Thermal diffusion effects in shock tube boundary layers

Thermal diffusion effects in shock tube boundary layers

Citation for published version (APA):

Dongen, van, M. E. H. (1978). Thermal diffusion effects in shock tube boundary layers. Technische Hogeschool

Eindhoven. https://doi.org/10.6100/IR154998

DOI:

10.6100/IR154998

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Published: 01/01/1978

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THERMAL DIPFUSION EFFECTS IN

SHOCK TUBE BOUNDARY LAYERS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN ,OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF.DR.P.VAN DER LEEDEN, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

DINSDAG 21 MAART 1978 TE 16.00 UUR

DOOR

MARINUS EDUARD HERMAN VAN DONGEN

DIT PROEFSCHRIFT IS GOEDGEKEURD

DJOR DE PROMOTOREN

Prof. dr. :ir. G. Vossers en

.Abstract List of 1. INTRODUCITON page 3 5 9

1.1. Shoak tube flow; the visaoua boundary layer; the 9

the:rmal boundary layer at the end wall.

1.2. Subjeat of the present investigation. 13 1.3. Related boundary layer investigations in shock tubes. 15

2. 1HE THERMAL BOUNDARY I.AYER IN A SINGLE MONATOMIC GAS 19

2.1. A simplified solution, 19

2. 2. Denvation of the boundary layer equations; Temperature 22

jump at the wall; Disturbance of the outer region.

2. 3. Comparison of the shoak reflection problem with the 30

the:rmal Rayleigh problem.

3. 'IHE 'IHERMAL BOUNDARY I.AYER IN A BINARY MIXTURE 3.1. Equations and boundary aonditions.

3.2. The transport coefficients

their mixtures.

3.3. A linearized solution.

3.4. The numerical solution.

argon, helium and

4. 1HE VISCOUS BOUNDARY I.AYER IN A BINAFY MIXTURE

35 36 40 42 44 49 4.1. Equations and boundary aonditions for the viseaus 49

boundary layer behind a shock wave.

4.2. Mirels' boundary layer solution. 53

4.3. Salution of the difjUsion equation. 55

5. EXPERIMENTAL ME1HODS 59

5. 2. The optical reflection method; internal reflections. 64

5.3. rne measurement of preesure and wall surface temperature. 70 5.4. Description the experimental set-up. 71

6.

EXPERI11ENTAL RESULTS .A..ND Cürvrr:'ARISON iiJI1H 1HEORY

75

6.1. Thermal bour.dary layer exper{ments. 76

6. 2. Viscov.s boundary layer experiments. 83

7.

DISCUSSION

AND

CONCLUSIONS

89

APPENDIX A

92

Rankine-Hugoniot relations for a perfect gas.

REFERENCES

95

ABSTAACf

Two different of shock tube bounclary layers in argon-helium mixtures are investigated, wi th special enphasis on the influence of mermal diffusion. Most attention is given to the ther.mal

boundary layer at the end wall. From the theoretical analysis it is found, that the relative argon ooncentration at the end wall increases in a step-wise rnanner at the instant of shock reflection. The

mgni tude of this change in concentration increases wi th the temperature ratio across the boundary and is independent of pressure. It is toa first of approxi.mtion proportional to the value of the therml diffusion ratio, and insensitive to variations of the other transpar't parameters involved.

The second boundary layer studied is the viscous one behind the incident shock wave along the side wall of the tube. The change in concentretion is considered as a perturbation and only the linearized equations are nunerically solved. The argon concentration at the wall undergoes a step-wise increase at the passage of the shock wave. This concentretion increase is approxi.mtely the same as for the thermal boundary layer for equal values of the temperature ratio.

A lii#lt reflection technique is applied to rreasure the refractive index of the gas at the interface with an optically transparent wall. From the rreasurerrent of the refractive index, at both the end wall and the side wall, the increase in argon concentration is derived for different shock strengths and initial pressures. A agreement between theory and experirrent is found, in particular for the thermal boundary situation.

As a side result of the experirrents, the exi.stence of a temperature jump between gas and wall is demonstrated. Approxi.mte values of the thei'llB.l accommodation coefficients for pure argon and for the argon-helium mixture are derived.

LIST OF SYMBOLS

a

A

~

A12'

)\

Ao'

B c f F F g h H k K

ï

m

~1

A

r

veloei i:y of sound À _{rer re} ,..o f/c _p

reference value of A for a binary mi:x.tu:re Wassiljewa coefficients

function of the ther.mal accommodation coefficient amplitudes of incident and reflected light waves parameter used in the descript2on of the entropy layer

heat

specific heat of a gas at constant pressure binary diffusion coefficient

reduced binary diffusion coefficient

relative concentratien of the heavy species

nonrBlized function, representing the boundary layer structure

ratio of ternperature- and pressure disturbances at a shock wave

ternperature jum:p distance reduced stream function stream function

Boltzmann1_{s constant}

wave number of a monochromatic light wave ther.mal diffusion ratio

Gladstone-Dale coefficient maan free path

I.ewis number: 5 Pre~ref/(2 ÀrefTref) atomie mass

n N

N

p

Pr

q r

s

₁ t T u

u

s V x y y z z

incident shock Mach number reflected shock 11::tch nurnber

number re fracti ve index

function of the thermal accommodation coefficient pressure

Prandtl number heat flux

total amplitude reflectivity

direct amplitude reflectivity for light reflection frcm a discontinuity

amplitude reflectivity for internal reflections gas constant

of the direct reflectivi ty for a Change

m the gas refractive index ti !re

ternperature

mass velocity

veloei ty vector; vector function shod< wave veloei ty

diffusion flux wi th respect to u velocity of the reflected shOQK wave

space coordinate in the direction of the shock tube axis space coordinate perpendicular to the shock tube axis relative change in reflectivity

thennal boundary layer coordinate viseaus boundary layer coordinate

y n 6 À T. J.r T ace Q 0 SUBSCRIPTS 1,2,3,1+,5 1,2 s Snell1 _{s invariant}

thermal accommodation coefficient

quantity used for describing optical reflection

exponent in the pONer law relation between binary diffusion coefficient and temperature

specific heat ratio

boundary layer thickness

exponent in the pClltler law relation between therms.l conductivity and temperature

I.agrangian coordinate

angle between direction of light propagation and direction of stratification

thermal conductivity

dynamic viscosity

time variable

response t:i.Jre for internal reflections

characteristic time far thermal accammodation effects at the end wall

internolecular cellision time

characteristic time

similarity coordinate for the thermal boundary layer; phase angle

phase factor

denote the different regions of shock tUbe flClltl according to Fig. 1. The use of these slibscripts is avoided in the text, unless specified explici tly.

heavy and light species in a binary gas mixture respectively.

