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ASYr.?TOTIC SOLUTIONS TO COMPRESSIBLE LAMINAR COtJIDARY-LAYER SOLUJTIONS

FOR DUSTY-GAS FLOW OVER A STIINIEFA LT

by

B. Y. 1tang and I. 1. Glass

UTIAS Peport 1,o. 310

C11 is

UNCLASSIFIEDTy CLASSIFICATION OF THIS PAGE (Whoen D-10f l'w,, ,

R[:FVRT DOCUMENTATION PAGE RLAD INSTRUCTIONSBEFORE COMPL .TING i'ORM

't. FLPOB NUMBER , GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

AFOSR-TR- 8 6- 2 1654. TITLE (and Subtitle) S TYPE OF REPORT & PERIOD COVERE!D

Interim

ASYMPTOTIC SOLUTIONS TO COMPRESSIBLE LAMINARBOUNDARY-LAYER EQUATIONS FOR DUSTY-GAS FLOW . PERFORMING ORG. REPORT NUMBER

OVER A SEMI-INFINITE FLAT PLATE UTIAS Report No. 3107. AuTHOR(s) e. CONTRACT OR'GRANT NUMBERts)

B. Y. Wang and I. I. Glass -F-AFOSR-82-0096

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TAS(

University of Toronto, Institute for Aerospace AREA a WORK UNIT NUMBERS

Studies, 4925 Dufferin Street,Downsview, Ontario, Canada, M3H 5T6 3 )_ /

II CON..OLLINbpfFICE N ISAND A DrESS 12. REPORT DATE

Air irce UTfice orrcientc Research/NA, E

Bldg. 410, Bolling Air Force Base, 13. NUMBER OF PAGES

DC 20332, U.S.A.14. MONITORING AGENCY NAME & ADORESS(if different from Controlling Office) IS. SECURITY CLASS. (of this report)

UNCLASSIFIED

5CX Q (2AS 150. DECLASSIFICATION/DOWNGRAD14r.SCHEDULE

16. DISTHIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the obstract entered in Block 20, It different from 'Report)

4

II. SUPPLEMENTARY NOTES -.

19. K WORDS (Continue on reverse side It necessary and identify by block number)

1. Dusty-gas flows; 2. Two-phase flows; Boundaryolayer flows$4 Partial-differential equations; 5') Numerical analysis. / r

20 ABSTRACT (Continue on reverse side If nec, isary end identify by block number)

An asymptotic analysis is given of the compressible, laminar boundary'layerflow of a dilute gas-particle mixture over a semi-infinite flat plate. Theanalysis extends existing work by considering more realistic drag andheat,transfer relations than those provided by Stokes. A more generalviscosity.temperature expression is also incorporated into the analysis. The

Continued

DD I JAN 17 473 UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE 014te.n De, Enrere .

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SECURITY CLASSIFICATION OF THIS PAGE(Whon Deta Enterod)

solution involves a series expansion in terms of the slip parameter of theparticles. The numerical results, including the zeroth and first'orderapproximations for the gas and particle phases, are presented for the twolimiting regimes: the largeslip limit near the leading edge and the small-sliplimit far downstream., Significant effects on the flow produced by the particleswith Stokes' and non4Stokes' relations are studied and clarified. The effectsof some nondimensional similarity parameters, such as the Reynolds, Prandtl andEckert numbers, on the two phase boundary'layer flow are discussed. ICAcoc.

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ASYMPTOTIC SOLUTIONS TO COMPRESSIBLE LAMINAR BOUNDARY-LAYER EQUATIONS

FOR DUSTY-GAS FLOW OVER A SEMI-INFINITE FLAT PLATE

by

B. Y. Wang and I. I. Glass

Submitted February, 1986

Dfl.4- co"l

August, 1986 UTIAS Report No. 310CN ISSN 0082-5255

Acknowledgements

We are pleased to express our thanks to Prof. J. J. Gottlieb andDr. W. S. Liu for their constructive review of the manuscript and manyuseful suggestions.

One of us (B. Y. Wang) is grateful to the Institute of Mechanics,Academia Sinica, Beijing, China, and to UTIAS for the opportunity to doresearch work during 1984-1986.

The financial assistance received from the Natural Sciences andEngineering Research Council under grant No. A1647, the U.S. Air Force undergrant AF-AFOSR-82-0096, the Defence Nuclear Agency under contractDNA 001-85-C-0368, and the Defence Research Establishment, Suffield (DRES),is acknowledged with thanks.

r-U"

ii

~. -

Abstract

An asymptotic analysis is given of the compressible, laminarboundary-layer flow of a dilute gas-particle mixture over a semi-infiniteflat plate. The analysis extends existing work by considering morerealistic drag and heat-transfer relations than those provided by Stokes. Amore general viscosity-temperature expression is also incorporated into the

- analysis. The solution involves a series expansion in terms of the slipparameter of the particles. The numerical results, including the zeroth andfirst-order approximations for the gas and particle phases, are presentedfor the two limiting regimes: the large-slip limit near the leading edgeand the small-slip limit far downstream. Significant effects on the flowproduced by the particles with Stokes' and non-Stokes' relations are studiedand clarified. The effects of some nondimensional similarity parameters,such as the Reynolds, Prandtl and Eckert numbers, on the two-phaseboundary-layer flow are discussed.

-J.

.p '.,

',p

,V, ii i

Table of Contents

-=I Page

Acknowledgements ii

Abstract iii

Table of Contents iv

Notation v

1.0 INTRODUCTION 1

1.1 Motivation for the Present Study 11.2 Previous Work 11.3 Present Study 21.4 Basic Assumptions 2

2.0 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS 3

2.1 Governing Equations 32.2 Boundary Conditions 9

3.0 LARGE-SLIP APPROXIMATION 10

3.1 Transformation of Boundary-Layer Equations 103.2 Expansion in Terms of the Slip Parameter (x*/XK) 123.3 First-Order Problem and Interaction Terms 16

4.0 SMALL-SLIP APPROXIMATION 19

4.1 Basic Equations in Terms of Slip Quantities 194.2 Transformation of Basic Equations 204.3 Expansion in Terms of the Slip Parameter (xtJx*) 22

5.0 RESULTS AND DISCUSSIONS 26

5.1 Large-Slip Limit 27

5.2 Small-Slip Limit 31

6.0 CONCLUDING REMARKS 35

REFERENCES 36

FIGURES

APPENDIX: RELAXATION PROCESS AND VELOCITY EQUILIBRIUM TIME

iv

Notation

a, coefficient in Eqs. (5.4) and (5.20)

a2 coefficient in Eq. (5.4)

bi coefficient in Eqs. (5.5) and (5.21)

b2 coefficient in Eq. (5.5)

cI coefficient in Eqs. (5.6) and (5.22)

c2 coefficient in Eq. (5.6)

C p specific heat of a gas at constant pressure

cs specific heat of a particle material

cv specific heat of a gas at constant volume

CD drag coefficient for a sphere in viscous flows

CDo Stokesian drag coefficient for a sphere in viscous flows

d particle diameter

V. D normalized drag coefficient

Dx x-coinponent of the drag force per unit volume acting on thegas

Dy y-component of the drag force per unit volume acting on thegas

Ec gas Eckert number based on freestream temperature

f transformation function for gas velocity

f p transformation function for particle velocity

F(I) first-order function defined in Eq. (3.34)

k heat conductivity of a gas

M Mach number

Nu Nusselt number based on particle diameter

p gas static pressure

Pr gas Prandtl number

V

4w rate of heat transfer at the wall

Q total heat transfer per unit volume to gas from particles

R gas constant

Rep particle Reynolds number based on freest eam velocity andparticle diameter

Res slip Reynolds number based )n particle slip velocity andparticle diameter

Re, flow Reynolds number based on freestream velocity andvelocity-equilibrium length

S Sutherland constant

T gas static temperature

T p particle temperature

Ts temperature defect between gas and particles

u x-component of gas velocity

up x-component of particle velocity

us x-component of particle slip velocity

U(1) first-order velocity of gas, defined in Eq. (3.35)

v y-component of gas velocity

Vp y-component of particle velocity

vs y-component of particle slip velocity

x horizontal coordinate along the wall

y vertical coordinate normal to the wall

Greek Symbols

a ratio of specific heats of two phases

P mass loading ratio of particles

y ratio of specific heats of gas

6 boundary-layer displacement thickness

n similarity variable for boundary-layer solutions

vi

first-order temperature of gas, defined in Eq. (3.36)

velocity-equilibrium length

dynamic viscosity of gas

v kinematic viscosity of gas

p density of gas phase

pp density of particle phase

Ps density of particle material

'v velocity-equilibrium time

TT temperature-equilibrium time

w shear stress at the wall

dissipation function due to the relative motion ofparticles in a gas

I@p stream function for particle phase

*Subscripts

w wall conditions

freestream conditions

o reference values

Superscripts

* dimensional quantities

(0) zeroth-order quantities

(1) first-order quantities

first-order derivative with respect to similarity variable

N.)

second-order derivative with respect to similarityvariable n

modified properties for gas-particle mixture

vii

1.0 INTRODUCTION

1.1 Motivation for the Present Study

Gas and solid-particle flows are encountered in many different fields.Typical examples occurring in nature are dust storms, forest-fire smoke andthe dispersion of solid pollutants in the atmosphere. Many processes inindustry utilize gas-particle flows, such as transportation of pulverizedmaterials in pneumatic conveyors, separation and classification of particlesin cyclone or other separators, fluidization in chemical reactors, and

combustion of powdered fuels in combustion chambers. In addition, gas flowswith suspended solid particles have various applications in science and

engineering, for example, satellite drag, ablation, MHD generators, solidpropellant rockets, laser-Doppler anemometry and blast waves moving over theEarth's surface.

For some applications in pipe or nozzle flows and 'lows over bodies,the behaviour of such two-phase flows at a solid surface is extremely

important. Hence, it is necessary to study boundary-layer flows of agas-particle mixture. From solutions of gas boundary-layer equations, it ispossible to determine the effects of solid particles on the boundary-layercharacteristics, say, shear stress, heat transfer and boundary-layer

growth.

The problem considered in this report is the laminar boundary-layer

flow over a semi-infinite flat plate in a compressible gas containinguniform, spherical solid particles. This study provides basic physicalinsight into the flow of such a two-phase system, even though the solutionis for a relatively straightforward problem. Moreover, as a parallel study,the asymptotic solution can be used to compare with finite-difference

solutions and to verify independently the correctness of the

finite-difference scheme [I].

1.2 Previous Work

Several authors have worked on the problems of two-phase boundary-layerflows. Most of these analyses were based on the assumption of an

incompressible fluid [2-16]. Singleton [17] first treated the case of acompressible dusty-gas boundary-layer flow. He derived the governingequations and obtained asymptotic solutions for two limiting regimes: thelarge-slip regime near the leading edge and the small-slip regime fardownstream. However, he assumed Stokes' relation for the drag force and

heat transfer, which is valid only for the case where the particle slipReynolds number is of order unity. He developed his governing equationsassuming that the gas viscosity-temperature relation has the special form of

1*/'* = VT*/T*, and gave his solutions for the case where the Prandtl andEckert numbers of the gas are equal to unity.

A "

1.3 Present Study

The present analysis will extend Singleton's analysis to the moregeneral problem of compressible laminar dusty-gas boundary-layer flows overa semi-infinite flat plate. It will present the basic equations underconditions that the drag and heat transfer Ibetween the two phases may havedifferent relevant forms instead of Stokes' relation and that the powerindex in the expression for the viscosity coefficient c;n have arbitraryvalues from 0.5 to 1.0. The paper will give the numerical results in thetwo limiting regions at several values of the Prandtl number, Eckert number,Reynolds number and the viscosity power in je<.

1.4 Basic Assumptions

The basic assumptions are as follows:

(1) The gas is perfect. The specific heats of the gas are constant. ThePrandtl number of the gas is constant. The viscosity and heatconductivity of the gas have a power-law relation with the gastemperature.

(2) The solid particles are rigid spheres of uniform size. The numberdensity of particles is sufficiently high to treat the particle phaseas a continuum. However, the particles are also sufficiently dilute toconsider them as non-interacting.

(3) The particles have no random motions and therefore the particle phasedoes not contribute to the static pressure of the two-phase system.

(4) The volume fraction of the particle phase is assumed as negligible.* This implies that the coefficient of viscosity for the gas-particle

mixture can be taken as the viscosity of the gas phase alone.

(5) The specific heat of the particle material is constant. Its thermalconductivity is much larger than that of the gas and hence thetemperature inside each particle can be assumed uniform.

(6) There is no radiative heat transfer from one particle to another.There is no chemical reaction, no coagulation, no phase change in thetwo-phase system. There is no particle deposition on the surface ofthe plate.

(7) Only the processes of drag and heat transfer couple the particles tothe gas. The drag coefficient and the Nusselt number for a singlesphere in a viscous flow are assumed valid for the particle cloud.Other force interaction terms, such as lift, buoyancy and gravity, areneglected.

