An elementary study of gas injection and sublimation into a simple shear layer

CoA Report No. 126

, D E L R

THE COLLEGE OF AERONAUTICS

CRANFIELD

4

_{»1 r t I}

AN ELEMENTARY STUDY OF GAS INJECTION AND

SUBLIMATION INTO A SIMPLE SHEAR LAYER

ty

EEF^ORT NO. 126 February , 1960. T H E C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

An Elementary Study of Gas Injection and Sublimation into a Simple Shear Layer

b y

-J . F . Clarke, B. Sc. , P h . D . , A. F . Ft. Ae. S. ADDENDA AND CORRIGENDA Fage 3.

Page 6.

In eq. 11 - X -v— should read - X

-r-dy -r-dy

Eq. 28 should read -q = -=- §L H^Le-DTA p-_{dy dy} Page 11, The phrase following eq.45 should read "where h = C T . "

o ^ It is important to note that q is the rate at which energy is

transported relative to a surface moving with the mass average or flow velocity. Thus -q is not the rate at which energy is transferred into the wall at y = 0. The latter quantity, written as -q say, is given by

:-/ '^'^b\

-q - rn h = -q = 1 X, -;— ) - m h,

'w w \ b dy /w i w

The second form of -q follows from the energy balance at the interface and is the energy flux within the body. Since -q = (xËZ) + m h - m h .

/ d l \ "^^"^ (for any Lewis No. value) it follows that( X ~ j is always equal to / d T A ^ ^/w

( '^b ——] in the case of gas injection, i . e . the conduction in gas and solid balance at the interface.

2

With t h e s e definitions eq. 45 is modified so as to r e a d :

-" -q bl 11 ^ {h - h ) - h A ' b ' - ( h + h ) b 7 2 (45a) s r o wo wo r o w

w h e r e h = C T . " wo p w

The second s e n t e n c e in the p a r a g r a p h follov/ing eq. 45 should be modified to r e a d : "Since b ' can be w r i t t e n a s m b fji a p p r o x i m a t e l y t h e l a s t t e r m in e q . 45a gives a reduction in -q equal to (h + h )m / 2

^ " s r o w F u r t h e r on in t h i s s a m e p a r a g r a p h the w o r d s " lead to an i n c r e a s e in heat t r a n s f e r r a t e if " should be modified to r e a d

lead t o an i n c r e a s e in -q if ^w

I I

The l a s t s e n t e n c e in the s a m e p a r a g r a p h should be deleted and r e p l a c e d by : "Even though -q m a y be i n c r e a s e d by injection of the ' w r o n g ' g a s , it can r e a d i l y be shown that injection always d e c r e a s e s -q . The amount of t h i s d e c r e a s e b e c o m e s s m a l l e r a s C , d e c r e a s e s , h o w e v e r " .

It i s worth pointing out that Stanton n u m b e r i s in fact m o r e c o n -veniently defined in t e r m s of -q than in t e r m s of -q . Also that

r e c o v e r y enthalpy r e f e r s t o the z e r o of -q and not to the z e r o of -q .

iV ' " " s

P a g e 14. The p a r a g r a p h following e q . 5 6 should be d e l e t e d , a s should eq. 57, and the following s u b s t i t u t e d .

"The e n e r g y t r a n s f e r r a t e s at the i n t e r f a c e must now be m a t c h e d , taking account of the fact that the solid m a t e r i a l a b s o r b s an amount of latent heat L p e r unit m a s s d u r i n g s u b l i m a t i o n , and a l s o accounting for convection of e n e r g y in both the solid and the g a s . T h u s :

. , . dT -q - m h =• X, :T— w w b dy - m C, T + m L , _ b w y = 0 -o r , using e q s . 45a and 56,

-q = m (L - C, T-) = m L ' , (57) ^s b b

In the line following eq. 57, change -q to r e a d " - q and

w s

" e q . ( 4 5 ) " to r e a d " e q . ( 4 5 a ) " . Then eq. 58 should r e a d :

-" -q ='(h - h ) 7 T / 6 - m h A' + (h + h )/2 } (58)" ^s r o wo ( w o r o wo ' F a g e 15. E q . 59 should now r e a d : -" m =• (h - h ) ( 7 r / 6 ) ( L ' + h A' + (h + h )/2 )-"-^ (59)-" r o wo wo r o wo In p a r a g r a p h following e q . 59, " -q " should be a l t e r e d t o r e a d " - q ". "^ ^s E q . 60 should r e a d / , w M „ v - 1 " - q = (h - h ) ( y / 6 ) L ( L + A ' h + ( h + h )/2) (60)" ^s r o wo wo r o wo

The s e c t i o n s t a r t i n g 3 l i n e s above eq. 61 and ending 3 l i n e s below eq. 62 should be deleted and the following substituted :

-"However, -q i s not of p r i m a r y i m p o r t a n c e h e r e . F a t h e r a r e we c o n c e r n e d with the heat flux into the w a l l , which e q . 56 and 57 show t o be given by : -., dT ' ' b ^ y=o-It follows that ., dT = m C , (T - T, ) » -q - m L + m C, T (61) b w b s b w (h - h )(^/6)C. (T - T, ) - ^ ^ b _ w b_ ^g2) y = 0 - L' 4- A' h + ( h + h )/2 "^ wo r o wo C l e a r l y X, d T / d y _^_ r e d u c e s s u b s t a n t i a l l y in magnitude a s L (and hence li ) r i s e s . As L •• co , X, d T / d y ^ •• 0; note that

D I y u

-although m •* 0 a s L •* oo the product m L r e m a i n s f i n i t e " . P a g e 17. In line 4, r e p l a c e pm / R T by pm / k T .

