The thermophoresis of a spheroidally shaped aerosol particle at small relative temperature differences has been considered with allowance for internal heat sources nonuniformly distributed in its volumeyesBS

Malai, N. V.

Pleskanev, A. A.

DSpace at Belgorod State University

INFLUENCE OF INTERNAL HEAT RELEASE  ON THE THERM OPHORESIS OF A SOLID  SPHEROIDALLY SHAPED AEROSOL PARTICLEN. V. Malai and A. A. Pleskanev UDC 533.72The thermophoresis o f  a spheroidally shaped aerosol particle at small relative temperature differences has been considered with allowance for internal heat sources nonuniformly distributed in its volume.Formulation of the Problem. We consider the steady-state motion of a solid aerosol particle of a spheroidalshape (oblate spheroid) with velocity U in the negative direction of the z axis inside which nonuniformly distributedinternal heat sources of density qp act. It is assumed that the gas is at rest at infinity and a small constant temperature gradient VT  is maintained with the use of external sources. The motion of the particle occurs at small relative tem­perature differences in its vicinity, i.e., when (7S -  7^)/7M «  1. When this condition is fulfilled, the thermal conduc­tivity and the coefficients of dynamic and kinematic viscosity can be assumed to be constants, whereas the gas can be considered as an incompressible medium. The particle size is much larger than the mean-free paths of the molecules of the gas mixture; therefore, we will disregard Knudsen corrections [1, 2].Description of the thermophoretic motion of the particles is carried out in the Stokes approximation in a spheroidal coordinate system (e, r|, (p) with its origin at the center of the spheroid, i.e., the origin of a fixed coordinate system in the instantaneous position of the center of the particle is selected. The curvilinear coordinates e, r|, 9 are re­lated to the Cartesian coordinates by the following relations [3]:x = c sinh e sin r| cos cp, y  = c sinh e sin r| sin (p, z = c coshe cos r | , (1)x = c cosh e sin r| cos 9 , y  = c cosh e sin r| sin (p, z = c sinh e cos r | , (2)where we have c = Vb2 -  a1 in the case of a prolate spheroid {a < b, formula (1)) and c = Va2 -  b2 in the case of an oblate spheroid (a > b, formula (2)); a and b are the semiaxes of the spheroid. The position of the Cartesian coordinate system is fixed relative to the particle so that the origin of coordinates is located at the center of the spheroid and the z axis coincides with the axis of symmetry of the spheroid.With the constraints considered above, the velocity, pressure, and temperature distribution are described by the system of equations [4]pgcpg (Ug'^0 Tg ^ gA 7 g , A7p <?p/^p (4)with boundary conditionscU  cosh e cU  sinh e v 1------~---------------—£ = £0 , Uz = - ---- —-------co s r |, U = — —----- sin r|-^T t s - f  (V r-e ,,) , Hz = v cosh' e -  sin r\ ,/7g /7g 1 gTg = Tp , Xg (VTg-es) = Xp (VTp-ee)-, (5)HgA£/g div U =  0 . (3)Belgorod State University, Belgorod, Russia; email: malay@bsu.edu.ru. Translated from Inzhenemo-Fizicheskii Zhumal, Vol. 77, No. 6, pp. 74-78, November-December, 2004. Original article submitted February 24, 2003; revision submitted March 15, 2004.e ^ -o o , £/g —>0 , 7g-> T„ + | VT\ c s in h e c o s r i , P ^ ^ P ^ ; (6)£ —> 0 , 7p ^  °° . (7)In boundary conditions (5) on the particle surface, we have allowed for: the slip condition for the tangentialcomponents of the mass velocity, the equality of temperatures, and the continuity of the heat fluxes on the particle surface. The particle surface corresponds to the coordinate surface with e = e0. Boundary conditions (6) hold at a large distance from the particle (e —> °°), whereas the finiteness of the physical quantities characterizing the particle for e —> 0 is allowed for in (7).Temperature Distribution in the Vicinity of the Particle. Determination of the Force and the Thermo­phoresis Velocity. In the problem, in addition to the dimensionless Reynolds and Peclet numbers, there is another con­trolled small parameter q = a \ V t \ /T „  «  1 characterizing the relative temperature difference on the particle size.Therefore, we will seek the solution of boundary-value problem (3), (4) in the form of an expansion in powers of q.To find the force acting on the particle and the thermophoresis velocity in the prescribed external field of the temperature