A numerical study of a structure protected by water film from an incident heat flux

INT. COMM. HEAT MASS TRANSFER Vol. 17, pp. 331-342, 1990 ©PergamonPtuss

0735-1933/90 $3.00 + .00 Printed in the United States

A NUMERICAL STUDY OF A STRUCTURE PROTECTED BY WATER FILM FROM AN INCIDENT HEAT FLUX

Wu-Shung Fu

Jenn-Der Lin Kuan-Chyvan Tu C h i n g - C h y i T s e n g D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g N a t i o n a l Chiao Tung U n i v e r s i t y H s i n c h u , 30049, Taiwan R . 0 . C

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT A f i n i t e d i f f e r e n c e method i s adopted to investigate the heat transfer mechanisms of t h e r m a l protection problem. This problem concerns with a structure protected by w a t e r f i l m s u f f e r i n g a s e v e r e h e a t flux by t h e r m a l radiation f r om s u r r o u n d i n g . The h e a t t r a n s f e r m echani sm s consist mainly of the sensible and e v a p o r a t i v e heat transfer of t h e w a t e r f i l m and s e n s i b l e h e a t t r a n s f e r o f the structure. I n t h e p r o c e s s of a n a l y s i s , t h e amount of h e a t a b s o r b e d by t h e w a t e r f i l m is calculated by B e e r ' s l aw , and evaporation occurring on t h e w a t e r film surface is c o n s i d e r e d . I n o r d e r t o compare t h e c o n t r i b u t i o n s o f t h e heat transfer m e c h a n i s m s , t h e v a r i a t i o n s of t e m p e r a t u r e distribution of t h e s t r u c t u r e and t h e water film are examined in detail and t h e results show t h a t the t e m p e r a t u r e of s t r u c t u r e d e c r e a s e s e f f e c t i v e l y due t o t h e p r o t e c t i o n of t h e w a t e r f i l m .

Introduction F o r an i n s t r u m e n t which has t o be e x p o s e d i n environment, maintain the layer

a

thermal

protective

a protected

material

body ( s t r u c t u r e )

the thermal protective

device.

condition.

(ablator)

to

Coating

a

on t h e s u r f a c e

of

i s one of t h e w e l l - k n o w n m e t h o d s f o r In t h e p a s t ,

. Institute

temperature

device is always utilized

i n s t r u m e n t at normal o p e r a t i n g

of thermal protective

high

of N u c l e a r E n e r g y R e s e a r c h 331

Landau[l],

Sunderland

332

W.-S. Fu, J.-D. Lin, K.-C. Tu and C.-C. Tseng

and G r o s h [ 2 ] ,

Zien[3]

analyze

phenomena of t h e h e a t e d a b l a t i v e

the

and

Chung[4]

c o n s i d e r e d as a s e m i - i n f i n i t e a structure.

Clark[5]

investigated

its

used

Vol. 17, No. 3

phenomena

Blackwell[7]

transfer

of

Teflon

using

by

film

of

a

material

the

structure. subject

layer

used as p r o t e c t e d m a t e r i a l

Chow[Ill, Schrippel[12]

infinitely

and

material and

and H a j i [ 1 3 ]

is

Yang[lO] mechanisms

on t h e f a c e of a also studied this

film, but the liquid

was

assumed

In o r d e r t o f a c i l i t a t e

analysis,

the assumption

of t h e h e a t f l u x from s u r r o u n d i n g t o be

absorbed

completely

the

thin.

heat

numerically.

