The steady motion of a spheroidal aerosol particle with inner nonuniformly distributed heat sources (sinks) that is placed in an external temperature gradient is theoretically studied in the Stokes approximationyesBS

Malai, N. V.

Shchukin, E. R.

DSpace at Belgorod State University

On the Thermophoresis of a Spheroidal Solid Aerosol ParticleN. V. Malai and E. R. ShchukinBelgorod State University, Belgorod, 308015 Russia e-mail: malayffi),bsu.edu.ruAbstract—The steady motion of a spheroidal aerosol particle with inner nonuniformly distributed heat sources (sinks) that is placed in an external temperature gradient is theoretically studied in the Stokes approximation. The mean temperature of the particle surface is assumed to differ slightly from that of the gaseous environment. An analytic expression for the force and rate of thermophoresis are found by solving the gas-dynamic equations in view of the motion of the environment.1. FORMULATION OF THE PROBLEMTo date, the thermophoresis o f  spherical aerosol par­ticles has been studied in great detail [1-4]. Many par­ticles occurring both in nature and in industrial plants are nonspherical, e.g., spheroidal. The problem o f  ther­mophoresis o f  a spheroidal aerosol particle has been considered in [5-7]. However, the convective terms in the heat conduction equation were neglected. Praud- man and Pearson [8] for the hydrodynamic problem, as well as Acrivos and Taylor [9] for the heat problem, showed that, away from the sphere, the inertial and con­vective terms became o f  the same order o f  magnitude as the molecular-transport terms. Therefore, normal expansion in a small parameter introduces an error, since it fails to rigorously satisfy the boundary condi­tions at infinity and find a unique exact solution that is valid throughout the flow region even in a second approximation. From the above, it follows that the issue o f  how the motion o f  the medium affects the force and rate o f  thermophoresis o f  a spheroidal particle is o f  the­oretical and practical interest.Consider the steady motion o f  a spheroidal (oblate spheroid) solid aerosol particle with a velocity U in the negative direction o f  the 0z axis. The particle contains nonuniformly distributed heat sources with a density qv The gas is at rest at infinity, and a small temperature gradient V77 is provided by external sources. The tem­perature drop in the neighborhood o f  the particle is assumed to be small; i.e., (Ts -  7'„)/7'„ <§ 1, where Ts is the mean temperature o f  the particle surface and /'„ is the gas temperature away from the particle. In this case, the thermal conductivity, as well as the dynamic and kinematic viscosity, can be considered as constants and the gas, as an incompressible medium. The particle size is considerably larger than the free paths o f  gas mixture molecules; therefore, corrections in Knudsen number will be neglected [3]. Hereafter, the subscripts e and i correspond to the environment and spheroid, respec­tively.We will describe the thermophoresis o f  a particle in the spheroidal coordinate system (e, r|, cp) with the ori­gin at the center o f  the spheroid; i.e., the origin o f  the fixed coordinate system coincides with the instanta­neous position o f  tihe center o f  the particle. The curvi­linear coordinates e, r|, and cp are related to the Carte­sian coordinates by the relations [10]x  = csinhesinrjcoscp, y  = csinhesinr|sin(p,( 1.1)( 1.2)z = ccoshecosr),x  = ccoshesinrjcoscp, y  = ccoshesinrisincp, z = csinhecosr),where c = J b  2 -  a 2 in the case o f  a prolate spheroidI 2 2(a<  b, formula (1.1)) or c = *Ja - b  for an oblate spheroid (a > b, formula (1.2)) and a  and b are the spheroid semiaxes. The 0 z axis o f  the Cartesian coordi­nate system coincides with the symmetry axis o f  the spheroid.In view o f  the