The Stokes and Hadamard-Rybchinsky formulas are generalized, making it possible to take into account the temperature dependence of viscosity in a wide range of temperatures and to calculate the force of resistance to motion and velocity of gravitational fall at arbitrary temperature differences between the particle surface and a remote regionye

Malai, N. V.

Shchukin, E. R.

Yalamov, Yu. I.

DSpace at Belgorod State University

Effect of Particle Surface Heating on the Sedimentation RateN. V. Malai, E. R. Shchukin, and Yu. I. YalamovBelgorod State University, Belgorod, Russia Moscow Pedagogical University, Moscow, RussiaReceived November 23, 2000; in final form, May 2001Abstract—The Stokes and Hadamard-Rybchinsky formulas are generalized, making it possible to take into account the temperature dependence of viscosity in a wide range of temperatures and to calculate the force of resistance to motion and velocity of gravitational fall at arbitrary temperature differences between the particle surface and a remote region.Determining the granulometric composition of disperse systems via their sedimentation is one of the most practicable and widely used methods of disper­sion analysis [1]. In cleaning of liquids to remove contaminating particles and dressing of minerals, it is important to accelerate sedimentation. This is com­monly achieved in various ways, depending on the kind of particles and their environment. Sedimen- tometry is based on the relationship between the velocity of particle motion in a viscous medium and the particle size. The velocity of particle motion can be markedly raised by heating the particle surface, since the viscosity of a fluid decreases exponentially with increasing temperature. Of practical and theoreti­cal interest in this regard is description of the gravita­tional motion of a heated particle in a viscous fluid.Let us consider the gravitational motion of a uni­formly heated hydrosol particle in another, viscous incompressible fluid, which is immiscible with the former and fills the entire space. At infinity, the fluid is at rest. By heated (cooled) particle is understood a particle whose surface temperature differs from that at a distance from the particle. The temperature differ­ence between the particle and the viscous fluid can be maintained steadily via, e.g., heat release in chemical reactions at the particle surface, radioactive decay of the drop substance, external irradiation, etc. In par­ticular, if a flux of electromagnetic radiation (wave­length a(). intensity 70) is incident on the drop, then the energy absorbed by the drop is nR21{)K,d. where R is the drop radius and K,d is the absorption factor [2]. If the heat conductivity of the drop much exceeds that of the external medium and A() »  R. then the absorbed energy is evenly distributed over the drop surface and, consequently, the drop can be considered uniformly heated.The heated particle surface affects the heat-transfer properties of the ambient fluid and, in the end, the distribution of the velocity and pressure fields in the vicinity of the particle.In contrast to previous investigations [3-7], the present study generalizes the Stokes and Hadamard- Rybchinsky formulas to the case of uniformly heated spherical drop in a viscous incompressible fluid at arbitrary temperature difference between the particle surface and remote region with account of the tem­perature dependence of viscosity.Of all fluid transfer parameters, the viscosity coefficient depends on temperature most strongly [8]. To take into account the temperature dependence of the dynamic viscosity, we use a formula that de­scribes changes in the viscosity of a fluid in wide range of temperatures with any required accuracy (at Fn = 0 this formula can be reduced to the known Reynolds relation [8]):ooM-e = 0^0 [1 + E w .  -  !)"] exP \-MTJT00 -  1)], (1)n  =  1where A and Fn are constants, |ax = |ac(7x ). and 7X is the fluid temperature far away from the particle.Hereinafter the indices e and i refer, respectively, to viscous fluid and heated particle; index oo denotes fluid parameters at infinity in an unperturbed flow; and index s refers to values of physical quantities at average surface temperature Ts.It is known that the fluid viscosity decreases exponentially with increasing temperature [8]. An analysis of the available semiempirical formulas dem­onstrated that expression (1) describes the variation ofEFFECT OF PARTICLE SURFACE HEATING ON THE SEDIMENTATION RATEviscosity in a wide temperature range in the best way, with any required accuracy. For example, we have A = 