘)exp[-(z’-z)secO-]sec&dz
I’- z
sin&cos8-de-d4-. (49)
S.T. THYNELL
Fig. 9. The slanted lines illustrate the (r’. 2) domain in the -‘-plane that contributes to the forward and backward radiation heat fluxes in the radial direction, y;t (r. z) and 4,; (r. z ). respectively.
Fig. 10. The slanted lines illustrate the (T’%z) domain in the =,-plane that contributes to the forward and backward radiation heat fluxes in the axial direction. (I,~ (r. -_) and y,,; (r. z). respectively
To transform these integrals from the angular domain of (0 +, C#I ‘) to the domain of (r’, x ), we employ the previous transformation given by Eqs. (19). For the surface integrals involving J,(r) and J3(r), and the volume integral involving the source function. we introduce the substitutions, respectively, given by (z
sin0’cosQ’dCI’d+‘=
_
=‘)Z
dd( 1. r’, rl, z - z’)
r’dr’da,
(50)
and sintI*
de* d4’
=
/z - Z’I
r’ dr’ da.
d-1(r, r’, a, z - z’)
For the surface integral involving the boundary the transformation given by Eqs. (31), yielding
intensity
J,(zl)
(51)
in Eqs. (48) and (49). we employ
(52) The limits of the integration for the z’ variable are readily established from Fig. 10, and we must have 0 d r ,< TCand 0 s r’ G R. The expressions for the forward and backward axial heat fluxes, respectively, are constructed as: n
R
J, (r’)(c
ql(r.z)=2
+ z)‘K,(r,
r’, CI,c + z)r’ dr’ dcr
s X=0 s ,'= 0 ’
+2 s.
s
J>(z’)(z
- z’)(R
- r cos x)K,(r,
R, CC,z - z’)R dcc dz’
z=O
')(z -
z’)K3(r,
r’, a, z - z’)r’ dr’ do! dz’,
(53a)
131
Integral form of the equation of transfer
and R
qP;(r,z)=2
n
J3(r’)(c
-
z)‘K,(r,
r’,
a,
c
z)r’dr’ dcc
-
s ;r=O s r'= 0 ‘
+2
n
SJ
J2(z’)(z’ - z)(R - r cos a)K,(r, R, a, z - z’)R dcr dz’
3-0
c
n
R
a=0
s r'=O
S*(r, r’, a, z, z’)(z’ - z)K,(r, r’, a, z - z’)r’dr’
+2 LS
dcc dz’.
(53b)
The resulting expression for the net radiant heat flux, which is defined by Eq. (4b), takes the form qt,(r, z) = 2
s’ sR
[J,(r’)(c + z12K(r, r’, a, c + z) - J3(r’)(c
Z=O
4
- z)‘K,(r,
r’, cx,c - z)]
r'= 0
c
x r’ dr’ dg + 2
l-S
J,(z’)(z - z’)(R - r cos a)K,(r, R, a, z - z’)R dcr dz’
0=0
S*(r, r’, LY,z, z’)(z - z’)K,(r, r’, LX,z - z’)r’ dr’da dz’. +~s:.-c~~~os.lo
n
(54)
It should be noted that there is also a discrepancy between the above Eq. (54), applied to the special case of isotropic scattering, and the corresponding one developed by Lin.” The difference between the two expressions is observed in the integrand of the integral involving the bounding intensity J,(z). In Lin’s work, the integrand contains the term (r - R cos a), whereas in this work it contains the term (R - r cos cc). The effect of Lin’s result is that it appears to produce physically unrealistic results; that is, for c( < cos~‘(r/R), a negative contribution to the net flux is predicted. However, such negative contribution to the net axial heat flux is possible only for z’ > z.
3. RADIATION
TRANSFER AXI-SYMMETRIC
IN
ONE-DIMENSIONAL, MEDIA
In numerous applications, the radiative energy transfer takes place in axi-symmetric media. In such media, the radiation transfer is independent of rotation about the z-axis as well as translation along the z-axis. In previous works, the integral form of the equation of radiative transfer in absorbing, emitting, isotropically scattering media has been formulated’0 and solvedI using the appropriate expansion functions. The objective of this section is to reduce the formulae developed in Sec. 2 for the incident radiation and the net radiant heat flux in the radial direction to the corresponding ones in the one-dimensional case. 3.1. Incident radiation G(r) We consider an infinitely long cylinder, in which the radiation transfer depends only on the radial variable r. Using Eq. (36) as the basis for the subsequent analysis, the integral equation for the incident radiation is written as 00 n G(r) = 45,
s &-‘=a s x=0
(R-rcosa)K,(r,R,u,z’)Rdccdz n:
n
R
r’, a, z’)r’ dr’ du dz’.