2

ILI

(8) There is no dry friction as the particles slide along the wall. Theslowing down of the particle motion is only due to the gas whosevelocity decreases to zero at the wall.

(9) The two-phase flow is steady. The flow Reynolds number is sufficientlyhigh so that a laminar boundary-layer forms on the surface of the flatplate, but lower than a critical value so that no transition toturbulence occurs.

(10) The usual boundary-layer assumptions are still valid for the two-phasesystem and consequently the variation of pressure across the boundarylayer can be neglected. In addition, for the flat-plate problem, thereis no pressure gradient in the external flow. The particle phase andgas phase in the external flow are in equilibrium.

2.0 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

2.1 Governing Equations

Let x* and y* be the distance along and normal to the wall,respectively. The origin is fixed at the leading edge of the plate. Thegeometry of the problem is sketched in Fig. 1. The conservation equationsfor steady two-dimensional laminar boundary-layer flows of a compressiblegas-particle mixture over a semi-infinite flat plate are as follows:

For the gas phase:

Continuity:

6 p*u* + k1. p'v* = 0 (2.1)()x* by*

Momentum:

p*(u* au_* + v* u*) - ( , *)+ Dx (2.2)

-6x* by* )y*

Energy:pc*u* 6T* + v* 6T*) k (k* ) + 2.

(k. + ) + Q (2.3)ox* 6y* )*y* 6y* (

State:

p* = p*R*T* (2.4)

3

For the particle phase:

Continuity:

x- u + v = 0 (2.5)

x-momentum:

P(u* _P + v =k) = -Dx (2.6)px* p y*

y-momentum:

p*(u* 'P + v* =v) -Dy (2.7)Px * Pay*

Energy:

*P sPFX ~ c (u* -a + V* MP. -Q (2.8)

In Eqs. (2.1)-(2.8), the interaction terms between the gas andparticles can be expressed by (see Appendix):

U*- U*

Dx = p* P D (2.9)p V

V* - V*

Dy = p* P D (2.10)lp T*

(u*- u*)Dx + (v* - v*)Dy (2.11)

Q = P * - T* Nulp s tt 2(2.12)

where

CDOD (2.13)

4'

V .. a ~ ~ ***% .-. . .

CD = 24 (2.14)

-p d*2 (2.15)18 *

12k* c: (2.16)

Here, D represents the real drag coefficient CD normalized by the

Stokesian drag coefficient C0 o and Nu is the Nusselt number based on theparticle diameter. According to the assumption (7), D and Nu determine the

gas-particle interaction. When the interaction law between the gas and

particles just has the Stokes form, D = 1.0 and Nu = 2.0. It is well known

that the Stokes relation is valid only for small slip Reynolds number oforder unity. In general, D and Nu are both functions of the Reynolds number

and Prandtl number. From the definition of slip Reynolds number and Prandtl

number,

p*/(u* - U*) 2 + (V* - v*) 2 d*Re-4 (2.17)

C*

Pr = _ (2.18)

The local equilibrium-time parameters, c and 4, are a measure of the

relaxation process. For example, the velocity equilibrium time - is thetime elapsed for a particle to reduce its relative velocity to e- of itsoriginal value if the force accelerating (or decelerating) the particle

toward the gas velocity is given by the Stokes drag. These two localparameters are functions of the local gas temeprature since the viscositycoefficient p* and the heat conductivity k* are functions of the gas

temperature. It is convenient to introduce an equilibrium length X,*, which

is based on the freestream parameters:

8 - d u* (2.19)184*

The two-phase relaxation process takes place throughout the equilibrium

length. Therefore, it is reasonable to choose X,* as the characteristiclength of the dusty-gas boundary-layer problem. Then the equilibrium-timeparameters can be expressed in the form

5

(2.20)

rt 3Pr S 2.12 C* ~L 2.1

Thus, the basic boundary-layer equations (2.1)-(2.8) become

a p*Uj* + -L- ..p*v* = 0 (2.22)

pi (* !L + v* 6u* ) = .- _ (4* - L-) + (U* u*) -*I 0 (.3ax Y* y* by* P p ~ (223

(y* PrT + * +~

pp

X* 3Py Pr pj* 6y

p* *) p* (2.25)Ut L

+ pp * D*+ . 0(.6C* p p iikp

+ __ p*(T -(u*- u* Nu!~ (2.27)

a v* u2 D(* * (2.28)* P * P(*

av* *

U* -p.+ v*~~P a ~(T* - T*) ~1Nu (2.29)P ax* p aY* 3Pr P 1(

6

where a is the ratio of specific heats of the two particles:

a = P (2.30)c*

For boundary-layer flows, the normal component of velocity is usually a

small quantity. It means that the contribution of the normal velocity is

often neglected compared with the tangential velocity. Then, the expression

for the slip Reynolds number, Eq. (2.17), becomes

p*Iu* - u*Id*Res = (2.31)

and the gas energy equation (2.24) reduces to

P _ ( + v ~) = L +~*~* j PK*)_ X -y*T Pr by* by* c* 6y

pp*u T x- T +v T)y r )y u** - T * u* *c _

+ -I (u* - u*)2 u* * D + L (T* - T*) u * Nu (2.32)C* P 3Pr PX*

In order to obtain a closed set of equations, it is required to specify

the expression for 4*(T). From the standpoint of kinetic theory of gases,

the most exact relation for the viscosity of a perfect gas is Sutherland's

form [18]:

* T+ S* T* 3/2 (2.33)

T*S* o*

where S* is the Sutherland constant. This form of Sutherland's relation,however, is not suitable for the series-expansion method which is used in

the present analysis. The other viscosity form used in many analytical

solutions to the boundary-layer equations is of power form, which is written

as

(* (2.34)

7

where w is the power index which lies between 0.5 and 1.0. This relationfor the gas viscosity is readily applied to the series-expansion method, asshown later.

From the basic assumption (10), the pressure is constant throughout the

boundary layer:

p* = constant (2.35)

With this condition (2.35) and the gas state equation (2.25), the gasdensity can be expressed in terms of the gas temperature:

p# = T * (2.36)

p# T*

Substituting Eq. (2.36) into the basic equations (2.22), (2.23),(2.26)-(2.29) and (2.32) with the power relation for the gas viscosity(2.34), the following equations are obtained:

* b * + av*) = u* 6T* + v* 6T* (2.37)

U*b* xau * Uy**xu * + v* _* ; v.(T*) _ [(T*)w 8] + (u* u*) u* (T*i D

-= y T ya p- (T - (2.38)

u* T* +v* -T* _V* (T*) 2 [(T* )" T*] + v* T* * Au* 2

xy* Pr T, y T*" y* cp " -

+ P2 (u* - u* ) 2 u* T* )w.1 1 _ T T* ) u * (T *.) lX + P + 2. 1 * Nu (2.39)

3Pr T

x P; u* + v* = 0 (2.40)

u* -P + v* -_ (u* - u*) - ( (2.41)P Ox p ay* p TXc T*CD

8

v+ v* = -(v - v*) U *)w D (2.42P bx* P Sy* pX,

uv + v P T - (T* - T*) u T- Nu (2.43)

Fx* P by* 3Pr P T"

Under the conditions of w 0.5, D = 1.0 and Nu = 2.0, the equations(2.37)-(2.43) reduce to those derived by Singleton [17].

Physically, the boundary-layer flow-field of a two-phase mixture can bedivided into three distinct regions (see Fig. 1). These regions are divided

according to the nondimensional slip parameter x*/X* as follows: thelarge-slip region (x*/X* << 1), the moderate-slip region (x*/X. - 1), andthe small-slip region (x*/X* >> 1). Following Singleton [17], a

small-parameter expansion method was used to solve the boundary-layerequations for dusty gases. Only the asymptotic solutions in the twolimiting regimes can be obtained by this perturbation technique: thelarge-slip approximation for the near leading-edge solution and thesmall-slip approximation for the far-downstream solution, respectively.Clearly, the large-slip regime is characterized by a frozen flow where the

gas and the particles move independently, while the small-slip regime ischaracterized by an equilibrium flow where the gas and the particles movetogether (see Fig. 2).

2.2 Boundary Conditions

The boundary conditions for the gas phase are:

(1) At the wall, there is neither slip in velocity nor jump in

temperature:

u*(x*, 0) = 0, v*(x*, 0) = 0, T*(x*, 0) = T* (2.44)

(2) As y* approaches infinity, the flow parameters must match those in theexternal flow or the freestream:

u*(x*, -) = ut, T*(x*, D) = T* (2.45)

The boundary conditions for the particle phase are:

(1) At the wall, there is no mass transfer

9

0v(x*, O) = 0 (2.46)

(2) As y* approaches infinity, the flow parameters must match theirfreestream values:

u*(x*, -) U* , T*(x*, *) Tp, P*(x*, -) = P* (2.47)

Since the particles and gas are assumed to be in equilibrium in the externalflow, the freestream parameters for the particle phase can be readilydetermined as

u* = ut,, = T*, p= P (2.48)P. P. p

where 0 is the mass loading ratio of the particles. Otherwise, thetwo-phase external flow must be solved first in order to obtain the outer

4 boundary conditions for the particle phase if the particles are not inequilibrium with the gas in the freestream.

3.0 LARGE-SLIP APPROXIMATION

3.1 Transformation of Boundary-Layer Equations

-For the large-slip region, it is convenient to define a stream functionSp for the particle phase:

prp up 'P = (3.1)

Pp*=~P - (3.2)p P x*

Then the continuity equation for the particle phase, Eq. (2.40), issatisfied automatically.

In this region, the following nondimensional flow variables andfunction transformation are chosen:

.1

. p4 l

x/ Y* (3.3)X* / V* Xx : , - 2v*®x*

u* 2x* v*, T p (3.4)u= , v *c* T* p*

f -nu p (3.5)

*,X *

Pp , Tp T (3.6)P p P= = T*

P~*

=(3.7)

With Eqs. (3.1)-(3.3) and (3.5)-(3.6), the particle velocities can be

expressed as

p pp b) n

v=k* ppc~ .x 2x2P4 - (3.8)

V* V2vt ut x* 1 bf a

p Pp 6x 2 x 2x by(.9

Substituting the above expressions, the basic equations (2.37)-(2.39)and (2.41-(2.43) are transformed into the following form:

T (AL + u L I =u -T f 6T (3.10)x 2x 2x o ax 2x -3.

u -u f i L T () (Tw Wu)+ 3T(L1(6p ppu)D (3.11)6x 2x 6) 2x - N n.1

4 11

au + u i a)f 2 T T0 a)+E T() ' 2D-1 (Tw 6T )+I w(- ) + Ec - - ppU

ax 2x 2x 3n 2xPr brn T 2x bnpp n

+ L PpTw(Tp - T)Nu (3.12)3Pr

Pp 2f (2f f 6P a~ PpC -x -. (-- - 2x =g)]fa~n ox bn 2x br 2 6) ox 2x bn

- (,P+ LE .. T, L (E-Pp _ _ _ox 2x 2x 3n 6T2 an on

( (3.13)

p ap

bf f 1 bf f 2f

P TI 6X 2 X ax 4x 2 2x U3x71 4x 2 6TI

+ (2!k 6p - E. 5Pp!RR + - L

an ax 6x byn 2x 6TI ax 2x 2x 6

(fp + .) 2 f 2ffp_)-___It_) pp2 T W(R + f I (f__2f_ = p2r('+LE-_T E + _a_ fp pp )D

P 3x xabn 2x ),2 2x 2x on 2x ppU -2x(3.14)

fp TE _Tp ( - ppT(Tp - T)Nu (3.15)

; bx n ax 2x 3Pr

where Ec is the gas Eckert number based on the freestream temperature,

u*2

Ec = C (y - I)M. 2 (3.16)

3.2 Expansion in Terms of the Slip Parameter (x*/X*)

The following expansion in terms of the slip parameter (x*/X,*) is

made:

12

f(x, n) f(O)(n) + xf(1 )(.i) +

u(x, rn) u(O)(n) + xu(1)(7) +

T(x, n) T(O)(r) + xT( 1 )(n) + ... (3.17)

fp(x, T f(p + xf p)(n) +

Tp(x, n) = T(O)(n) + xT(1 )(n) +

pp(x, n) p(O)(n) + xpl)(n) +p p

Putting the expansion (3.17) into Eqs. (3.10)-(3.15) and equatingcoefficients of (x)n where n = o, 1, ... , the zeroth and first-orderequations for the large-slip limit are obtained as follows.

The zeroth-order problem is:

f(O)' - T(0 ) f(O) - u(0 ) = 0 (3.18)T(0)T, (O)

u(0) + (f(O) + T(0)' )u(0 )' = 0 (3.19)T(O) (&I T _O

T(O),,+ f(O) + wT(O) )T(0) + Ec Pr (u (0)' )2 0( .0

T(O) + T(0 )

f(O) = (3.21)

T(O) = 1 (3.22)

p

13

I. .. .- ~ , w - . , , . . ". ..- .€ .--- -

P(O 1 (3.23)

with the boundary conditions

f~(O) )=0 u(0)(0)= 0; u()- 1; T(0)(O) =Tw; T(O)(-)= 1 (3.24)

f~0 )(O) =0. f(. ) =1; ()'-~

p( ()- (0)(-) = 1 (3.25)

In fact, the equations for the particle phase, (3.2l)-(3.23) , are algebraicexpressions which represent the zeroth-order solution for the particles.The boundary conditions (3.25) were already used in Eqs. (3.2l)-(3.23).

