Radiative heat transfer in plane participating media

Radiative heat transfer in plane participating media

INT. COMM. HEAT MASS TRANSFER 0735-1933/83/030191-09503.00/0 Vol. I0, pp. 191-199, 1983 ©Pergamon Press Ltd. Printed in the United States RADIATIVE ...

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INT. COMM. HEAT MASS TRANSFER 0735-1933/83/030191-09503.00/0 Vol. I0, pp. 191-199, 1983 ©Pergamon Press Ltd. Printed in the United States

RADIATIVE

HEAT

TRANSFER

IN

PLANE

Giampiero

Spiga

PARTICIPATING

MEDIA

Laboratorio di Ingegneria Nucleare, Universit~ di Bologna,

Italy

and Marco

Spiga

Istituto di Fisica Tecnica, Universit~ di Bologna,

Italy

(Cc~aunicated by E. Hahne) ABSTRACT The problem of radiative heat transfer through a nonisothermal scattering, absorbing and emitting grey medium between reflecting, absorbing and emitting plates is analytically investigated. The solution technique is based on a projectional procedure, equivalent to a variational approach. The results concern the most meaningful physical quantities, and get an improved accuracy with respect to the data available in literature, with extremely low computational time. Introduction Although many papers were devoted in the past to radiative heat transfer in participating media, and several monographs and textbooks already exist ~ - 3 ] , in the recent literature E 4 - ~ ,

a renewed interest can be noticed to provide an improved accuracy

for those predictions that are susceptible to numerical errors. This is promoted by the rigorous safety and efficiency requirements in every applications of advanced heat transfer, such as porous materials,

rocket performance and nuclear engineering.

191

192

G. Spiga and M. Spiga

Vol. I0, No. 3

This latter area will be here emphasized, tackling the problem of radiative heat transfer in the gap between fuel and cladding in a LWR (or LMFBR) cylindrical fuel rod, where a considerable temperature jump takes place through the gap, in spite of its extremely small width. The effects of curvature are quite trifling, because of the ratio between gap width and fuel rod radius, so that the radiation exchange takes place in a grey plane participating medium

(absorbing,

ing the nonisothermal

scattering and emitting),

represent-

gas filling the gap, bounded by grey emitt-

ing and reflecting opaque surfaces. If the optical abscissa z, measured in units of the mean free path (namely the inverse total extinction coefficient), ranges from -a to +a, the transfer equation and boundary conditions for a homogeneous

isotropically

scattering and emitting medium

read as

I

~I i-~ n2o T4(T) + ~ I ( ~ , ~ ) d ~ ~-~ + I(~,~) - ~ 2 1

-a<~
-i~i

d

(i)

=~i 2 4 s Pi ; I(;a,l~) 2~ n oTihi(~)+ PiI(ga,;~)+ ~-~ gi(~)q (;a) ~ , where hi(gi) account for the angular distribution emitted

(diffusively reflected)

q(~)

= q+(r)-

In c o n d i t i o n s

q-(z)

,

of radiative

i=1,2

of radiations

by the walls with ei+p is+pi -id-" and

+ q-(T)

= 2=~I(z,tu)dv

equilibrium

Eq(1)

(2)

has to be c o u p l e d

to

1

dq = (l_~)[4n2oT4(x)_ dx so that the net radiative eliminated

from Eq.(1)

2~

l(T,~)d~]= 0 ,

(3)

-I

flux q is a constant,

and T(x) can be

itself. The resulting equation can be sol-

ved rigorously and effectively by using the integral method proposed in [6] and applied later to several radiative problems for non emitting media in plane geometry [7- 4 . In this paper the method will be applied to the particular case of isotropic emission and diffuse reflection by the walls into a participating medium. It is then known ~4] that

Vol. i0, No. 3

RADIATIVE TRANSFER IN PARTICIPATING MEDIA

T4(T) -T24

O(~) + (E21_I)Q

T4-1 T42

l+(gIl+g21-2)Q

193

(4)

2 4 4 n o(TI-T2) Q q =

(5) i+ (ell+E21-Z)Q

where Q = I

fa 2 ~E2(a+z)O(T)dT

(6)

and O is a universal function, solution to the linear integral Fredholm equation

-a

The evaluation of O and Q constitutes a standard benchmark problem in radiative transfer, and an useful comparison test for various approximate techniques commonly used for more realistic problems in the field. The most accurate results seem to be those reported by Heaslet and Warming

~], who resorted to the Chandra-

sekhar's X and Y functions. We want to show here that very accurate results can be obtained straightforwardly and rigorously by solving Eq.(7) via a Bubnov-Galerkin projection procedure

DO] ,

which is equivalent to using Ritz's method in connection with a standard variational principle for evaluating Q

~i],

the parame-

ter of outstanding physical meaning. Analysis It proves convenient to set first

so that a

Q = ~+E3(Za) - ~ ~2(a+T)-E2(a-z)~(r)dz

(9)

and ~(z) = ~ ~E l([~-T' I)~(r')dr'+!E4 2(a+z)-nE2 71 (a-T) -a

'

(10)

194

G. Spiga and M. Spiga

showing that w is an odd function,

Vol. i0, NO. 3

namely W(-r)= W(T).