0 c w ref SUPERSCRIPTS (o),(1) (j)

initial state of the test gas or uniform reference state

state at the contact surface between boundary layer and wall

state at the outer edge of the boundary layer

undisturbed state outside the boundary layer, for the viseaus boundary layer i t refers to region ( 2) of Fig. 1; for the thermal boundary layer i t refers to region ( 5) of Fig. 1.

wall

reference state

zeroth and first order solutlon after expansion in pcwers of

k.r

differentiation of order i

disturbance with respect to the Rankine-Hugoniot reflected shock region

SPECIAL NOTATION

VBL

TBL

d.irrensionless variable, reduced wi th respect to a reference quantity

viseaus boundary layer

CHAPTER 1

INTRODUCTION

1.1. Shoak tuhe flow; the visaous boundary layer; the thermal boundary layer at the end wall.

During the last decades the shock tube has become an important resean::h facility. In a shock tube it is possible to generate a well-defined abrupt change in the state of a gas , a flCMI of high uniformi ty in a wide velocity range, and temperatures up to JJPre than 104 K.

Shock tube operation is relatively easy and inexpensive, and nearly all gases and gas mixtures can be examined. There is a variety of different shock tube applications, covering several scientific disciplines. Shock tubes are used for aerodynamic purposes, to study the kinetics of chemical reactions , vibrational and rotational energy transfer, plasmas, spectroscopy, vapour bubble dynarnics in two

flow and even fertilizer production (Her 1975).

The principles of shock tube operation can be found in many textbooks, of which those by Glass and Hall (1959), and by Oertel (1966) are mentioned here.

In this introductory paragraph we shall give a concise description of the principal phenomena involved.

A shock tube is essentially a short duration facility. Initially, the tube is separated by a diaphragm in two parts , the lCMI pressure test gas section and the high pressure driver gas section. The energy, stored in the driver gas, is suddenly rèleased after the collapse of the diaphragm and transfered to the test gas by means

of pressure waves. The fast expansion of the driver gas generates strong cornpression waves propagating into the test section and accelerating the test gas. These compression waves overtake each other and form a flat shock front with a thickness of the order of the molecular mean free path. The situation is elucidated by means of the space-time diagram of 1.

\

\ \ .,.c. \ \

.i"

\~

\ è:' _\ (5) ~0 _\

"

\~

\ \

l>

\ \ (;' \~ 't \ B' B ~'C \~·

·~

\ ~'(. _,". \ '-a (2) _\ (3) \~ ,':f _\

..

\ \ E \

-

\ \ \ _\ A A' (1) (4)

test sectien sectien

, - - - 1 temperature I - - - 1 I I r -1 I L _ _ _ / pressure

cross sectien AA'

temperature pressure / / , I cross sect 10n BB

In this idealized picture it is assUJred, that the shc:ck front is fOl:'IJ:ed instantaneously after the collapse of the diaphragrn. The shock wave propagates with a constant velocity. The test gas is accelerated, compressed and heated at the shock front from the undisturbed ini tial state ( 1) to the new uniform pressur'e and high temperatur'e state of region ( 2). A contact surface the high temperatur'e test gas of region (2) and the expanded, low temperatur'e driver gas of region ( 3) . The ini tial driver gas state is indicated as region ( 4) . The shc:ck wave reflects from the rigid end wall of the tube, and thereby causes a second compression, temperature- and pressure increase of the test gas. The stagnant state between the reflected shock wave and the end wall is indicated as region { 5). The t:wo dashed lines are examples of paths. Below the space-time diagram, pressur'e and temperat:ur'e profiles are shown at t:wo instants of time, one before and one after shock reflection.

The key pararreter in shock tube operation is the shock Mach nurriber, Ms, defined as the ratio of the incident shock wave veloei ty, and the velocity of sound in the undisturbed test gas. The state variables of regions ( 2) and ( 5) are related to the ini tial state of the test gas and the shock Mach nurriber by the Rankine-Hugoniot equations. For a perfect gas wi th a constant specific heat ratio, these equations take the form of the simple algebraic expressions, that are listed in appendix A.

The shc:ck Mach nurriber and the corresponding terrperature ratios T2/T_{1 and T5/T1}, can be varied over a wide range, varying the initial pressur'e ratio P₄/P1' or by taking different driver gases. The important regions (2) and (5) are uniform, except close to the wall. There, boundary layers are formed in which the state variables vary strongly with distance to the wall.

Two different types of boundary layers are considered, the viscous compressible boundary layer along the side wall of the tube (VBL), and the .thermal boundary layer, which is formed at the end wall after shock reflection (TBL).

The VBL is two-dirrensional, can be considered as steady in a shock-fixed coordinate frarrE. The si tuation is illustrated in Fig. 2a, and differs from the usual flat plate compressible

VBL

.. I

·c=-q:

(1) (a) (1) o====t> (2) h ~ utyl~ I t--i (b) VBL ... ,. ~--r:::;·,:::

Fig. 2a. Velocity and temperature profiles in region (2). Shock tube reference frame (a); shock wave reference frane (b).

TBL

( 5) (2)

Fig. 2b. Temperature and density near the end wall of the tube after shock reflection.

boundary layer in the sense, that the highest veloei ty in the shock-fixed reference frame is attained at the wall. The tube wall is almost isotherma.l, so that terrg;Jerature differences across the boundary layer occur. For an observer in the frame, the boundary layer thickness increases wi th the square root of time, as long as the boundary layer remrins laminar. The trans i ti on to turbulence occurs aftera transition time, which is strongly dependent on the smoothness of the wall (Har 1960).

Only those si tuations are considered, in Which the boundary layer is laminar, wi th a thickness much smaller than the tube width.

The second

type

of boundary layer is formed at the end wall of the tube after reflection of the shock wave. I t is the transition region between the shocl<:-heated gas of region (5) and the cold end wall. This thermal boundary layer is one-dirnensional and time dependent.

T

and density p vary with distance to the wall as is illustrated in Fig. 2b. Viscous effects are of minor :importance, so that the TBL is easier to describe than i ts viscous along the si de wall of the tube.

So far, we have described a somewhat idealized situation. In practice, the boundary influence the main flow. The viscous boundary layer, displacement thickness , generates e:xpansion waves, .,_,~. '"u.c=-'- decrease of shock (Kam 196 2) .

condi ti ons , the reflected shock wave can interact so the viscous boundary along the side wall, that a growing bifurcation of the foot of the shock wave occurs. This may disturb the of region ( 5) (Dav 196 7).

For monatomic gas es , this phenomenon does not occur. For these cases, the reflected shock region is very uniform and well-defined.

1. 2. the present investigation.

Thenral diffusion, the diffusion of -,...·--··-- in a mixture due to a

mn~~+~n+ process in compressible terrperature gradient, can be an

boundary layers i..rl which temperature d~fferences exist.

The~l diffusion effects can LDfluence the spatial distribution of the ligtt component, injected to hypersonic boundary layers to enhance coolir.g of the wall El64). It also a role in boundary layers with cr~mical reactions (Tam 1977).