as t h e r m a l p r o t e c t i v e

of a p l a t e w i t h l i q u i d

and

mechanisms of a b l a t o r

n u m e r i c a l method t o a n a l y z e t h e h e a t t r a n s f e r liquid

of

method. R e c e n t l y ,

a n o t h e r w e l l - k n o w n method. C h a n g [ 8 ] , Shembharker[9] utilized

which was

investigated

numerical

of s t r u c t u r e

water

to

mechanisms by b o t h e x p e r i m e n t a l

studied the heat transfer

the temperature variation Besides,

material

a d o p t e d T e f l o n as t h e a b l a t i v e

heat

method

s o l i d and c o a t e d on t h e s u r f a c e

and n u m e r i c a l m e t h o d s , and H o l z k e n e c h t [ 6 ] transfer

numerical

surface

of

liquid

film

was

made

However, s i n c e w a t e r i s s e m i - t r a n s p a r e n t , perfect

for

radiation The

aim

difference

the

of

this

this

assumption is

of h e a t f l u x t r a n s f e r r e d

study

is

to

method t o i n v e s t i g a t e

by

water

semi-transparent,

employ

to

the structure film

the

on

heat

an

not

by t h e r m a l

implicit

the heat transfer

problem. A s t r u c t u r e

transferred

protected

t h e above s t u d i e s .

from s u r r o u n d i n g .

thermal protection which

condition

in

on

the

finite

mechanisms of

absorbs severe heat by t h e r m a l r a d i a t i o n

surface.

absorbed

by

Since the

water

flux and i s

water film

is is

Vol. 17, No. 3

WATER FILM FROM AN INCIDENT HEAT FLUX

calculated

by

distribution order

to

of

heat

transfer

law.

the

examine

sensible

being

Beer's

structure

the

of

of t h e s t r u c t u r e .

considered. difference

Both t h e r e s u l t s

of

temperature

and w a t e r f i l m a r e c a l c u l a t e d

water

on

of

and

water

and

the sensible

heat

with the increase

the

heat

film surface

different

in

evaporation

t h e model o f

the

are

the

film

Besides,

absorbed

decreases

variations

distributions

transfer

completely

The

333

initially,

flux

is also but

the

of time.

Governing Equations As

shown

in Fig.1

is a structure

is the physical

in a quiescent

air

with

with length H is water film outside an

incident

film

by t h e r m a l r a d i a t i o n

water

physical the

film

is

sensible

heat

transfer

heat transfers

of

mechanisms

X=L

structure

X',,L+H

and

environment. Since heat

absorbed

by

law. A c c o r d i n g t o t h e consist

mainly

of

body i and t h e e v a p o r a t i o n

and

~XI

FIG,I Phys;ca[ node[

2

body 1 f r o m

of body 2.

) XlO

the

t h e amount of

model, the heat transfer

sensible

to

by B e e r ' s

Body

at x = 0 is adiabatic

from o u t s i d e

calculated

T .

body 1 t o p r o t e c t

transferred

water film is semi-transparent, the

temperature

h i g h h e a t f l u x q. The s u r f a c e

and t h e h i g h h e a t f l u x q i s water

m o d e l . Body 1 w i t h l e n g t h L

AX2

;-I I+1 Fig,2 Numerical g r l d s

334

W.-S. Fu, J.-D. Lin, K.-C. Tu and C.-C. Tseng

Vol. 17, No. 3

In order to facilitate the analysis, the following

assumptions

are made: (I) The

properties

of

the

structure

and

the water film are

constant. The water film is stationary and the interface

of

the water film and the surrounding air is saturated. (2) The

structure is black, then the incident heat flux through

the water film is absorbed by the structure completely. (3) The atmosphere pressure of environment maintains

at

i

arm

and the maximum temperature of water is IO0°C. Based

upon

these

assumptions,

the governing equations and the

boundary conditions are as follows: In t h e

structure

0 T1 PlCpl (9 t

-

0 2TI K1 0 X2

(1)

and water film T2 P2Cp2 b t the

initial

t= O,

0 2T2 -

K2

0 X2

conditions

0 qr +

(2)

0 X

is

(3)

T I = T 2 = constant

The boundary conditions are 0 T1 X=O,

=0 OX

a T2

TI X = L

KI

= K2 OX

= T2

aX

r

+ qx=L

(4)

