Subregion of validity of the lumped-based model for transient, radiative cooling of spherical bodies to a zero temperature sink

Vol. 23, No. 6, pp. 855-864, 1996 Copyright © 1996 Elsevier Science Ltd Prinl~d in the USA. All fights reserved 0735-1933/96 $12.00 + .00

Int. Comm. HeatMass Transfer,

Pergamon

PII S 0 7 3 5 - 1 9 3 3 ( 9 6 ) 0 0 0 6 8 - 1

SUBREGION

OF V A L I D I T Y

FOR TRANSIENT, BODIES

OF T H E L U M P E D - B A S E D

RADIATIVE

COOLING

OF S P H E R I C A L

T O A ZERO T E M P E R A T U R E

Antonio Nuclear Idaho

MODEL

SINK

campo

Engineering

State university

Pocatello,

ID 83209

Rafael Villase~or Instituto

de I n v e s t i g a c i o n e s

62000

Cuernavaca,

El~ctrioas

Morelos

MEXICO

( C o m m u n i c a t e d b y C . L . T i e n a n d P.F. P e t e r s o n )

ABSTRACT

Transient radiative cooling of small spherical bodies having large thermal conductivity has not been critically examined in the open literature from a demarcation point of view as compared with the transient convective cooling. For a sink temperature at zero absolute, a comparative study involving the distributed- and the lumped-based models have been presented where the controlling parameter is the r a d i a t i o n - c o n d u c t i o n parameter, N~. The family of curves for the relative errors associated with the surface-tocenter temperatures followed a normal distribution in semilog coordinates. Viable quantitative thresholds can be extracted from the figures and for a maximum allowable 5% relative error, the magnitude of N~ must be 0.15.

Introduction

From a physical a dimensionless convection

group

standpoint; that

the Biot number,

characterizes

the

Bi,

heat

liberation

from the surface of a body to a surrounding 855

constitutes by

fluid via a

856

A. Campo and R. Villasefior

ratio

of t h e r m a l

resistances,

by h e a t convection. conductive meaning Bi

the b o d y has to be n e g l i g i b l e

becomes

small

that have

very

The

in

variation

or

b)

show

of

i.e.

that

transients

the simple

lumped-based

is

than

0.i,

Bi

the

the c r i t i c a l for c o n v e c t i v e

the

ratios,

whenever

maximum

Tw/Tc, w i t h i n

cooling

and

needs

to

be

Bier = 0.i, serves two d i s t i n c t

lumped-based

it is w o r t h w h i l e

is m a t h e m a t i c a l ,

as long as the

between

the

In this

the B i o t n u m b e r

model

Biot number,

heat

models,

mentioning

whereas

that

the

the e m p h a s i s

of

is physical.

Contrary bodies

lumped-based criterion

distributed-

At this point, of the former

latter

to

cooling

5% a p p r o x i m a t e l y .

distributed-based

a borderline

orientation

[1,2],

model can be a d o p t e d

Consequently,

the

one-dimensional

0.i

In contrast,

to d e f i n e

namely:

materials

to c o n v e c t i o n

temperature

utilized.

respectively.

<

is less t h a n

is less t h a n 0.i.

models,

from

or c) are e x p o s e d

subjected

whenever

Biot number greater

by c e r t a i n b o d i e s t h a t

constructed

transient,

solid b o d i e s

of the s u r f a c e - t o - c e n t e r

the b o d y d u r i n g

are

conductivities,

solutions

for simple

consistently

to w h a t h a p p e n s where

is

(Bi < 0.i),

clearly

the

treated

that

the d i s t r i b u t e d -

in o c c a s i o n s

latter.

using

for

educational

arena,

convective

regular it

the bodies

is

by

radiative

a sui g e n e r i s

or the

the

cooling

References to

number

of b o d i e s This means have

been

to the l i m i t a t i o n s

distributed-based

convenient

models

mention

model [3-7]. that

of

and the

Biot

approach.

lumped-based

in

cooling

the d i s t r i b u t e d -

specified

without paying attention

Specifically,

implemented

between

transient,

been

either

in transient,

the t h r e s h o l d

models

has u s u a l l y

the

[1,2],

environments.

analytical

conduction

adopted

small

can be s a t i s f i e d

size,

large t h e r m a l

convective

solid

very

(i)

this c r i t e r i o n

the

the

~ O(i)

very

regard,

within

number

are

weak

and the other

in order to impose this criterion,

resistance

the Biot

As expected, a)

one by h e a t c o n d u c t i o n

Clearly,

thermal

that

Vol. 23, No. 6

has From the

of

been the heat

Vol. 23, No. 6

transfer DeWitt

textbooks

[i0]

applied these

RADIATIVE COOLING OF SPHERICAL BODIES

by

Mills

[8],

Thomas

[9]

are the only ones that explain

to transient, authors

do

radiative

not

cooling

enlighten

the

and

857

Incropera

the lumped-based

of bodies. appropriate

and

model

Surprisingly, subregion

of

validity.