Apart from these aspects, therm3.l diffusion is an i..rlteresting phenomenon from a

coefficients of thermal diffusion are

of view. The transport on -:he

involved. A'î

deterll1L'îation of these coefficients is important as a supplementary verification of kinetic transport (Fer 1972).

In th~s thesis, the of the shock tube to

the therm3.l diffusion prooess are examined. 1his is done by analyzing the shock tube layers for an mixture of argon and

helium, with on the therm3.l layer. There

are reasons to assLJJre a , that for such a system large ir. concentration are to be expected. As a rough estinate the change ~ relative concentration in a mixture wi t'1 extreme temperatures T1 and is of the order ln C/T₁), where is the so-called therm3.l diffusion ratio. For the argor.-helium mixture,

~ is about 0. 1; temperature rat~ os of the order of 10 are attainable in a shock tube.

'L'1e reaso:~s for using as t:'1e test mixture to thermal diffusion effects are the followi..rlg:

i 'L'1e transport of argon, helium and their mixtures are fairly well-knCMn.

i~ '.l'he absence of internal degrees of freedom facilitates the theoretical analysis.

iii The of argon and helium a..vt very different. This opens the possibili ty of an experimental detennination of t'le local species concentrations.

iv The reflected shock lS uniform and well-defined for monatomic gas es.

The the:nnal boundary layer in the binary mixture is treated in chapter 3. The mathematical problem is formulated and the salution of the exact boundary layer equations is compared wi th the resul ts of a linearized theory. The sens i ti vi ty of the salution for the transport parameters involved is examined. In chapter 4, the more complicated viscous boundary layer is treated. Far this case, only the linearized equations are solved. The results for both types of boundary layers are cornpared.

Concerning the experiments ,

application of light reflection to measure the local refractive index of a gas at the interface with a transparant wall. Sirree the refractive index of an argon-helium mixture is mainly determined by the argon number density, it should be to derive the argon concentration from the refractive index, the pressure and the temperature of the gas at the wall. The question arises then, whether the temperature near the surface can be considered as a continuous function or not, which necessitates a further analysis of the bouncla:ry condition

for the temperature. The large temperature at the wall can result in a temperature jump at the wall, which has then to be taken into aCC()unt.

In order to the importance of a temperature j ump

condition, a.'ld to lay a foundation for the boundary layer equations of chapters 3 ru1d 4,we shall first discuss in chapter 2 the thermal bounèary layer formation in a monato~ic

T'ne experi!TI2ntal methods are discussed in chapter 5. In chapter 6 experimental re sul ts are compared wi th theory, both for the TBL a.'1d the VBL.

1.3. Relateà boundary layer inveetigations in shock tubes.

The the:nnal boundary at the end wall of a shock tube has been extensively studied, mainly to obtain experimer,tal information on the thermophysical properties of gases.

at the instant of shock reflection. The of this temperatu.re step is related to

the magnitude of the heat flux to the wall, which depends parametrically on the relation beLWeen thermal conductivity and temperatu.re. In this way the temperatUI~ of the thermal conductivity in a wide temperatu.re range arrl for different gases has been obtained (Srr~ 1957, Lau 1964, Col 1966). Matula (1968) determined the thermal conductivity of an argon-helium IT~xtUl:'e with the same tecrW!ique, neglecting the possible influence of thermal diffusion. Saxena (1972) discussed the possibili ties and of the shock tu:::Je rrethod, and presented new experimental results on the thermal conductivities of argon, heliurr,, nitrogen and carbon monoxide.

A different approach was followed Smeets (1965), who examined the boundary layer structure rreans of differential interferometry. He deduced the therrral conductivities of air from an analysis of the experirrentally obtained

Bunting and Devoto (1967) investigated the boundary layer structu.re in t1ach ZerJlder Interferometry. Recently, Vrugt ( 1976) detemined the structu.re

in several m::matomic gases by r:1ear.s of a laser schlieren technique. He was able to obtain from his data the thermal

conductivity-temperature any parametrie

relation a priori.

All these studies are based on the of continuurn theory. From these continuurn equations , an estimate can be made of the boundary layer thick:ness as a function of time. 'Ihe thickness can be expressed in terws of the intermolecular collision tirre T col and the mean free path Ï as è f\; For values of t/ T

1, the continuurr, approximation is valid. For moderate values co

of this ratio, the continuurn car, still be but the boundary conditions have to be modified to take into account the temperatu.re jump between gas and wall at their interface.

A theoretical discussion of the thermal boundary wi th a -reiTl!)<en:;-rlrr:B boundary condition was given by Clarke (1967) and Fiszdon et al. (1970). The influence of this temperature jump effect on the heat

transfer to the wall was detennined experimentally by Busing and Clarke (1967) and Hanson (1970).

For even smaller values of t/Tcol' the concept of a boundary layer is no longer meaningful. The bounclary layer becomes part of the reflecting shock wave. The description of the shock reflection process and the subsequent heat transfer to the wall has to be given on the basis of kinetic theory. A m..urerical solution of this problem based on the BGK and E.S. kinetic model equations was obtained by

Van Dongen (1969). Monte Carlo solutions of the full Boltzmann equation were presented by Deiwert (1973). His results were compared wi th narmal stress and surface temperature measurements by Hans on

(1973).

The time and space dependent density profiles for the shock reflection process in argon at lew densities were determined experimentally by Piva and Sturtevant (1969). They observed an apparant vialation of the mass conservation law and concluded, that a strong adsorption at the wall must have occurred. Meldau (1977) carefully repeated their experiments, and found no significant adsorption.

At higher temperatures, the bounclary layer structure becomes strongly affected by ionization and recombination processes , the energy

exchange between electrans and heavy particles, and the presence of an electric sheath layer adjacent to the wall. The heat transfer from such a partially ionized boundary layer to the end wall was investigated for argon by Willeke and Bershader (1969), the structure of the boundary layer was examined by means of two wave length

interferometry by Kuiper (1968). A theoretical treatment of the problem was given, among others, by Camac and Kemp (1964).

and, recently, by Hutten Mansfeld (1976). The latter also proposed a two wave length laser schlieren method as a suitable experimental technique to determine the electron and heavy partiele density profiles.

The influence of the internal degrees of freedom on the bounclary layer structure was investigated by Jongen ( 1971). From an analysis of the heat transfer to the wall in vibrationally relaxing N

it was found, that inside the a state of strong non-equilibrium bet1.;reen the different energy :rncxles must exist.

The viscous been explored as the thermophysical A theoretical by l1irels ( 1955 ,

gases was treated Th,onY~T

the side walls of the tube has not the thermal boundary layer to study

op was given

flow in partially ionized Knöös (1968) and Liu (1975). The numerical aspects of were discussed by Honma and

Komuro (1976). Brimelow (1974) examined experirrentally the structure of the viscous behind an ionizing shock wave in argon by ::-.eans of

CHAPTER 2

THE THERMAL BOUNDARY LAY'ER IN A SINGlE MONATOMIC GAS.