Vol. 17, No. 3

WATER FILM FROM AN INCIDENT HEAT FLUX

335

0 T2 X = L+H

K2

+ qL = 0

Ox

i n which qL = ~ . L a represents by

mass e v a p o r a t i o n r a t e

of w a t e r which

is

obtained

s o l v i n g t h e f o l l o w i n g c o n c e n t r a t i o n e q u a t i o n and La i s l a t e n t

h e a t of w a t e r . 0C

02C = D

the solution C - CN C-

(5)

at 0 of Eq. (5) i s o b t a i n e d from R e f . [16]

2(Trr_)

(6)

= err(X(L+H)

CN

- D ~Ix=L÷H

= (% - C).~

D/(~-t)

(7)

N~m~rical Method An i m p l i c i t governing grids

finite

equations.

difference Based

method i s a d o p t e d

upon

the

and

every

cCntrol

lumped-capacity. A brief

outline

volume of t h e

solve

numerical tests,

10 shown i n F i g . 2 a r e u s e d i n t h e s t r u c t u r e

domains

to

is

uniform

and w a t e r assumed

solution

the

to

procedures

film be is

d e s c r i b e d as f o l l o w : ( 1 ) S e t t h e h e a t f l u x q, t h e i n c i d e n t e n e r g y a b s o r b e d by t h e i t h c o n t r o l volume i s o b t a i n e d from E q . ( 8 )

q[ = q.{e-ke[(L+H)-xi] - e-ke[(L+H)-xi-i]}

(8)

In which x i , xi_ 1 are positions shown in Fig.2. ke

=

~kA'IbA(T)'dA IbA(T)

residual

heat

flux

and k A i s computed from [ 1 3 ] a n d [ 1 4 ] . which

a b s o r b e d by t h e s t r u c t u r e

passes

through

the

at x = L completely.

water

film

The is

336

W.-S. Fu, J.-D. Lin, K.-C. Tu and C.-C. Tseng

(2)Decide interface Combine

t h e t i m e s t e p h t and g u e s s t h e t e m p e r a t u r e TN of t h e

of water Eqs.(1)

conditions Thomas

Vol. 17, No. 3

fihu

and

Eq.(4)

method

arid air

to

(2)

of every

control

to

form

tri-diagonal

to solve

a

it. The

solve

~

shown

in

Eq.(7).

with

the

boundary

volume

criterion

matrix

for

and

convergent

utilize

condition

Tn+I_T n

~N

is

n

~N

< 10 -4 , in the meanwhile,

the other

convergence

TN

condition

[

at

every

q-(qSl q+ qs2 + qL)

(3)The t h i c k n e s s

time

step.

volumes

(4)According c o n t r o l volume

be

satisfied,

of w a t e r f i l m w i l l d e c r e a s e due t o e v a p o r a t i o n

in

of t h e w a t e r f i l m and

air,

the water film are re-divided

The t e m p e r a t u r e of t h e r e - d i v i d e d

interpolation

must

] < 10 -3 , In which qs = EpCphT.dx.

occurring at the interface control

step

method w i t h t h e o r i g i n a l

grids are

then

the

at every time calculated

by

ones.

t o t h e a s s u m p t i o n ( 3 ) , when t h e t e m p e r a t u r e of t h e reaches

100°C,

the

heat

transferred

to

this

c o n t r o l volume w i l l be added t o t h e i n n e r n e i g h b o r c o n t r o l volume (in direction t o w a r d t h e

(5)Continue

the

structure).

calculation

until

the

t e m p e r a t u r e of whole

w a t e r f i l m i s r a i s e d t o IO0°C. R e s u l t s and D i s c u s s i o n The t e m p e r a t u r e of s u r r o u n d i n g a i r i s 60 and t h e p r o p e r t i e s

i s 30°C, q i s 50000 W/m2, ke

of s t r u c t u r e

and w a t e r f i l m a r e

structure(copper)

water

D e n s i t y p (kg/m 3)

