COMPARISON OF THE RADIATIVE TWO-FLUX AND DIFFUSION APPROXIMATIONS Charles M. Spuckler NASA Glenn Research Center 21000 Brookpark Rd. Cleveland Ohio 44135 Abstract

Approximate solutions are sometimes used to determine the heat transfer and temperatures in a semitransparent material in which conduction and thermal radiation are acting. A comparison of the Milne-Eddington two-flux approximation and the diffusion approximation for combined conduction and radiation heat transfer in a ceramic material was preformed to determine the accuracy of the diffusion solution. A plane gray semitransparent layer without a substrate and a non-gray semitransparent plane layer on an opaque substrate were considered. For the plane gray layer the material is semitransparent for all wavelengths and the scattering and absorption coefficients do not vary with wavelength. For the non-gray plane layer the material is semitransparent with constant absorption and scattering coefficients up to a specified wavelength. At higher wavelengths the non-gray plane layer is assumed to be opaque. The layers are heated on one side and cooled on the other by diffuse radiation and convection. The scattering and absorption coefficients were varied. The error in the diffusion approximation compared to the Milne-Eddington two flux approximation was obtained as a function of scattering coefficient and absorption coefficient. The percent difference in interface temperatures and heat flux through the layer obtained using the Milne-Eddington two-flux and diffusion approximations are presented as a function of scattering coefficient and absorption coefficient. The largest errors occur for high scattering and low absorption except for the back surface temperature of the plane gray layer where the error is also larger at low scattering and low absorption. It is shown that the accuracy of the diffusion approximation can be improved for some scattering and absorption conditions if a reflectance obtained from a Kubelka-Munk type two flux theory is used instead of a reflection obtained from the Fresnel equation. The Kubelka-Munk reflectance accounts for surface reflection and radiation scattered back by internal scattering sites while the Fresnel reflection only accounts for surface reflections.

https://ntrs.nasa.gov/search.jsp?R=20060051747 2020-07-11T21:56:09+00:00Z

1G

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Com

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adia

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Tw

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App

roxi

mat

ions

Cha

rles

M. S

puck

ler

NA

SA G

lenn

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2G

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Obj

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Det

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ine

the

erro

r in

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ng th

e di

ffus

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com

pare

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the

Miln

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ddin

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two-

flux

sol

utio

n

Det

erm

ine

if u

sing

a r

efle

ctan

ce f

rom

the

Kub

elka

-M

unk

2-fl

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eory

cou

ld r

educ

e th

e di

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solu

tion

erro

r

3G

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Res

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ente

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ewis

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App

roac

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A p

aram

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of a

one

dim

ensi

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sem

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pla

ne la

yer

and

a no

n-gr

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yer

on a

sub

stra

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as

perf

orm

ed

Fres

nela

nd K

ubel

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refl

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e us

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= 0

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cm-1

and σ s

= 94

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cm-1

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for

the

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App

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Dif

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t●

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Fre

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6G

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Res

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ente

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a σ

s

T s1

= 20

00K

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250

w/m

2 K

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= 20

00K

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2.1

q r2o

T s2

= 80

0K

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110

w/m

2 K

T g2

= 80

0K

1 m

m

ρo

ρik =

0.8

w/m

K

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odel

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Percent

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two

⋅ ⎟⎟ ⎠⎞⎜⎜ ⎝⎛

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−

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Scat

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Scat

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Percent difference in temperature

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-1

11G

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23

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567

Scat

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-1

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-250255075100

125

150

175

200

225

Percent difference in heat fluxso

lid li

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brok

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12G

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a σ

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w/m

2 K

T g1

= 20

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n =

2.1

q r2o

T s2

= 80

0K

h 2=

110

w/m

2 K

T g2

= 80

0K

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m

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0.8

w/m

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Scat

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Percent difference in temperature

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-1

solid

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itybr

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Kub

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Res

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Scat

terin

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effic

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-1

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Percent difference in temperature