gradient it is necessary to know the velocity, pressure, and temperature distribution in the vicinity of the spheroid. In solution of the problem, we restrict ourselves to corrections of first order of smallness in q. Leaving the terms proportional to q in Eqs. (4) and solving the resulting systems of equations by the method of separation of vari­ables, we finally obtain for the zero approximations (q = 0):tg'* (X) = 1 + arccot X , (8)x  x xtp°^  (X) = D + —** jXq arccot X + J arccot X f d X -  arccot X J f d X , (9)P \  \where X = sinh e, Xq = sinh e0, and y = ;s -  1 is the dimensionless parameter; ts = 7S/ 7 M. and 7S is the average tem­perature of the spheroid surface, determined asT1 s^ r - =  1 + — ----------  \ q j v - ,Si . t  J ’T„ 4 KXgcXQT J v ^  ’ (10)£ » = 1  +/  t \  2 + 1c f 2 2y^oarccotX0 ; / =  -  I q„ (X + x ) d x ;  x = c o s r |.In formula (10), we integrate over the entire volume of the particle, and for the first approximations ( ~q) wehave(X, x) = cos r| \ —  + Tc {X arccot X -  1) + co® ClA- 2 A 2 (arccot X -  -  arccot X) +^ 1  2 + (arccot X - X  arccot X) ( ID(X, x) = COS T| .A,BcX + — ^arccot J q^ Z(j y _  ^  J arccoi ^  _ i) -4 n c \ T r(A arccot A -  1) J kf\dk ( 12)Here co = Pr jko/(ac) and Pr is the Prandtl number.The general solution of the Stokes equation in the oblate coordinate system has the following form (see, for example, [3]):UF (e, r|) = ------—----- cos r| \kA2 + [A,-  (1 + k 2) arccot k] A, + c (1 + A2)}.c  cosh 1 1 ' (13)U .(e, r|) = -  -j-j- sin r| -j—  + (1 -  k  arccot A) A 1 + c k (14)2 2P (e,ri) = P 00 + c - ^ - x ( ? i  + x ) A 2 .B HZ(15)The integration constants r ,  B, A\,  and . I2 appearing in (11)—(15) are determined from the boundary condi­tions on the spheroid surface (5). Integrating the stress tensor over the particle surface, we obtain the following general expression for the force, which is additively made up of the viscous force of the medium /'’Ll and the force I,(]F  = Fil + F*.(1) (16)whereF „  =  - 8 tthbC/-P [Aq + (1 -  k 0) arccot A0 I(17)^ l)  „ s T,  vg *0 - ( l  + ^o) arccot k 0 5F- = -  8tc|x cKt s ----------------- z-----------------------2"ts p [A0 + (1 -  Aq) arccot Aq] (1 + Aq) A T,V T I1 -3 a (Aq arccot Aq -  1) 4 % c \T J ^2r l + ^o l + ^o arccot A0 (2 -  A0 arccot Aq)J  q jd V +  Pr — —  AoT  ------ 2-------------- -- ---------Y 2 Aq -  (1 + Aq) arccot A0(18)n , . . .  *0 Pr 8 A0 - ( 1  + Aq) arccot A0P = l ~ 2Kt s — j ---------- 2 ---------------2------------1  ^ A 1 + Aq Aq + (1 — Aq) arccot Aq1 - Aq + — arccot A0 2arccot Aq -A q  -  (2 -  A q  arccot A q )  A q  arccot A qA0 -  (1 + A0) arccot Aq; A = (1 -  5) arccot Aq + 5-An1  +  A q  A q a:In the general case, the force / ,n 1 consists of the sum of three forces stemming from respectively: a purely thermophoretic force (first term), a force proportional to the dipole moment of the density of the heat fluxes nonuni- formly distributed in the particle’s volume (second term), and a force due to the influence of the motion of the me­dium, i.e., allowing for the convective terms in the heat-conduction equation (third term).Equating F  to zero, we have the following expression for the velocity of motion of the solid spheroidal par­ticle in the prescribed external field of the temperature gradient:1 sb v„ 1 -  (A0 + 1/Aq) arccot A0a(1 -  5) arccot A0 + 5 -Aar 3a—  I 1_— 3------11L 4ncarccot Aq -  —A a: J  q^dV+Pr1 + A0 AqJ 21 + Xq l + ^o arccot Aq (2 -  X0 arccot A0)■hyX0 -  (1 + X0) arccot X0V7lT(19)To obtain the thermophoresis velocity for a prolate spheroid we must replace X by iX and c by - ic  (/' is the imaginary unit) in (19).Analysis of the Results Obtained. If the influence of the medium’s motion and of the internal heat sources is disregarded, expression (19) becomes an expression (coincident with formula (9) in [5]) for a pure velocity of ther­mophoresis of the spheroidal particle.To evaluate the contribution of internal heat release to the velocity of thermophoresis of the spheroidal parti­cle we must specify the nature of the heat sources nonuniformly distributed in its volume. We consider as an example the simplest case where the particle absorbs radiation as a black body. The radiation is absorbed in a thin layer of thickness 5e «  £o adjacent to the heated part of the particle surface. In this case the density of the heat sources inside the layer of thickness 5e