In the p a r a g r a p h beginning j u s t before e q . 6 9 the condition should be " L » h A' + (h + h )/2 - C . T , . "

REPORT NO^A26 February,. "1.960

T_H E C O L L E G E p.P._ A^ E R _0 N J L ^ J L ^ . H3 C R A N F I E L D

An Elementary Study of Gas I n j e c t i o n and Suhlimation i n t o a Simple Shear Layer

t y

-J . P. Clarke, B . S o . , P h . D . , A.P.R.Ae.S,

SmftiARY

The i n j e c t i o n of a forcsign gas i n t o a siniple shear layer (Oouette flov?) i s examined and tlie rc;sults extended to the case for t/hich the w a l l m a t e r i a l suhlimes under the a c t i o n of heat t r a n s f e r r e d through the l a y e r . I t i s hoped t h a t t h i s simrilif ied a n a l y s i s v/ill serve t o emphasise some of tlie processes and parametoji's involved i n more rigoroxis treatments of t h i s type of a b l a t i o n .

COlflTEMTS

iÈm

Summary L i s t of Symbols 1 . I n t r o d u c t i o n 1 2 . Gas I n j e c t i o n a t t h e Lov/cr W a l l 1

3 . Recovery E n t h a l p y and Heat T r a n s f e r R a t e 8

4 . S u b l i m a t i o n 12f 5 . R e f e r e n c e s 18

LIST OP SïlffiOLS b , b " 0 %

°f

0 P

c .

p i D h k L Le A P P r

4

So T

u

V y 6 A,A' P /i X T Defined i n e q s . 30 and 33 F o r e i g n gas c o n c e n t r a t i o n (mass f r a c t i o n ) S p e c i f i c h e a t of v / a l l m a t e r i a l F r i c t i o n c o e f f i c i e n t S p e c i f i c h e a t of e x t e r n a l g a s S p e c i f i c h e a t of i n j e c t e d gas D i f f u s i o n c o e f f i c i e n t E n t h a l p y p e r u n i t mass Boltzmann' s C o n s t a n t L a t e n t h e a t of s u b l i m a t i o n Levrf-s Number Mass i n j e c t i o n r a t e p e r xsnlt s u r f a c e a r e a P r e s store P r a n d t l ntmiber Energy f l u x Schmidt number Temperature V e l o c i t y p a r a l l e l t o w a l l s V e l o c i t y p e r p e n d i c u l a r t o w a l l s C o - o r d i n a t e p e r p e n d i c u l a r t o w a l l s D i s t a n c e b e t \ ï e e n ï / a l l s D e f i n e d i n eq^s. 12 and 33 D e n s i t y V i s c o s i t y C o n d u c t i v i t y S h e a r s t r e s s

L i s t of Symbols Continued Suffixes b e i o r w 8

LCT,7er wall material value External gas

Injected gas

Zero-injection rate value Recovery value

Evaluated at y = 0 Evaluated at y = 6

'CHOOL

1

-"^» .introduction

There is a great deal of ciirrent intci-est in "tlie ^processes of ablation into h^ipersonic boimdoay layers and several rigorous theoretical treatments have appeared in the published literature.

Notable amongst these v.-e may mention the v/orïc of Lees (1958) and Scala. The latter has given a siJETiiarj'' of a great VOIUIIB of hisv/ork in this field (Scala, I960).

The processes involved are quite complex an.d the number of variables Tjhich can affect then is large. The follovTijij^ represents an

ultra-simplified treatriient of gas injection and gaseous ablation into a sin-^ple shear layer (Couette f la:) in an attempt to er^Tphasise some of the concepts and parar.ieters involved.

First the injection of a perfect gas into anotlior perfect gas is discussed and later on these results are used to establish ablation, and the corresponding heat transfer rates. Homogeneous chemical reactions are not considered. The variations of sublimation •te:jperature with pressure are examined briefly, via the kinetic theory. M attenrot is made to confine the treatment to the barest essentials, and no attempt made to give a copious list of references. As far as possible the work concerns the general situation.