The first-order problem is

f(l)' _ T(O)' f(1) - 3uI + (2u(0) + T(O~f0)T 1 ()T 1 0)

T(0 ~T(O) T() T(O) (3.26)

UM1+ () + W r(0)')u(1). - 2 u 1O u(l)-c*) f(O )u(O)'

T()T(O) T(O) T(+

+ T0 (O) 'L ] (1 ) + 61 T( O) I( ) + u O) f+ l

-2 P(1 - u ())D (3.27)

TO"+ (Pr f() + 2w T(O)' )T(W) - [(&*1I)Pr + W (C)'

()* ]T() TE~ ~(O)'ulr 0 ' fl

+2Pr __ ]T1___cru)u' + Pr fl

-2f3EcPr(1-u(0 JD - 20(i T(O) )Nu (3.28)

3

14

TI2 -~ 2,lp( 1 n (1) 3f (1) 2T( 0 )(- - f ( 0 ) )D (3.29)pp p p

In2 3nf (1) + 3f(l1 -2T(0 )'rnj(0 ) -f (0 ) D (3.30)

TTl.- 2T( 1 ) 2a TP (O)w(r(O) - I)Nu (3.31)

where the normalized drag coefficient D and the Nusselt number Nu are givenby their zeroth-order approximations. The boundary conditions are:

f~')(0) =0; u(1 )(0) = 0; u(,)(-) = 0; T(l)(0) = 0; T(1)(-) =0 (3.32)

f~l)(0) = 0; f(l)(-) = 0; T~(),- 0; P( 1 )(_) = 0 (3.33)p p p ' p

From the first-order approximate equations, Eqs. (3.26)-(3.31), it isseen that the loading ratio of the particles p appears oly in the equationsfor the gas phase, i.e., Eqs. (3.26)-(3.28). In addition, it is possibleto yet a more general solution which is suitable for any value of theloading ratio, by introducing the new variables:

T)-f(l) ( n)I/ (3.34)

u~'~(n)- u()(T)/p(3.35)

(1)(r) - ( 1~(~)/p(3.36)

Substituting the expressions (3.34)-(3.36) into Eqs. (3. 26) -(3. 28) , thefirst-order equations for the gas phase become

F(1)' - T(1 FM = 301) - (2u(0 ) + T~0 (O) 202) (1 f(O) (1). (3.37)T (0) T(O) T(0 f T(O)

U1 + fO)W+ + __ u (0))UI' W1+ \ T + FOF U T(U) u~)Continued

15

% A....... . , . .

. . ' .. ..... . .

[(wI) f(O+u(OT_(+ ) u(0)' (1) - (O)' E(1)

T(O)w 2 T() 2 T (0 )

- u(O)'F(1) - 2(1-u(0))D (3.38)

T (T) ( 1)'

E (1) + (Pr f(O) + 2w() (1) - [(.+l)Pr f(O)T(O)- +T(O) ft) *2

+ 2Pr u(O) E)(1)

- -2EcPr(u(O)'u (I) ' ) - Pr T(0)' FM 2EcPr(l-u(O))2D _ 2 (IT(O))Nu•)', TO 3 r ~ F 1 iTO N

T (3.39)

Similarly, the boundary conditions (3.32) take the form

SF(I)(o) = 0; U(1)(0) = 0; U(1)(-) =1 0 ; E)(1)(0) = 0; 8)(1)() = ?3.40)

3.3 First-Order Problem and Interaction Terms

00;From Eqs. (3.18)-(3.23) and (3.26)-(3.31), it is found that theinteraction terms between the gas and particles appear only in thefirst-order approximation. In other words, the zeroth-order equations canbe solved without knowing the interaction relation between the two phases.However, in order to obtain the first-order solution, appropriateexpressions for the drag and heat transfer between the two phases should be!i!!given.

As mentioned before, when solid particles move through a gas at verylow relative velocities, that is, when Res < 1, the Stokesian form can beapplied. For the case where Stokes' relation applies, then D = 1.0 andNu = 2.0. When the relative, or slip, velocities between the particles andthe gas increase to a higher value, Stokes' relation is not valid.Therefore, a relevant form for the interaction should be assumed for thenon-Stokesian case. In the case of larger slip velocities, it may be

16

V °

reasonable to apply the following drag and heat-transfer relations[19, 20]:

C0 = 0.48 + 28 Res 0 " 8 5 (3.41)

Nu = 2.0 + 0.6 Pr1/3 Res 1/2 (3.42)

The drag coefficient CD and Nusselt number Nu for the non-Stokesiancase in this analysis, given by Eqs. (3.41) and (3.42), are functions of theslip Reynolds number Res as well as the Prandtl number Pr. Comparing thenon-Stokes and the Stokes relations indicates that these two cases are inagreement only for the slip Reynolds number of order unity or less. As theslip Reynolds number becomes larger than unity, the Stokes relationunderestimates the drag and heat-transfer between the gas and the particles.In general, the non-Stokes relation agrees with the standard drag curve [21]much better than the Stokes relation. Correspondingly, the normalized dragcoefficient is

D Re + Z Re 0.15 (3.43)50 6 s

In the series-expansion method, the slip Reynolds number Res should beexpanded just as the other quantities. Neglecting the first-order smallquantities, the slip Reynolds number can be expressed as

Re= Re I - u (0 ) (344)PT( 0 ) U+ 1 (.4

where Rep is the particle Reynolds number based on the freestream velocity

pt u* d*Re = (3.45)

The zeroth-order approximations for D and Nu can be obtained bysubstituting Eq. (3.44) into Eqs. (3.42) and (3.43). Then the first-orderequations can be solved numerically. The equations for the yas phase, Eqs.(3.26)-(3.28), consist of second-order, ordinary-differential, simultaneousequations with two-point boundary values, similar to the case of thezeroth-order equations for the gas phase. The solution to the equations forthe particle phase, Eqs. (3.29)-(3.31), can be obtained in the integralform:

17

p(1) = -2n 2 f (x - f(0) )Ddx - 2f (( 0 ) - f(O) )Ddxxo 4h C X 4

f T(0) xu( 0 ) - f(O) )Ddx (3.46)

f 1)..3.n2 f I ( xu(0) f(O) )Ddx + f T(O) (xu(O) - f(O))Ddxp x4 ( x2 (3.47)

f(l) = _13 f I0 (xu(O) - f(O) )Ddx + Y f r) T2 (xu( 0 ) - f(O) )Ddxx D

O x2 (3.48)

2a f1n T=.02 (T(O) - 1)Nu dx (3.49)3p Pr x

The quantity f(l)' df()/dn, given by (3.47), can be used to give the

p p ,gie

fir t-order approximation ofthe tangential velocity for the particle phase,UM1 . From Eq. (3.8), the x-component of particle velocity can be given bythe derivative of the transformation function fp(n):

Up : 1 fp (3.50)p p anl

In addition, in the large-slip limit, the nondimensional density of thep ricle phase, p, is of order unity since the zeroth-order solution ispp 0 = I, i.e., Eq. (3.23). Substituting Eqs. (3.17), (3.21) and (3.23),Eq. (3.50) yields the series-expansion form as

up =1 + x(fC1 )' - PMl) )+* (3.51)pp p

4 Clearly, the first and second terms in Eq. (3.51), represent respectivelythe zeroth and first-order approximation of the tangential particle velocityu P*

4.0 SMALL-SLIP APPROXIMATION

4.1 Basic Equations in Terms of Slip Quantities

For the small-slip region, it is convenient to employ slip quantities

as dependent variables since they are small quantities of first order withrespect to the slip parameter (kt/x*). The slip quantities are defined as

u* = u* - u*, v* = v* - v*, T* = T* - T* (4.1)s p s p s p

Putting Eq. (4.1) into the basic equations (2.37)-(2.43) and making

some algebraic manipulation results in

T. (* + bv*= u* 61* + v* 6T* (4.2)-x* by - bx by -

-bu vu* ,, . v , u* P.P + v* = Q (43)

P*(-S +-s+_ +6 p* -6p* +U p+V 43

by* b xy* 6x* by* x* by*

u * u*u* b uu uuU 6 + v* +*-- v* + u+* + v*P** s y* & * 'y*- s 5- vs by*

+ (1 + P )(u* +V u* t) [(IT* -w u* = (44)px* 6y p* oy* T, by*

Ju* +v* bu* * bu u* bu * _ [ (T_)W J* ]

a x* y x* by* S y* pS Ey* , y*

u + u T* cw * (4.5)= t 0- IT' p. Us

u y+ vb 64* u* - bv* * a *U L +-, v* s+ u* + v* _+,. *s x* 6y* bx* 6y* 6 y x* y

Continued

19

u* T Wv s(4.6)

p~c*" - Ts- 5 s

_____ , v _

ax* y* O* y* )

+ (1 + -*u 3T* + v* 6TT*) + T u 2_p c xy* Pr p* *y* "T* y* cS p T ";yu 6 * 6T.x,.* 4 7

u* C* FJ* * y - *

Z -" ' . ~~ ~ ~~ * 6 (T* ' )T] W IT* .*2

= _ __ ~~~u* (T )~. 2- u* (L (TIT 48

*c b-* T Pr p* by Cp bT-y cs * * b

sml-l p rein

- *, us (T.7

P* *. T ~ * s~;

Pr p*' + u* * p Tc* T, 6T*

u* T v* + u* ST +* T*

20 2

", "

-T U* ()"u2 _ 2 U (1 + fpil") (T* )WT* (4.8)p** * T* 3Pr X* p* c* T*

The conditions of D =1.0 and Nu =2.0 are already employed in Eqs.(4.2)-(4.8), since the slip velocities are always small quantities in thesmall-slip region.

4.2 Transformation of Basic Equations

Let

u v,~ 2(1+p)x* v*, T T* P p* T (4.9)

20

s U*' vs = 2(1i0)x* v*, Ts L p _L (4.10)

and

2* / *Xx= (--, q: 2 ux y* := - v (4.11)

(4.12)

As in the large-slip limit, substituting Eqs. (4.9)-(4.12) into Eqs.

(4.2)-(4.8), the dimensionless basic equations can be obtained:

T 2 + u T - T f -u T + L -T 0 (4.13)x 2x 2x )n x 2x 3n

.Us _ s 1 Vs au u I + P -_j + s - us

2x )n 2x n x 2x 2x ) Tx 2x 6n

6- u = - : 0 (4.14)2x 3) U) 2x 3Tn

aU T uU T) u LS + uUs- i

appT u s _- u u s + u+ s us -

Us 2x 3n 2x 3n ax 2x 3tl ax 2x l

vs s (_ f L A -' 'TL (T u) 0 (4.15)

+ s-~ + (1 + P )(2x -1 p N 2x )n 2x 3T 1 n

_ s u + s f Js + u t q u s

x 2x 3n 2x n x 2x3n U 2x n

+ + - ) = -11+0pT)T' s (4.16)2x )n 2x )T " Tp

21

T v _ s vs "s v svs

u~ ~~ Uu -- 2 - nq- f +-- L f f + us Sus -D- + ss

ax ax 2x 2x bn 2x OTI Sx 2x byn 2x n 2x

+ s U f 2 + n bu-ff +u L4 f -n +Lx 2x us A + x u L - - uus (4.-x 2x a xn 2x 2x 2x 2x T

Vs Vf + u (= -T s (4.17)2x bq ax 2x TP 6n

SuT u L+ T + u('T s - L T-*+u OT++u - + 2) ( 4.1

oxx S TX an

x 2-x san- 2x 6-n ax 2x on ax

Vs -- )T +T ( + {pp T) (u 61 fL kT)- + T -L (T W-

2x 2r xx 2x 6- Pr 2x 6n

-Ec+P T"l (b )2 = Ec PppT)T'lus2 (4.18)

Us T_ _us T + _V T+ u .!- - f- Ua s __ Vs ST 2-s f ST TsT-s + Vs Tsax 2x b--l 2x a)n Ox 2x b'n o)x 2x s-r-) 2x byl

+ 1+p3 T a (Tw OT) + Ec !+1P T4'*'I(_ILI) 2

2x TPr , ' T) CY 2x b'n

= ECop pTulu s2 _ 2 ( + ppT) TOTS (4.19)3Pr

4.3 Expansion in Terms of the Slip Parameter (X ./x*)

In the small.-slip lilmit, perturbation expansions are made in terms ofX /x* :

f(x, n)) = f (0)(i) + I f(1)(1) +x

22

u(x, TI) = u(0)(T) + ! u(1)Cn) +x

T(x, T)= r(O)( ) + I T(1 )(n) +X

Us(x, T) = Iu Us () + ... (4.20)x

Vs (x, T) v ) .+x

Ts(X, n) 1T (1 ) +x

p(x, n)= p (0) 1 -- ) pp () +

where x = x*/,* is the nondimensional slip parameter.