5.779, Fj = -2.318, F2 = 9.118 for water in the temperature range 0-90°C, with relative error not exceeding 2% (7X = 273 K).It is assumed that the densities, heat conductivities, and specific heats of the fluid and particle are con­stant, the heat conductivity coefficient of the particle much exceeds that of the medium, and particle mo­tion is rather slow (small particles by Reynolds and Peclet).It is convenient to introduce a reference system associated with the center of a moving particle (the problem is reduced to analysis of how an infinite plane-parallel flow, whose velocity is to be deter­mined, moves past a particle). The distributions of velocity and temperature is axially symmetric with respect to the 0Z axis passing through the drop center in the direction of the velocity vector of the incident flow. Therefore, the analysis uses a spherical system of coordinates in which radius is reckoned from the drop center, and angle 0, from the velocity direction of the incident flow.In terms of the assumptions made, the equations and boundary conditions for the velocity, pressure, and temperature inside and outside a drop can be written in Stokes approximation in the spherical sys­tem of coordinates r, 0, r| in the form [9]VPe = ^eV2t/e + 2(V^e) W e + (Vne) x <yue), divt/e = 0,(2)Hi AU; = VP;, divC/j = 0, (3)ATe = 0, (4)r = R, If, = U[r = 0, U% = U\, Te = Ts, (5)r -> 00, Ue -> Ux cos0er -  Ux  sin0ee, Te -> Tx , Pe -> Px ,—>■ 0, I Uj I Ф 00, Pi ф(6)(7)where |a. and A, are, respectively, the dynamic viscosity and heat conductivity; P is the pressure; Ur and Ut) are the radial and tangential components of the mass velocity of the fluid, U, in the spherical system of coordinates; Uœ = |Uœ|, Uœ is the velocity of the in­cident flow, which is to be determined from the condi­tion of vanishing total force acting on the particle (t/00 > 0 if this velocity is directed along the z axis,and Uœ < 0 otherwise); er and eg are the unit vectors in the spherical system of coordinates; and 7S is the mean temperature of the drop surface.Conditions (5), assuming impermeability and con­tinuity of the normal and tangential components of the mass velocity and constant particle surface tempera­ture, are taken for the particle surface. As boundary conditions at infinity, i.e., far away from the particle, are taken conditions (6), and the finiteness of the physical quantities characterizing the drop at r —> 0 is accounted for by conditions (7).To obtain a closed problem, the boundary condi­tions on the surface of a uniformly heated drop are supplemented with the condition of continuity of the stress tensor components (normal and tangential) on the drop surface [9]:d l ß r  d u lr-P e + 2(j,e—— = -P { + 2(X;-dr1 dUt dU% U%\\ r  39 drMirdr1 dill dui и {Л(8)\ r  39 dr■ (9)r /The boundary conditions (5)-(9) make it possible to separate variables and reduce the system for per­turbed quantities to a system of ordinary differential equations. Solving Eqs. (2)-(4) with boundary condi­tions (5)-(9) gives the distribution of velocities, pres­sures, and temperature in the fluid and the drop. After these distributions are found, we can calculate the force acting on a uniformly heated drop and the velocity of its drift.At Reœ = TLfA U J \ iao «  1, the incident flow exerts only perturbing influence, and, therefore, the solution to the equations of hydrodynamics and heat transfer is to be sought for as an expansion in the Reynolds number Reœ. In determining the force acting on a uni­formly heated drop and its drift velocity, we restrict the consideration to corrections of the first order of smallness. To find these corrections, it is necessary to know the fields of velocity, pressure, and tempera­ture in the vicinity of the particle. The general solu­tion to the heat conduction equation (4), satisfying the relevant boundary conditions, has the formte = 1 + j/y ,  У = (Ts -  T J /T K (10)where y is a dimensionless parameter characterizing the temperature difference between the particle surface and remote region; tc = Te/T(X), y  = r/R.MALAI et al.Substituting (10) in (1), we obtainooMe = M l  + E F „ ( f / / ) ] e x p M y / j ) .  (11)n  =  1Formula (11) is further used for determining the velocity and pressure fields in the vicinity of a uni­formly heated drop.Since the viscosity depends only on the coordi­nate y, the equations of hydrodynamics are solved by separation of variables, with the velocity and pressure fields expanded in Legendre and Gegenbauer poly­nomials [10]. In particular, the following expres­sions were derived for the components of the mass velocity U:lfr (y,Q) = Ux  cos0 [l + A & t y )  + A 2G2m  (12)U§(y,0) = - U r s i n e  [1 + A f o t y )  + A 2G4( y ) l  (13) t 4 (y ,0 )  = Ux  cos 0 (A 3 + A 4V2), t4(y,0)where= - [ / „ s in © ^  + 2A4y2), (14)G.iy) = - y - ^ A ^ K n  + i V r 1,n  =  0G\ = dGj /Ay, G\ = dG2/dy,G2(y) = -y-1 Z  4 2)[(« + 1 )ynT ln  =  0oo-  «y-3 Z  [(” + 3)lny_1 -  1]A W[(h + 3)2/ T \n  =  0aW  = [n(n + 5)] 1 2  {(« + 4 -  ^)[a[1}(« + 5 -  A:) -  a[2)]/fc = 1+ o^ V aW * (» > 1). (15)G3(y) = Gj(y) + O.SyG1!, G4(y) = G2(y) + 0.5yG^,nA^ 2) = [(» + 3)(h -  2)]-1[ - 6 4 V  + Z ( ( «  + 2 -  *)k =  1nx [a^(w  + 3 -  &) -  a^2)] + a^}y^A^2l  k + a  ^  [(2« + 5k  =  0-  2Jc)a^ -  a f> \yk6 $ l k _ 2 (n > 3). (16)In calculating the coefficients A(J  by meansof recurrence formulas (15) and (16), account should be taken that4 1) = -3, A^ 2-* = -1, = -1, 4 1) = af?) = -1,a [)2) = 4, af,3) = -4, a ®  = Fn, a ^  = (4 -  ri)Fn + AFn _ haf3) = 2AFn _ 1 -  2(2 + n)AF„, a™ = A n/n\,a  = -y{6y2a ^  -  [3 (4 a^  -  a^2)) + a^]A ^2)+ [2(30$) -  a ^ )  + a^]y}/15,A^ 2) = -y [6 a ^  + 2 (3 a ^  -  a ^ )  + af>]/4.The integration constants A \-A4. appearing in Eqs. (12)—(14), are found upon their substitution in the corresponding boundary conditions on the drop surface. With this done, the force acting upon the par­ticle is found by integration of the stress tensor over its surface [9] and has the following form:F = - 4nR (x^ C/ ^ 2  exp {-^ 4y}n.whereA 2 = -(7V3 + iV4 n|/3M?)/A, A(17)n aN,y=i= 2 G\ + G1/,y = iI ^IK  a. n \= G2(2G\ + Gf) -  Gj(2G^ + G2),= G,Gl2 -  G2G\, No = GÎ,The index s denotes the values of the physical quantities for the mean surface temperature of a uni­formly heated drop, Ts; nz is the unit vector in the direction of the z axis.It is noteworthy that the force F is calculated on the assumption of a uniform drop motion, which is only possible if the total force acting upon the particle is zero. Since the force (17) is proportional to the velocity and vanishes together with the latter, then, for uniform motion of a heated drop to occur, it is neces­sary to assume the presence of a certain extraneous force counterbalancing the force (17).Substituting the explicit form of coefficient A2 in (17), we obtain the expressionF = 6iiR\ix Ua/ n ,H Z> (18)EFFECT OF PARTICLE SURFACE HEATING ON THE SEDIMENTATION RATE^  = 2  exp {-Aj[N3 + A^4 ^ / (3 ^ ) ]} / [3 (^  +Let us consider as an example the gravitational fall of a uniformly heated drop. A spherical drop falling under the action of gravitational force in a viscous medium ultimately starts to move at a constant veloc­ity at which the action of the gravitational force is counterbalanced by hydrodynamic forces. The gravita­tional force acting upon a particle is given, with ac­count of the buoyancy force, byF = (Pi -  pe)gT 3^nzx4/3, (19)where g is the gravitational acceleration.Equating expressions (18) and (19), we obtain the fall velocity of a uniformly heated spherical particle (analog of the Hadamard-Rybchinsky formula):U = hp nz, (20)where= -RlP i  P eM-oo/ liIf |o,g/|j| ^  0 in expressions (18) and (20), then we obtain a formula for the resistance force and the veloc­ity of gravitational fall of a solid uniformly heated particle (analog of the Stokes formula).Thus, formulas (18) and (20) allow evaluation of the force acting upon a uniformly heated drop and the velocity of its gravitational fall with account of the temperature dependence of viscosity, represented as an exponential-power series at arbitrary temperature differences between the particle surface and a remote region.In the case when the drop surface is heated to a relatively low extent, i.e., the mean surface tempera­ture differs only slightly from the temperature of the ambient at infinity [y = (7's -  7X)/7X 0], the tem­perature dependence of the dynamic viscosity coef­ficient can be neglected and G\ = 1, (i\ = -3. G1/  = 12, G2 = 1, G\ = -1, G?,1 = 2 ,N x = 2, N2 = 6,N 3 = 3, and N4 = 6. In this case, formulas (18) and (20) are transformed into the known expression for a sphere[9, 1 0 ].Mention should be made of some problems con­cerning the motion of a uniformly heated spherical drop, whose solutions can be found directly from the results obtained here. Let us consider motion of a particle containing uniformly distributed heat sourcesFig. 1. Contribution to the resistance force, ® =I y -  273 K ’ vs' mean temperature Ts o f the mer­cury drop surface.