2 • Gas Iiyiectign at jtlie Jjq\:er Wall

The assumption is made that both gases, namely that of the external f la? and the injected gas, are ideal. Then the res-peotive specific enthalpies, h and h. oon be written as

* e 1

h = C T : h. = C . T , (I) e p 1 pi * ^ '

v/here the s p e c i f i c h e a t s a t constant p r e s s u r e , C and C . , are both

c o n s t a n t s . T i s tlie absolvite temperature» Since vro exclude the p o s s i b i l i t y of chemical r e a c t i o n s betireen the t\70 species tlicre i s no need t o r e f e r energies t o a coiiTaon zero l e v e l . I f the i n j e c t e d gas i s p r e s e n t a t

c o n c e n t r a t i o n (mass f r a c t i o n ) c , the s p e c i f i c entlialpy of the mixture, h , i s given by

h = (1 - c)h^ + c h. =^C + (0 . - C )o2 T , (2) ^ ' e i I p ^ p i V J

Por plane parallel Couette f lev: with no x-v/ise pressure gradient, the usual conservation eauations become

2

-4f^ = o , (3)

„ du d / du\ / , \ P v ^ = ^ ( / ^ a ^ ) , ' (4) oy d y d y \ 5 c l y / ' ^^ dy - ^ dj' - - dy + 3 VdyJ ' ^i^dyj ' ^^^

P , ^ and p Êire the mixtvire d e n s i t y , v i s c o s i t y a.nd tliermodynamic pressure r e s p e c t i v e l y , and u and v are the gas v e l o c i t i e s p a r a l l e l and perpendicular to the p l a t e s . 4 i s the j"-ooiiTOOnent of the energj^ flvix vector vjhich, i n the presence of i n t e r d i f f u s i o n of the e x t e r n a l and i n j e c t e d g a s e s , must be v/ritten

jam

4 = "* '^ dy + P° v^ l^i + PC"! - O)VQ \ . (7)

V. and V are the appropriate diffusion v e l o c i t i e s ajid \ i s the c o e f f i c i e n t of thenTial conductivity. The diffxosion v e l o c i t i e s i n the siniple binary mixture under consideration are sin^ily r e l a t e d t o the concentration g r a d i e n t s j

c V. = - D f = - (1 -

c)v^

,

(8)

vAiere D is the binary diffusion coefficient for the particular mixture, In addition to the overall mass conservation equation (eq. 3)» continuity equations for each separate species ca^i be written down, These are

h f ^ ^ v . P D ^ i = 0 , (9)

öy 1^ - - - dy

^ ^ ( 1 - o ) v + p D ^ i = 0 , • (10) dy j^^^' - -/" - A^- ^

for the injected and external gases i^espectivelj/-, and use has been made of eqs. (8) to eliminate the diffusion velocities.

The energy flux is better expressed in teri.is of concentration gradients, namely'-, (usajig ecis.(7)f (8) and (l)),

3

-= - X # , - P D A T ^ , , (11) öy dji _r

vAiere vre have v . r i t t e n

C . - C = A . (12) p i p ^ '

The species contin'uity equations (9 and IO) can be integrated at once to give

PC V - p D -^ = A , (13)

p(l - c)v + P D ^ = 0 . (14)

Here we have made use of the boimdary condition a t tlie lavtev w a l l to evaluate tlie i n t e g r a t i o n c o n s t a n t s . The l e f t hand side of eq.13 i s equal t o pc(v + V.) and i s therefore equal to the mass f l a ? r a t e of i n j e c t e d gas per i m i t a r e a , y/hich i s 'v^ritten as m a t the lov/er \7all. Similarly the l e f t hand side of eq. l i f i s p(l - c ) ( v + v ) , and since no e x t e r n a l gas i s allaved to e n t e r the lower wall the appiropriate value for t h i s quantity a t y = 0 i s z e r o . I t immediately f o l l a / s tliat

PV = A , (15) ( a r e s u l t v^hich could also be deduced d i r e c t l y from eq. 3.) f and the

equation s a t i s f i e d by c n a / becomes

P D - | = - A ( 1 - c ) . (16)

The momentum equation parallel to the wall can also be integrated directly and, using the condition ^{^.•ü/^y) = r when y = 0 (r = shear stress at tlie wall), we have

^ ^ = ±u + T , (17)

dy w ^ ' ' Using eq. 17, a l l d/dy d e r i v a t i v e s can na7 be v/ritton i n terms of d/du d e r i v a t i v e s , ( t h i s i s tlie Couette f l a r version of Grocco's laminar boundary l a y e r transformation). I n p a r t i c u l a r tiiis transformation applied t o eq^.l6 y i e l d s

4

-for the r e l a t i o n betvveen o and u . The q-uantity Sc i s the Schmidt number,

So = j u / p D , (19)

and in what follaTs we shall assume it to be a constant throughout the gas layer. It is not likely to vary too much for most gas mixtures of interest and, in any case, v/e can assvmie it to have some suitable m-ean value for the purposes of the present, hexoristio, analj^sis. This being so, eq. 18 integrates to give

So

1 - c = (1 - cj(l + Au/^^P , (20)