8954

999.2

Heat C a p a c i t y Cp ( J / k g ° C )

383.1

4195

Vol. 17, No. 3

WATER FILM FROM AN INCIDENT HEAT FLUX

K (W/m2 °C)

Conductivity

386

F i g u r e 3 shows t h e v a r i a t i o n s

of

0.585

temperature

distribution

the structure

and t h e w a t e r f i l m w i t h t i m e e v o l u t i o n .

temperatures

of

the

structure

to

The s o l i d

Beer's

l a w ( c a s e A), w h e r e a s t h e d a s h l i n e s

heat

flux

represent

and 1

film.

the

near

However,

the

Because

the

the

water

difference incident

between

film

decreases

so

the

temperature

so d r a s t i c a l l y

thicker,

is

that

time,

and

temperature

incident

Since

the

A

the

and

B

initial

increasing

time.

of

time.

of

near the surface

temperature

is

by

water doesn't

the control

reaches

lO0°C,

c a u s e s t h e r e g i o n of w a t e r f i l m which can Therefore,

water

the

difference

of the structure

the

water

then the temperature shown i n F i g .

tendency of variation

of v a r i a t i o n s

decreases

of temperature

film with increasing

of time.

i s 80°C and t h e

water

The film

f i l m absorbs h e a t not only from t h e

h e a t f l u x b u t a l s o f r om t h e

the situation

cases

of t i m e .

structure

30°C.

of

t o t h e whole r e g i o n

F i g u r e 4 shows t h e o t h e r s i t u a t i o n

initial

mean t h e s i t u a t i o n

in

with

distribution

i n which

a b s o r b h e a t t o be t h i n n e r .

the

water the

as t h e c a s e B. As t i m e i n c r e a s e s ,

volume o f w a t e r f i l m ,

in

is

h e a t a b s o r b e d by w a t e r f i l m c a l c u l a t e d

film,

with evolution

cm

of considering

the

surface

law ( c a s e A) i s d i s t r i b u t e d

becomes

1.1

the situation

Beer's

rise

The i n i t i a l

a b s o r b e d c o m p l e t e l y by w a t e r f i l m a t t h e s u r f a c e

x=L+H ( c a s e B ) . The d i f f e r e n c e apparent

in

and w a t e r f i l m a r e 30°C. I n t h e

r a n g e o f 0 t o 1 cm i s t h e s t r u c t u r e lines

337

structure

of water film rises 3. However, as

of t e m p e r a t u r e

time

distribution

in

the

initial

more r a p i d l y

than

increases,

the

will

be s i m i l a r

338

W.-S. Fu, J.-D. Lin, K.-C. Tu and C.-C. Tseng

to that

shown i n F i g .

Figure

5

structure. without that

3.

shows t h e e f f e c t

of w a t e r f i l m on t h e t e m p e r a t u r e

As shown i n t h e f i g u r e , being

protected

by

the temperature

water film

o f c a s e s h and B. T h e r e f o r e ,

structure

f r o m an i n c i d e n t

Since

the

surface,

Fig.

evaporation

4,

larger

the

transfer.

structure,

and e f f e c t i v e .

decrease

thickness

of

The

factors

the

are

(1) s e n s i b l e

heat transfer

and

(3)

(3),

rise

In the beginning,

contributed

to

heat transfer

of

evaporation

i s a b s o r b e d by w a t e r f i l m which c a u s e s t h e to

film.