a =

0.00

13 c

m-1

a =0

.013

5 cm

-1

a =

0.13

46 c

m-1

a =

1.34

6 cm

-1

a =

13.4

6 cm

-1

a =

67.2

8 cm

-1

solid

line

Fre

snel

refle

ctiv

itybr

oken

line

Kub

elka

-Mun

k re

flect

ivity

Per

cent

Dif

fere

nce

in T

empe

ratu

res

betw

een

Tw

o-fl

ux

and

Dif

fusi

on S

olut

ions

for

Fro

nt o

f Sub

stra

te

λ c=

5.0μ

m a

nd ε

bc=

0.7

Kub

elka

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unk

Fre

snel

15G

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Res

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ewis

Fie

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Per

cent

Dif

fere

nce

in H

eat F

lux

betw

een

the

Dif

fusi

onan

d T

wo-

flux

Sol

utio

ns fo

r a

Lay

er w

ith

a Su

bstr

ate

λ c=

5.0μ

m a

nd ε

bc=

0.7

100

101

102

103

23

45

67

23

45

67

23

45

67

Scat

terin

g co

effic

ient

cm

-1

a =

0.00

13 c

m-1

a =

0.01

35 c

m-1

a =

0.13

46 c

m-1

a =

1.34

6 cm

-1

a =

13.4

6 cm

-1

a =

67.2

8 cm

-1

-20

-1001020304050

Percent difference in heat flux

solid

line

Fre

snel

refle

ctiv

itybr

oken

line

Kub

elka

Mun

k re

flect

ivity

16G

lenn

Res

earc

h C

ente

r at L

ewis

Fie

ld

Eff

ect o

f Bon

d C

oat E

mis

sivi

ty o

n th

e P

erce

nt D

iffe

renc

e in

T

empe

ratu

re b

etw

een

Tw

o-F

lux

and

Dif

fusi

on S

olut

ions

F

ront

Sur

face

of L

ayer

F

resn

elR

efle

ctiv

ity

λ c

= 5.

0μm

100

101

102

103

23

45

67

23

45

67

23

45

67

Scat

terin

g co

effic

ient

cm

-1

-505101520 Percent difference in temperature

solid

line

ε

bc =

0.7

brok

en li

ne ε

bc =

0.3

a =

0.00

13 c

m-1

a =

0.01

35 c

m-1

a =

0.13

46 c

m-1

a =

1.34

6 cm

-1

a =

13.4

6 cm

-1

a =

67.2

8 cm

-1

17G

lenn

Res

earc

h C

ente

r at L

ewis

Fie

ld

Eff

ect o

f Bon

d C

oatE

mis

sivi

ty o

n th

e P

erce

nt D

iffe

renc

e in

T

empe

ratu

re b

etw

een

Tw

o-F

lux

and

Dif

fusi

on S

olut

ions

F

ront

Sur

face

of L

ayer

K-M

Ref

lect

ivit

y

λ

c=

5.0μ

m

100

101

102

103

23

45

67

23

45

67

23

45

67

Scat

terin

g co

effic

ient

cm

-1

-5-4-3-2-10

Percent difference in temperature

solid

line

ε

bc =

0.7

brok

en li

ne ε

bc =

0.3

a =

0.00

13 c

m-1

a =

0.01

35 c

m-1

a =

0.13

46 c

m-1

a =

1.34

6 cm

-1

a =

13.4

6 cm

-1

a =

67.2

8 cm

-1

Kub

elka

–M

unk

18G

lenn

Res

earc

h C

ente

r at L

ewis

Fie

ld

Max

imum

Cha

nge

in P

erce

nt D

iffer

ence

fo

r a C

hang

e in

Bon

d C

oat E

mis

sivi

ty

from

0.7

to 0

.3

-4.0

4-1

.90

-1.1

10.

86K

-Mre

flect

ion

-11.

33-2

.94

-2.9

8-1

.96

Fres

nel

refle

ctio

n

Hea

t flu

xB

ack

ofsu

bstra

teFr

ont o

fsu

bstra

teFr

ont o

fla

yer layer

substrate

layer

substrate

layer

substrate

19G

lenn

Res

earc

h C

ente

r at L

ewis

Fie

ld

Con

clus

ions

The

per

cent

err

or u

sing

the

diff

usio

n so

lutio

n is

low

est

for

all s

catte

ring

con

side

red

whe

n th

e ab

sorp

tion

is h

igh

The

larg

est e

rror

s oc

cur

mos

tlyfo

r hi

gh s

catte

ring

and

lo

w a

bsor

ptio

n w

hen

the

Fres

nels

urfa

ce r

efle

ctio

n is

use

d

The

err

or a

t hig

h sc

atte

ring

can

be

redu

ced

by u

sing

a K

ulbe

lka-

Mun

kre

flec

tanc

e

For

a la

yer

with

a s

ubst

rate

the

Kub

elka

-Mun

kre

flec

tanc

e is

abl

e to

han

dle

a ch

ange

in b

ond

coat

emis

sivi

ty b

ette

r

20G

lenn

Res

earc

h C

ente

r at L

ewis

Fie

ld

Gen

eral

ized

Con

clus

ions

For

the

heat

ing

cond

ition

s pr

esen

ted

here

the

abso

rptio

n th

ickn

ess

(a∙D

) no

t the

opt

ical

thic

knes

s [(

a + σ s

)∙D

] is

the

impo

rtan

t fac

tor

in d

eter

min

ing

the

use

of th

e di

ffus

ion

appr

oxim

atio

n. N

eed

a la

rge

abso

rptio

n th

ickn

ess.

The

wor

st c

ondi

tions

to u

se th

e di

ffus

ion

appr

oxim

atio

nar

e fo

r hi

gh s

catte

ring

thic

knes

ses

(sσ∙

D)

with

no

abso

rptio

n or

low

abs

orpt

ion

thic

knes

s an

d on

ly s

urfa

cere

flec

tions

acc

ount

ed f

or.

Kub

elka

-Mun

kre

flec

tivity

can

red

uce

the

diff

usio

n so

lutio

nap

prox

imat

ion

erro

r fo

r la

rge

scat

teri

ng th

ickn

esse

s