TsAiere c i s tlie foreign ge-s concentration a t the l a 7 e r xrsll. Note t h a t i f m = 0 , c = c, everyv7herej i f A > 0 ( i n j e c t i o n ) then c < c . everywhere and vice versa i f m < 0 ( s u c t i o n ) . Some siinplification v.dll r e s u l t l a t e r i f v/e take eg , the foreign gas concentration a t the upper v/all, t o approach zero. This i s c o n s i s t e n t with the A > 0 requirement for i n j e c t i o n cind a l s o corresponds most c l o s e l y 'VTitli tlie p r a c t i c a l , boundary l § y e r , problem. The assumption Cg -• 0 precludos consideration of the suction case however. I t can be seen from eq,8 t h a t as cg ^ 0 the product ( c V. )g must remain f i n i t e and non-zero ( i n otiier words the diffxjsion v e l o c i t y v . a t y = 5 i n c r e a s e s \->dthout l i m i t ) . The rfiean v e r t i c a l v e l o c i t y of the i n j e c t e d gas i s v + v . and i t s mass flea-: r a t e p e r u n i t area i s

therefore pc(v + v . ) , as remarked p r e v i o u s l y . The l i m i t i n g case cg -«0 then implies t h a t the i n j e c t e d gas i s c a r r i e d av/ay through the upper w a l l purely by diffusion (since pv = m = c o n s t , and pcv ^ 0 as c -» O),

Eq 20 shov7s t h a t o. i s d i r e c t l y r e l a t e d t o A once cg i s f i x e d , because,

1 - C ^ = ( 1 - C g ) ( l ^±\Jy^r^° . (21)

Purtherroorc, even though vre let cc -• 0, c. / 0 as long as A > 0, The

injected gas therefore enters into the flcra region by a mixture of diffusion and convection.

The transport of foreign gas across the shear layer by a mixture of convection and diffusion is precisely v/hat happens in the practical case of a boundary layer flow, and it is for this reason that we study it here, As is already apparent the inclusion of diffusion introduces the Schmidt number into the analysis, a number \7hich must be inportant also in boundary layer flov/s since it determines the relative tlxiclaiess of the momentum and concentration layers (i.e. Sc = ^ / D ) .

7

-for b r e v i t y , eq. 29, can be vnritten dh

du,

(Le - 1) A(1 - c )bSG(ub + 1)^°-'' Gp+ Ah -(1 - o^^(ub + i )

~ - P r (u + 1 A ) " ^ (u + l A ) V 2 - l/2b^ + ^ /m + h^ (31)

Eq. 31 can be i n t e g r a t e d t o y i e l d , u

>-1

f(u) h - f(o) h^ = - P r (q^/m + h^^ + g(u)) f(u) (u + l A ) " " d u (32) ''o

where

f(u) = b ^ ^ (ub + 1 ) - ^ ^ j C^ + A [l -(1 _ cg)(ub + l ) ^ 7 ( U b + 1)^° J 5-1

g(u) = ( l / 2 b ^ ) [ ( u b + 1)^ - 1 I

(32a) (32b) (Eq. 21 has been used to eliminate 1-c i n teriis of cg, U and b i n eq. 32a,

T /

« 8

-3» Reoqyory Enthalpy_ and .Heat Transfer _R_ate

Some general results can nov7 be derived from eq. 32. First we note that v/hen u = U, h = hR and if, in addition, a = 0 then h = h , the recovery enthalpy. It follo\7s therefore that w

\ =[_ f(u)h5 + Prj g(u) f(u) (u + lA)"'' duy

j f(o) - Pr f f(u)(u + lA)""" <3u

o

and i n terras of t h i s q u a n t i t y ,

-q^.^= (1/Pr) [ ( f ( o ) / j f ( u ) ( u + lA)""^ a u ) - 1 j ( h ^ - h p b r ^

These tvro q u a n t i t i e s can be re\yritten i n a more convenient form by defining dimensionless functions f and g v;hicli are r e l a t e d t o f and g i n e q s . 32a and 32b. Thus v/e vnrite

f ' ( u ' ) = ( u ' b ' + l ) - ^ ^ [ 1 + A ' [ l -(1 - C6)(u'b' + l ) ^ ° / ( b ' + i ) S ° J L e - 1 (33a)

(33-b) = f ( u ) / b ^ ^ C p ^ ^ - ' '

g'(u') = (u'b' +1)' - 1 = 2b'g(u)/V

•where u' = u A , h' = bU, A' = A/c . Whenos i t follov7s t h a t

h^^ = [ f'(l)lfe +(Prlf/2) ] ( g ' f ' ) b ' - \ l +u'b')-^ d u ' ] /

r f'(o) - P r b ' f f ' . d + u'b' )-''du'J (34)

•'o

9

-These expressions for h and q, are veiy •unvd.eldy and little can be gained from an examina.tion of them as they stand. Consequently v/e must resort to approximation in order to gain some insight into the physical picture and it seems natural to attenipt solutions for b << 1 . Referring

to the definition of b' , this assumiption is equivalent to setting

AU/T << 1 . Physically this group of variables has a simple interpretation w

since mU is proportional to the fl\ix of x-vvdso momentum induced in a direction av:ay from the \7all by the act of blov/ing into the shear layer, vAiilst T indica.tes the magnitude of the SDIÏB flux taking place ta7ards y = 0 as a result of the microscopic, molecular motions. Setting

b << 1 is equivalent to assuming that the amount of blovdng is such as to cause only small decreases of skin friction,

I t i s worth noting t h a t b/ i s almost always m u l t i p l i e d by u' i n the i n t e g r a l s v/here i s c r e a t e s d i f f i c u l t i e s , so tliat a reasonable approximation should be found f o r quite l a r g e values of b' j a t l e a s t the physics of the s i t u a t i o n should be preserved i n such circxriistances.