It is ea-

sy to prove that the known term of the integral equation belongs to the Hilbert space L2(-a,a), rator K generated by the kernel

(i0)

and that the integral ope

~El(Iz-z'l)

is compact in that

space since a

a

%dT%E~(Iz-z'l)dz' ~- a

The s p e c t r a l

~--a

< 8alog2 <~

radius

of K is less f

than unity,

but a l s o

[1~ this

Eq.(lO) solution

nce o f a p p r o x i m a t e

because

a

[IKll ~ max ~ I E I ( I T - ~ ' I ) d ~ Therefore

(11)

~

' = 1 - E 2 ( a ) <1

not only has a unique is the limit

.

solution

(12) ~eL2(-a,a),

in t h e L2-norm o f t h e s e q u e -

solutions !

~N(~)

!E (a+r) i , N 4n-i 2 = 4 2 -TE2(a-z)+~n%~--~a) ~nU2n_l(r)

(13)

where a

Un(r)

= 5 E I ( I T - T ' [ ) P n ( ~ )dT'

(14)

-a

and t h e s c a l a r algebraic

coefficients

E

are

n

solutions

o f t h e NxN l i n e a r

system 1

Z N

~

n=l

1

!

= _ ~ 4m-I z (4m_l)a(4n.l)a ] m~ 4a C2m-l,2n-I ~n (--~a-) D2m_l

with

a

(15)

a

-a

-a

a

Dm = 5 E 2 ( a - ~ ) P m ( ~ ) d z

(17)

-a

In E q s . ( 1 4 )

through

(17) P

denotes

the n-th

Legendre polynomi-

n

als,

whose c o l l e c t i o n

i s an o r t h o g o n a l

basis

v a l u e o f Q i n t h e a p p r o x i m a t i o n of o r d e r N QN = ~ + E 3 ( 2 a ) - 4 Z B g , n=l n n

in L 2 ( - a , a ) .

The

N is then (18)

and t h e s e q u e n c e o f t h e QN c o n v e r g e s t o t h e e x a c t v a l u e o f Q. The o n l y t a s k

to be a c c o m p l i s h e d n u m e r i c a l l y

of the linear

algebraic

s y s t e m (15)

is thus the solution

f o r any a p p r o x i m a t i o n

order

Vol. i0, No. 3

RADIATIVE TRANSFER IN PARTICIPATING MEDIA

N, s i n c e a l s o a l l sion functions

matrix

elements,

can be e v a l u a t e d

known c o e f f i c i e n t s

analytically

195

and expan-

as

m+n+1

C = 4a2|E"k (2a)6 6 + mn I mo no

~ 6mnvv v( 2 a ) | v=O

m (-2,)k(Zk-1)!!(m+k)T D = 2a ~ " in k=O (2k) [ (m-k) ! n Un(r):(a+r) Z '(-l)k(2k-l)H k=O (k+l) !

[E2(2a)~_

]"~

+

~(k+l)!

for m+n even

2aEl(2a)

+

2aVk+l(2a)

(k+2) !

]

k+2

(19)

z k (k+l)( r (l+a) Cn_ k a) [E1(a+r)+k!Vk(a+r)]+

n

r kc(k+ r [E1(a-r)+k!Vk (a-r)] +(a-z) Z (2k-l)!!(l-~) n-k ~) ( a) k=O (k+l) ! where Vk(X)=y(k+l,x)/(k!xk+1 ) roduced by Kschwendt considering

[12~

and 8 mn are the coefficients

The above procedure

the variational

functional

is equivalent a

¢ ( r ) d r + 4~ ¢ 2 ( x ) d r -

Q*(¢)=I+E3(2a)-2 ~ [Ee(a+r)-Ee(a-r)] -a a

a

-a

-a

-a

which i s an u p p e r bound f o r Q, and a t t a i n s e of Eq.(10).