In this chapter an description is of the therrna.l boundary layer at the end wall for a rnanatomie gas with a thermal conductivity, Which is proportional to temperature. For values of the t!t

1, the temperature is continuous at the co

layer does not disturb the outer region. In wall,

§2.1- the layer equations for this

situation is with the numerical solution using a more realistic thermal conductivity-temperature relation. In §2.2. the boundary layer are derived from the conservation

laws in the continuum approximation. At the wall a temperature jump boundary condition is introduoed. A'1 solution of the equations is obtained, assuming the reflected shock region outside the boundary to be of infinite extent. The consequences of this approximation are finally discussed in §2. 3.

The treatrrent of this basedon a paper by Clarke (1967).

2.1. A simplified solution.

The temperature

reflected shock wave, the outer reflected shoCk velocity. If this gruwth rate of the boundary

between the end wall and the , extends in time wi th the

is much larger than the to the wall, the process of shock reflection and subsequent boundary layer fonnation can be

in good approximation the so-called thennal

problem: A gas of uniform and pressure p"" is brought

in contact at the time t

=

0 with a solid heat conducting wall of temperature Both gas and wall are of serni-infinite extent. The validi ty of this representation is discussed in §2. 3. we

assume, that the outer region is not disturbed by the and that the pressure is constant.

layer

Then, the energy equation in written as: coordinates ( t, n ) can be T wit'"! A and

n

f x p dx' 0 2.1.1. 2.1.2.

In these equations T denotes the , p the pressure, p the densi ty, R en the specific gas constant and specific heat at constant pressure, À the thermal conductivity. Reference values are in<Lcated with the subscript "ref". Pressure, and temperature are related by the ideal gas law.

The temperature distribution in the wall is described by the energy fora solid:

2.1.3.

where a subscript "w" is used to denote the wall properties, which are assumed to be constant.

The initial and boundary conditions are:

t

=

O,

n >

0: T

=

Tw,

x<

0:

T

=

T

0

t > 0, n ~ oo: T + ) X ~ -oo; T ~

x 0:

=

(T) • (À

.i!!)

=

(À

i!:)

0 x=-O' Pan n=+O w a:x :x=-0

2.1.4.

For À to T, the energy equation ( 2 .1.1. ) becorres linear

and an analytical solution can be obtained. the coordinate z as:

n

z

= ..___,..

2.1.5.

the sol ut ion for x > 0 can be wri tten as: T ; T + (T - T )erf z c "" c 2.1.6. _...;.;x;_"..; _{{T z + (T} -c "" 1 [z erf z - (1 - e )]} 2.1.7. The temperature of the contact surface, separating gas and wall, Tc, is found

from:

2.1.8.

The value of this quanti ty is in mest si tuations much smaller than one. Therefore, Tc will be taken

theoretical analysis.

The expressions (2.1.6-?.)constitute

l

temperature as a function of x/t2 •

to T

0 in the further

an implicit salution for the

In 3 an is shcwn of a thermal boundary layer profile in argon. The temperature and pressure of the outer region, and p

00, correspond to a shock Mach mmiber of 3. 5 and an ini tial

test gas pressure p

0 of 2.66 kPa. I I !

~t

Q 2000~ I. I. I I / / / ...

-,.,., / / argon M • 3.5,

f6 •

2.66kPa, 10 • 2 93K P-·189kPa l-·2B65K --~·~reflllo, ----~. '-ref ( l f l₀J0·72 t-rel ·0.0175 wm·1 K-1

Fig. 3. Temperature profile of a therrral boundary layer in argon. Camparison between the analytical salution far À ~ T and

The artalytical salution for À "' T is compared with a numerical salution for a more realistic temperature dependenee of the therrnal conductivity for argon, i.e. À is proportional to T0·72. The

consequences of the assumption À "' T are , that the boundary layer thickr>.ess becol:'es somewhat larger, and that a temperature gradient and heat flux are found at the wall. Nevertheless, the analytical salution appears to give a good qualitative idea of the boundary layer structure and is very sui table for analyzing and estimating purposes.

2. 2. Derivation of the bou:nda:r:y Zayer equations; Temperature jump

at the waU; Disturbanae of the outer region.

In order to ar.ri ve at the simple analytical solutions ( 2 .1. 6-7.) some ad hoc assumptions were made, the pressure was considered tö be constant, the outer region undisturbed, and the temperature continuous

at the wall.

Hcwever, a temperature decrease in the boundary layer at constant pressure corresponds to a density increase, which is only if a mass flcw towards the boundary layer is induced. The temperature

- l gradient at the wall was found to decrease in proportion to t 2

• In

the ini tial , the temperature so large, that a temperature jump exists at the wall.

To investigate these phenomena in ITIOr'e detail, and to derive criteria for the validi ty of the different approximations , we start from the general conservation laws for continuous media. In order to be able

to make an order of magnitude estimation of the different terms, the

physical variables in these equations are scaled. '!'.his is done with

-nc'0""'"'+ _{to their asymptotic values as x -+ "'. A problem is, that a}

there is no macroscopie we take the veloei ty of sound, and scale time with respect to T

00,

scale available, Therefore , as the characteristic velocity, a characteristic time for the solution. The characteristic length is then a

00 T"'.

Derloting the reduced by "~", the conservation laws take the follcwing form in l.agrangian coordinates:

(mass)

- = - -

au

a

T

2.2.1.

at

p

(monentum) 2. 2. 2.

(energy)

=

Kn !__

(~

+ -(y-1)\.IKn 4 ~ Fr~(---:::) D dU 2

an

T an

The heat ratio is denoted by y,

Two dinensionless parameters appear, the Prandtl 3

T

an

2.2. 3.

by IJ•

defined as ll c I À and a parameter Kn

=

À I ( yp c T ) • The latter · can be

"'P"' "' "'P"'

interpreted as the ratio of the mean free time between collisions of molecules and the mR<'Y~""'<YlTll

of a Knu:isen nurnber:

Kn= T _co

time T"', and has therefore the meaning

2.2.4.

At the wall, a teriiDE~ratc

In dimensional form:

jump

boundary condition is introduced.

aT

_2.2.5.

gp

The ternperature distance g is of t~e order of the molecular mean free path, and is therefore not scaled wit~

but to the mea.'1 free a"' T col' The ini tial and condi ti ons are sumrnarized as:

~

t

o,

n

>

0: T

p

=

1,

u

= o

t > 0,

n

0: u _0'

1' -

Kng~-: ~

E

aT

2.2.6.

T an

n + "': T+

1, p

+

u

+

0

The Knudsen number has to be a small quanti ty in the continuum apprcx:ination. Therefore, we shall neglect te:r'llE of order Kn. Then the conservation laws, (2.2.1-3.),reduce to the Euler equations of gas

au

au

1 0 2.2.7.

-

+ at y

i1i?.