Then, the small-slip approximation for the zeroth-order problem leadsto

f(O) _ T(O)' f(O) - u(O) = 0 (4.21)T(O)

U(O)" + + +(' +W T(O)' u(O) = 0 (4.22)T(O)" + 1 T(0 )

"_ _ _ __ _ _ T(0 ) T(0)' (0 )

T(O) + (Pr 1 + P/ f(O) + .T )T( + EcPr(u 2) 0 (4.23)1+ P T( 0 ) +I T( )

(0) =( (4.24)

23

J -x'.- .

with the boundary conditions

f(O)(o) = 0; u(0 )(o) = 0; u(O)(-) = 1; T(O)(0) = Tw; T(O)(-) = 1

(4.25)

For the small-slip approximation, the zeroth-order solutions of the particle

velocity and temperature are the same as those for the gas and thefirst-order solutions are given by the first-order slip quantities. Thezeroth-order density for the particle phase is given by Eq. (4.24). Theboundary condition for the particle phase density

P (0, ) (4.26)

has been used during the derivation of Eq. (4.24).

The first-order equations for f(l), u( 1 ), T( 1 ) and p(1) are given by

S., ),uOl ( ) f O I ' u 1

f(l) _ T(O)' f(1) : T(O)' f(O)T(1) + 2 U,+ _- T -

(0T(O) T()2 TO T(O) (4.27)

u(1),,+ f(O) + WT(O)WT(O) u(1), + 2u(O) (1)

T(O) w 1 T(O)

I [(0)2U(O) _ 2 f(O)u(0)u(O) + ( f-) f(O) (0)u(O)

I T(O) * 2T(O)' T(O)w T(O)

f(O 0 1) f f(O)u(O)' (T(1)+ T(O)p I ) )

TO)2 T(O)t.+ 2 T(0)

TM + (Pr 1 + f(O) + 2 w)(0) T (0) + [2Pr 1 + B/au(O)-+P T(O)I w+1 1+B

Continued

24

N.. .. ... .. .... ..... . .. .. .... ..

-rw(l + P/ a) + 1 T(')' f(O) - ]+0 T() T(0)1- T(0 )0+

-Pr O/C 1 [nu( 0),T( 0 ). . (~()() + 3(2w-1)Pr/ax-2 f_____2_T()_1_

9Pr f(0)u(0 )T(0 )' + 3Pr 2 1 + P/, f(0)3 T(0)'+ 3EcPr 2 f_____2_____

a- T(0) "1 a 1+0 T (0) 2t4la T(0 )w

-EcPr 0 f (0) 2U(0) , - Pr 1 + P/a T(O). f(l) - Pr _ /a f(0)T(0 )' (1)

- 2EcPr(u( 0),u( 1) ) (4.29)

f(0) p(1)' 2uO + T(O, ____(I

f(0)u(0), rmj(O)u(O)' + -2 f( 0)u( 0 )T(O) 0) _______ -_2

T(O) (*1 ()W1 2 T(O w- TO L

+ J~[Pr 1 + P/ f(O)Tr(0 ' + EcOr(u(O). ) 2 f (0) 2 *- w41 O _______

2 1+0 (),l T(O) u 2 2 T0'+

+ f(O)T(O) TM - 2 u(0) TM1 f(') T~l). (4.30)

T(O)' () T( 0)2

with the boundary conditions

f(1 )(O) 0 ; u(l)(Q) =0; u(l)(-) 1; T(l)(0) =0; T(l)(,,) =0

(4.31)

25

% . -

The first-order problem in the small-slip limit is determined by thesecond-order, ordinary-differential, simultaneous equations with two-pointboundary values. In this aspect, it is similar to the first-order problemin the large-slip limit. But the solution to the above equations(4.27)-(4.30) is difficult to obtain, since these equations are too complexand have high coupling. In addition, Eq. (4.30) has a singular point at n =

0 Moreover, al lqinted out by Singleton [17], regard ess of the choice ofu I ) (0) and T (0), the resulting solutions to u((ii) and T (1)n) fromEqs. (4.28) and (4.29) always approach zero as n approaches infinity, makingit impossible to pick out the correct solutions. Therefore, in thisanalysis, it was not attempted to obtain the first-order solutions to Eqs.(4.27)-(4.30) but just the zeroth-order solutions to Eqs. (4.21)-(4.24)which are of more practical interest. The first-order problem for theparticle velocity and temperature are readily obtained. With the seriesexpansion (4.20), the slip quantities of first order are given as

1..-,,(1) f(O)u(O)

us = fT(u~w (4.32)

v (1) = 1 (r(O)2 + Tf(O)u(O) - f(O)u(O) - T(0 ) f(0) 2 (4.33)

s 2T(O)w T(0)

TO) -(i) f(O)T(O)'

s 4a TO)") (4.34)

Equations (4.32)-(4.34) are not differential but 1 ebraic equations. Thevalues of three slip quantities us ' vs and T - at any given point aredetermined from the zeroth-order solutions for the gas phase.

5.0 RESULTS AND DISCUSSIONS

The zeroth and first-order equations for the large-slip and small-sliplimits can be solved numerically. They are a system of nonlinear,second-order, ordinary-differential equations. The corresponding boundaryconditions are specified at the two end points, i.e., at the wall and theouter edge of the boundary layer. Mathematically, it is a two-pointboundary-value problem and it can be solved by Gear's method [22].

*,1.."

5.1 Large-Slip Limit

From Eqs. (3.18)-(3.20), the zeroth-order problem for the gas phase in

the large-slip limit is as simple as that for the boundary-layer flow of apure gas without particles. Therefore, as for the conventional viscousflows of a pure gas, the Reynolds number, the Prandtl number, the Eckertnumber and the viscosity power index are important controlling parameters in

the analysis of compressible, laminar, boundary-layer flows of a

gas-particle mixture. The numerical solutions of Eqs. (3.18)-(3.20) aregiven in Figs. 3 to 5 and the influence of the parameters Pr, Ec and w on

the flow properties are shown in these figures and it is seen that their

effects are relatively small. Here, it is not necessary to discuss thezeroth-order solutions for the gas phase in detail since it is the same asthe similarity solution for the flat-plate boundary layer of a pure gas.Similarly, the zeroth-order solutions for the particle phase are readily

obtained. Equations (3.21)-(3.23) indicate that, in the zeroth-orderapproximation, the particle motion in the boundary layer remains uniform.All the zeroth-order flow quantities for the particle phase (density,

velocity and temperature) are the same as those in the freestream or the

external flow. This is due to the fact that both the gas and the particles

move independently of each other in the zeroth-order problem. The influenceof the particles on the flow properties is prevalent only in the first orhigher order solutions. It is a major feature of the two-phase

boundary-layer flows in the large-slip region.

Figures 6 to 8 show the first-order solutions for the gas phase, i.e.,

the solutions to Eqs. (3.37)-(3.39). They are the numerical results for the

Stokes case and the effects of Ec, Pr and w are significant this time. For

the first-order problem, the same value of the flow parameters Pr, Ec, w andTw were chosen as in the zeroth-order problem where Pr = 0.69-1.0, Ec

0.1-1.0, w 0.5-1.0 and Tw 0.5. For the non-Stokes case, the numerical

results with Pr = 0.69, Ec = 1.0, w = 0.67 and Tw = 0.5 are presented in

Fig. 9. They cover quite a wide range of the particle Reynolds number(Re = 0.1-100.0) and the changes are very significant. For the particlephase, the first-order solution can be obt3ined by numerically integrating

Eqs. (3.46)-(3.49). The results for the Stokes case with a = 1.0 are giy

in Figs. 10 to 12. It is seen hlS significant changes occur in pp and T ' j

with w, Pr and Ec and in f 1 with Pr and Ec. The results for ehep

non-Stokes case are shown in Fig. 13 where the changes with Rep are even

-. more significant. The computations for the particle phase is carried out

under the same conditions as those for the gas phase.

By comparing the results for the non-Stokes case with those for the

Stokes case, it is seen that the results based on the Stokes relation arereasonable qualitatively. They present the similar tendency of variations

in the flow properties, such as velocity and temperature. However, they are

not correct quantitatively, especially for the large particle Reynolds

number, as expected. Nevertheless, this comparatively simple case of the

Stokes relation is still considered in many analyses of dusty-gas flows

27

since it is useful for understanding the main characteristics of two-phase

flow phenomena.

From the solution to the first-order problem, it was found that thereexist significant differences in the first-order flow profiles between thetwo phases. For instance, the first-order velocity of the gas is positive

and, while passing across the houndary layer from the outer edge to thewall, it increases first to a maximum value and then decreases to zero (seeFig. 7). By -ontrast, the first-order velocity of the particles is negativeand its magnitude increases monotonically from the outer edge of theboundary layer to the wall (c.f. Fig. 11). This arises from the fact thatthe mechanisms of motion for the two phases are not the same. There are twokinds of forces exerted on the gas: the viscous force by the gas and thedrag force by the particles. For the particle phase, however, only the dragforce of the gas influences its motion. Therefore, after entering theboundary layer at the leading edge, the gas decreases immediately itstangential velocity from the freestream value at the outer edge to zerovelocity at the wall due to viscosity. Since the density of the particlematerial is much greater than the gas density, the particles cannotaccommodate this rapid deceleration but tend to slip through the gas as theydecelerate. It takes some time for the particles and gas to adjust to anequilibrium state. It implies that in the large-slip region near theleading edge, the gas has small deviation from the pure-gas boundary-layer

flow while the particles have small deviation from their original state ofuniform motion in the freestream. The particles are 'frozen'. This

situation is justified by the zeroth-order solution, which represents thecomplete frozen-flow limit. The relaxation process takes place throughoutthe equilibrium length X*. In the meantime, owing to the slip velocity, thedrag force arises between the two phases and then the first-order flow is

induced by this gas-particle interaction. The gas is accelerated and theparticles are retarded. This is the reason why the two phases have theirfirst-order velocities in opposite directions. While traversing theboundary layer from the outer edge to the wall, the slip velocity increasesand then the first-order velocity for the two phases both increase first inthe region near the outer edge, since the drag force is proportional to theslip velocity. The first-order velocity of the particles continues toincrease in magnitude on approaching the wall, since the particle motion isdriven only by the drag force. However, for the gas phase, in the regionnear the wall where the velocity gradient for the gas is great, the viscous

force prevails and the no-slip condition at the wall forces the gas velocityto go to zero. Thus, the first-order velocity of the gas decreases in theinner boundary layer and vanishes at the wall. A similar argument is valid

for the first-order temperature profil ' by employing the correspondence ofthe temperature to the velocity, the heat conductivity to the viscosity andthe heat transfer to the drag force.

Finally, for boundary-layer analyses, there are three characteristicquantities of interest: the shear stress at the wall Tj, the heat-transfer

rate at the wall 4, and the displacmeent thickness 5*. As usual, they are

28

. ' 'wS . l J lJ i W l r , , m ~ J ' m " m".. . ........ ' " - , " - ' " " , " " " - ' , " m w "

. " " , " " " " , " • " ' ' " -

determined from the flow profiles of the gas phase:

00

by :qw = -(k* T*) b 6* = f (1 - P* u* )dy* (5.1)W0 pt U

It is convenient to introduce the following nondimensional characteristicquantities:

Tw R- w_ : 6* /e (5.2)

where Rex is the flow Reynolds number based on the freestream velocity u*,

and the velocity-equilibrium length X*,

Re,= P Xwc (5.3)

Then, the nondimensional boundary-layer characteristics can be expressed

as:

w = a1(1 + x"a2 + ... ) (5.4)

w= bj(1 + x~b 2 + .. ) (5.5)

6 = Vx c1(1 + x~c 2 + ... ) (5.6)

where is the nondimensional viscosity of the gas at the wall

6)

'w : W rw(5.7)

29

" CI I"4 W w , ~ - , a • • . •/ ..

and the coefficients a,, a2, bl, b2, cl and c2 are given by

a, = u(0)'((), a2 = u(0 )'(O) (5.8)

T(1)'(O) (5.9)

b i = T(0 ) '(0), b 2 = T(O)(O)(59

u ( ) " Go u (O )T (1 ) - T (O )u ( 1 ) d n 5 . 0c= f (1 - 9-"i)dn, c2 = f (5.10)

o T(0 ) 0 T(O) 2

From the above relations, it is known that the coefficients a, b, and c,

are determined only by the zeroth-order solution and that the coefficient

a2, b 2 and c 2 depend on the first-order solution as well as the zeroth-order

solution. In fact, the first three coefficients, i.e., al. b I and cl, give

the zeroth-order approximation of the three characteristics which is the

same as for the similarity solution of a pure gas. The other three

coefficients a2, b2 and c2 represent the first-order modification owing to

the presence of the particles. These coefficients can be estimated from the

numerical results using eqs. (5.8)-(5.10). In Table 1, the listed values

are the coefficients for the case where the flow parameters are Pr = 0.69,

Ec = 1.0, w = 0.67 and Tw = 0.5.