After a c e r t a i n amount of algebra i t can be shovm t h a t h <^ h - b '

r r o (Le - 1)SC A' hg + ( P r l f / 6 ) ( 2 ( L e - l ) S c A'+ 1 > Pr) . . . (36) - ^.^ UPrA,., =^ (h^^ - h.) [1 - ( b ' / 2 ) ( ( L e - l)So A'+ 1 - Pr)J (37)

- b ' (Le - l ) S c A ' h , + (Pr UV6)(1 - P r - ( L e - l ) S c A')

vAiere h i s the recoveiy enthalpy vdth zero mass i n j e c t i o n r a t e ,

\ o = h + P ^ U V 2 . (38)

When both Pr and Le (and hence Sc also) are equal to unity the results 37 and 38 siniplify to

h = h , (39)

r ro ' ^•'•^y

- (5 HPr = (h - h ) T . (40) ^ ^ ro w' w ^ '

In this case then, injection of a foreign gas through the la7or wall has no inflxience on the recovery enthalpy (since Cg = O) and the _form of

üie "Reynolds analogy" expression in eq, 4-0 is exactly like the no-injection one. Ho\7ever, it is clear from eq. 2 that at a given wall temperature, h will become greater or less than the no-injection value of C 2 depending on whether C . > C or <C (i.e. whether A' > 0 or <0),

10

-Eq. 21 sho\7S t h a t an approximation t o o which i s equivalent to those made i n deriving e q s . 3é and 37 i s

c^^ - So. b ' (42)

Then it follows that

or, in the case of Le = Pr = Sc = 1, the fractional increment in the T(ra.ll enthalpy as a result of foreign gas injection is ^' b' ,

A further modification to the heat transfer rate arises via the reductions in r, brougtit about by gas injection. Assmdng that A^ in eq, 17 can be repla.ced by a properly weighted constant mean value, U

say, it folla7s that

m U ^ rr U .^ U / , , N *

exp(mS/ iï) -1

%, = ~~Ï7^:rr ' ^^ - * I ' ('^^

to a reasonable degree of approximation„ Aside from any variations v:hioh may arise in the value of Jl due to the pi^esenoe of the foreign

gas, the term Jlu/b is the zero injection skin friction, Eq. Iji). shows that injection reduces skin friction by an amount equal to the mean rate of upv7ards transport of x-wise momentum which results from blovdng, a plausible-looking first estimate.

* Eq. 2+4 sho\7s that significant skin friction reductions occur virhen A is coniparable vdth AT/S , Simple kinetic theory gives /ï ~ p 0^3, n = mean molecular speed ( ~ a, speed of sound) and •£ = mean free path, whence condition is m/pU ~ (•&/6)M'' v/here M = U/a = Mach number at outer edge of shear layer. This illustrates ha? very small rf/pU values are effective in reducing T_, since •&/6 << -J at reasonable altitudes (pressures). It also suggests tlia.t blovdng is more effective at higher Mach mombers. The present theory is only valid if A 6 / jü <<1 , although this condition need not be too strictly observed, see the remarks preceeding eq, 36,

/

V

1 1

-S t i l l c o n s i d e r i n g tlie sirrtple c a s e v:here Le and P r a r e u n i t y , we c a n nav w r i t e - a 0 / M =^ ( h - h ) - h A' t/ - ( h - h j ^ ( 4 5 ) ^ ' ^ r o V70^ viTO ^ r o w' 2 ' where h _ - C T . _{wo p V7} The f i r s t t e r m on t h e r i g h t - h a n d s i d e c o r r e s p o n d s t o z e r o - i n j e c t i o n h e a t t r a n s f e r , t h e r e m a i n i n g terras e x p r e s s t h e e f f e c t of i n j e c t i o n . S i n c e b' caai be v.Tdtten a s A 5 / ? a p p r o x i m a t e l y , tlie l a s t terai i n e q . 45 g i v e s a r e d u c t i o n i n - 5 e q u a l t o ( h - li ):V'2. T h i s t e m i i s w h o l l y a n a l o g o u s t o t h e term A u A ii^ e q , 244, and r e p r e s e n t s t h e mean upvrards t r a n s p o r t of e n t h a l p y a s a r e s u l t of i n j e o t i o n . The second term i n e q , 45 r e p r e s e n t s a p o t e n t i a l l y s i g n i f i c a n t s o u r c e of h e a t t r a n s f e r r a t e r e d u c t i o n b y i n j e o t i o n . The r e d u c t i o n i n - 5 from t h i s source i s (C . - C )T ril, a q u a n t i t y v:hich e x p r e s s e s t h e l o s s of e n t h a l p y d i f f e r e n c e a c r o s s the l a y e r v/hich r e s u l t s from t h e i n j e c t i o n of a gas v d t h a l a r g e h e a t c a p a c i t y , C . , I t i s i n t e r e s t i n g t o o b s e r v e t h a t t h i s q u a n t i t y may be n e g a t i v e and i t f o l l a v s from e q . 45 t h a t t h e e f f e c t of i n j e c t i o n as summarised i n t h e r i g h t - h a n d s i d e of t h i s e q u a t i o n may l e a d t o an a n c r e ^ s e i n h e a t t r a n s f e r r a t e i f 2 ( l - C ,/C ) > ( T / T - 1 ) , v/here C f " = h " . ( P o r example, _{^ p r p ' ^ rcf w ' ' p r o r o ^ -f »} i f C ./C = 1 / 2 t h e h e a t t r a n s f e r r a t e i s i n c r e a s e d djf T / T < 2 ) . _{p i ' p ' rcr vv '} That i s t o s a y , i t seems q u i t e p o s s i b l e t h a t t h e i n c r e a s e i n h e a t f l u x d r i v i n g f o r c e a r i s i n g from i n j e c t i o n of t h e 'v/rong' gas c a n more t h a n c o u n t e r b a l a n c e t h e e f f e c t s of c o n v e c t i o n of energy avray from t h e s u r f a c e by b l o v d n g .