0 at y p

In view of the uniform ini tial condi tions , the solutión of these equations is of the simplewave type (Owc 1964),

u

-It is that this solution can never conditions at the wall, Eqs. (2.2.6.),

2.2.8.

the boundary to the conclusion that the order of magnitude estimation is apparently not correct close to the wall. a boundary layer has to be

1

introduced with a characteristic dirrension of order Kn2, since close to the wall, the conduction term in the energy equation must be of order one.

A 1

a boundary layer coordinate 1}! 11/Kn2 , the conservation are rewri tten in the stretched coordinate system.

A

au

a

T

at

p

2.2.9. 1 l

{-

a~ + .:: Pr

a

<.!!li.

~)}

::: J<n2 y a~ at 3 al}; T Cllji 2.2.10. - y-1

T

_.21:

a

_(~

4 _{Pr g(d~)2} A + 1(y-1)Jl

at

y p at olJ! T ol/J T o1Ji

From the mass conservation law ( 2. 2. 9. ) , i t follcws that the

A 1

velocity u is of order Kn2 • Then, neglecting all terms of order Kn, the final boundary equations are obtained:

au 2.2.12. 2.2.13. aT _ y-1

~

lÊ_

= 1._

(21_

aT

y

p at 31jl 'Î'

a;

2.2.14.

For tjl + "', the boundary layer salution is rratched to t~e salution of the outer region, so that the boundary conditions become:

-

<1'Î' A

-E

tjl= 0: u = _0' T- _{Tc = g -}

_.,.,

2.2.15. 31jl .L tjl+ "': T+ 1 + (y-1) u+ ( y-1)2 A2 u 2. 2.16. 4

These conditions can be somewhat simplified, consistent vJi-th the procedure . to terms of order Kn. This removes the quadratic

term froTIJ (2.2.16,), The preSSure is isentropically c;UIJcU_t=ll

to the temperature outside the boundary layer; so that from (2.2.16.) it follcws:

-p = 1 + yulj;=oo = 1 + 0(Kn)2 1 2.2.17.

Since g is a function of the state variables, the deviation of

g

and

T

at the wall from their asymptotic values for large

t

is also of order . Therefore, t~e boundary condi tions ( 2. 2.15-16.) take the form:

,p

₌

0 u =

o,

T- T _c = _Kn~:::-~A 1 aT T

aw

c 2.2.18.

-tjl +«> T+ 1 + (y-1)u 2.2.19.

1:'1e mathematical tormulation of the problem is nru complete. In order co arrive at an analytical solution, we ass1.liTE the therrnal conductivity to be prcportional to temperature, ~=

'Î'.

1he (2.2.12j and (2.2.14.) can be brought ln a more attractive forrn defining two na.J variables.

t

f p dt' 0

2.2.20.

Tne suffix 11₆11 _{denotes the state at the outer edge of the boundary}

layer, where temperature and pressure are related according to:

1

₌

_{_Y_ :}

2.2.21.

Inserting the variables

e

and' into Eqs. (2.2.12 -14.),using the relation (2.2.21.), ar.d neglecting termsof order Kn

2.2.22.

aû

--

2.2.23.

This is a surprisingly simple result, the energy equation takes the forrn of the Fourier equation. The effect of the time dependent

pressure is incorporated as a slight reduction of the time coordinate The continui ty equation describes the mass flru truards the wall, due to tr.e decreasing temperature and density in the

boun-from (2.2.22.) into (2.2.23.)

af

ter integration:

CombiDing this equation wi th boundary condi tions ( 2. 2.18 -19.) results in:

-

A

+~

1/1

=

0:

e

=

Tc

+ {(y-1) }

T

2.2.25.

c

A 1/1 _,.(XI:

e

-+ 1

with the initial condition:

-

,= o: e

= 1

Equation (2. 2. 22.) wi th initial and bounda:t:y condition (2. 2. 25.) is solved by Laplace transforrnation. We give here the result for the pressure, and for the temperature of the gas at the wall.

(y-1) (1-Î' ) {1 + A c (y-1)

T(O,;)

=

p

y -+ Cy-1)T c A 1 + (1-T ) exp;... erfc (2....)2 } c

with a characteristic time defined as:

2.2.26. 2.2.27. Kn

c:C

+ (y-1)

'Î' )