Table I

Coefficient Values for Boundary-Layer Characteristic Quantities

Stokes' non-Stokes' Case

CaseRe=0.1 Re=1.0 Re=10.0 Re=100.0

al 0.5472 0.5472 0.5472 0.5472 0.5472

a2 1.888 1.599 2.312 3.728 9.870

b 0.4382 0.4382 0.4382 0.4382 0.4382

b2 2.275 2.058 2.827 4.487 11.21

c 1 1.101 1.101 1.101 1.101 1.101

c2 -0.6672 -0.4515 -0.7152 -1.148 -2.748

30

,, ' ,,: ...<-.'./.,'.',.-'..'.':+.,'..'? '.-"-. .';..,.v, ,,r.-. ,",.. .', -+ ---.',..-, .. .' .'.-. ',....; .,.- A,.-..>.A..+.'

From the values given in the above table, it is seen that the shear stressand heat-transfer rate at the wall in the case of dusty gases become greaterthan those in the case of pure gases and the displacement thicknessthinner.

5.2 Small-Slip Limit

The zeroth-order equations for the gas as well as for the particles inthe small-slip limit, Eqs. (4.21)-(4.23), are similar to the conservationequations for a pure-gas boundary-layer flow but with modified properties.Physically, the small-slip approximation represents a quasi-equilibrium flowand the zeroth-order problem constitutes the exact equilibrium limit wherethe particles are 'fixed' to the mass of the gas so that the gas andparticles move together like a perfect-gas mixture. For the dilutetwo-phase system with the approximation of negligible volume fraction of theparticles, the particles contribute to the mixture density but not to theviscosity [23]:

(I+ P) p* (5.11)

= (5.12)

The other thermodynamic properties are given by

c= c* 1 + 3/a (5.13)

k* = k* (5.14)

Then the modified similarity parameters can be expressed as

Pr =p. = Pr 1 + 3/a (5.15)I+Pk*

Ec- c - Ec (5.16)

li..

c* T*Cp TO

31

.................................. ...........

Substituting the modified Prandtl number and Eckert number into Eqs.(3.18)-(3.20), which are exactly the same as the boundary-layer equationsfor pure gases as mentioned before, the resulting 'modified' equations arejust the zeroth-order equations for the small-slip limit, Eqs.(4.21)-(4.23). It implies that in the zeroth-order approximation, thegas-particle mixture behaves like a pure gas with modified thermodynamicproperties. In this paper, the numerical solutions in the small-slip limitare calculated for the condition of a = 1.0. Under this condition, thezeroth-order equations (4.21)-(4.23) for the small-slip limit reduce tothose for the large-slip limit, Eqs. (3.18)-(3.20). The results are givenin Figs. 3-5. Clearly, in the small-slip limit, the zeroth-order velocityand temperature for the particles are the same as the ones for the gas.From Eq. (4.24), it is found that the nondimensional density of the particlephase p( ) is equal to the nondimensional density of the g phase p(O) (orthe reciprocal of the nondimensional gas temperature 1/TUv). It is seenfrom Fig. 14 that p(O), or p( , varies monotonically from its maximum valueat the wall to its freestream value at the outer edge. In this report, theconstant wall temperature Tw is specified as Tw = 0.5 and then thedensity at the wall is equal to 2, as shown in Fig. 14. From Eqs. (4.9) and(4.10), the densities in the dimensional form are given by

p* = p* p, p = pp* pp (5.17)

Neglecting the small quantities of first order, the above gas and particledensities are approximated by

p, = p (0) (0), p(O ) = * 0 p (5.18)

pp

Therefore, at all points of the boundary layer in the small-slip region,

- =(5.19)

p *

It means that the constant loading ratio of the particles holds across thewhole boundary layer in the small-slip region. In other words, the solidparticles remain attached to their original gas mass and always movetogether with this gas mass. The two-phase system behaves like a gaseousmixture. It is a major feature of the gas-particle flow in the small-slipregion.

The slip quantities u( 1 ) , v(1) and TO1 ) are given in Figs. 15 to 17,where the effects of the flow parameters Pr, Ec and w on the first-order

32

~ ,~% %'I V A~ .. r V* N,

flow of the particles are shown. In fact, the slip quantities represent thefirst-order approximation for the particle phase. It is seen that theprofile of the normal slip velocity v ( 1) is different from that of thetangential slip velocity us At the outer edge of the boundary layer, thetangential slip velocity becomes zero but the normal slip velocityapproaches a finite value, since the boundary conditions for the tangentialand normal velocities are different in boundary-layer analyses. As in theusual boundary-layer problems, the tangential particle velocity at the outeredge of the boundary layer should be equal to the freestream value and thenthe slip velocity should become zero at the outer edge. However, no similarboundary conditions at the outer edge can be specified for the normal

velocity. The unique boundary condition for the normal velocity is that itis equal to zero at the wall. Owing to the continuity equation, the normal

.velocity is induced and approaches its maximum value at the outer edge.Therefore, the normal slip velocity at the outer edge takes a finite valuewhich is the difference between the normal velocities of the two phases atthe outer edge. In addition, by comparing Fig. 15 with Fig. 11, it is found

that there exist significant changes in the first-order profile for thetangential velocity of the particles in the two limiting regions. In thesmall-slip region, the particle slip velocity at the wall is equal to zero,while in the large-slip region, the first-order velocity of the particleshas its maximum value in magnitude at the wall as mentioned earlier. As a

result of the maximum slip velocity, the interaction term between the gas-and the particles has its maximum value at the wall and then the maximum

deceleration of the particles takes place along the wall. Hence, at some

distance from the leading edge, the particle velocity at the wall reduces tozero and is equal to that of the gas. After this point the particles keep

their zero velocity at the wall because of the zero slip velocity. Thisspecial point, where the particle velocity becomes zero at the wall, isdefined as the critical point for the gas-particle boundary-layer flow. Atthe critical point, the dusty-gas boundary layer essentially fulfills thetransition from the quasi-frozen flow to the quasi-equilibrium flow. Thetwo-phase flow in the small-slip limit is a typical example of aquasi-equilibrium flow. By comparing Fig. 17 with Fig. 12, the samesituation happens to the first-order temperature of the particles in the twolimiting regions: at the wall, the first-order temperature has its maximumvalue in the large-slip region and the temperature defect vanishes in thesmall-slip region. A detailed discussion is omitted here, since it issimilar to the above case of velocity. Similar to the large-slip case, itis interesting to obtain the expressions for the boundary-layer

characteristic quantities. However, in the small-slip case, since there areno solutions available to the first-order equations of the gas, they can beexpressed only in zeroth order. Similarly, the three boundary-layercharacteristics are given in the nondimensional form:

w = -a- (/+_a ) (5.20)Vx

33

4w (4+0b1) (5.21)/x

6 = X (ci (5.22)

/I +

The coefficients a,, b, and c1 in Eqs. (5.20)-(5.22) have the same values asin Table 1. By contrast with the large-slip limit, the zeroth-orderexpressions for the boundary-layer characteristics in the small-slip limit,Eqs. (5.20)-(5.22), involve the effects of the particles. In fact, the term

,/1+0 in Eqs. (5.20)-(5.22) represents the alteration of the boundary layerby the particles. In the small-slip limit, the two-phase system acts like asingle gaseous system with modified properties as pointed out before. WithEqs. (5.9)-(5.10), the 'modified' similarity variable 1 becomes

-/ u* , ! (l+ )U*y

1 = y* * /I+ (5.23)

2 * 2v*x*

This implies that the boundary-layer flow of a dusty gas in the small-sliplimit corresponds to a similarity solution with the normal scale modified by

the factor Y1+0, owing to the particles. Consequently, the shear stress andheat-transfer rate at the wall increase and the displacement thickness

decreases by the same factor of /I+3. Therefore, it can be concluded thatthe presence of particles enhances the shear stress and heat-transfer at thewall and thins the boundary layer in the two limiting regions. Thistendency can be seen in Figs. 18 and 19, where the shear stress and

*heat-transfer rate are shown as functions of the nondimensional distance xfor the cases with and without particles. As expected, the results for thelarge-slip limit and the small-slip limit coincide with the pure-gas resultsin the limits x + 0 and x + -, respectively. Note that the large-slip

*. results when x > 0.1 and the small-slip results when x < 10 are meaninglesssince the aysmptotic solutions are not valid. Physically, the changes inthe characteristics caused by the particles can be explained as follows.The gas-flow profiles with and without particles are schematically shown inFig. 20. As a result of the interaction, the gas velocity and temperature

* increase in the cold-wall case (say, Tw = 0.5). Then the derivatives ofthe gas velocity and temperature with respect to the normal coordinate y* atthe wall become greater than those without particles. These changes resultin an increase in the shear stress and heat-transfer at the wall, since theyare proportional to those derivatives. In addition, the boundary layer

34

" . '"* -'.~N %

becomes thinner since the velocity approaches its freestream value morequickly.

The numerical results for the asymptotic solutions using theseries-expansion method were compared with the finite-difference solutions

. in the two limiting regions. The agreement between the asymptotic solutionsand the difference solutions was excellent [1].

6.0 CONCLUDING REMARKS

Some general conclusions obtained from the asymptotic solutions of theflat-plate boundary-layer flow of a dilute gas-particle mixture aresummarized as follows:

(1) The asymptotic solutions to the dusty-gas boundary-layer equations canbe obtained using a series-expansion method. They describe thelimiting properties of two-phase flows in the large-slip region and thesmall-slip region, which are characterized by a frozen flow and anequilibrium flow, respectively. The asymptotic solutions are inexcellent agreement with the finite-difference solutions.

(2) The interaction between the gas and particles determines the flowproperties of the particle phase, and influences strongly the flowproperties of the gas phase in addition to the viscosity. When theparticle slip Reynolds number is high, a proper expression for the dragand heat transfer between the two phases should be specified instead ofusing Stokes' relations. The results when using Stokes' relations arereasonable qualitatively but not correct quantitatively for a Reynoldsnumber greater than unity.

(3) For a given gas-particle system with specified values of the massloading ratio and the ratio of the specific heats of the two phases,similar to the case of high-velocity viscous flows of a pure gas, allof the following parameters are important for the analysis ofcompressible laminar boundary-layer flows of gas-particle mixtures:Reynolds, Prandtl and Eckert numbers as well as the transport

properties (viscosity and heat conductivity).

(4) For compressible, laminar, !)oundary-layer flows of dusty gases, theshear stress and the heat-transfer rate at the wall increase and thedisplacement thickness decreases when compared with the correspondingresults for a pure gas. Owing to the presence of the particles, thegas velocity and temperature increase on a flat-plate boundary layerwith a cold wall. As a result, the velocity and temperature gradientsat the wall for the gas phase increase so that the shear stress andheat transfer are enhanced and the velocity achieves its freestreamvalue at a shorter distance from the wall so that the displacementthickness of the boundary layer is decreased.

35

'4,. - A7----

REFERENCES

1. Wang, B. Y. and Glass, I. I., "Finite-Difference Solutions forCompressible Laminar Boundary-Layer Flows of a Dusty Gas Over aSemi-Infinite Flat Plate", UTIAS Report No. 311, 1986.

2. Marble, F. E., "Dynamics of a Gas Containing Small Solid Particles",Combustion and Propulsion, 5th AGARD Colloquium, Pergamon Press, 1963.

3. liu, J. T. C., "Flow Induced by the Impulsive Motion of an InfiniteFlat Plate in a Dusty Gas", Astronautica Acta, Vol. 13, No. 4, 1967,pp. 369-377.

4. Soo, S. L., "Non-Equilibrium Fluid Dynamics - Laminar Flow Over a FlatPlate", ZAMP, Vol. 19, No. 4, 1968, pp. 545-563.

5. Zung, L. B., "Flow Induced in Fluid Particle Suspension by an InfiniteRotating Disk", The Physics of Fluids, Vol. 12, No. 1, 1969, pp.18-23.

6. Otterman, B. and Lee, S. L., "Particulate Velocity and ConcentrationProfiles for Laminar Flow of a Suspension Over a Flat Plate", HeatTransfer and Fluid Mechanics Institute, Monterey, California, June

10-12, 1970 Proceedings, Stanford University Press, pp. 311-322.

7. Lee, S. L. and Chan, W. K., "Two-Phase Laminar Boundary Layer Along aVertical Flat Wall", Hydrotransport, Vol. 2, 1972, A4.45-A4.58.

8. DiGiovanni, P. R. and Lee, S. L., "Impulsive Motion in a Particle-FluidSuspension Including Particulate Volume, Density and MigrationEffects", Journal of Applied Mechanics, Vol. 41, No. 1, 1974, pp.35-41.

9. Soo, S. L., "Fluid Dynamics of Multiphase Systems", BlaisdellPublishing Co., 1967.

10. Tabakoff, W. and Hamed, A., "Analysis of Cascade Particle Gas BoundaryLayer Flows with Pressure Gradient", AIAA Paper No. 72-87.