(55)
Here, we arbitrarily set z = 0 and note that the integrals are symmetric in the z/-direction. source function S&(r, r’, a, z’) in the one-dimensional case is defined by
The
+4
S?,(r,
.?‘=!I r-0 sss
r’, a, z’) = (1 - o)Z,[T(r’)]
Sy,(r, r’, a, z’)K,(r,
r’=O
+ z
G(r’) + aq+(r’)
(rcosa-r’) d, (r, r ‘, a, z’)
1’
(56)
S. T.
131
THYNELL
It is now desirable to reduce the number of integrals such reduction, we consider first the integral defined
By introducing
in Eq. (55) from three to two. To perform by
the transformation (58)
d, (r, r’, a, z’) = [I, (r. r’. 2. O)sec 1~. we find dz’ = o’,(r, r ‘, z, 0)sec’ 1~d/c Substitution
of this transformation
(5%
into Eq. (57) yields
(r cos x - r’)_ G(r’)Ki, [d, (Y, r’, a, 0)] + aqCr(r’)Ki2[d, (r. r’, CC.O)] --~---~ n, cr. r’. L-i.01
F,(r)=4
x -~--~~~ ri, (r. r’. 1. 0) where Q(x)
is the Bickley
function14.‘” defined
(60)
by
nz! K,(s)
exp( - .Y set P)COS’ ’ p dp.
=
(hl)
s0 By using its relation
to the modified
Bessel function Ki, (s) =
and the addition
theorem”
applied
to r’ < r given by
F,(r)
that the function
(03,
K,(u) du,
I’
K,[d, (r, r’, x, O)lexp(il$) = it may be shown
K,,(x), namely.
ci
,I:
K,+,Cr)ln(r’)exp(inu),
can be written
(63)
as
I F,(r) = 47~
T ,=n u=l SS[
G(r’)K,,(ru)Z,(r’u)
+ aqGT(r’) k K,,(ru)l,(r’u)
ss[ R
+ 4n
du r’dr’
1
I
T=T ti=I
G(r’)Z,(ru)&(r’u)
Here, i = v; - 1 and the angle $ is located opposite the surface integral in Eq. (55) and define %
- aqGr(r’) i &(ru)K,(r’u)
du r’ dr’. (64) 1 to r’ for r’ < r and vice versa. Next we consider
*
F?(r) = 4J,
(R - r cos a )K,(r, R. r, z’)R dcc dz’.
(65)
1: -01 z=O Using
the transformation
given by Eq. (58), Eq. (65) becomes
F,(r) = 4J, The use of the addition
’ (Rs z=o
theorem
r cos r)K&[d, (r, R, x, O)]
yields the desirable
R dcc d2 (r? R, u, 0)’
(66)
result given by
1 F:(r) = 4nRJ, By introducing
the kernel
L,,,(r,
% s
r’) defined
L,, (r, r ‘) = 47t
L K, (Ru)Z,(ru) du. s [,=I u
u=I
by
K,(ru)Z,(r’u)
du ~ t?1+?7’ r’ < r, u
%
L,.,(r,
r’) = 4n( - l),+’
(67)
i u= I
I,(rU)K,,(r’u)
du z, u
(6W
r’ > r,
(68b)
Integral
the integral
equation
form of the equation
for the incident
G(r) = RJ&,,,(R
radiation
133
of transfer
is represented
by
r)
s[i
1
R
+
(1 -w)lb[T(r’)]
r’=O
3.2. Net radiative heat flux
r’) r’dr’.
L,,,(r, r’)+azqgT(r’)L,,,(r,
+zG(r’)
(69)
q;,(r)
The development of the integral equation for the net radiative heat flux follows the procedure used above for the incident radiation. Using Eq. (46) as the basis for the analysis, we express the net radiative heat flux in the form q2, (r) = 4J,
n (r-R x s I=0 s a=0
cos
T. f4 By defining
n
a)(R - r
F,(r)
function
a)K4(r, R, Z, z’)R da dz’
R
:‘=0 z=o /=O sss
a working
cos
S*(r, r’, E, z’)(r - r’ cos cc)K,(r, r’, 2, z’)r’ dr’ dcr dz’. as
x (r -r’cosa)K,(r,r’,cc,z’)dz’dccr’dr’, and using the transformation
given by Eq. (58)
G(r ‘)I& [d, (r. r ‘. a,
&:7(r) = 4
this function
WI+ q,rWG
is expressed
F3(r) = 471
ss “’
x
r=O
u=l
(rcostl
Finally,
we introduce
3
7
, = r’ u=l
of the modified
dz(r, r’, LX,0)’
(72)
Bessel function
and
1
cr’dr’
1
-G(r’)Z,(ru)K,(r’u)+uq,~(r’)~Z,(ru)K,(rIU)
of the surface
’
(r-R
s a=0
cr’dr’.