2 2.2.28. c

Tc

1

The function exp x erfc x2 _{can be approximated for large values of}

x

(nx)-~

(Abr 1965). Therefore, for large values of

~~~a'

the salution becornes asymptotically:

A 1

p -+ 1 - y Kn2_{(1 - 2.2.29.}

T(O,t)-+

T + (1- T)

c c 2.2.30.

The asymptotic behaviour of the pressure disturbance is determined uniquely by the bounda:t:y layer induced mass fla-v, whereas the temperature jump at the wall depends on the gas wall interaction parameter

g .

From the asymptotic resul ts ( 2. 2. 29 -30. ) , we can ncw formulate the con di ti ons for the validi ty of the

(2.2.29.) it follows, that the constant and undisturbed by the

(2.2.30,) we find, that the

salution of §2.1. From can be regarded as

1

if Kn:2" « 1. From

~ l - 2

continuous if ge Kn2_{/Tc « 1. The latter condition is the strengest,}

particularly for high !'-1ach nUJ:'.bers of

and small values

The temperature jump expression ( 2. 2. 30.) could have been obtained from the simplified salution of §2. 1.

The temperature gradient at the wall

aT 1

n

=

0: -

=

(T - T ) __]2_---;

an oo c RT~ (nAt)~

this result into the temoe~at1

(2. 2.5.) yields:

n

0: T-which is

=

(T - T L1Z_

-iL..,.

oo c'~ (TIAt)2 witr. (2.2.30.). to Eq. ( 2 • 1. 6 • ) : 2.2.31. boundary condition 2.2.32.

a few remarks will be added to the temperature

dist~~ee g, and the gas density close to the wall. In o~er to findan

Knudsen

for g, it is necessary to the so-called adjacent to the wall, whicr. has a thickness of the

o~er of the molecular r.ean free path. In terros of the reduced variables of this paragraph' the Knudsen has a characteristic dirension of the o~er Kn. Such an analysis was by

Yamamoto (1975) on the basis of the BGK wi th a Maxwell boundary condition for the velocity distribution function witr. thermal accommbdation coefficient ~ at the wall. A similar procedure was followed Sone and Ya=.oto (1971) to describe the flcw of a rarefied gas over a plane, perfectly "'"'~nn"Til!'\ti"'·hT1cr wall. For the

"-' Îf... 2-CI. ~ (2RT)-~ g " -;;;- 5pR Cl. 2.2.33. with

A"

1.3024 + 0.1781

C~-1)

+ .••.

The reduced value of g een then be wri tten as:

2.2.34.

The temperature T(O,t) and the temperature gradient (CIT/<!n)n"O' as they appear in the temperature jurnp boundary condition for the boundary layer equations, (2.2.5.), are in fact hypothetical, and have no real physiçal meaning. They have to be interpreted as quantities, found by extrapolating the temperature end its gradient from the

outer edge of the Knudsen layer to the wall, as is clarified in Fig. 4.

p

1

Knudsen layt>r I

I

I

T _I

i

T ...-::

I

boundary la yt>r /

---

I Ho, I)

---

I I Tk I I I Tc I q/Kn

Fig. 4. Schematic representation of temperature end density

The experimental methcd, that will be discussed in chapter 5, enables the experimental determination of the real gas density at the wall, pk in Fig. 4. Yamamoto ( 1975) found the follc:wing relations between pk, the boundary layer pressure, and the extrapolated temperature T(O,t):

pk _T(O,t) 'V'v

(T(~,t)

n = 0: = 1 +

l2....

_{- - y -}

+

aN

_{- 1) 2.2.35.}

Pc Poo _c 'V _{A c}

with

fr

0,3040- 0,0312 c';;:-1) +

...

_'

and pc p IRT • 00 c

2.3. Comparison of the shoak refleation problem with the thermal Rayleigh problem.

The process of shock reflection and the subsequent boundary .layer developrnent differ from the thermal Rayleigh problem in this respect, that the reflected shock region is not semi-infinite, but bounded by the moving reflected shock wave. This has two consequences:

First, the pressure waves, generated by the densi ty increase in the boundary layer, reflect from the shock wave. Fortunately, shock waves are very good absorbers of pressure waves, the reflection coefficient, i.e. the ratio of reflected and incident pressure wave arrplitudes is rather small for the present situation (Cla 1967). Much more important is the fact that the pressure waves modify the reflected shock strength in the initial stage after shock reflection. Since the entropy change across a shock wave is dependent on shock strength, a layer of varying entropy outside the boundary layer is for.med. In this entropy layer, a position dependent density and temperature persist, even after the pressure disturbances have becorne zero.

We shall investigate this effect, starting from the assumption, that the reflection of pressure waves from the shock wave is negligible, so that the pressure disturbance follaws from the salution described in 132.2.

'!he equa.tions and bOillldary condi ti ons for the outer region are

linearized, assuming the temperature, pressL!r'e and shock wave velocity to differ only slightly from their Rankine-Hugoniot values:

...

'

...

'

T

=

1

+

T ,

p

=

1

+ p 2.3.1.

'!he linearized shock wave trajectary and the characteristics are straight lines in the (

~,

t)

of 5, wi th slopes equal to the final reflected shock veloei ty ;;R and to one, respecti vely.

5. ~-t diagram of the reflected shock region.

For the pressure distL!r'bance at the boundary layer, p 0

1

, we take

the asymptotic result of

Eq.

(2.2.29.): !

-P.s (t) _{- 1} 2. 3. 2.

( 7ft) 2

These pressure disturbances propagate along the characteristics:

Along a partiele , the entropy is constant, so that the

temperat\.lr€ and pressure at (t,n) are related to their values at the shock front (tR,n) the linearized equation of state.

'

~ ~ ~

'

~ ~

'

~ ~

T (t,n) (tR,n) + (t, n) - p (tR,n)} y

The tirre depends on the of the shock front n

= n/wR

coupled by the linearized shock relations (App. A).

wi th as the reflected shock Mach nur:lber and wi th

F(M) _{= - -}y-1

y

The solution for the temperature disturbance at any arbitrary point (t,n) is obtained after combining Eqs. (2.3.2 -6):

with I -

-T

(t,n)

WR

1 B

=

(F _Y_- 1)(-~- )2 y-1 1-wR 2.3.4. 2.3.5. 2.3.6. 2.3.7. 2.3.8. 2.3.9.

The last term between the brackets of Eq. ( 2. 3 . 8. ) represents the i sentropie expansion, whereas the first term between the brackets is the tirre independent temperature disturbance of the entropy layer.

An impression of the importance of this entropy layer for the bounda:ry layer solution is obtained by calculating the temperature disturbance at the outer of t"'le boundary layer, n

0 , which we define as:

2.3.10.

The temperature disturbance at n

8 is found after a substitution of ( 2 • 3. 10 • ) into Eq. ( 2 • 3 • 8 • ) :

2.3.11.

on the incident shock Pach number, and varies for manatomie gases frcm zero at Ms

=

1 to 0.66 for M _s

=

5.

The isentrcpic term is dominant in the initial i.e. for t/T₀₀₁ « (2/B)\ but for times it is vanisring much more rapidly than the entrcpy layer disturbance. The presence of the entrcpy does not significantly alter the conclusion of §2.2., that the outer region can be considered as undisturbed if

1

CHAPTER 3

THE THERMAL BOUNDARY lAYER IN A BINARY MIXTURE

In a thermal boundary in a binary mixture the species concentration will vary with position as a consequence of thermal diffusion. Thermal diffusion is a complicated molecular process, different from viseaus rromentum exchange , heat transfer and ordinary concentration diffusion in the sense that cannot be understood in a simple manner by considering the molecular collisions as such. For a physical it is necessary to take the velocity na·~a,nn<>nr=