11. Hamed, A. and Tabakoff, W., "The Boundary Layer of Particulate GasFlow", Zeitschrift fur Flugwissenchaften, Vol. 20, 1972, pp. 373-381.

12. Jain, A. C. and Ghosh, A., "Gas-Particulate Laminar Boundary Layer on a

Flat Plate", Z. Flugwise Weltraumersch, Vol. 3, 1979, pp. 379-385.

13. Hamed, A. and Tahakoff, W., "Analysis of Nonequilibrium ParticulateFlow", AIAA Paper No. 73-687.

36

14. Prabha, S. and Jain, A. C., "On the Use of Compatibility Conditions inthe Solution of Gas Particulate Boundary Layer Equations", AppliedScientific Research, Vol. 36, No. 2, 1980, pp. 81-91.

15. Osiptsov, A. N., "Structure of the Laminar Boundary Layer of a DisperseMedium on a Flat Plate", Fluid Dynamics, Vol. 15, No. 4, 1980 pp.512-517.

16. Prabha, S. and Jain, A. C., "On the Nature of Gas-Particulate Flow",13th International Symposium on Space Technology and Science, Tokyo,Japan, June 28-July 3, 1982, pp. 517-522.

17. Singleton, R. E., "The Compressible Gas-Solid Particle Flow Over aSemi-Infinite Flat Plate", ZAMP, Vol. 16, 1965, pp. 421-429.

18. Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-UniformGases, Cambridge University Press, 1961.

19. Gilbert, M. Davis, L. and Altman, D., "Velocity Lag of Particles inLinear Accelerated Combustion Gases", Jet Propulsion, Vol. 25, 1965,pp. 26-30.

20. Knudsen, J. G. and Katz, D. L., Fluid Mechanics and Heat Transfer,McGraw-Hill, 1958.

21. Schlichting, H., Boundary-Layer Theory (Seventh Edition), McGraw-HillInc., 1979.

22. Gear, C. W., Numerical Initial Value Problems in Ordinary DifferentialEquations. Prentice-Hall, Englewood Cliffs, 1971.

23. Rudinger, G., Fundamentals of Gas-Particle Flow, Elsevier ScientificPublishing Co., 1980.

,4

37" X "":',," "

mIMP"

APPENDIX

RELAXATION PROCESS AND VELOCITY EQUILIBRIUM TIME

In gas-particle flows, generally speaking, the gas and the particlesmay have different velocities and temperatures, and as an immediateconsequence the two phases must interact. Every particle experiences a dragforce and exchanges heat with the gas. The same happens to the gas but inthe opposite direction. Consequently, the velocities and temperatures tendto approach each other, and the instantaneous rate of this approach dependson the instantaneous velocity and temperature differences between theparticle and the gas. This phenomenon, which is usually known as therelaxation process, is a major feature of gas-particle flows. A discussionof the viscous drag between the gas and the particles and of the velocityrelaxation process will illustrate the relaxation processes in a two-phase

system..4

For the case of a single spherical particle in a viscous flow, if the

particle velocity Up (here the unstarred quantities denote dimensional flow

properties) is different from the gas velocity U, the viscous drag exertedby the gas on the particle depends on the relative, or slip, velocity

(Up - U). As usual in fluid mechanics, the drag F is expressed by means ofa drag coefficient CD which is defined in terms of the dynamic head of therelative flow and the frontal area of the particle:

F = CD{ P(U -1 IU - UpI d} (A.1)

-.9where p is the gas density and d is the particle diameter. On theright-hand side of Eq. (A.1), the square of the relative velocity is writtenin this manner to ensure that the drag force always has the correct sign.The drag coefficient CD is a function of the particle slip Reynoldsnumber

CD = CD(Res) (A.2)

where Res is the slip Reynolds number based on the particle diameter

Res PU - Up d (A.3)

A.1

: -- ' ' '- .A ; I n .I. , } ~, ,'. ; . -. .- " ,. - a-- , ,,-. -r.

where p is the gas viscosity.

For the Stokes case, with a low-slip Reynolds number less than aboutone, the drag coefficient takes the simple form

* CD = 24 (A.4),', o Res

This is the Stokes relation for the drag force. For a higher slip Reynoldsnumber (Res > 1), there are several empirical relations for the dragcoefficient given by different investigators. It is convenient to write

CD = CD 0 D(Res) (A.5)

.'.- oror

D = CD (A.6)

COO

"w where D is the so-called normalized drag coefficient in this analysis. Itis clear that D(Res) = 1 corresponds to Stokes' drag.

The above discussion deals only with isolated particles. Of course, ifthere is an appreciable number density of particles, the effective, orapparent, drag coefficient may be different from that for a single particlebecause of some forms of interaction between the particles, such as directcollisions and particle-wake interaction. At present, there are no adequateanalytical or experimental results available for these interactions. Inthis report, the case of a dilute gas-particle mixture is considered, and it

is reasonable to assume that the drag coefficient for a single sphere isvalid for the particle cloud. Therefore, if the drag coefficient takes theform of the Stokes relation, the drag force on every particle is

F'-= 24 {1 p(U - UpU- d2Re. 2 - Up4

24 In 2>u .u.p Up-Ul d! 8 p' P

3iTpd(U - Up) (A.7)

A.2

Let np denote the number Jensity of the particles, pp the densityof the particle phase, Ps the density of the particle material and m themass of a particle. Then

m P . d (A.8)6

np : = PP_ (A.9)m . d 3

~6

The force per unit volume of the gas-particle mixture acting on the gas,

D iOx + jDy, can be given as

D =-np • F

P 3pd(U U)"S n d 3

p p (A.10)pV

where TV is defined as the particle velocity-equilibrium time

'T Psd 2 (A.11)

The velocity-equilibrium time Tv depends on the density of particlematerial, the particle diameter and the gas viscosity. Some typical valuesof the velocity-equilibrium time are given in Table 2, where the particlematerial density pp = 2.5 g/cm 3 (typical of crown glass).

A.3

Table 2

Typical Values of the Particle Velocity Equilibrium Time (ms)

Gas Particle Diameter d (m)Temperature -- IT (K) 10 20 40

300 0.755 3.02 12.08400 0.605 2.42 9.68500 0.510 2.04 8.16

Physically, Tv is the time required for a particle to reduce the slipvelocity to e- I of its initial value. To examine this criteria, consider aflow field with constant gas velocity and temperature. For the particles

moving in this flow field, Newton's law of motion becomes

m -- = F (A.12)dt

Substituting Eqs. (A.7) and (A.8), Eq. (A.12) becomes

dt m

3nd(U - U.5 *id 3Ps

: 18 (U- Uppsd 2

+"=" U U

(A.13)TV

A.4

gig.

Since the gas velocity and temperature are assumed as constant, i.e., UJ andTv are constant, Eq. (A.13) is readily integrated with the result

t

.6 + +V

U - U U U)e (A.14)

where U Po is the initial velocity of the particles at t = 0. This resultindicates a typical velocity-relaxation process in which the slip velocity

decays exponentially. From Eq. (A.14), it is known that the velocity-equilibrium time Tv is the time elapsed for a particle to reduce itsoriginal relative velocity by e-1. It should be pointed out that, in fact,the velocity relaxation process is not completed in the velocity-equilibrium

time Tv, since the slip velocity still has an appreciable value, i.e.,about 37% of its original value. In addition, if the gas velocity and

temperature are not constant, or Stokes' drag does not apply, Eq. (A.13)becomes more difficult to solve, and the solution is no longer a simpleexponential decay of the slip velocity, Eq. (A.14). Nevertheless, the

parameter Tv remains a convenient measure for the tendency of the

.p particles to reach velocity equilibrium with the gas.

For the non-Stokes case, the drag force on every particle is given by

+ +* + +

F CD{! p(U - Up)U - U 4 2}P

3nTEd(U - Up)D (A.15)

The drag force per unit volume exerting on the gas by the particles is

obtained as

D =-n p F

p-

U -UPp 0 (A.16)

or

Dx p T D, Dy P p v .... D (A.17)lV %V

.1A.A.5

% §.~ c:§§:~~i.~:i-: I

where u and v are the velocity components for the gas, and uD and Vp forthe particles:

-w

+- + - 4 +.-t

U = iu + jv, Up = iu + jV (A.18)

The dissipation function due to the relative motion of the particles is

. = D * (Up - U)

(u u)Dx + (v - V)Dy (A.19)

The heat transfer from a particle to the gas has the form

q = k(imd 2) TP_. T - Nud

where k is the heat conductivity of the gas, Nu is the Nusselt number, andTp and T are the particle and gas temperatures, respectively. Then the

total heat transfer to the gas per unit volume is given by

W n p q

= P k • ndl2 -P- Nu

Ps d 3 d

: Pp 12_k (T - T) Nu

ps 2 2

- T c5 - Nu (A.20)'T

where cs is the specific heat of the particle phase and TT is the

temperature-equilibrium time:

A.6

'" V "% ---" -" ' ,Iw,' "" " "J "" " " ' ,il,;t ;, " M ' "" '. .,ii w .i ',. . .i" "" "-- " " ''",,f.. '.. ,.;..-w'''-"_%" ;0% '4 . ' ' - . .'.% '. w-.:-. ," .. . .,'.-,-.-' -'''w""",= -- .-.. .

T= pd2 c (A.21)

The temperature-equilibrium time -rT can be expressed in terms of thevelocity-equilibrium time 'rv:

=PSd 2 18p.cT 18p 12k C

3 Pr C v (A.22)2- cp

where Cp is the specific heat at constant pressure for the gas and Pr isthe gas Prandtl number:

Pr = (A.23)k

-

A.7

4 it"

i*

FIGURE 1. SCHEMATIC OF A DUSTY-GAS BOUNDARY-LAYER OVER A SEMI-INFINITE FLATPLATE. REGION I: LARGE-SLIP REGION. REGION II: MODERATE-SLIPREGION. REGION III: SMALL-SLIP REGION.

4,

-4

b1

I

I

D LA jot

II

II

Tw rc TT

FIGURE 2. ASYMPTOTIC FLOW PROFILES IN THE TWO LIMITING REGIMES.

()FROZEN FLOW; (b) EQUILIBRIUM FLOW.GAS; -- PARTICLES.

N.I

,

.00

0-~ A

4

0 0 T T

FIGURE 2. ASYMPTOTIC FLOW PROFILES IN THE TWO LIMITING REGIMES.

i (a) FROZEN FLOW; (b) EQUILIBRIUM FLOW... -- GAS; --- PARTICLES.

10.0 T_ - - -- T~T T

9.0 I.I(-0.500S 2. co-0.6668.0 3.&e)-0.770

7.0 [ u =~o

36.0

3~.c PRi:OGO0

3.0 L002.0

1.0(a)

0.c.0 1.0 2.0 3.0 [.0 5.0 6.0 7.0 8.0 9.c 10.0

10. 0~ T r

9.0 1.PR.0.690[ 2.PFR=0.720

6.0 3.PR-0.7504.PR=I .000

7.0- a,

6.0

f 015.0[

~.0

W - .6a. nI EC- 1.000

2.0 7

1.0(b)

a0.c 1.0 ?.0 3. 0 q . C 5.0 6. f 7.09 8.01 9.0 10.0

FIGURE 3. THE ZEROTH-ORDER FUNCTION f(0) FOR THE G;AS PHASE.

(a') Pr = 1.0, Ec =1.0 and 0.5, 0.67, 0.77 AND 1.0,RESPECT IVE LY.

(h) Ec =1. 0, , 0. 67 and Pr 0. 69, 0. 72, 0. 75 AND 1. 0,RESPECTIVELY.

(c) Pr =0.69, =0.67 AND Ec 0.1, 0.4 AND 1.0, RESPECTIVELY.

10. 0t ,

9.0 I.EC-0.1002.EC-0.qO0

8.0 3.EC=1.000

7.0

6.0 -

f 10 5.0

5.0

3.0 4)-0.666PR-0.690

2.0

1.0(c

0- . 0 .- L'i' 5 6 --

0.0 1.0 2.0 3.0 4.O 5.0 6.0 7.0 8.0 9.0 10.0

FIGURE 3 - CONTINUED

1.0 -

0.9 .(0-O.500-i 2. W-0.666

0.6 3.C-0.770- . q.4()-I.000

0.7 34

0.6

u0)0.5

0.14

0.3 PR-1.000

EC-! .000

0.?

0. 0 . . .0 q 0 L5.0 6. 0 1. ) fl() .. 0 9. .1

FIGURE 4. THE ZEROTH-ORDER VELOCITY u(U) FOR THE GAS PHASE.

(a) Pr = 1.0, Ec = 1.0 AND w = 0.5, 0.67, 0.77 AND 1.0,RESPECTIVELY.

(b) Ec = 1.0, w = 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,RESPECTIVELY.

(c) Pr = 0.69, , = 0.67 AND Ec 0.1, 0.4 AND 1.0, RESPECTIVELY.