Por this reason it seems advisable to choose a light gas for heat transfer reduction. Yfithin limits the molar heats of gases do not vary a great deal and a large C is perhaps more easily obtained by selecting

Pi

a gas of small molecular vreight, rather than one of coniplex structure vdth many modes of communicable energy storage.

So far we have overlooked the possibility that /Ï majr be favourably or unfavourably affected by injection. So long as the amounts of

injection are small, so that c remains small throughout the layer it is possible that \i is not much affected, althou.gli it is difficult to

generalise here. Suffice it to say that liglit gases have la7er viscosities than the heavier ones in the main,

A further point óf some iniportance v;hen attenpting an appraisal of bo\mdary layer behaviour via the predictions of the present

ultra-simplified analysis is that heare the thickness 6 remains constant. In a boundary layer 6 vdll incixjase vdth increasing m and vdll result in reductions of - q_ and r over and above those considered to date.

12

-We n o t e from e q s , 39 and 40 t l i a t d e f i n i n g S t a n t o n number and f r i c t i o n c o e f f i c i e n t i n the u s u a l v:ay, namely

- a 2 T

S t = . ^ 3 : - . - . _ . G_ = ^ ^ ^ X . , (46) PoU(h - h ) ^ p . U ^

o ^ r o w' 0

results in the relation

St = ^ C^ (47)

One iTJxy e x p e c t a simple Reynolds analogy e x p r e s s i o n of t l i i s t y p e t o h p l d a l s o i n boimdarj''' I f y e r f l a v s f o r which Le = P r = 1 , Since

b = AUA i't f o l l a 7 s from t h e s e r e s u l t s t l i a t

\'f

•b' = ^ * _ = ( 2 ^ )

P g U . S t

and t h i s groiip of v a r i a b l e s may be e x p e c t e d t o c o r r e l a t e boundar^j'- l a y e r f l a v s t o o . I t s h o u l d be n o t e d ttiat S t i s tlie S t a n t o n number wj-tji

i n j e c t i o n , ( s e e , f o r e x a m p l e . Lees ( 1 9 5 8 ) ) .

Eq, 35 shovrs t h a t e q , 47 c a n be g e n e r a l i s e d t o i n c l u d e t h e e f f e c t s of Le and P r d i f f e r e n t from u n i t y . V/ith

St = —-"1^^:--=^^.^^ (2,9)

^b ^ ( ^ - V

and C„ a s i n e q , 4 6 , t h e approxim^ate foarm of e q . 35 y i e l d s

St ^ 2-lr

1 + b / _ (A' Sc(Le - 1) + P r *- 1) " , C . . (50)

72

f'

The f a c t o r m u l t i p l y i n g C„ i n e q , 50 i s the m o d i f i e d Reynolds ana.logy f a c t o r a c c o r d i n g t o C o u e t t e f lev: tlie c r y . Eq, 50 can be r e v / r i t t e n a s

r 1 „ 1 . ^ A _ I A'd -Le"'') + 1 -Pr-^ 1 =C. (51)

C ^ p.TT S t •- J J I 2 S t P r 1

-and i t seems t h a t t h e groiip l ï / p g U S t w i l l s t i l l be u s e f i i l i n the more o o n p l i c a t e d c a s e s v/here Le ;^ P r ;^ 1 ,

13

-Returning to e q s . 49 and 50 v:e find t h a t

^ .^'LJI . ^ . (1 + | ' ( A ' Sc(Le - 1 ) + P r - 1 ) )

h - h ^f o ^ ro vro

where suffix o indicates zero injection value, A reasonable estimate for Gf/C^ is 1 - b'/2 and using the results in eqs, 36 and 37 i^e find after some manipulation that

q P r A' h +(Pr U^A)(1 - P r -(Le - l ) A' Sc) O ^ h - h

T70 r o vro

- I (2 - P r +A' Sc(Le - 1)) , (52)

This r e s u l t reduces t o the equivalent of eq, 45 when Le = P r = 1. The e f f e c t s of Le and P r d i f f e r e n t from u n i t y are d i f f i c u l t t o assess i n g e n e r a l , c e r t a i n l y the l a s t term i n eq. 52 i s favotirably affected by the s i t u a t i o n Le > 1 , P r < 1 provided A' > 0 . (Note Sc(Le - 1) = P r - Sc and P r > Sc i n t h i s case) . This does not follow i n the case of the second terrii hov/evor and the s i t u a t i o n de^Dends on the r e l a t i v e magnitudes of a l l the parameters p r e s e n t .