Thermal occurs

in

radiation

is the most common heat transfer

specialized

high

technological

thermal sprays for rapid solidification liquid

droplet

thermophysical levitated

radiators

in outer space

of

spherical

note

to

elucidate

distributed-based transient, absolute

model

radiative

[5,12], solid

as

[Ii],

determination

or

liquid

of

droplets

[13,14].

In view of the foregoing technical

such

of advanced materials

of spacecrafts

properties

mode that

applications,

statements, and

vs.

this

lumped-based

cooling

zero temperature

it is the intent of this

clarify

of

confusing

model

spherical

issue

arising

bodies

exposed

in to

of the an

sink.

Analysis of Radiative Coolinq of Spherical Bodies Consideration

is given to the transient,

a solid or liquid spherical a uniform

temperature,

is idealized radiation 0°K.

The

sphere

T = T i initially.

to dissipate

radiation

shape

cooling of

The surface of the sphere

as gray with a total emissivity

begins

radiative

body of radius R which is m a i n t a i n e d at

6. For t > 0, heat by

to a sink whose temperature

factor,

and the sink is assumed

Fw.,, between

the

surface

is T, = of the

to be one.

Distributed-based model Under the premises medium

is described

by the dimensionless

T *

that the thermophysical

are temperature-invariant,

= ~,

r n

= 7"

st ~ -

R2

the transient

properties radiative

of the cooling

variables (2)

858

A. Campo and R. Villasefior

in c o n j u n c t i o n

with

Vol. 23, No. 6

the r a d i a t i o n - c o n d u c t i o n

parameter:

EaRT~ N~c

The

-

reciprocal

called

(3)

k

of the

conduction-radiation

the Stark number,

[2]).

Conceptually,

which

governs

Sk,

N~ is the c o u n t e r p a r t

an e q u i v a l e n t

The distributed-based heat

in the R u s s i a n

conduction

equation

transient

parameter

is n o r m a l l y

literature

(see L u i k o v

of the B i o t

convective

model for the transient, is g i v e n

number,

one-dimensional

by:

a@ = ~@ + 2 a@

aT

~2

The i n i t i a l

Bi,

cooling.

(4)

~ ~q

and boundary c o n d i t i o n s imposed on eq.

(3)

are:

(s) ~ :i,

I;=0

---~ : 0, @q

~

:

0

(6)

in w h i c h which

~, = T,/T i d e s i g n a t e s

in this

case

Fundamentally, formulation evolution

is o b t a i n e d accuracy,

that

for t h o s e

dimensionless

the d i s t r i b u t e d - b a s e d

provides

nonlinearity

knowledge ~(~,T),

of eq.

of

model

temperature,

the

radial

is a m a t h e m a t i c a l

temporal

inside any b o d y

(6) the s o l u t i o n

finite-difference

a fine m e s h c o n s i s t i n g in the

sink

as zero.

by the e x p l i c i t

constructed

sionless

is t a k e n

the

of the temperature,

the i n h e r e n t

was

(7)

"q=l

dn

of eqs.

method,

of i00 u n i f o r m l y direction.

and

spatial

[1,2].

(3) - (6)

for m a x i m u m

distributed

In addition,

the

t i m e step e m p l o y e d was 0.001 in m o s t cases r e s e r v i n g cases

characterized

by large

values

Due to

of N~ [6].

nodes dimen0.005

Vol. 23, No. 6

RADIATIVE COOLING OF SPHERICAL BODIES

For purposes

of establishing

a realistic

859

comparison

between

the distributed- and the lumped-based models; the natural thermal quantity

common to both models

is the volume-average

temperature

(quantity of global character):

(8)

i/v

(I], x) dV

the

volume

4@(z) = --~

where

V

is

of

the

sphere.

On

the

other

hand,

the

temporal deviation between the center and the surface temperatures (quantities of local character)

produced by the distributed-based

model may serve as an indicator also.

Lumped-based model The

lumped-based

which essentially

model

is a physically-oriented

relies on the idealization

formulation

that transient

heat

conduction takes place in a spatially, quasi-isothermal body [1,2]. In other words, the conductive mechanism inside the body is totally overshadowed by the radiative mechanism operating between the body surface and the surrounding medium. This simplified model relies on the volume-average variable

temperature,

in a transient,

#

(see eq.

(7)) as the dependent

overall energy balance.