----- -* 'P . ,, r -;., , - :.-.:-,-..,..:.. -> ,-.....-.-.' ....e. 7. ..

'IR

0.8 1 .PR-0.690

2.PR-0. 7200.8 3. PR-0. 750

1. PR-I. 000

0. 7

U 0.6~

01

0.3 W-0. 666

~ [ EC-1. 000

(b)

.0 1.0 2.0 3.0 q.0 S.0 6.0 7.0 8.0 9.0 10.0

ym.0

*0.9 E-.0

0.9-040

0. 7

0.6

u 10 0.5

0.11

0.3 to-0.666PR-0. 690

0.?

0.1

0. 0.0 1.0 2.0 3.0 q.0 5.0 6.0 7.0 8.0 9.0 10.0

FIGUIRF 4 -CONTINUjEo

1.3

1.2

1.1

1.0

0.9,, 1 c oo-3 2. CO -0. 666

0.A 3.4)-0.7700.8 .W-.00

T (0) 0. 7

0.6

0.5

0.3

0. 3 PR-I.O000

EC-! .000

0.2

0. 1 (a)

0 .0 I I I , I ,0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

FIGURE 5. THE ZEROTH-ORDER TEMEPRATURE T(0 ) FOR THE GAS PHASE.

(a) Pr = 1.0, Ec = 1.0 AND w 0.5, 0.67, 0.77 AND 1.0,RESPECTIVELY.

(b) Ec = 1.0, w = 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,RESPECTIVELY.

(c) Pr = 0.69, w = 0.67 AND Ec 0.1, 0.4 AND 1.0, RESPECTIVELY.

.,V

1.1.

1.3

1.2

1.1

1.0

0.9 !.PR-0.69033 2,PR-O, 720

0.8 3,PR-0,750

(00.74PR-1.000

0.6

4! 0.5

4 0.14

C. 0.3 - (j-0.666EC1.O000

0.2

0.1(b)

V.. 0 .0I I , , , , 6 , .0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

i "

l.1.1 - , ,. . .., , .

1.3

1.2

1.1

1.0

0.9 .EC=0100

20EC.084000.8 3 EC-1 000

0.7T) 0.7

0.6

0.5

0.14

0.3 co-0.666PR-0.690

0.2

0.1(c)

0. .0 1.0 2.0 3.0 q.0 1.0 6.0 7.0 8.0 9.0 10.0

FIGURE 5 - CONTINUED

' - , ' r - . , • , , % % % t . " , , , , ,, % % , , , , , . • . . _

2.5

2.0

1.5

Fill

1.0

1. W-O.5002. ca-0.6663. 0-0.770

0.5 4. cj-1.O00

PR-. 000

(a) EC-1.000

0.0 0.0 1.0 2 .0 3 .0 4.0 51.0 6.0 7'.0 8.0 9.0 10.0

2.5

2.0 , _

1.5

1.0

I.PR-0.6902.PR-0. 7203. PR-0. 750

0.5 '.PR-1.000

-4-0. 666EC-I.000

(b)

0.0 1.0 2.0 3.0 '4.0 5.0 6.0 7.0 8.0 9.0 10.0

FIGURE 6. FIRST-ORDER FUNCTION F(1 ) FOR THE GAS PHASE IN THE LARGE-SLIPLIMIT FOR THE STOKES CASE.

(a) Pr - 1.0, Ec = 1.0 AND w = 0.5, 0.67, 0.77 AND 1.0,RESPECTIVELY.

(b) Ec = 1.0, w= 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,RESPECT[ VELY.

(c) Pr - 0.69, w = 0.67 AND Ec = 0.1, 0.4 and 1.0, RESPECTIVELY.

''' " % ' " '" ""- .'"" " '."" ' JJ"' "".'.;'". .. ' . ''.

2.5

2.0

1.5 2

F

1.0

I.EC-0. 1002.EC-0.4003.EC-I .000

0.5

6) .0.666PR-0.690

(c)

0.00 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

71FIGURE 6 - CONTINUED

0. 6

0.5- t

2 1. -0. 50032. -0. 666

0.4 4 3. w-0.7704. (A-I1.000

.3' 0.3

0.2

PR-I. 000EC- .000

0.1

0.0 .0.0 1.( 4. ~r L0 5.0 G. C, I .e CL 0.

77

FIGURE 7. FIRST-ORDER VELOCITY U(1) FOR THE GAS PHASE IN THE LARGE-SLIP

LIMIT FOR THE STOKES CASE.

(a) Pr = 1.0, Ec = 1.0 AND , = 0.5, 0.67, 0.77 AND 1.0,RESPECTIVELY.

(b) Ec = 1.0, (w = 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,RESPECTIVELY.

(c) Pr = 0.69, w. 0.67 AND Ec = 0.1. 0.4, AND 1.0,

RESPECTI VELY.w"7-p

0. 6 -- - --- - -r --

0.5F

4

0.4

U D. 3

0.2

I. PR-0. 6900.1 2.PR 0 .720

3. PR-0. 7500, ] .PR-I.000

(,W-0.666EC-=.000

(b)

0.0 1.0 2.0 3.0 q.0 5.0 6.0 7.0 8.0 9.0 10.0

0.50-3

0.4

II

I.EC0.300.?