of the callision frequency into account. Such a phenomenological tion was given by Manchick and M3.son (1967) and in condensed farm by Ferziger and (1972).

In §3. 1. the basic conservation equations for a thermal boundary in a binary mixture are forrrulated and transformed into a set of two ordinary differential equations for the temperature and the concentration. The transport coefficients, therma.l conducti vi ty, binary diffusion

coefficient and thermal diffusion ratio are specified for the argon-helium mixture in §3.2.

In §3.3. a salution is based upon a linearization procedure. The numerical salution of the complete set of and boundary conditions is discussed in §3.4. Numerical results are presented for the argon concentration

and

temperature as functions of the boundary coordinate for a temperature ratio. In addi ti on in argon concentration at the wall is as a function of the temperature ratio. The influence of the transport parameters on the salution is

3.1. Equations and boundary conditions.

The equations and boundary conditions for the thermal boundary layer

in a binary rrixture are rarefaction effects are

under the assumption that all

From the of 2 ,

it was found that this assumption is valid for sufficiently values of t/ 'col' So, we assume the tenperature to be continuous at the wall, the outer region to be undisturbed, and the pressure to be constant. Then the boundary layer equations have the follcwing form (Hir 1954, Fer 1972):

dp +

au

dt

pä.X =

0 3.1.1.

+ n.

~+!..en.

V.) ~ l

ax

ax

l l 0 i

=

1,2 3.1. 2. 3.1.3. · d . . d . f' ed

a

a

The materlal erlvatlve dF ls de ln as ät + u

ax;

p denotes the

mass density of the mixture, is the number density of i and n the tot al nur.tber densi ty, u the mass averaged vel=i ty, T the temperature, p the pressure, l<: Boltzmann1_{s constant, V. the}

l

diffusion vel=ity with respect to u, and q the heat flux. The are not all independent; the follawing relations exist:

p :nkT

The heavy component is denoted mass by the syrnbol "rn".

The diffusion veloei ty of

a subscript 11_{1", the molecular}

1 is specified as:

The dimensionless parameter

Kr

is known as the ther.mal diffusion ratio, D₁₂ is the binary diffusion coefficient.

The last term in this express ion is in the present case the dri ving "force"; a temperature gradient causes a diffusion flux and hence a concentratien gradient. The first term between the brackets represents the ordinary concentratien diffusion. For a mixture of

argon and helium the ther.mal diffusion ratio, kT' is a posi tive quanti ty in the temperature regime of interest. I t is strongly concentratien dependent, and has a maximum value of about 0. 1.

The expression for the heat flux consists of conduction and diffusion terms:

3.1.5.

i. substituting 8u/3x from

Eq.

(3.1.1.) into Eqs. (3.1.2 -3); ii.

trans

forming the equations in l.agrangian coordinates ( t, n) , wi th

x

n

=

f P dx1_•

0

These two steps eliminate the velocity since d/dt becOJJ:es ( 3/8t) n; the mass conservation equation ( 3 .1. 1. ) then needs no further consideration.

The next steps are the introduetion of the relative ooncentration f

=

_{n 1/n and the temperature T as the only dependent variables, and} rewriting the transformed • (3.1.2 -3) in terms of f and T. Before doing so, it is convenient to define an auxiliary quantity

m:

m

=

1 One then finds:

The of conservation of species 1 becomes:

af

n

a

- m

2 (1 + w~)L (PJV1)

ät

3.1.6.

T.1e energy equation is cowbined with (3.1.6.) and takes the form:

(1+ 3.1.7. with k

c.2f

+..!

oT)

an

T

an

3.1. 8. and q - ., filf) 5 3.1.9. The ini ti al and boundary condi tions are:

t 0 , : ; > 0 : .

t > 0, n 0: T :JT 3.1.10.

The contact temperature Tc is only slightly different from the initial temperature T

0, and ca.'1 be regarded as time-ir.dependent. For the Purpose, _ is taken equal toT • The temperature outside c 0

the boundary , is the undisturbed reflected shock temperature; is the initial concentration of t~e

From the set of equations and boundary con::li tio:tS ( 3 .1. 6-1J. ) , i t follows that f and T are unique functiors of

similarity coordinate z is introduced, defined as:

z

=

_...;.n_".. (4A t)2

m

Therefore , a

with

Reference values are denoted by a Inserting the reduced quantities,

n

=

~12

,

~

Pref ref ~

T

T=~ ref 3.1.12. "ref".

into ( 3. 1. 6 -9. ) and perforrning the similari ty transforrration, the are obtained in their final form:

1 _{+ tî\fo> 2 df} ( ~ ) 2z

az

= -

Le 1 + mf ) } (1 +

~

)2

~

* *

--~""-

2z

.2!

=

È:9.._ +

T

Le d

(~)

1 +mf dz dZ

az

T with *

= -

i.

(1 +

;rr)

ciT

+

3.

Le v*cL

~- ~

;rl)

q

T

dz 5 -'l' 1 - r 2

The Lewis number,

Le

₂

5

Pre~ref

ÀrefTref

is defined as**:

Tne bounclary condi ti ons are:

z

=

0: T

=

T ,

0

z + oo: T + T"', f +

*""

The Lewis riurnber is frequently defined as the inverse of expression (3.1.17). 3.1.13. 3.1.14. 3.1.15. 3.1.16. 3.1.17. 3.1.18. 3.1.19.

ordir.ary differential salution parametrically or. the temperature ratio T

00/T ,

~ ~ ~ ~ 0

foo, le, kT and the functions D(':') and À(T). Since the lewis number and are independer.t of pressllr€, the same property holds for the solution. Tr.e character of the salution that the value of f at the vlall, z

=

0, is of time after a

step-3. 2. 'l'he tra:nsport aoeffiaients of argon, he and their mixtures.

i. Tl:e the:cmal conduct i vi ty

For the thernal conducti vi ties of the pllr€ components , the values recomrrended by Touloukian et al. (1970) are used as a 1ne temperature depe.11dence is approxi.r:-ated a power law relation:

3.2.1. wi-:h À n;f ,1

=

0.0175 W m (argon) . -1 -1 À

=

0.1507 W m K (heli urn) ref ,2 T ref 295 K 6

o.

72

T.~e relative deviation of the values according to

Eq.

(3.2.1.) from the nocomrrended values far ü1e therrral conductivity is less than 5% for -:he temperature range 290-4DOOK. 1ne shock tube experiments of Vrugt ( 19 76) rendered lower values of ~., which are better described a power law coefficient ö

=

0.66.

1he concentretion dependenee of \ is described by mear6 of the approximation,

3.2.2.

The l.vassiljewa coefficients, A₁₂ a.;1d , are assuned to be te.'11.peratllr€ independent (Jai 19 77). T.~e numerical va.lues, obtained

A

12

=

0.36; = 2.77.

The sensitivity of the salution to the power law exponent will be examined in § 3. 4.

ii. The bina:ry diffusion coefficient

The dependenee on temperature and concentration of the bina:ry diffusion coefficients for argon-helium mixtures was investigated experimentally by Van et al. (1968) for the temperature range of 80K to 400K. Experirrental values for the diffusion coefficient between

300K and 1400K were obtained by Hogervorst (1971). The concentration dependenee is rather weak and becomes slightly more pronounced at increasing temperature. The effect is neglected in our calculations; this rnay cause a maximum relative error in D₁₂ of 3%.

The temperature dependenee of pD₁₂ is also approximated by a power law representation: with = 10 5

Pa,

f

=

0.5. 3.2.3. " -5 2 = 295K, D f

=

7.34 10 m /s,

B

=

1.70, at . re

The power law with

B

=

1.70 is in good agreement with Van Heijningen' results. Far the temperature regime, a power law coefficient of 1.66 gives a better fit. The sensitivity of the solution to the value of the exponent