factors

heat flux

transfer

water

of

of

film

rapidly,

most of t h e

therefore

maximum t e m p e r a t u r e

the

of w a t e r f i l m i s l i m i t e d

and

the

becomes d o m i n a n t . stationary,

structure

Because

so t h e r a t i o

the

increases. water

of e v a p o r a t i o n

film

heat

assumption

t h e h e a t a b s o r b e d by t h e w a t e r f i l m

by

incident

temperature

As t i m e i n c r e a s e s , absorbed

heat

the sensible

of w a t e r f i l m i s dominant. According to

the

the

of c a s e A

(2) s e n s i b l e

water

and

water film of case B is

transfer

of

a

o c c u r s on t h e w a t e r f i l m

of w a t e r f i l m w i l l

F i g u r e 7 shows t h e v a r i a t i o n s heat

protect

6. A c c o r d i n g t o t h e r e a s o n m e n t i o n e d i n

evaporated

than that

structure

using water film to

heat transfer

a r e shown i n F i g .

of

of

( c a s e C) i s h i g h e r t h a n

heat flux is available

then the thickness

results

Vol. 17, No. 3

t o lOO°C. decreases

Then t h e f a c t o r is

assumed

heat transfer

(1)

to

be

is small.

Conclusions When

using

a

thermal protection,

numerical

method t o i n v e s t i g a t e

the following conclusions

(1)Using water film to protect

a structure

the problem of

are drawn. from a

severe

heat

Vol. 17, No. 3

WATER FILM FROM AN INCIDENT HEAT FLUX

100

100

~o""

339

i~l'l'

'

i. !

0 0

65

'

65

SO

30

0

1.0

1.1

,

,

,

0

,

I

i

,

,

1.0

,

1.1

x(cm) FIG.3 The v~riations of t e m p e r a t u r e distribution with time evolution 110

,

i

,

i

|

,

,

i

FIG.4 The variations of temperature distribution with time evolution

|

0

70

0

27.5

55

~sec)

tOO

v tran~r 6O

M

®:latent

of

-I

h~t

-I

trander of water film

.

.

.

.

,

.

.

.

.

27.5

55

t(seo)

l~G.5 The e f f e c t of w a t e r film on t h e t e m p e r a t u r e of s t r u c t u r e

E

97 i 0

r

0

FIG.7 The variations of the f a c t o r s c o n t r i b u t e d to h e a t t r a n s f e r

4 4

FIG.6 The variations of w a t e r film t h i c k n e s s with time evolution

340

W.-S. Fu, J.-D. Lin, K.-C. Tu and C.-C. Tseng

flux is available (2)The

of

of c o n s i d e r i n g

considering (3)Water transfer

and e f f e c t i v e .

difference

situation

it

Vol. 17, No. 3

temperature

Beer's

is apparent

film

is

law

variation

and

the

in t h e i n i t i a l

stationary,

between

situation

of

the

sensible

mechanisms of w a t e r f i l m and t h e s t r u c t u r e

case B

t h e model of n o t c o n s i d e r i n g

case C

a structure

C

concentration

(kg/m 3)

Cp

heat capacity

( J / k g k)

ke

P l a n c k mean a b s o r p t i o n

D

mass d i f f u s i o n

g

thickness

IbA

spectral

K

thermal conductivity

L

thickness

La

latent

intensity

evaporative

(m2/s)

(W/m k) (m)

rate

heat flux

(J/kg)

of w a t e r ( k g / m 2 s )

(W/m2)

heat transfer

of w a t e r

(W/m2)

h e a t a b s o r b e d by w a t e r f i l m in c a s e A (W/m2) sensible time

(s)

Subscripts 1

by w a t e r f i l m

(m)

h e a t of e v a p o r a t i o n

qL qr

law

of b l a c k b o d y

of s t r u c t u r e

incident

Beer's

coefficient

coefficient

of s t r u c t u r e

q

law

without being protected

mass e v a p o r a t i o n

qs t

Beer's

structure

heat transfer

(W/m2)

heat

are dominant.

Nomenclature t h e model of c o n s i d e r i n g

not

time range.

therefore

case A

the

Vol. 17, No. 3

WATER FILM FROM AN INCIDENT HEAT FLUX

2

water film

N

interface

oo

surrounding air

341

of w a t e r f i l m and a i r

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