Approximate solutions of stationary diffusion and thermal conductivity equations are obtained for the power-law dependence of the molecular transfer coefficients (thermal conductivity, diffusion) and the density of a binary viscous non-isothermal gas medium on temperatur

Malai, N. V.

Malai, V. V.

Shostak, Yu. I.

DSpace at Belgorod State University

ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2022, Vol. 43, No. 6, pp. 1135–1140. c© Pleiades Publishing, Ltd., 2022.Boundary Value Problem of Mass and Heat Transferof an Evaporating Spherical DropletN. V. Malai1*, Yu. I. Shostak1**, and V. V. Malai1***(Submitted by A. B. Muravnik)1Belgorod National Research University, Belgorod, 30015 RussiaReceived January 30, 2022; revised March 05, 2022; accepted March 10, 2022Abstract—Approximate solutions of stationary diffusion and thermal conductivity equations areobtained for the power-law dependence of the molecular transfer coefficients (thermal conductivity,diffusion) and the density of a binary viscous non-isothermal gas medium on temperature.DOI: 10.1134/S1995080222090190Keywords and phrases: mathematical physics equations, diffusion equation, heat conductionequation.1. PROBLEM STATEMENTTransfer phenomena are nonequilibrium processes that occur in a thermodynamically nonequilibriumsystem—the spatial transfer of energy, matter, momentum or some other physical quantity. The reasonfor this transfer is disturbances that violate the state of thermodynamic equilibrium. For example, thepresence of spatial inhomogeneities of temperature, composition, or average velocity of the particlesof the system. The transfer of a physical quantity occurs in the direction opposite to its gradient, asa result of which the thermodynamic system isolated from external influences approaches the state ofthermodynamic equilibrium. Transfer phenomena occur stationary if external influences are maintainedconstant.The significance of heat and mass exchange processes in production, nature, etc. is determined by thefact that the properties of bodies depend most significantly on their thermal state, which, in turn, is itselfdetermined by the conditions of heat and mass exchange. These conditions have a significant impacton the processes of changing the state of matter, mechanical, thermal, magnetic and other properties ofbodies. This explains the intensive development of the theory of heat and mass transfer and the extremelyimportant importance that they are given in industry, medicine, energy, agriculture and nature.As a rule, the transfer of heat (matter) in solids, liquids and gases is subject to conditionally acceptedlinear dependencies. For example, heat transfer—Fourier’s law: the density of the heat flux (specificheat flux) is proportional to the temperature gradient; molecular transfer of matter—Fick’s diffusionlaw: the density of the flow of matter is proportional to the concentration gradient (the difference indiffusion chemical potentials). Based on these linearized laws, the corresponding differential equationsare derived.When describing the behavior of particles suspended in thermodynamically nonequilibrium in tem-perature and concentration of viscous gaseous media, a dimensionless parameter Θ is introduced,characterizing the relative temperature difference between the average surface temperature of the particle(TS) and the temperature of the gaseous medium away from it (T∞) to the latter, i.e. Θ =(TS − T∞)/T∞.The relative temperature difference is considered small if the inequality Θ  1 is satisfied and significantif Θ ∼ (1). When the first condition is met, the coefficients of molecular transport (viscosity, thermal*E-mail: malay@bsu.edu.ru**E-mail: juliashostak@mail.ru***E-mail: malay_v@bsu.edu.ru11351136 MALAI et al.conductivity, diffusion) and density of a viscous gaseous medium can be considered constant values,and the medium itself isothermal. This condition significantly simplifies the mathematical procedure forfinding expressions for the fields of temperatures, concentrations, pressures, etc. in a thermodynamicsystem (see, for example, (see, for example, [4–6]). If Θ ∼ (1), the particle is called heated (heatingof the particle surface can be caused, for example, by the course of a volumetric chemical reaction,the process of radioactive decay of the particle substance, absorption of electromagnetic radiation by theparticle, etc.), and the viscous medium is non-isothermal. In this case, it is already necessary to take intoaccount the dependence of the molecular transfer coefficient and the density of the gaseous medium ontemperature, and the