Hence,

assuming

that the thermophysical properties of the material are unaffected by temperature,

the

pertinent

ordinary

differential

equation

is

given by

dz I together with the initial condition : i,

~'=

0

(I0)

Rote t h a t a m o d i f i e d d i m e n s i o n l e s s t i m e T' has been i n t r o d u c e d i n eq.

(8) and t h e r e b y a m o d i f i e d r a d i a t i o n - c o n d u c t i o n parameter, N'~,

has emerged a l s o :

860

A. Campo and R. Villasefior

¢I _

a

t

Vol. 23, No. 6

- 9 ~--~t = 9 r

(V/A) ~

a s

(il)

~,

_

~ o ( v / A ) ~ ]

rc

The

3

dimensionless in t e r m s

the

between

ratio =

R/3,

radius,

of

3

for

a modified

the v o l u m e

instead

of

the

=

(1

(9)

surface

characteristic

area

are

formed

of t h e

sphere,

length,

the

model.

@, = 0, t h e a n a l y t i c a l

solution

3~'~ ~ )-~/~

+

by

of

is:

(12)

The numerically-determined the distributed-

Mean

model

length

and t h e

Discussion

presented

lumped-based

characteristic

R in the d i s t r i b u t e d - b a s e d

(8) and

the

natural

For zero s i n k t e m p e r a t u r e , eqs.

~,c

_

k

quantities

redefined

V/A

I ~oRT~

_

k

of

Results

temperature results calculated from

and l u m p e d - b a s e d m o d e l s for the s p h e r i c a l b o d y are

in two parts:

temperature

The plotted

relative in Fig.

parameterized

errors

for

by

the

that the overall

extreme

for

of the

errors

large

the

intensified approximately

of

are w i t h i n

relative until

N~

reaching

[ = i0. A t t h i s

Nr: r a n g i n g

it is o b s e r v e d

of the

dimensionless

time.

and

small

=

a

increase a

of N~,

shows an i n v e r t e d bell s h a p e d - c u r v e

a 1% margin.

errors

are

T. T h e c u r v e s are

parameters

of the m a g n i t u d e

logarithm

value

temperatures

time,

radiation-conduction

error pattern

each

relative r,

volume-average

1 a g a i n s t the d i m e n s i o n l e s s

f r o m 0.i to 0.5. R e g a r d l e s s

as a f u n c t i o n

the

peak

At

time the

other

gradually

and

large time,

~

thereon

with

T as

one the

extreme

decrease

the r e l a t i v e

At

0.01,

N~

for is

until

errors

are

Vol. 23, No. 6

RADIATIVE COOLING OF SPHERICAL BODIES

c o n f i n e d a g a i n to a 1% band. same limiting

For

small

satisfied to

the

e r r o r as • i n c r e a s e s

values

of

N~

corresponding

value

stabilizes relative

All the c u r v e s s e e m to c o l l a p s e to the

the

error

maximum

of N~ tested,

around

3%.

criterion

error

band

of

1%

is

i.e. N~ < 0.2. As N¢ i n c r e a s e s relative

N~ = 0.5,

Moreover,

b e y o n d a v a l u e of I0.

relative

as long as N~ s t a y s small,

0.33,

largest

relative

861

the

it m a y

error

is 2%.

maximum

be

is p r e - s e l e c t e d ,

For

relative

inferred

that

the

error

if

a

5%

N~ m a y r e a c h a v a l u e

of

0.7 or e v e n higher.

0 i_ kID

-1

._> "4--'

ID

-2

Nrc

01

" ......................

(D -3

-4

---

Nrc = 0.15

.....

Nrc = 0.2

.....

Nrc = 0.:33

.....

Nrc = 0.5

-5

"',. ,,

...... ,..............

. . . . . . . .

'

0,010

1.000

0.100

10.000

7FIG. 1 Temporal variation o f the relative errors for the volume-average temperatures for various Nrc

Local temperatures Fig. for t h e

in

indication occur

the temporal variation

surface-to-center

parametric tures

2 displays

values Fig. of

between

i. the

temperature

of N~ e m p l o y e d From

a

magnitudes

the center

for

physical of

of t h e r e l a t i v e

ratios, the

~w/~c,

volume-average

standpoint,

the

temperature

and the s u r f a c e

for

Fig.

2

errors

the

temperagives

gradients

of the s p h e r i c a l

same

an

that

body

as

862

A. Campo and R. Villasefior

Vol. 23, No. 6

a f u n c t i o n of the d i m e n s i o n l e s s t i m e T and t h e p a r a m e t e r N=. It can be

observed

bell

shape

side

of

fixed from

the

r = 5%.

family

only.