(74)
given by Eq. (58) into Eq. (74) to obtain R da
cos cr)(R - r cos a)Ki3[d, (r, R, a, 0)]
(75) 4
Expressing the Bickley function in terms of repeated using the addition theorem, we find I;,(r) = -4nRJ2 ‘Then, based on the developed
(73)
in Eq. (70) as
cos cc)(R - r cos a)K,(r, R, u, z’)R du dz’.
’ (r-R m s :‘=0 s ol=lJ
the transformation
integral
1
dcr r’ dr’
m
a definition
F4(r) = 4J2
integrals
3
ss[
47c
F4(r) = 45, Next, we substitute
-r’)
k4 (r, r’, @,O)ld,cr r, c1oj
G(r’)K,(ru)Zo(r’u)+oq,T(r’)~K,(ru)Z,(r’u) R
+
in terms of repeated Eq. (63), we find
(71)
as
x (r -r’coscr)
Expressing the Bickley function applying the addition theorem,
(70)
expressions
integrals
(r,
of the modified
R,
a,
0)’
Bessel function
CC 1 1 K, (Ru)Z, (ru) du. s u=l n for F,(r)
and F4(r), we express
and
(76) the net radiative
heat
S. T. THYNELL
134
flux as q,r(r) =
-RJ2L,,(R, r)
sci R
+
“’= 0
(1 - w)I~[T(Y’)] + 2
L,,o(r, r’) + a z
G(r)
qpr(r’)L,,,
(r, r’)
r’ dr’.
(77)
I
To eliminate the boundary intensity J2 in Eqs. (69) and (77), we use Eqs. (2b) and (4a) applied to the one-dimensional case for r = R. This yields 1 -tz
nJz = T& + ___(z
Evaluation
q,;,(R).
(78)
of Eq. (77) at r = R and the use of Eq. (78) to eliminate
q;,(R)
yield
~27143 RJ2=(zn
+p2
+
P2
RL,., (R,R) (1 -co)1,[7’(r’)]+~G(r’)
I
L:,(R,r’)+a~~q,~(r’)L:,(R,r’)
r’dr’,
(79)
where
L&CR, r’) = Substitution
RL.,(R r’) (271 + P~RL,.,(R
of Eq. (79) into Eqs. (69) and (77) produces
(80)
RI’
the desirable
result given by
(1 -o)Ih[T(ri)]+zG(r’)
G(r) = c27db2LTn(R,r) +
x [Lo,o(r,r’) + P~L~,~(R, r)L?, UC r')l +~~g,~(r’)~Lo,~~r.r’)-~2~l,o(R,r~L~l(R,r’)l
r’dr’
(81)
1 and
s[i R
qB,(r)= - ~2~~b2L~l(R r>+
r =0
(1 - wN,[T(r’)]
G(r’)
+ z
x LL,(r, r’) - PAL,.,CRrWiYoCRr')l
+~~q,,(~‘)[LI.,(r,r’)-~2LI,,(R,r)L~l(R,r’)l
r’dr’.
(82)
1
To obtain an accurate solution to the coupled integral equations given by Eqs. (81) and (82), it is used to construct an initial estimate to G(r) and qi,(r). is suggested that the P,-approximation” A significantly improved value is expected to both of these quantities if the P, solution is substituted into the right-hand side of Eqs. (81) and (82). The integrals over r’ can be performed analytically.” thereby reducing the number of required numerical integrations in Eqs. (55) and (70) from three to one. Such reduction in the computational effort represents a significant achievement, and it is of particular interest to those workers who analyze the interaction of radiation with other modes of energy transfer. 4.