above assumptions, the distributions o f the velocity Ue, pressure P e, and temperatures Te and 7] are described by the set o f  equations (1.3) and (1.4) with boundary conditions (1.5)—(1.7):V P e = | ieAU e, divU e = 0, (1.3)PeSe(U e • V ) r e = XeA T e, A r ; = - ? A ,  (1.4)TT c U  coshe U£ = --------  c o s t ],8ct/s in h e  . „  Ve/ Y7 t7 \ (I-5)= — —----- sinri - K tc— ( V T e - e n),T e = T„ Xe( V T e • eE) = X t V T i  • eE) for e = e0, U e — -  0, T e — T x + IV T \ csinhecosr)-1\,  for e -( 1.6)ON THE THERMOPHORESIS OF A SPHEROIDAL SOLID AEROSOL PARTICLEfor 0. (1.7)Here, ee and er) are the unit vectors o f  the spheroidal coordinate system; X is the thermal conductivity; U =I 2 2|U|; H,. = a!cosh e -  sin r) is the Lamé coefficient; cpe is the specific heat; K tc is the thermal creep coeffi­cient, which is calculated from the kinetic theory o f  gases; and V is the Laplacian. Today, the most rigorous expression for the coefficient Ktc is known for a spheri­cal particle [3]. The gas-kinetic co e ff ic ie n t^  = 1.152 when the accommodation coefficients o f  tangential momentum, a.T. and energy, a E, are equal to unity [3,4]. In numerical calculations, we assume that the coeffi­cient K tc for a spheroid differs insignificantly from that for a sphere [6].Boundary conditions (1.5) on the particle surface (e = e0) allow for creep for the tangent component o f  the mass velocity, temperature equality, and the conti­nuity o f  heat fluxes on the particle surface. Away fromthe particle (e --- ► oo ), boundary conditions (1.6) arevalid, and the finiteness o f  the physical quantities char­acterizing the particle at e — ► 0 is taken into account in (1.7).The resultant force acting on a spheroidal particle from the environment is given by the formula [11]cost! -  S^ g a ETisinr|\iSl,(1.8)= Jf-^eCOSTl + CEwhere dS  = c2 cosh e sinr|<ir|<i(p is a differential element o f  area, and c ee and c eri are the strain tensor compo­nents in the spheroidal coordinate system.2. TEMPERATURE DISTRIBUTION IN THE VICINITY OF THE PARTICLE, FORCE AND VELOCITY OF THERMOPHORESISWe make Eqs. (1.3) and (1.4) and boundary condi­tions (1.5)—(1.7) dimensionless by introducing the dimensionless velocity, temperature, and pressure: Ve = UJU, tk = TJT . a n d  p k = P J P ^  (k  = e, i). Here, the spheroid major semiaxis is taken as the unit length; U. as the unit velocity; = \xJJIa. as the unit pressure; and as the unit temperature (U  ~ |ic|V7'|/(pc7'„)).Expressions (1.3)—(1.7) have the single controllable small parameter q = a|V7'|/7'„ <§ 1. Therefore, we will look for a solution to the boundary-value problem in the form o f  expansion in powers o f  q:' ^ V g 1"1 +  . . . ,  t  =  / 0) +  ^ (1) +  . . . ,ve =  +  +v„ = v i 0)-We will restrict our consideration to the first-order terms in q when calculating the force acting on the par­ticle and the velocity o f  its thermophoretic motion in the given external temperature gradient field. In orderto find these quantities, one has to know the distribu­tions o f  the velocity, pressure, and temperature both outside and inside the spheroid. Substituting (2.1) into (1.4), leaving terms and solving the sets o f  equa­tions found by the method o f  separation o f  variables, we will finally find in the zero approximation (q = 0)t f \ X )  = 1 + yA0arccotA (A, = sinhe), (2.2)4 0)(A) = D  + ^ y A 0arccotA A;|  arccot Xfcfk -  arccotÀ J  fcfk.(2.3)Here, X0 = sinhe0 , J  = ts -  1 is the dimensionless parameter, ts = TJT^,  and Ts is the mean temperature o f  the spheroid surface given byT—  = 1 + „ 4nX,XfD  = 1 + [ 1 -  ^  ]yà0 arccot A0, (2.4)+i/  = ~ 2 f - f ~  J 0 ?  + %2)d x ■> x cosr).In (2.4), the integral is taken over the entire particle volume. In the first approximation (~^),t{ç \ X ,  x)  = cosr|<{ —  + rc(À arccotÀ - 1) [ a\cX  n 00 A 2( arccotX -  ^arccot2 (2.5)-j-( arccotX -  X arccot2X)t[V)(X , x )  = cosrJ.BcA + — — A. arccot A.)I 471C Jx (2-6)-  (A arccotA- 1 ) f ldX + (AarccotA - 1 ) ^ X f ldX I.X0 X0Here, co = PryXJ(ac)  and Pr is the Prandtl number. The constants o f  integration^! and^2 appear in expressions for the components o f  the mass velocity and pressure. These expressions are found by solving Stokes equa­tions (1.3) in the oblate coordinate system and have theMAL AI, SHCHUKINform [10]C 4(e,ii) =Uc c o s h e H .cosr)(2.7)x  { X A 2 + [À -  (1 + A ^arccotA ]^ + c2( 1 + À2)} ,U \A^ (er i) = ——  sinr| + (1 _ AarccotA)^P e(£,T\) = P ^  + c ^ - x ( k 2 + x 2) A 2. HZ■ c \ \(2 .8)(2.9)The constants T  and B  enter into expressions (2.5) and (2.6) for the temperature fields inside and outside the particle from the corresponding boundary condi­tions on the spheroid surface. Since an expression for the coefficient T  will be o f  interest to us, we write it in explicit form:1 - 8A a  47tc3Air ooAA0(l+ A q )j« , z d V1 + AnAn■ Aq ^0arccot An1 -  Ô 2^  —~— arccot A02Aré l2 1 + An(1 -  ô) arccot A0(2 .10)2 -  j -  ] arccot A01 + A,A = ( 1 -  ô) arccotA0 + 8-An1 ■A28 - Ï .f : = 4 M a »x I -3a(A0 arccot A0 -  1) 4 n c3AP7700AnJ?,.z d V(2 .11)The coefficient A 2 is found from boundary condi­tions (1.5) in view o f  expressions (2.7), (2.8), and (2 .10):A 2 = -2c- 2 K ,A)S 2v ec(3[A0 + (1 -  A0) arccot A0]A0 -  ( 1 + Aq ) arccot A0 | V T\U \  + Aq(3[A0 + (1 -  Ao)arccotA0]Prwhere1 + A0. 1 -  A0 arccot A0( 2 -  A0 arccot A0)A0yA0 -  ( 1 + Aq) arccotA0n , 1 f Aq Pr § A0- ( l  + A 0)arccotA0p = l - 2 A tc—Y --------- -------------------------------s A 1 + A0A0 + ( 1 -  A0) arccotA0x 1 -  A, -  arccot A0 j arccot A0A0 -  (2 -  A0arccotA0)A0arccotA0 A0 -  ( 1 + Aq) arccotA0In view o f  the explicit form o f  the coefficient A 2, we find a general expression for the force acting on a sphe­roidal particle. This force is the sum o f  the viscous force F„ and the force I ,u >F  = F^ + F O)where(3[A0 + ( 1 -  A0)arccotA0]c<i) o s v  Ve A0 -  ( 1 + Aq)arccotA0 F  = -8 7 t|iecA:tc------------------------------------(2 .12)(2.13)xHereafter, the superscript 5 denotes the values o f  physical quantities at the mean temperature Ts o f  the spheroid surface, which is given by formula (2.4).Substituting (2.7)-(2.9) into (1.8) and integrating, we arrive atPr-jVTl 1 +1 -h  (3[A0(1 -  Aq) arccotA0] ( l  + Aq)A 3a(A0arccotA0 -  1)- J  qtz d V■a;A0y4 nc  Ae^Aoe u y1 -  A0 arccot A0( 2 -  A0 arccot A0) A0 -  ( 1 + Aq) arccotA0(2.14)In the general case, the force I ,u> is the sum o f  three forces: the thermophoretic force, the force proportional to the dipole moment o f  the density o f  heat sources nonuniformly distributed over the particle volume, and the third term due to the motion o f  the medium (i.e., the force component taking into account the convective terms in the heat conduction equation).Equating the resultant force /■' to zero, we arrive at a general expression for the drift (thermophoretic) veloc­ity o f  a solid oblate spheroidal particle in the external temperature gradient field:U* = - a K ‘ T *ON THE THERMOPHORESIS OF A SPHEROIDAL SOLID AEROSOL PARTICLEX ■1 -  (A0 + l/A 0)arccotA0V i + a , (1 -  8) arccotA0 + 8An(2.15)x 1 -3 a4 n c  A„7\,arccotA0- z d VPr1 + A0 1 -  A0 arccot A0 (2 -  A0 arccot A0)A0yA0 -  (1 + Aq) arccotA0|V71In order to find the rate o f  thermophoresis for a pro­late spheroid, one has to substitute /A for A and ic for c (i is the imaginary unit) in (2.15).Thus, formulas (2.12) and (2.15) have the most gen­eral form and make it possible to estimate the resultant force acting on a solid spheroidal aerosol particle and its drift velocity in the external temperature gradient field for the case when heat sources (sinks) are nonuni - formly distributed inside the particle. In this approach, the motion o f  the