system of gas-dynamic equations describing such a medium becomes essentiallynonlinear. There are few works in the scientific literature devoted to the study of this case, for example(see, for example, [7, 8]).2. STATEMENT OF THE PROBLEM: BASIC EQUATIONSAND BOUNDARY CONDITIONSThe steady-state diffusion process is considered between an evaporating spherical drop of radius Rwith density ρi, thermal conductivity λi, viscosity μi and a viscous non-isothermal binary gas mixturewith density ρe, thermal conductivity λe, diffusion D12 and viscosity μe during its flow a plane-parallelflow directed along the Oz axis with a velocity of U∞. Heat sources with a density of qi operateinside the particle. This leads to the fact that the average surface temperature of the particle differssignificantly from the temperature away from it, i.e. Θ ∼ (1). The gas medium consists of twocomponents C1 and C2. The component C2 is considered the main (carrier), and the boundary surfaceis impervious to it. The substance of which the drop consists experiences a phase transition at thespherical interface of the drop-binary gas mixture. We will assume that this is a component of C1, i.e.in its physicochemical composition it coincides with the substance of a liquid drop and the boundarysurface for it is continuous. Here C1 = n1/ne, C2 = n2/ne are the relative concentrations of the firstand second components, ne = n1 + n2, ρe = ρ1 + ρ2, ρ1 = m1n1, ρ2 = m2n2, m1, n1 and m2, n2 massand numerical concentration of molecules of the first and second components of the mixture. From thedefinition of relative concentrations, it follows that C1 + C2 = 1 and the first component satisfies thecondition C1  C2. This inequality means that the evaporation of the droplet proceeds in a diffusionmode, i.e. when the main influence on the process of heat and mass transfer in the vicinity of a drop isdetermined by molecular diffusion, described by the equation (1) [4, 5].div(n2em1m2ρeD12∇C1)= 0. (1)The steady-state diffusion process is considered in a spherical coordinate system (y = r/R, θ, ϕ)associated with the center of mass of the particle. With the specified choice of the origin of the coordinatesystem, the distributions of velocities, pressures, concentrations and temperatures outside and insidethe evaporating droplet have axial symmetry with respect to the Oz axis, i.e. they are functions of twovariables y, θ. The indices “e” and “i” hereafter refer to the binary gas mixture and the evaporatingdroplet, and the index “∞” are physical quantities characterizing the gas medium in an undisturbedflow.A TASK. A viscous non-isothermal binary gas mixture occupies an unlimited region Ω ⊂ R3. Youwant to find the distribution the relative concentration of C1 in the binary mixture that satisfies thediffusion equation (1), and distribution outside temperature (Te) and in (Ti) of the evaporating liquiddroplets are described by the heat conduction equations (2) [5]div(λe∇Te)= 0, div(λi∇Ti)= −qi, (2)with boundary conditionslimy→∞Te = T∞, limy→∞C1 = C1∞, limy→0Ti = ∞. (3)Here qi is the density of heat sources, due to which the surface of the drop is heated.In the boundary conditions (3) away from the droplet, the temperature, the relative concentration ofthe diffusing component is set and the finiteness of the temperature in the center of the particle is takenLOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 43 No. 6 2022BOUNDARY VALUE PROBLEM OF MASS 1137into account. As for the boundary conditions for the diffusion equation on the surface of the evaporatingdrop itself, we do not give. They are determined by a specific physical problem, which allows us to obtaina solution of the equation (1) in general form. The problem is solved by the hydrodynamic method, i.e.the equations of gas dynamics with the corresponding boundary conditions are solved.When describing the properties of a binary gas mixture, the power-law form of the dependence ofthe molecular transfer coefficients and the density of the medium on the temperature [2] is taken intoaccountμe = μ∞(TeT∞)β, ρe = ρ∞T∞Te, λe = λ∞(TeT∞)α,D12 = D∞(TeT∞)1+ω, λi = λi0(TiT∞)γ, (4)where μ∞ = μE(T∞), ρ∞ = ρE(T∞), λ∞ = λE(T∞), D∞ = D12(T∞), λi0 = λi(T∞), 0.5 ≤ α, β, ω ≤1, −1 ≤ γ ≤ +1 [2].Taking into account the power-law dependence of the molecular transfer coefficients and the densityof the medium on temperature, we obtain a rather complex nonlinear system of gas-dynamic equations,and, naturally, the question arises about the method of solving this nonlinear system of equations andabout the assumptions justified, for example, from a physical point of