14

-4 . Subljjnatiqn

I f the conditions are r i g l i t , i t may happen t h a t tlie lovrer v/all reaches a teniperatu3re a t v/liich sublimation of the vraJ.1 m a t e r i a l takes p l a c e , r e s u l t i n g i n the " i n j e c t i o n " of a foreign gas i n t o the shear

l a y e r . I n t h i s event the quantiiy m i s no longer a f r e e , or independent, v a r i a b l e b u t must be r e l a t e d i n some way t o the heat t r a n s f e r r a t e and

the l a t e n t heat of the siibliming m a t e r i a l .

To i l l u s t r a t e t h i s e f f e c t i n as siniple a fashion as possible we w i l l assun^e t h a t P r and Le f o r tlie r e s u l t i n g gas mixture are unity and use e q s , 39 and 4 0 . To achieve a steady s t a t e i t i s necessary t o allow the lower v/all m a t e r i a l to move upv/ards a t a r a t e j u s t s u f f i c i e n t t o keep the BUbliming i n t e r f a c e a t tlie p o s i t i o n y = 0. Alla7ing for t h i s

"convection" of s o l i d m a t e r i a l uprards tOT7ards y = 0, the energy balance for the s o l i d i s

^ n ^ =

7^^^^

(55)

"^ b dy b ^ ^-"-^^ (Specific h e a t C, and conductivity "K are assumed c o n s t a n t ) . Then

\ § = * S ^ ^ - V (54)

vdiere T, i s the s o l i d ' s temperature a t a positicai so f a r inside the v/all t h a t dT/dy i s sensiblj'- zero. I t f o l l a 7 s a t once t h a t

^ - ^b = (^^w - V ^ ^ ( * % y/ V > (55)

and the heat transfer rate into the solid just belav the subliming interface is

\ (f) = ^\K-\'> (56)

y 0

-This r a t e must be matched to the value of -3 a t y = 0+, taking accoxmt of the f a c t t h a t the m a t e r i a l absorbs an amount of l a t e n t heat L v/hilst imdergoing the ch.ange of phase from s o l i d to g a s . I n other v/ords

- ^ , = A ( L + C ^ ( T ^ ^ - T ^ ) ) = A L ' , (57) say,

ITith the constant mean v i s c o s i t y a.ssuniption - 5 can be viodtten (see eq. 45) as

15

-I t follov/s a t once from e q . 57 t l i a t

,-1

, m = ( h - h ) ( ? / 6 ) ( i ; + h A' + ( h - h ) / 2 ) " . {59)

• • ^ r o \K>' ' '^ v/o ^ r o v/o" ' ^ '

This result exemplifies tlie self-regulating character of the sublimation process. The mass lost tlirough sublimation is seen to decrease as L' and A' increase, other things being equal. Prom a structural viev/point then, it is best to choose an ablating material v/hose latent heat of sublimation is high and which degrades into a gas v/ith as high a specific heat as possible. This v/ould suggest that the reductions in - ó

arising from abla.tion may be come small, since A itself is small, Putting 59 into 58 gives

- 4. = (h ^ - h

J ( ? / ) L ' (L

+ h ^ A' +(h - h

^/l)"^ '

(60)

TV ^ ro v/o'^^' ' ^ v/o ^ ro v/o'' ' ^ '

So l o n g a s A' > 0 t h i s v a l u e i s alv/ays l e s s t h a n t h e n o - a b l a t i o n v a l u e , b u t i t w i l l n o t be much b e l a 7 i t i f L becomes v e r y l a r g e .

HaTCver, - S i s n o t of p r i m a i y i n p o r t a n c e h e r e . R a t h e r a r e v/e c o n c e r n e d v d t h t h e h e a t f l u x i n t o t h e v / a l l , v/liich e q s . 56 and 57 shew t o be g i v e n by - q^,^ - m L = - q^ ( 6 1 ) s a y . Thus n r L + h A' + ( h - h ) / 2 -)

- a = (h - h ) ^ [ 1 - -^^:sR .:.£<i^.sp!L. (62)

v/o ^ r o v/o"

Clearly - a reduces substantially in magnitude as L rises. (As L ^ CO , - a •* 0 ; note that although A "* 0 a.s L -» co the product mL remains finite and non-zero),

It seems reasonable to suggest that a high L and a higli C . vdll

result in the most effective and least v/asteful type of surface sublimation. The siniple results just presented are not tlio whole story, ha7ever, because it has been tacitly assun.icd throughout that T is knov/n. In

v/

o r d e r t o f i n d - 4 f o r exai'jple i t i s , of covirse, n e c e s s a r y t o knov/ T and t h i s must be a f u n c t i o n of t h e pj^essure i n t h e s h e a r l a y e r .