environment is taken into account for small temperature differences in the vicinity o f  the par­ticle.3. RESULTS AND DISCUSSIONIf one does not take into account the motion o f  the environment and internal heat sources, (2.15) is reduced to an expression for the purely thermophoretic velocity o f  a spheroidal particle:&th = ^ tcv e8 / ^Ab) =  _ b ________J th a ,-------1 -  (A0 + 1/A0)arccotA0(1 -  8) arccot A0 + 8-An1 + Aq A,(3.1)which coincides with formula (9) in [5].In the case o f  a sphere, (2.15) turns into an expres­sion for the thermophoretic velocity o f  a solid spherical particle o f radius R  that includes the flow o f  the envi­ronment and internal heat sources:U ^ \ a = b = R ) =  Ktcv l & f sph) IV T\T ,(3.2)whereDisregarding the flow o f the environment and inter­nal heat sources yields the conventional formula for the thermophoretic velocity o f  a large spherical particle [1,2]Uth(a = b = R )  = (3.3)teIn order to estimate how the motion o f  the environ­ment affects the thermophoretic velocity o f  a spheroi­dal particle, one has to specify the nature o f  heat sources nonuniformly distributed over its volume. As an example, let us consider the simplest case when the particle absorbs radiation as a black body. In this case, radiation is absorbed in a thin layer o f  depth 8e <  e0 that is adjacent to the heated particle surface. The den­sity o f  heat sources inside the layer o f  depth 8e is equal to [12, 13]7i(£> Tl) =coshecosr) T2 2 c*c(cosh  e -  sin r |)8e  e0 -  8e < e < e0 (3.4)where 70 is the intensity o f  an incident radiation.The integrals J (// d V  and | (//, zd V  appear in theexpression for the thermophoretic velocity. Substitut­ing (3.4) into these integrals in view o f the fact that 8e <  e0 and performing integration, we findj q . d V  = 7t/0c2A ofl + —2  ^ Anjq iZ d V  = ~ ^ n I 0c3X l (  1 + \  |.(3.5)a;In view o f  (3.5), expression (2.15) takes the formU l  = K ty eS f ? J ^ ,  (3.6)where_  b 1 -  (A0 + l/A 0)arccotA0J  t h  ~  .Vl + Aq t , (1 -  8) arccot A0 + 8An 11 + An VTo =14rci?AX■jqidV, 1 +Aq a 2AI 0{  1 + A l[A 0 arccot A0 -  1 (3 -7)a ;y O p h )  _  _4 (1  + 28)1 + ------7------4 k R 2K T ,  I2To • + ^ 7 1 + ^ o1 -  A0 arccot A0( 2 -  A0 arccot A0) A0 -  (1 + Aq ) arccot A0Table 1MALAÏ, SHCHUKINЫа = 0.1/ 0 х 102, W/m20.5 2 5 10/ X) / 2) / X) / 2) / X) / 2) / X) / 2)15 0.52 0.32 2.08 1.30 5.20 3.29 10.40 6.7120 0.69 0.43 2.77 1.74 6.93 4.42 13.87 9.0825 0.87 0.54 3.47 2.18 8.67 5.56 17.33 11.51Table 2Ыа = 0.370 x 102, W/m20.5 2 5 10/ X) /2) / X) /2) / X) /2) / X) /2)15 0.39 0.20 1.59 0.81 3.97 2.04 7.95 4.1620 0.59 0.27 2.12 1.08 5.30 2.74 10.60 5.6325 0.66 0.33 2.65 1.35 6.62 3.44 13.25 7.13Table 3Ыа = 0.5I0 x 102, W/m20.5 2 5 10/ X) /2) / X) /2) / X) /2) / X) /2)15 0.32 0.12 1.26 0.48 3.15 1.20 6.31 2.4520 0.42 0.16 1.68 0.64 4.21 1.61 8.41 3.3225 0.53 0.20 2.10 0.80 5.26 2.03 10.51 4.20In the case o f  a sphere, (3.6) is recast as7-7-Oph) _  ТГ A|SX /(Sph)|V71U t h  th  y  ’wherey ( s p h )4 ( 1 + 2 0 )1 - Rio6АРГ 1 + 5 '(3.8)(3.9)The mean temperature o f  the spheroid surface is related to the incident radiation intensity I 0 asС Ant = t „  + - ^ /n  i + 4  .4A,1a;(3.10)In order to illustrate the contributions o f  the form- factor (ratio o f  the spheroid semiaxes), flow o f  the envi­ronment, and internal heat release (nonuniform distri­bution o f  heat sources over the particle volume) to thethermophoretic velocity (3.6), Tables 1-4 list the numerical estimations for particles o f  borated graphite(A- = 55 W/(m K)) suspended in air at /'„ = 280 К and P e = 10s Pa.The numerical analysis showed that, at a given ratio between the semiaxes, the relative contribution o f  the other factors increases with increasing incident radia­tion intensity I0. This effect depends significantly on the equatorial radius o f  the spheroid (a). For instance, in Table 1 (a = 15 |am),/W  = 0.52 a t /0 = 0.5 x  102 W/m2and/W  = 10.40 at I 0 = 10 x  102 W/m2 (/W = (|/ *  -f th  |/ f th  ) x  100%). Such behavior o f  the fu n ction /(1) is due to the fact that, as follows from (3.10), (3.7), and the numerical estimations, the major contribution is from the terms proportional to the dipole moment o f  the density o f heat sources nonuniformly distributed over the particle volume. In (3.7), this is the termON THE THERMOPHORESIS OF A SPHEROIDAL SOLID AEROSOL PARTICLETable 4b!a = 0.8I0 x 102, W/m20.5 2 5 10/ X) / 2) / X) / 2) / X) / 2) / X) / 2)15 0.24 0.04 0.94 0.15 2.35 0.38 4.69 0.7720 0.31 0.05 1.25 0.20 3.13 0.51 6.26 1.0425 0.39 0.06 1.56 0.25 3.91 0.64 7.82 1.32A0arccotA0 -  1. The dimensionless term related to the motion o f  the environment (see the heat conduction equation) is proportional to the Prandtl number. In a gas, this number is on the order o f  unity; therefore, the contribution o f  this term differs from that o f  the first one by one order o f  magnitude. This fact may be used to separate particles by size, finely purify gases from aerosol particles, estimate translucent zones appearing in clouds and fogs when they are probed by laser radi­ation, etc. The influence o f  the factors mentioned above will increase with increasing radiation intensity. How­ever, the mean temperature o f the spheroid surface will also increase (see (3.10)). In this case, we cannot con­sider the coefficients o f  molecular transport to be con­stant. Therefore, expressions (2.15) and (3.6) must involve the mean values o f the physical quantities at a given temperature o f the particle surface, which is determined by (2.4) and (3.10), to avoid large errors. It is also o f  interest to compare the thermophoretic veloc­ity with that for a spherical particle with a radius equal to the equatorial radius o f  a spheroid, i.e., with formula(3.8). The numerical analysis showed that in this case, too, the relative error increases with increasing incident radiation intensity and equatorial radius. For example, in Table 1 (a = 15 |im ) ,/ (2) = 0.32 at70 = 0.5 X 102 W /m2and / 2> = 6.71 at I 0 = 10 x  102 W/m2 ( f n  = (| _x  100%). However, this increase is approxi­mately 1.5 times smaller than that in the former case.REFERENCES1. S. P. Bakanov and B. V. Deryagin, Dokl. Akad. Nauk SSSR 142, 139 (1962).2. Yu. I. Yalamov and A. S. Sanasaryan, Inzh.-Fiz. Zh. 28, 1061 (1975).3. Yu. I. Yalamov, A. B. Poddoskin, and A. A. Yushkanov, Dokl. Akad. Nauk SSSR 254, 1047 (1980) [Sov. Phys. Dokl. 25, 734 (1980)].4. A. B. Poddoskin, A. A. Yushkanov, and Yu. I. Yalamov, Zh. Tekh. Fiz. 52, 2253 (1982) [Sov. Phys. Tech. Phys. 27, 1383 (1982)].5. K. H. Leong, J. Aerosol Sci. 15, 511 (1984).6. Yu. I. Yalamov, V P. Redchits, and M. N. Gaidukov, Inzh.-Fiz. Zh. 39, 538 (1980).7. A. A. Gukasyan, Thermophoresis of a Moderately Coarse Spherical Pareticle, Available from VINITI, 1981, Moscow, No. 5320-81.8. I. Praudman and J. R. A. Pearson, J. Fluid Mech. 2, 237 (1957).9. A. Acrivos and T. D. Taylor, J. Phys. Fluids 5, 387 (1962).10. J. Happel and H. Brenner, Low Reynolds Number Hydro­dynamics (Prentice-Hall, Englewood Cliffs, 1965; Mir, Moscow, 1976).11. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Gos- tekhizdat, Moscow, 1958; Oxford, Pergamon, 1959).12. C. F. Bohren and D. R. Huffman, Absorption and Scat­tering o f  Light by Small Particles (Wiley, New York, 1983; Mir, Moscow, 1986).13. S. Bretshnaider, Properties o f  Gases and Liquids: Engi­neering Methods o f  Analysis (Moscow, Khimiya, 1966).Translated by M. Fofanov

On the thermophoresis of a spheroidal solid aerosol particle

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On the thermophoresis of a spheroidal solid aerosol particle

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