O., Li I.EC-O. 100

4-0.666PR-0.690

c.r . 8 ... . .. . .. , I

~FIGURE 7 CONTINUED

I I ... .II . . .. . .I . . .. 2 . . ... I, - .. ... . .- ... . ... . . .-.. -- . .. . . ... ... . . .....---.

~~~~~1.2 I ., ,

1%~iNI .

"" 1.0

0.9 1.(0-0.500

2. w-0. 666

0.8 3. (00.770q. W-I. 000

0.7

0.5

0.3 __ PR- 1.000EC-I. 000

0.2 -

0. 1 (a)

0.00.0 1.0 2 .0 3.0 q.0 5.0 6.0 7.0 8.0 9.0 10.0

I.

1 .0

0.9

0.6

0.7

t1J. F,

,, 0.

% 0. 4

I PR-0. 6900. 3 2. PR-O. 720

3.PR-0.7500.2. PR.1.0000.?

w-.0.666EC-1.000

0.1 (b)

00.0 3.2. l0.O 5.0 6.0 7.0 6.0 9. T0.0

77

FIGURE 8. FIRST-ORDER TEMPERATURE (1) FOR THE GAS PHASE IN THE LARGE-SLIP

LIMIT FOR THE STOKES CASE.

(a) Pr = 1.0, Ec I.U AND = 0.5, 0.67, 0.77 AND 1.0,RESPECTIVELY.

(b) Ec = 1.0, w 0.67 AND Pr - 0.69, 0.72, 0.75 AND 1.0,RESPECTI VELY.

(c) Pr 0.69, ,, = 0.67 AND Ec = 0.1, 0.4 AND 1.0, RESPECTIVELY.

WI

1.21---

1.0

0.9

0.8

0. 7

9'0.6

0.5

I. EC-O. 1000.3 12.EC-O.4OO

7 3.EC-1.000

co-0.666

0.1 PR-O.690(c)

0.0 1.0 2.0 3.0 4.0 S.U 6.0 7.0 8.0 9.0 10.0

FIGURE 8 - CONTINUED

-~~ .1.FE-. 1002. RE- I. 000

) 3. RE- 10. 00L.RE100.0

Wa-0. 663PR-.4390EC- I .000

I! (a)

SFIGURE 9. FIRST-ORDER SOLUTIONS FUR THE GA S PHASE IN THE LARGE-SLIP LIMIT

FOR THE NON-STOKES CASE WITH Pr = 0.69, Ec =0.b7 ANU Rep= 0.1,1.0, 10.0 AND 100.0. RESPECTIVELY.

(a) FUNCTION F~l); (b) VEL0CIT U(') (c) TEMPERATURE

2.0

1.5 r I.RE-O. 1002.RE-1.O000

3.RE10.00 14. RE-100.0

U'" 1.04

3

W-0.6660. -PRO.o690

EC-I.000

(b)

0.00 .0 2580 0 1.0 2 . ....0 3.0 q-0 5.0 6.0 7.0 8.0 9.0 10.0

2.0

I.RE-0. 1001.5 ~2.RE-1.000

3.RE-10.004.RE-100.0

%3

8&)-0.666

z PR,,O.690O..1, EC=l.000

(c)

. 2. ,.o q. 6. 5.0 7.2 P 1 e .P .q

FIGURE 9 - CONTINUED

Ii, sI0 1

O. 0 :< -- . wo-0. 500

2.o)0-0.6663. 6)-0. 770

-0. I 4.WA-1.O00

-0.2

PR-1.000

EC-1.000

-0.3 -

(a)

0 .0 n r .C 7.C E.0 r, I

FIGURE 10. FIRST-ORDER DENSITY p(1) FOR THE PARTICLE PHASE IN THEp

LARGE-SLIP LIMIT FOR THE STOKES CASE.

(a) Pr = 1.0, Ec = 1.0 AND w = 0.5, 0.67, 0.77 AND 1.0,RESPECTIVELY.

(b) Ec = 1.0, w = 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,RESPECTIVELY.

(c) Pr = 0.69, w = 0.67 AND Ec = 0.1, 0.4 AND 1.0,RESPECTIVELY.

0.0

-0 1 R0 9

-0.1 4C-1. 000

-0.3

(b

C-. 1.7. . . S 0 L.0 -. 6. 0 1. 3. r ljn

0.0 /I.EC-0. 1002. EC .0.4100

3. EC 1. 000

-0.?

t40. 666PR -0 .690

(c)

f.C 1.0 2.0 3.0 4.0 ,.G 6.0 7.0 6.n 9.0 10.[

FIGURE 10 - CONTINUED

- .

-0._0

2.0 0.6663.w.0.777

0 '40- '.€.0 1.000

PR-I.000

EC-1.000 I

(a)-1. i

--- __

1' 01 2'..0 350 4. .0 6.0 7.o 8.0 9.0 10.0

"1

i o~~. 00 ,---- .. p

. i 1.PR-0.690

2.PR-0.7203. PR-0. 750

0.40 4.PR-1.000

fjpt

-0.60

W=0.666EC-I.O00

-0.80IE

- . 0

(b)

-l~~~~~~~o~~ 0 Lo ". o s o ,o 7 o 8 o 9 o ~

FIGURE 11. FIRST-ORDER VELOCITY f(l)' FOR THE PARTICLE PHASE IN THEp

4LARGE-SLIP LIMIT FOR THE STOKES CASE.'4

(a) Pr = 1.0, Ec = 1.0 AND w 0.5, 0.67, 0.77 AND 1.0,RESPECTIVELY.

(b) Ec = 1.0, = 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,RESPECTI VELY.

(c) Pr = 0.69, ' 0.67 AND Ec 0.1, 0.4 AND 1.0,RESPECTI VELY.

L 0 f. =" . .%,

0.00 - "

-0.20

I 1.EC-0. 1002.EC-0. 4003. EC- 1. 000

-0. qo

fp

-0.60

w-0.666PR0. 690

-0.60

(c)

.0 2.0 3.0 q0 0 6.0 .0 6.0 0.0 TO.0

FIGURE 11 - CONTINUED

0. 30 , r-- --

I. a-0. 500

0.20 2. w -0.6663.6c, 0. 770q. lI,000

0.10

41ilT 0.00

0.10,

PR- I. 000

ECi! .000

-0.20

(a)

-0. 3n t.-t -.- . _J. -. 0 2.0 .0 q .(I ) ,. [0 7.0 8.0 9.0 10.0

1

FIGURE 12. FIRST-ORDER TEMPERATURE T(I) FOR THE PARTICLE PHASE IN THEP

LARGE-SLIP LIMIT FOR THE STOKES CASE.

(a) Pr = 1.0. Ec 1.fl AND w 0.5, 0.67, 0.77 AND 1.0,RESPECTI VELY.

(b) Ec - 1.0, ( ) 0.67 AND Pr 0.69, 0.72, 0.75 AND 1.0,RESPECTIVELY.

(c) Pr - 0.69, = 0.67 AND Ec - 0.1, 0.4 AND 1.0,RESPECTIVELY.

kr 1%

1. PR-.6900. ~2. PR..720

1 ~3. PR-0. 750d. PR- I .000

c.::

-0. S

1C -0. 666

3ECl .000

0.20

__1 0. 00

0.3069

-0.20 2 C-.0

-0.0

FIUR 12-CNTNE

5 .0 --------

I. RE-O. 1002.RE - . 000

3. RE -10. 004.RE-100.0

P~l I

&)-0. 6663 PR-JO. 690

2.0

1.0

I.RE-O. 1002. RE-1.0003. RE.10. 00

fpf

PR-0. 690

EC-1. 000

-3. 1 -(b)

0 1.0 2.0 3.0 4'0 5.0 6'.0 7'.0 8'.0 9'.0 10.0

~7)FIGURE 13. FIRST-ORDER SOLUTIONS FOR THE PARTICLE PHASE IN THE LARGE-SLIP

LIMIT FOR THE NOR-STOKES CASE WITH Pr - 0.69, Ec - 1.0, w - 0.67AND Re~ p 0.1, 1.0, 10.0 AND 100.0, RESPECTIVELY.

(a)DESIY p) (b) VELOCITY f(l) ; (c) TEMPERATURE TI

0.0

/ RE-I. 0-1.0.RE-.00

w-0.666PR-0.690

EC-1.000

(c)

-2.07 , 2, .0 c --.r r: e 7.0 8.0 9.0 10.0

FIGURE 13 - CONTINUED

S2 . - --- - - - - -----

1. 1 .(o-0.500 4

2.c-0.666. 3. 4) =0. 770 J

.4.0)-1.000

i 34

' ,'

P-oI

.

000

r PR I.OO0 0

EC-.000- -'(a) 4

I

I77 j?~7 6Q 7

FIGURE 14. ZEROTH-ORER DE1NSITY p(O) FOR THE PARTICLE PHASE IN THESMALL-SLIP LIMIT.

(a) Pr = 1.0, Ec = 1.0 AND w 0.5, 0.67, 0.71 AND 1.0,RESPECTI VELY.

(b) Ec = 1.0, w 0.67 AND Pr 0.69, 0.72, 0.75 AND 1.0,RESPECTI VELY.

(c) Pr - 0.69, = 0.67 AND Ec = 0.1, 0.4 AND 1.0,RESPECTI VELY.

2 . 0 . ..- r ...

1.8 1.PRO.690

2.PR-O. 720

1.6 3.PR-O 7504.PR-1.000

1.4

1.2

1.0

0.8

0.6 -,0. 666EC-1.000

04

0.2 (b)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

2. 0

1.8 I.EC-0.100

2.EC-O.400

1.6 3.EC-1.000-1f

Jq_

1.2

to ] 1

0.8

0.6 60-0.666PR-0.690

0.4

0.2 (c)

0.c'. 0 1.0 2.C 3.' 4.0 S.C, . 0 7.0 8.0 S. 10.0

FIGURE 14 - CONTINUED

0 . I I

0. 13-

0. 12

3

0. 1.1)=0.500

2.(4-0.666

_ , 0.08!- 3. Ci) -0. 7704. -1.000

I0.07-

0.06-

0.05-0. Oq

0.03 PR=I.O000

EC-I.0000.02-

(a)

3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

FIGURE 15. TANGENTIAL SLIP-VELOCITY u 1) IN THE SMALL-SLIP LIMIT.

(a) Pr = 1.0, Ec = 1.0 AND w = 0.5, 0.67, 0.77 AND 1.0,RESPECTI VELY.

(b) Ec = 1.0, w = 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,RESPECTIVEI Y.

(c) Pr = 0.63, w = 0.67 AND Ec : 0. 1, 0.4 AND 1.0,RESPECTI VELY.

0. 14.

0. 13

0.12-

0.11-

0.10-

0.09-

0.08

0.07-

0.06-

0.05-

0.04

1. PR-O. 6900.03 2.PR-0.?20

3. PR-O.750

0.02 'J.PR-i .000

a)-0. 6660.01 (b)EC-l. 000

0.'06o 1.0 2'.0 3.0 4.0 5'.0 6.0 7.0 8.0 9.0 10.0

03. 1 w

0. 131

0 .i~ 11

0.09k 1.EC-0. 1002.EC-0.400

0.083.EC.'1.000

0.06-

0.05

0.03[

w-0. 6660.01-L PR-0. 690

0-001.L0 2.0 31.0 q4. 0 5'.0 6'.0 7'.0 8.0 9'.0 10.0

FIGURE 15 -CONTINUED

1.0

0.9 1. W-0.500

2. C.-0.666

0.8 3. , -0. 770

4.(A)-l.OOO

0.7

0.6 F-V5 0.5

0.4

0.3 PR= .000EC-1.000

0.2

0.1 (a)

0 . 0 --- ---- ----L- ----L0.0 1.0 2.0 3.0 q4.0 5.0 6.0 7.0 8.0 9.0 10.0

,..

1 .0 T r rI...--r - Fr- r -r

0.9 I.PR-0.6902,PR=O, 720

0.8 3.PR=0.7504.PR=I.000

0.7

0.6

0.2

0.1

0 .0 --- - -. - -I - -t_ _ j_- -- -- --

.0 1.0 2.0 3.0 q*0 0.0 ) .0 7.0 C .4 U .0 e ..0

FIGURE 16. NORMAL SLIP-VELOCITY IN THE SMALL-SLIP LIMIT.

(a) Pr = 1.0, Ec = 1.0 AND w = 0.5, 0.67, 0.77 AND 1.0,

RESPECTI VELY.

(b) Ec = 1.0, w = 0.67 AND Pr = 0.69, 0.72, 0.75 AND 1.0,

RESPECTI VELY.

(c) Pr = 0.69, w = 0.67 AND Ec = 0.1, 0.4 AND 1.0,

RESPECTI VELY.

- , ~ - - -IW O =,f'

1 .0 1 1 . . . . . .

0.9 I.EC-O.IO0

2.EC-O.YO0

-1.8 3.EC-1.000

0.7

0.6

V 0.5

O.4

0.3 O. 3 n R,0.660

0.2

o.I (c)

O'0.0 1.0 2.0 3.r q'.0 5'.0 6.,0 71.0 B6.0 9 .0 10.0

FIGURE 16 - CONTINUED

0 . 10 ------

o.ce 1.0=0.500

2. 4) -0. 6660.06 3. W, =0. 770

' I

0.02

-0.02

-0. Cq PR,,1.000EC-1.000

-0. 0CC-

-0. 08-

(a)

0. - 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

72FIGURE 17. TEMPERATURE DEFECT IN THE SMALL-SLIP LIMIT.

4%

(a) Pr = 1.0, Ec = 1.0 AND w = 0.5, 0.67, 0.77 AND 1.0,RESPECTI VELY.

(b) Ec = 1.0, w =0.67 AND Pr = 0.69, 0.72. 0.75 AND 1.0,RESPECTIVELY.

(r) Pr = 0.69, w = 0.67 AND Ec = 0.1, 0.4 AND 1.0,RESPECTIVELY.

0. 08 ..PR 0.690

2. PR-0. 7200.06 3.PR-0.750o o _ . . ._ -. ,, .PR-1.O000

0. 02 4

T_0. 001s

-0.0 u)-0 .66 6

EC-1.000

-0.06

-0.08 4(b)

- 0. 1 o 3.0 q.o 5'.0 6'.0 7.0 8.0 9.0 10.C

0.10

0.08 I.EC-0.100

2.EC-O.oO0

0.06 3EC- 1. 000

0.04

0.02

T 0.00

-0.02

-O.Oq cj-0.666PR-0.690

-0.06

-0.08 (c)

- 0 . 8 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

FIGURE 17 - CONTINUED

5.01

4 .0 1 t !L I t I r I

3.0

2.0 ,,,,

1.0

0.0 10-2 10-1 i0 101 102

xFIGURE 18. NONDIMENSIONAL SHEAR STRESS AT THE WALL AS A FUNCTION OF THE

NONDIMENSIONAL DISTANCE x.

- PURE-GAS SOLUTION; --- LARGE-SLIP SOLUTION FOR THE DUSTY

GAS; SMALL-SLIP SOLUTION FOR THE DUSTY GAS.

4.0 L

3.0

I-II

* I

2.0 U

*I

1.0 I ,

0.0 10?' 10lO c 101 10?

FIGURE 19. NONDIMENSIONAL HEAT-TRANSFER RATE AT THE WALL AS A FUNCTION OFTHE NONDIMENSIONAL DISTANCE x.

- PURE-GAS SOLUTION; --- LARGE-SLIP SOLUTION FOR THE DUSTYGAS; - SMALL-SLIP SOLUTION FOR THE DUSTY GAS.

1 6 ! - - M-

(b)

TT T

FIGURE 20. COMPARISON BETWEEN THE FLOW PROFILES OF THE GAS WITH AND WITHOUTPARTICLES.

(a) VELOCITY; (b) TEMPERATURE.WITHOUT PARTICLES; - WITH PARTICLES.

)Cca g-~c- ci>, CC*LL'~ a

C CCCC 'LL in 0 ~ CCCL CCCCCU Q

2 UZ~C-Z~ a ;tCnXt2 . ai-iCLCOCCCU U 2 3;tc-c-~ u

in CCCC C OCuECCOCLC..CC Ci 41 CLC... IL 41

.~ ~ ' -ci~~tX ~

CCciI-~.~.,g~i -~ g' 3~C L ,nCLO 41

~>' a-0.u i- C .,-~ ci,

CLO '~ L ~C

0 CCSCCt 0L ~ "Cci" 3.

jCi L cC C4*CiA'C _tin inCC~~~ C Li- >~~C C*C CCnCCCCCC CL CC JCC

4 *C, .3

CC4*C-C~,i CX ?~XC rCC'0

CC ~r~2 c~j~ >,n,,! *t- 0 j~: vi'0c. *a"%ffCC < - C~L

LciC CCCCCL - CCaO'CUCX

- S2'S~ Zt~g" DCC ICin 1 C ~ - -- C C Xc

0L

-- CC~~u~ CC- ~I in~~-Z- *

L Mr.,ci Cc-~ ;ciC~.sC; a

0 - a ~ .c-. c a

C t C~CiCS 0. Cin CC CS CiC ~~oS.5~I~ci ~"-* C CC - C'~C4*C0 - . , ~ CCC iii )~in CL CCCCci C CC - C - inC CC Co~ t'

C - ~ C .C Cn'-4 1

r- CS iC- - ~ AS CCC C' n'iC Li-iCiCssCCC C- >~C nci"a~ESa -i C LU ~ CS iji- 44'-'I- C LU . i-~ 4' -J C C C - CL

CC -~ EL - ' C C CL C'C CQLC ~L S

C i CL CCC 0 0

L 4*0 C- Lc-Ccic-C,C' g~ ; :~a~~tCCCCc- ~3 : ta" C ci'~C

L ~a SC .CD ' C0~Cin.~ C *i-iC CC, at I,!ci 41 ~* jul CCI C ~tin C ~ 41

- CCC" - RCiciin acCi ~ ,-i CCC C ,, - ain Cit U -Lu" ~ 0J tn,~n

CCC CC ' CC U-C~C' *c CCC- - C

ciC CCC. * c-C ~CiE ~ '*~ isin ciC 0

Z3CC0

-oSC C Co ~c-

0 41 .. ~ n. rciLn-

4 *. cci -

CC - C- ~ - 44 aL) CC CiCC, tCa~ C .45

ats C ~it ~ C a in ' C -J ' CC C LC C, * .CC..-CCCL C 0 -~ o CC c-C~CL~CCC- -it C CC Ci~.C'-i

0

LCC *~ LO ~ inzciCi-~aCCC-n- CL ~-iC~ - CC CCCCci~ ! =

CL ~SC- - CC ~ C 41 41 C -C Ci~ C CCC *C tC, win

Li-i win ci ci7,,ciC CC .0 41C !C~C~!CLLCC C~ ,. ci CC >-~-C~Eait8XC,

CC C,"' C in Cti-0C~C, i-ti Cit C *~*41C CciC~" ,.L

CCC Cit -0 0

a "'in C C ' 4 *.iL inc-C ',

''in CO CC~oi-C ,L ~ ~ CCC. inC - in~ Ci-a-'-.inc~.C C- C,3 ~ I aC - C C C CcULU

~ 01 ~ Ciin C C *rjtt 0 inC CS"i ~LCCiCCCCJ 0

CC iS--' - "~LC"CC,4*CCS 0.. C,.- -r C, CL C i-~i aor CC, CL C i--- 41 '> aC, C C CCL c-uX 41C-iC CC in CCL c--uC ~ ~'c C- 'LUCC~ ci

-> C-C CC 0 C..- CCCC 0 CC oi-.

0c-'. -

0C ~ i-SC ' .r C,!wLC ~CC in

CC SCC LUi-iC" ~ - c-i -CO cioCCco''~ CCCCLC

0

CC ' C 00 - C C'-'A o~ " ~Ci-.aiCX~CC~ ..C - i-w C - -cC' C - - ci, '1-iC-CS C - C -C~C uCC C CC, CCC C, Lci

in. - - C, COCCC C~.ciC ii 'C Ci- in C * i-)CL ~o CC CCC C, tCCCCi-c- ~ 0 ~ Cci CO C C C C C, C~~C

0 ~C- 0

- CC in - C .CLC CiL~ LL -o in ~ ~ inC, Ci- Din in Ci- C - C-a

LL -o C 3L in Cii-, C~ CC, -C, C Cci C"i- CiCtOOC, . in* CC, CC, C Ci- C,4*Cii-CCCCC, * 41 in C-C Ci- - ~it C, CiL~.rCC~in 41

~ a C, C" c-c CCC ~ Ei- U~i--

tCC flit - ,~.itCCjt NCCC -~j iC i-c- L 9~- ~Ci- C, i-C

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