(20)

i t s minimum ( e q u a l to

t h e e x a c t v a l u e o f Q) i f and o n l y i f ¢ c o i n c i d e s

with the exact

When ¢ i s chosen in t h e N-th d i m e n s i o n a l

s p a c e spanned by t h e f i r s t form

to

[Ii]

a

solution

int-

N odd Legendre p o l y n o m i a l s ,

in t h e

N

@N(r) =

.4n-l. ! r Z [--~--J2~nP2n_l(~)

,

(21)

n=l

and Q

i s t h e n minimized w i t h r e s p e c t

e n t s ~n' t h e same a l g e b r a i c

system Eq.(15), w

and t h e o p t i m a l v a l u e o f Q i s j u s t ticular

QN c o n s t i t u t e s

increasing

to t h e e x p a n s i o n c o e f f i c i is obtained

g i v e n by QN' E q . ( 1 8 ) .

a monotonically

decreasing

[13], In p a r -

sequence for

N. Results

Results optical

f o r b o t h Q and O a r e r e p o r t e d

thickness

In t a b l e

2a r a n g i n g

in t a b l e s

1-3 f o r t h e

from 0.OO1 t o 500 mean f r e e p a t h s .

1 the convergence is verified

to be a l w a y s e x t r e m e -

196

Vol. I0, No. 3

G. Spiga and M. Spiga

ly good,

and at least

six stable

achieved with a p p r o x i m a t i o n slabs

significant

orders

(like the ones occurring

to 4-8 for i n t e r m e d i a t e - l a r g e

the Optical

for Q are

in the gap o£ a nuclear

fuel rod)

1

Thickness:

and Converged 2a

figures

front 1 for very thin

layers. TABLE

Q Versus

varying

First

4 Approximations

Value.

QI

Q2

Q3

Q4

Q

O.001

0.999004

0.999004

0.999004

0.999004

0.999004

O.01

0.990275

0.990275

0.990275

0.990275

0.990275

O.I

0.915703

0.915703

0.915703

0.915703

0.915703

0.5

0.704183

0.704170

O.704169

0.704169

0.704169

i,O

0.553450

0.553409

0.553406

O.~53406

O. 553406

2.0

0.390147

0.390068

O.390061

0.390060

O. 390060

5.0

0.207736

0.207674

0.207661

0.207658

O. 207657

iO.O

0.116780

O.i16760

0.116750

0.116747

O.116745

The fast convergence ressions make

ve in terms of computer very p r o f i t a b l y or steady

time.

dered cases. ly in terms

for a~O.l,

It is remarkable of exponential

The first order

bound for Q accurate

and to 4 digits

integral

functions

a ~ o,

to

in all other consi-

that Q1 can be evaluated

analytical-

as

-I

(22)

of Q are given by

Q = ~a + °(I/a2)

for

a ~ ~

,

(23)

~ ~T/2a for a ~ ~. This explains why Q1 is very close

to Q even for very thick slabs, vergence

fuel rod.

Z

The limiting values

exp-

and unexpensi-

- ~ + ~E3 (2a) + E4 (2a) 2!- 1 2 + E3(2a) + 2 E4(2a) + % Es(2a) 4a a a

=

while ~(z)

of analytical

effective

codes dealing with the transient

of a nuclear

Q1 gives already an upper

~+E3(2a)-4

for

quite

So this method could be introduced

in the numerical

at least 6 digits

Q ~ i

technique

state behaviour

approximation

Q1

and the a v a i l a b i l i t y

the present

becomes

In table

in spite of the fact that the con-

slower and slower

for increasing

2 the deduced numerical

values

a.

of Q are compared

to

Vol. i0, NO. 3

RADIATIVE TRANSFER IN PARTICIPATING ME)IA

the ones reported

in [9], available

those obtained by the asymptotic

to 4 figures

197

for 2a~3,

and to

formula Q = 4 / 3 ( 2 a + l . 4 2 0 8 9 2 )

valid

for 2a > > i . TABLE Q Versus

the Optical

Thickness:

2

Comparison

of Different i

2a

Q

Ref.9

o.i

QI 0.91570

0.91570

O.9157

0.2

o. 84918

0.84918

O. 8491

0.3

0.79358

0.79358

O. 7934

0.4

0.74586

0.74585

0.7458

0.5

o. 70418

0.70417

O. 7040

0.6

o. 66732

0.66730

0.6672

0.8

0.60477

0.60474

O. 6046

I.O

o. 55345

0.55341

0.5532

Results.