16

-I n other v/ords, the f j n a l solution of the a b l a t i o n problem must depend on a knaTledge of the v a r i a t i o n of the vapour pressure and l a t e n t h e a t of tlie w a l l m a t e r i a l vdth temperature, as we s h a l l s e e ,

The k i n e t i c theory of the vapour pressure assuifï3s. ( i ) t h a t tiie process of condensation r e q u i r e s no a c t i v a t i o n energ;'" (so t h a t every molecule of the condensing substance v/hich s t r i k e s the w a l l retioms t o

the s o l i d phase) and, ( i i ) t h a t the molecules i n the s o l i d phase behave as a nijmber of multiple o s c i l l a t o r s , n per u n i t area of s u r f a c e ,

v i b r a t i n g v/ith a frequency v. The ntunber of molecules vapourising ( o r subliming) from the s o l i d phase i s then given by tiie product of n ,1^ and the p r o b a b i l i t y tliat they posses s u f f i c i e n t energy, ^^ , I f 2s i s the nxjmber of quadratic momentum or displacement co-ordinates v/hich contribute t o t h i s energy, t h i s l a t t e r p r o b a b i l i t y i s given by Bertlioud's r e l a t i o n

. -X /kT

( X / k T ) ^ - ^ e ^ / ( s - 1 ) !

(see Moelwyn - Hughes, loc,cit). Then if v/e v/rite n for the number of molecules of the subliming material v/hich exist in vaiit volume of the vapour phase at the interface, n for their mean velocity and m for their

g

mass, it folla/s that the rate of sublimation, A, is given by (X/kT)^-'' -X/kT • ^

A = m n V ..^ ^.^ e - m n r {^3)

e ° (s-1)! g 4

The v e l o c i t y fi i s given by

n = (8 kT/Trm^''' , (64)

o r , since the vapour pressure p i s given by n k T, we can va-ite

n ^ = p ^ ( 2 7 7 m g k T ) - ^ . (65)

At equilibrium A = 0 and the equilibrium va.pcur pressure p follov7S directly from eqs. 63 and 65. If n is the corresponding number density in tlie equilibrium state, the expression for A can be vadtten sljrply as

A = m r(ii « n ) = p r ( o - c ) (^C\ g 4 ^ eq ' ^ 4 ^ eq ' ^ '

* The treatment presented here i s a p r e c i s of the a.ccount given by Moelv;yn-Hughes (1957) of vapourisation from the l i q u i d ph3.se,

17

-Vidiere c and c are tJie aorresponding mass fractions. A relation like eq, 66 has been given by Scala (1958). It should be noted that P

is itself a function of c at given p and T, but if c << 1 v/e may reasonably estimate P as pm /RT v/here m is the mass of an external gas molecule.

Specialising the result 66 to apply at the la^er v/all, v/e laiov/ that o !a b (when Sc = 1, see eq. 42) and for present purposes v/e can v/rite

b' - ifiJ, - IVPO U st^ , (67)

7r

v/o o

( sToffix o = no-injection case) , Then

A . i P V ^ v P i " v A 2 ° w e q _ (gg)

~ 1 -. (p m/kT^p(ny2^gUStJ

One can nOT7 equate 68 and 59, yielding a relation for T in terms of p, c (which is a function of p and T ) and the latent heats etc,

^* v/eq ^ w

(Clearly X must be related to the latent hea>.t IJ at the temperature T ) ,

The jresulting equation is obviously not one for which an analytical solution can be obtained, but by stripping it down to the essentials some estimate of the trend of T v/ith p can be made.

w ^

Assuming that X ^^ L m very roughly, and that L >> C, ( T , - T, ) + h A' + (h - h )/2 (v/hich is consistent wdth the analysis taking

wo ^ ro vro" -v e b << 1) v/e can v/rite via eqs, 68 and 59 etc,

-Lm/kT s - 3/o -r

A e ^ ^-^ c {Yi - C T J T /^ (l + Bp T " ^ (69) ^ ro p vr w ^ -^ w ' ' where A and B are constants v/hich depend on the niaterials involved.

Then any rise in p can be counterbalanced by a relatively small rise in T in general (via the exponential term). The number of oscillators in each molecule v/hich take part in the vapourisation processes is usually greater than 3, the value for monatomic molecules (i,e, s > 3 ) .

In practice eq, 69 irrrplies that T. vdll increase (at a stagna.tion point, say) vdtli increa.sing Mach number and decreasing altitude. By how much must depend on the particular material viiich is involved.

Finally, it must be remembered that a vdde group of materials pass tliroijgh the liquid phase before vapoiurising, and that a liqmd layer will exist

between gas and solid. It goes v/ithout saying that the presence of the liquid film vdll influence the final heat transfer rate to the interior of the v/all, the rate of ma.ss loss, etc. Hov/ever a nurrber of the broad

- 18 ~ 5 . R e f e r e n c e s 1 . L e e s , L . (1958) 2 . Moe Ivjyn-Hughe s , E.A, (1957) 3 . S c a l a , S,M, (1958) 4 . S c a l a , S,M. ( i 9 6 0 ) Combustion and P r o p u l s i o n , 3 r d Agard Colloquium. 2f.51 - 4 9 8 . Pergamon P r e s s . London, P h y s i c a l C h e m i s t r y , Pergamon P r e s s , London. J o u r n a l A e r o , S c i e n c e s , 2 5 , 1 0 , 63^J656

Journal Aero/S-oace Sciences, 27, 1, 1-12