0.01, As

a

3%

curves

As the

factor

retain

but

N~ m o v e s

for

a

from

5%

error,

the

the to

range

around

maximum

characteristic

0.i

times

positive 0.5

for

a

respectively

all

the

relative

r = i0. T h e r e f o r e ,

relative

in the v i c i n i t y of

relative

the

occupy

errors

large

This occurs of

now up

relative

expected,

of N~ has to be

stringent

of

coordinates,

zero again.

within

magnitude a

of to

approach

stay

the

semi-log

quadrant

value 1.25%

errors to

that in

0.15.

error,

the

Similarly,

permissible

value

of

for

N~

is

r e d u c e d to 0.i. 15 . .-----.. .,"

"...

,/ I... ~-

~D

--

Nrc

= 0.1

---

Nrc

= 0.15

........

Nrc

= 0.2

.....

Nrc

= 0.33

.....

Nrc

= 0.5

10

,o ,,"

°_

.-

r~

5

,-"

0 0.010

/'

,,.

i

,.

/¸

,.....'"

//

.....'"

,

,

i

, ,.

.......

_ -.

.......

, , ,,,I

.,. %-,..

~

i

i

t

\.~,

. . . . .

0.100

i

i

i

.

.

.

.

.

1.000

.

10.000

7" FIG. 2 Temporal variation of the relative errors for the surface-to-center temperature ratios for various Nrc

Conclusions

In

synthesis,

lumped-based

model

it m a y in

be

concluded

transient

that

radiative

the

validity

cooling

of

of

b o d i e s can r e s t on a c r i t i c a l r a d i a t i o n - c o n d u c t i o n p a r a m e t e r , as the d e c i d i n g c r i t e r i o n . engineer

is

facing,

the

the

spherical (N~)cr

D e p e n d i n g on t h e n e e d s t h a t the t h e r m a l criterion

must

be

based

either

on

the

Vol. 23, No. 6

RADIATIVE COOLING OF SPHERICAL BODIES

volume-average center

temperature

former,

N~ can

around

0.15.

variation all

temperature

Of

ratio up

(ratio

to

0.7,

course,

other

of two

whereas

these

are a s s o c i a t e d

fairness,

imposed

go

(global quantity)

numbers

errors

Bi

Biot

Cv

specific

hc

mean

k

thermal

N~

radiation-conduction

N'~

area

of the s p h e r e

number,

hcR/k

heat

convective

conductivity parameter

in t e r m s

of R, eq.

modified

radiation-conduction

in terms

of

r

coordinate

R

radius

t

time

(V/A),

(3)

eq.

parameter

(ii)

of a s p h e r e

T

absolute

¥

volume-average

V

volume

Greek

Letters

G

coefficient

temperature temperature,

eq.

(8)

of the s p h e r e

thermal

diffusivity

surface

emissivity

Stefan-Boltzmann

constant

dimensionless

T, eq.

(2)

dimensionless

r, eq.

(2)

T

dimensionless

t in terms

~'

modified

dimensionless

terms

(V/A),

p

density

of

eq.

of R, eq.

t in

(ii)

(2)

For the

N~ has

a wide

5% r e l a t i v e

different

Nomenclature

surface

later

yielding

as well.

A

quantities).

the

with a pre-selected

relative

or on the s u r f a c e - t o -

local for

863

than

to

be

margin

of

error. 5%

may

In be

864

A. Campo and R. Villasefior

Vol. 23, No. 6

Subscripts c

center

i

initial

s

sink

w

surface References

i.

V. Arpaci, MA,

2.

Addison-Wesley,

Conduction Heat Transfer,

Boston,

(1966).

A.V. Luikov, Heat Conduction Theory, Vyzszaja Szkola, Moscow, Russia,

(1967).

3.

D.L. Ayers,

4.

A. Campo, Intern. Comm. Heat Mass Transfer 4, 291-298,

ASME J. Heat Transfer 92,

5.

R. Siegel,

6.

R.

ASME J. Heat Transfer 109,

Villasefior,

A

linear

integral

equation

boundary

conditions,

multistep

applied

180-182, 159-164,

method

to heat

(1970). (1987).

for

diffusion

the

with

Proc.

Intern.

Symposium

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Cancan,

M~xico,

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nonlinear

Heat

Vol.

and Mass

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(1993). 7.

C.W. Groetsch,

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A.F. Mills, Heat Transfer, Richard Irwin, Boston, MA,

9.

F.P.

NJ,

D. Apelian,

(1992).

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D.P.

DeWitt,

Fundamentals

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Froes,

F.H.

and

Savage,

S.J.),

ASM,

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et al., In Aerospace Heat Exchanger Technology,

(Eds.

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Khodadadi,

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Received January 8, 1996