,

ii,

Asymptotic

1.5

0.45739

0.45732

0.4572

2.0

o. 39015

0.39006

O. 3900

O. 38976

2.5

O. 34006

o. 34027

0.34017

O. 3401

3.0

o. 30174

0.30164

O. 3016

4.0

o. 24606

0.24597

O. 24596

O. 30160

5.0

o. 20774

0.2O766

O. 20766

8.0

0.14158

0.14153

0.14153

io.o

0.11678

0.11674

0.11674

lOO. o

0.013147

0.013147

0.013147

500.0

0.0026591

0.0026591

0.0026591

There

is a difference

git with respect

of one or two units

to the results

mula turns out to be accurate Table for 2a=l.

3 shows

to four figures

that the convergence

than for Q, n e v e r t h e l e s s

as apparent

in the fourth dithe asymptotic already

finally the function O versus

It is obvious

spectacular good,

of [9~, while

of a temperature

for 2 a ~ 3 .

the ratio ~/a

is n e c e s s a r i l y

less

it turns out to be pretty

from Table 3 itself.

we]l known o c c u r r e n c e

for-

It can also be noticed

the

slip at the boundaries.

Acknowledgement This research was Council

(C.N.R.).

supported

by the Italian National

Research

198

G. Spiga and M. Spiga

Vol. i0, No. 3

TABI, ti 3

The Function 0 Versus when

z/a

for

Approximation

Different

Orders

2a=1. @I

@2

@3

~4

-1.0

0.7578

O.7581

O.7581

O.7581

O.7581

-0.8

0.6948

0.6947

[email protected]

O. 6946

O. 6946

-0.6

0.6436

0.6429

0.6428

0.6429

0.6429

-0.4

0.5949

O.5941

0.5942

0.5942

0.5942

-0.2

0.5473

0.5467

0.5468

O. 5468

O. 5468

0.0

0.5000

O.5OOO

O. 5000

O. 5000

O. 5000

0.2

0.4527

0.4533

0.4532

0.4532

0.4532

0.4

O.4051

0.4059

0.4058

O. 4058

O. 4058

0.6

0.3564

O.3571

0.3572

O.3571

O.3571

0.8

0.3052

0.3053

O. 3054

O. 3054

O. 3054

O.2419

O.2419

O.2419

O.2419

~/a

1.0

0.2422

Nomenclature a

optical

Cn En

Gegenbauer

half thickness polynomials

exponential

integral

I

angular

func-

tions n

Greek symbols: y

6mn Kronecker E

radiation

refractive

incomplete

gamma function index

emissivity

of the walls

intensity

cosine of the angle between

fluxes

positive

direction

index

of radiation

and

+

q-

partial

radiative

q

net radiative

T

absolute

temperature

Superscripts: s d

flux

p

reflectivity

o

Stefan-Boltzmann

T

optical

of the walls constant

coordinate

albedo of the medium

specular diffuse

~ axis

Mathematical

symbols:

Subscripts:

T

factorial,

n!=n(n-l)...l

1

left b o u n d a r y

!!

semifactorial,

2

right b o u n d a r y

(2n-l) !!=(2n-l)(2n-3)...l

Vol. I0, NO. 3

RADIATIVE TRANSFER IN PARTICIPATING MEDIA

199

References 1. 2. 3.

S. C h a n d r a s e k h a r , Radiative Transfer, D over, New York ( 1 9 6 0 ) . V.V. S o b o l e v , A T r e a t i s e on R a d i a t i v e T r a n s f e r , Van N o s t r a n d , Princeton (1963). M.N. Ozisik, Radiative Transfer, Wiley, New York (1973).

4.

W.H. Sutton and M.N. Ozisik, J. Heat Transfer IO1, 695(1979).

5.

J. Janata, Lett. Heat Mass Transfer 6, 365(1979).

6.

V.C. Boffi and G.Spiga, J. Math. Phys. 18, 2448(1977).

7.

G.Spiga, F.Santarelli and C.Stramigioli,

Int. J. Heat Mass

Transfer 23, 841(1980). 8.

F.Santarelli, C.Stramigioli,

G.Spiga and M.N.Ozisik,

Int. J.

Heat Mass Transfer 25, 57(1982). 9.

M.A. Heaslet and R.F. Warming, Int. J. Heat Mass Transfer 8,

979(1965). 10.

M.A. K r a s n o s e l s k i i Equations,

11.

et al.,

Approximate Solution

Wolters-Noordhoff,

of Operator

Groeningen (1972).

12.

S.G. M i k h l i n , Variational Methods in Mathematical Pergamon, Oxford (1964). H. Kschwendt, Atomkernenergie 15, 122(1970).

13.

R.J. Cole and G.Spiga, Q. J. Mech. Appl. Math. 42, 233(1979).

Physics,