B

will also be examined in

§3. 4.

iii. The thermal diffusion ratio kT.

of the therrnal diffusion ratio was investigated, both experirrentally and theoretically, for the noble gas mixtures by :Lar>anjeira (1960). He predicted, on the basis of mean free path argurrents, that there should be a linear relation between the quantity f(1 - f)/~ and the concentratien f. This type of dependenee was confiTI'Ied experimentally to a high degree of accuracy

(lar 1960, Sav 1975). There is li ttle experirrental inform:J.tion on the temperature dependenee of kT in the temperature range of interest. The Chapman-Enskog theory for a Lennard-Jones intermolecular potential

(Hir 1954) prediets for an equimolar mixture of argon and helium a relative variation of of about 1% in the entire temperature range of 300-4000K. Therefore, will be considered as temperature

independent.

The concentration dependenee is described by l.aranjeira's empirical relation, obtained for the temperature range of 300-600K:

3. 2. 4.

3. 3. A lineaY'ized aolution.

If the change in concentration is rather small, the thermal diffusion effect can be treated as a perturbation. We consicter the thermal diffusion ratio kT to be the perturbation parameter, and

the temperature and concentration in powers of

Kr·

(f,,) :

f f₀₀ + kT f (1) +

1,(1) + ... _3.3.1.

After inserting these expressions in the governing equations and boundary conditions . (3.1.13 -19.), and arranging the terrus in zeroth and first order in kT' we obtain:

zeroth order: f

=

f 00

ér(o)

ct 2z~ + ;:::,( 0) Z

=

Q; L :: À (o) dT(o) (;ror~)

=

0

T

0·, z + "" T(o)+ T

_'

_""

3. 3. 2.

first order:

df(1) d { _DCo) df(1 ) 1 dT(o) ~

2Z - d z + l.e -d - - 2 <-=--z + ~o ~z )

z T(o) az T~oJ u~ 0 3.3.3.

z

=

0: ;z-+oo, 3.3.4.

The first order change in concentration depends on the zeroth order temperature solution.

~

The equations becone linear if À is taken proportional to T, and D

proportional to

r

2, i.e. ö 1 and B

=

2. Wé then obtain the analytical salution

Eq.

(2.1.6.) for the temperature:

T(o)

=

~ _3.3.5.

The diffusion equation becomes: 2z df(1 ) + - - -_{l.e dz}

=

3.3.6. or written otherwise, + 2z

~)(f(1)

+ ln T(o)) 2z d 1n T(o) l.e dz

=

ïë

dz 3. 3. 7. Th is by a factor

exp( By inserting the boundary conditions into the solution, we obtain for the first order change in concentration at the wall:

T 1 )(0)

=

ln

_:.-T

0 f dz' 0 f 0

z'

_(z1_2-z11_{2)/l.e 2z}11 _{d 1n T(o)} e ~ dz" l E dz" 3.3.8. The salution consists of two contributions: the logari thmic term can be interpreted as the linearized steady state solution, the

a correction for the behaviour. of Eq. (3.3.8.) was evaluated nunerically for the temperature profile of (3.3.5.). The ratio of f( 1)CO) and

1.0 0 5 10 Le-1.18 p ·2.0 6 ·1.0 15

Fig. 6. Ratio of the first order changes in concentratien for the thermal bounclary layer and for the steady state, as a function of the ternperature ratio.

ln(T",IT

0) has been plotted in

ratio T /T . I t varies between 0. 55 and 0. 70 for ternperature ratio 1 _s oe 0

between 2 and 15. As a rough estimate, the change in concentratien at the wall can be approxirrated by:

(Lif) % 0.6 c

3. 4. The nwnericaZ soZution.

3.3.9.

The set of and bounclary condi ti ons ( 3. 1. 13 -19 . ) was

solved numerically by mear.s of a finite difference method. 1he equations farm a set of two non-linear ordinary differential equations, written forrrally as:

with boundary conditions:

z

=

0: F(u u )

=

0;

- - ' -z z +

co,

u+ 3.4. 2.

The outer boundary condition is applied at a finite value of z.

After sorne numerical trials a value z

=

3. 5 was found to be a sui table upper bound of the domain. • (3.4.1.) were solved iteratively. First an approxima.te solution, e.g. the zeroth order solution of §3. 3., was used to calelilate

E

and

Q·

The difference equations corresponding to Eqs. ( 3. 4.1.) were solved, and the solution was inserted into the expressions for

E

and

Q.

The procedure was repeated until the des ired accuracy was obtained.

TBL 0.10 T..,/lQ= 7.22 M =3.00 Le -1.18 6=0.72

P=

1.70 10 5

Fig. 7. Change in argon concentration and reduced temperature as functions of the boundary layer cooroinate z.

In 7 an example is shown for the salution of the temperature and the concentration as functions of the boundary layer coordinate z. In the outer part of the boundary layer a depletion of argon is found, wherèas in the inner part adjacent to the wall the argon concentratien increases. The concentration profile should an integral conservation law:

co

f n(M) dx= 0 with llf = 0

This also can be ~Ti tten as ;

00 f - f

! "" dz :: 0

0 1 +

3. 4. 3.

This relation was used as a check on the accuracy of the nl.lllerical results. The integral was divided in ~o parts, one with a positive

. + - . .

and tl-J.e other Wl th a llf, denoted by I and I . The ratlo (I+ + I-)/(I+ - I-) was evaluated for each solution. It never exceeded a value of 3.10-3•

The largest change in concentration is found at the wall. This wall value, denoted by (Lif) c, has been plotted as a function of tne temperature ratio in 8. The solid curve is the "recommended" solution, based on the transpar't coefficients specif ied in § 3. 2. The upper curve represents the linearized solution, discussed in §3. 3. , for

o

1. 0 and J3 = 2. 0. Two other "exact" solutions are shcwn, namely those for

o::

1.0, J3 = 1.7 and for

o

= 0.72, J3

=

2.0. The parameter o for the ternperature dependenee of the thermal

conductivity appears to have a minor' influence on the value of (Lif)c. The binary diffusion coefficient parameter J3 affects the solution in a slightly more pronounced marmer, the dependenee of (Lif)c on the value of J3 is still rather weak. The linearized solution appears to give quite accurate results. It almost coincides with the exact solution for ternperature ratio s smaller than 5. The parametrie dependenee of the concentratien at the wall on the Lewis number is shown in 9 • Rather large variations of the Lewis number only have a srnall effect on (Lif)c.

0.15 TBL 0.10 0.05 0 -+-+-+ Ar- He t.., ~o.s Le =1.18 5 0.72 1.00 0.72 1.00

p

1.70 1.70 2.00 2.00 Lin. Sol.

8 . Change in argon concentration at the wall as a function of terrperature ratio across the boundary layer. The

6 and

s

are the pCMer law exponents that determine the terrperature dependenee of thennal conducti vi ty and diffusion coefficient,

in concentration (llf) _c is most strongly determined the kT' the dependenee of (

one. An experimental deterrrünation temperature ratio will therefore diffusion ratio.

The terrperature profile T(z) is

on kT being almost a of ( óf) c as a function of the

infonnation on the thennal

affected by the thennal diffusion process, which is consistent wi th the validi ty of the linearized solution. Also the heat flux to the wall is insensitive to thermal

0 . 1 5 . , -TBL 0.10 0 / / Ar-He f= =O.S Le -0.68 Le =1.18 Le = 1.68 ó -0.72 p =1.70

Fig. 9. Change in argon concentration at the wallas a function of tençerature ratio across the boundary

of the Lewis number.

q (0)

=

(4-A

m

for various values

3.4-.4-.

The product H 1 + mf) is weakly dependent on f. The increase of the argon concentration at the wall apparently causes a decrease of À