SUMMARY
A rigorous approach has been implemented to develop expressions for the radiation intensities, incident radiation and the forward and backward radiation heat fluxes in the radial and axial directions, respectively, in absorbing, emitting, linear-anisotropic scattering, two-dimensional
Integral form of the equation of transfer media bounded for the incident
135
by diffusely emitting and diffusely reflecting surfaces. Subsequently, the expressions radiation and the net radiant heat flux in the radial direction have been reduced
to integral equations in axi-symmetric, one-dimensional media. A solution to the formulated integral equations may be constructed by various numerical and semi-analytical means. ~C~~~wZe~ge~e~r-The author is grateful to acknowledge the partial support from the Office of Naval Research under contract No. ~~~4-86-K-~8. REFERENCES I. R. Viskanta. Fortschr. Yer- Tech. 22, 51 (1984). N. D. Sze, JQSRT 16, 763 (1976). S. A. Elwakil, J. Appl. Phys. D: Appl. Phys. 13, 339 (1980). G. Spiga, F. Santarelli, and C. Stramigioli, Int. J. Hent Muss Tran~/kr 23, 841 (1980). M. N. &isik and Y. Yener, J. Heat Transfer 104, 351 (1982). M. M. R. Williams, lM,4 J. Appl. cash. 31, 37 (1983). 7. S. T. Thynell and M. N. &igik, Appl. Phys. 60, 541 (1986). 8. M. G. Smith, Proc. Camb. Phil. Sot. 60, 909 (1964). 9. G. C. Pomraning and C. E. Siewert, JQSRT 28, 503 (1982). 10. S. T. Thynell and M. N. Ozisik, JQSRT 36, 492 (1986). 11. J. D. Lin, JQSRT 37, 591 (1987). 12. M. N. &isik, Radiative Transfer, Wiley, New York, NY (1973). 13. S. T. Thynell and M. N. t)zigik, JQSRT 38, 413 (1987). 14. W. G. Bickley and J. Nayior, Phil. Mug., 7th Ser., No. 20, 343 (1935). 15. W. W. Yuen and L. W. Wong, JQSRT 29, 145 (1983). 16. I. S. Gradshteyn and I. M. Ryzhik, Table oflntegrafs, Series, and Products, p. 979, Academic Press, FL (1980). 2. 3. 4. 5. 6.
NOMENCLATURE a = Linear anisotropic
scattering coefficient c = Half the optical axial length of the enclosure F;(r) = Functions defined in Sec. 3 G(r, z) = Incident radiation G:(r, z) = Components to the incident radiation, Eqs. (17) f(r, z, 0,4) = Radiation intensity &(T) = n2BT4/lr, the Planck function & = Emission from the ith wall I,(X) = Modified Bessel function of the first kind J, (r) = Diffuse intensity of bounding surface at z = -c Jr(z) = Diffuse intensity of bounding surface at r = R J3(r) = Diffuse intensity of bounding surface at z = c K,,(r, r’, CI,u) = Kernels of integral equations, Eq. (26) Kn(x) = Modified Bessel function of the second kind Xi,(x) = Bickley function .L,,,(r, P’) = Kernei for one-dimensional case, Eqs. (68) L,&(R, I’) = Kernel for one-dimensional case, Eq. (80) n = Index of refraction q#r(r, z) = Net radiation heat flux in r-direction q$ (r, t) = Radiation heat flux in forward ( + ) and backward ( - ) r-direction qez(r, z) = Net radiation heat flux in z-direction qt (r, z) = Radiation heat flux in forward (+) and backward ( -) z-direction P(0) = Scattering phase function r = Optical radial variable r,i = Radius of circle, Eq. (25f) t-2 = Radial distance, Eq. (25e) r&, = Radial distance, Eq. (2%) rfmax= Radial distance, Eq. (25d) rl = Radial distance, Eq. (IOd) r, = Radial distance, Eq. (15). R = Optical radius S(r, z, 8, #) = Source function, Eq. (8) S*(r, r’, GI,z, 2’) = Source function, Eq. (24) %p(r, r’, tl, 2’) = Source function, Eq. (56) f (r, R, rb *) = Distance defined by Eq. (10~) T(r, z) = Temperature
S. T.
THYNEL.~.
T, = Wall temperature : = Optical axial variable Z? = Axial distance, Ey. (I 3) z& = Axial distance, Eq. (2Sa) Greek letters 2,: = Angle, Eq. (Xb), illustrated in Figs. 6 zi = Angle. Eq. (25g). illustrated in Figs. 6 6(s) = Dirac-delta function C,= Emissivity of bounding surface 0 = Polar angle (I= = Forward and backward polar angles. Eqs. (7a) and (7b) Q = Scattering angle # = Azimuth angle 4’ = Forward and backward azimuth angles. Eqs. (7~) and (7d) 6 = Stefan-Boltzmann constant \jJ = Diffuse reflectivity of bounding surface $ = Angle defined in the addition theorem, Eq. (63) (1)-1 Single scattering afbedo