(drains) with constant power and density q^, i.e., the motion of a particle with uniform internal heat release. In this case, it is necessary to supplement Eqs. (2)-(4) with an equation describing the distribution of tem­perature within the drop (A'/\ = q j k x) and to take into account the equality of temperatures and heat fluxes in the boundary conditions. Then the mean surface tem­perature of a uniformly heated drop is given by the relationTJTX = 1 + qiR2/ (K T J . (21)It is noteworthy that, in formulating the problem, the boundary condition (8) for normal stresses, un­necessary in analyzing the flow around a uniformly heated drop, was left out. It can be shown that, upon substitution of the obtained solution for velocity and pressure components, the given boundary condition reduces to identity. This means that, in the approxima­tion employed, the drop remains strictly spherical and its shape should be analyzed in terms of higher-order approximations.To illustrate the contribution of particle surface heating to the resistance force and the fall velocity of a drop, i.e., to take into account the temperature dependence of viscosity, represented by formula (1), Figs. 1 and 2 show curves relating the values ® =| f  = 2 7 3  K ^*1 _  T = 273 Kfor large mercury drops with radius R = 2 x 10-5 m, moving in water at 7X = 273 K. The curve ®* is plotted using the formula for small relative tempera­ture differences (y —» 0) [9, 10], but with the molecu­lar transfer coefficients taken for 7C = Ts. As seen from the presented curves, the drop surface heatingMALAI el al.Fig. 2. Fall velocity of mercury drop, = h^/h^ ^ _2?3 K,vs. the mean temperature Ts of the mercury drop surface.strongly affects the resistance force and the velocity ofgravitational fall of the drop. This result can be usedin practice for accelerating the sedimentation process[11]. Let us consider selective sedimentation of coalparticles in a water flow with temperature of 20°C ina rectangular chamber with length L = 100 cm, widthd = 80 cm, and height h = 5 cm. In the absence ofheating, the particle velocity in this chamber is1.109 x 10-2 cms-1. To this velocity correspondsthe volumetric flow rate of cleaned water 0  = 3 —10.319m h [11]. The formulas derived in the pres­ent study yield Q = 0.356 m3 s-1, which corresponds to sedimentation accelerated by a factor of 1.12, and, with the particle surface heated to 50°C, we obtain Q = 0.588 m3 s-1, which already gives an acceleration by a factor of 1.84. Consequently, the use of the derived formulas enables acceleration of particle sedimentation.CONCLUSIONS(1) An analytical solution of the problem of a fluid flow around a uniformly heated particle at small Reynolds numbers is described. In solving the hydro­dynamics equations, account is taken of the tempera­ture dependence of dynamic viscosity, represented as an exponential-power-law series. The Stokes and Hadamard-Rybchinsky formulas are generalized, which makes it possible to calculate the resistance force and the velocity of gravitational fall at arbitrary temperature differences between the particle surface and a remote region. It is shown that particle surface heating markedly affects the resistance force.(2) The results obtained can be used in evaluating the settling velocity of spherical hydrosol particles in channels, in designing experimental installations in which a directed motion of hydrosol particles is to be ensured, in developing methods for fine cleaning of fluids to remove hydrosol particles, etc.REFERENCES1. Khodakov, G.S. an Yudkin, Yu.P., Sedimentatsionnyi analiz vysokodispersnykh sistem (Sedimentation Analysis of Highly Dispersed Systems), Moscow: Khimiya, 1981.2. Bohren, C.F. and Huffman, D.R., Absorption and Scattering o f Light by Small Particles, New York: Wiley, 1983.3. Kuzikovskii, A.V., Izv. Vyssh. Uchebn. Zaved., Fiz., 1970, no. 5, pp. 89-94.4. Acrivos, A. and Taylor, T.D., J. Phys. Fluids, 1962, vol. 5, no. 4, pp. 387-394.5. Naidenov, V.I., Prikl. Mat. Mekh., 1971, vol. 38, no. 1, pp. 162-166.6. Leonov, A.I., Prikl. Mat. Mekh., 1962, vol. 26, no. 3, pp. 33-40.7. Shchukin, E.R., Malai, N.V., and Yalamov, Yu.I., Teplofiz. Vys. Temp., 1988, vol. 25, no. 5, pp. 1020- 1024.8. Bretsznajder, S., Wlasnosci gazow i cieczy, Warsaw: Naukowo-Techniczne, 1962.9. Landau, L.D. and Lifshits, E.M., Gidrodinamika (Hydrodynamics), Moscow: Nauka, 1986.10. Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics, Leiden: Nordhoff, 1973, 2nd ed.11. Fajita, H., Mathematical Theory o f Sedimentation Analysis, New York: Academic, 1969.

Effect of particle surface heating on the sedimentation rate

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Effect of particle surface heating on the sedimentation rate

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