is equal to [6]4n (E, ri) =cosh e cos r| T2 2 0 ; c (cosh e -  sin r|) oe0 .2 <r ) <7i ,  e0 - 8 e < e < e 0 ; 0 < n < § . (20)With account for (20) the expression for the thermophoresis velocity takes the form* s * I v r lt f t .p h  =  * t . s V B 8 / t .g^t.ph T (21)A.ph1 -  + 1/A0) arccot X0r A aVl + Aq 4 1 (1 -  arccot X0 + 8L i + x.i l2 „ I1 + *0°2 A„7L1 + ^2 Xo20!", , Pr , / T ^ T  1 “  K  arccot A0 (2 -A 0 arccot A0) l  |: | X0 arccot Aq -  1 + — vI + Aq--------------------- ô----------------------- 1 >Ap h -  „4 A0 -  (1 + Aq) arccot A01 -  (A0 + 1/Xq) arccot A0V l + Aq 4 1 (1 -  arccot Aq + 5 01L ! + A0 A0To illustrate the contribution of the shape factor (ratio of the semiaxes of the spheroid) and of the influence of the medium’s motion and internal heat release on the thermophoresis velocity (21) Fig. 1 gives the curves relating the values /  = 7t.ph//tph 17;, =30ok to the intensity of incident radiation for borated-graphite particles (Ap = 55 W/(mdeg) with spherical (curve 1) and spheroidal (curve 2) shapes of the surface with an equatorial radius of a = 35 jam forFig. 1. Function /  vs. incident-radiation intensity for the semiaxes ratio b/a =0.2 (a) and b/a = 0.5 (b). I q, W/cm2.different ratios of the spheroid semiaxes: b/a = 0.2 (see Fig. la) and b/a = 0.5 (see Fig. lb) (the particles are sus­pended in air at = 300 K and 1 \  = 105 Pa). A numerical analysis has shown that, when the ratio of the semiaxes is fixed, the total contribution of the medium’s motion and the internal heat release leads to a monotonie decrease in the thermophoresis velocity (see Fig. 1) with increase in the incident-radiation intensity I q, and this decrease substan­tially depends on the equatorial radius of the spheroid a.NOTATIONa and b, semiaxes of the spheroid; c/)g. heat capacity of the gas; and ee, unit vectors of the spheroidal co­ordinate system; i y ,  viscous force of the medium; I q, incident-radiation intensity; K ts , coefficient of thermal slip; qp, density distribution of the heat sources inside the particle; Pr, Prandtl number; / 'g. pressure of the gas; 7g and Tp, tem­peratures of the gas and the particle respectively; 7's. average temperature of the spheroid surface; 7'^ and P tempera­ture and pressure of the gas at a large distance from the particle; UE and U^. components of the mass velocity of the gas Ug; U = | U | , velocity; V, volume of the particle; A.g and Âp. thermal conductivities of the gas and the particle respectively; vg, |ig, and pg, kinematic and dynamic viscosities and density of the gaseous medium; ç = a | V7' | /7'^. small parameter characterizing the relative temperature difference on the particle size; e, t|, cp, spheroidal coordinate system. Subscripts and superscripts: g, gas; 0, values of the quantities on the particle surface; p, particle; s, quantities at the average temperature of the spheroid surface; » , values of the physical quantities away from the particle (at in­finity); w, quantities without allowance for internal heat release and the medium’s motion; e and T), components of the physical quantities in the spheroidal coordinate systems; t.s, thermal slip; t.ph, thermal phoresis (thermophoresis).REFERENCES1. Yu. I. Yalamov, A. B. Poddoskin, and A. A. Yushkanov, Boundary conditions in flow of a nonuniformly heated gas past a spherical surface of small curvature, Dokl. Akad. Nauk SSSR, 254, No. 2, 1047-1050 (1980).2. A. B. Poddoskin, A. A. Yushkanov, and Yu. I. Yalamov, Theory of thermophoresis of moderately large aerosol particles, Zh. Tekh. Fiz., 52, No. 11, 2253-2262 (1982).3. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics [Russian translation], Mir, Moscow (1976).4. L. D. Landau and E. M. Lifshits, Mechanics o f  Continua [in Russian], Gos. Izd. Tekh.-Teor. Lit., Moscow (1954).5. K. H. Leong, Thermophoresis and diffusiophoresis of large aerosol particles of different shapes, J. Aerosol Sci., 15, No. 4, 511-517 (1984).6. C. F. Bohren and D. R. Hoffman, Absorption and Scattering o f Light by Small Particles [Russian translation], Mir, Moscow (1986).

Influence of internal heat release on the thermophoresis of a solid spheroidally shaped aerosol particle

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Influence of internal heat release on the thermophoresis of a solid spheroidally shaped aerosol particle

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