view, which need to be done tosolve the problem. The study showed that for most gaseous media, the thermal conductivity coefficientof the particle (λi) is much larger in magnitude than the thermal conductivity coefficient of the gas (λe),i.e. λe  λi. There is a weak angular asymmetry of the temperature distribution in the “particle–gas”system. This leads to the fact that the viscosity is related only to the temperature T (0)e (y): μe(Te(y, θ))≈ μe(T (0)e (y)). At the same time, Te(y, θ) = T (0)e (y) + δTe(y, θ), where δTe(y, θ)  T (0)e (y); T (0)e (y),δTe(y, θ) are determined from the solution of the thermal problem. This assumption leads to the factthat the diffusion equations and the thermal conductivity equations make it possible to obtain analyticalsolutions.3. SOLUTION OF THE DIFFUSION AND THERMAL CONDUCTIVITY EQUATIONSThe general solution of the first equation (2) satisfying the first boundary condition (3) taking intoaccount (4) has the formte(y, θ) =[1 +N0y+∞∑n=1(Nnyn+1+Dnyn)Pn(cos θ)]1/(1+α), te = Te/T∞.Here Pn(cos θ) is Legendre polynomials; N0, Nn,Dn are constant integrations that are determined fromthe boundary conditions on the surface of the evaporating droplet.In our problem there is a small parameter ε = Re∞, i.e. when solving (1), (2), we can use theperturbation theory [1]. From the defining parameters of our problem, we can make the Reynoldsnumber Re∞ =(ρ∞RU∞)/μ∞  1. Here U∞ = |U∞|. The incoming flow has only a disturbing effect.Therefore, the function te(y, θ) is in magnitude proportional to the perturbation and hencete(y, θ) = te0(y) +t−αe0 (y)1 + α∞∑n=1(Nnyn+1+Dnyn)Pn(cos θ), te0(y) =(1 +N0y) 11 + α ,and accordinglytωe (y, θ) = tωe0(y) +ω1 + αtω−α−1e0∞∑n=1(Nnyn+1+Dnyn)Pn(cos θ). (5)The general solution of the second equation (2) satisfying the second boundary condition (3), i.e. theextremity of temperature at y → 0 has the formti(y, θ) = ti0(y) +t−γi0 (y)1 + γ∞∑n=1[Hnyn+1+DnynLOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 43 No. 6 20221138 MALAI et al.+12n+ 1(yny∫1qinyn−1dy − 1yn+1y∫1qinyn+2dy)]Pn(cos θ),ti0(y) =(B0 +H0y+ yy∫1yqi0dy −1yy∫1y2qi0dy) 11 + γ,whereHn =12n + 10∫1yn+2qin(y)dy, qin(y) = −1 + γλi0T∞R22n+ 12+1∫−1qiPn(x)d(x), x = cos θ,Bn are integration constants, which are determined from the boundary conditions on the surface of theevaporating droplet. Substituting the expression (1) into the diffusion equation (5) getdiv(tωe0∇C1)= − ω1 + αdiv[tω−α−1e0∞∑n=1(Nnyn+1+Dnyn)Pn(cos θ)∇C1]. (6)We will search for the solution of the equation (6) in the formC1(y, θ) =∞∑n=0ψn(y)yn+1Pn(cos θ).Function C1(y, θ) is proportional in magnitude to the perturbation, i.e.C1(y, θ) = C10(y) + C11(y, θ), C11(y, θ)  C10(y),C10 =ψ0(y)y, C11(y, θ) =∞∑n=1ψn(y)yn+1Pn(cos θ). (7)After substituting (7) into the equations (6) and using the orthogonality condition of Legendre polyno-mials, we obtain two equations in a spherical coordinate system1y2ddy(y2tωe0(y)dC10dy)= 0, (8)1y2ddy[y2tωe0ddy(ψnyn+1)]− n(n+ 1)yn+3tωe0ψn= − ωy1(1 + α)ddy[y2tω−α−1e0(Dnyn +Nnyn+1)dC10dy]. (9)The general solution of the equation (8) satisfying the second boundary condition (2) has the formC10(y) = C1∞ +M(t1+α−ωe0 − 1). (10)Here M is a constant determined from the boundary conditions on the drop surface. Substituting (10)into (9), we obtain the following equation for finding the function ψn(y)y2d2ψndy2− y(2n+ s)dψndy+ s(n+ 1)ψn=ω(1 + α− ω)(1 + α)2Mtωe0[Dny2n+1(n+ ) +Nn(− n− 1)]. (11)LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 43 No. 6 2022BOUNDARY VALUE PROBLEM OF MASS 1139Here s = ω/(1 + α),  = N0/(N0 + y). The partial solution of the equation (11) has the formΦn(y) =1 + α− ω1 + αMtωe0[Dny2n+1 +Nn].Let’s find a solution to the homogeneous equation (11), for this we go to the new variable z = y/N0,we get(1 + z2)z2d2ψndz2− z [2nz + 2n+ s] dψndz+ (n+ 1)zψn = 0,and we are seeking for a solution to this equation in the formψn(z) =(1 +1z)δnfn(z),here δn = −12(2n + 1 + s) +12√(2n+ 1)2 − 2s + s2. The function fn is determined from the solutionof the equationt(t− 1)f ′′n (t)−[2(n + 1) + t(2δn + s− 2)]f ′n(t)− (n + 1)(2δn + s)fn(t) = 0, t =11 + z. (12)If in the equation (12) enter the notation γn = 2(n + 1), αn = −δn, βn = 1− δn − s, then the equation(12) takes the formt(1− t)f ′′n (t) +[γn − (αn + βn + 1)t]f ′n(t)− αnβnfn(t) = 0,and its solution is the hypergeometric function Fn(αn, βn; γn; t) [3].Theorem. The general solution of the equation (1) satisfying the boundary conditions (3)has the formC1(y, θ) = C10(y) +∞∑n=1Pn(x)yn+1[Φn(y) + Ln(1 +N0y)δnFn(αn, βn; γn; )],C10(y) = C1∞ +M(t1+α−ωe0 − 1), te0(y) =(1 +N0y) 11 + α ,Φn(y) =1 + α− ω1 + αMtωe0[Dny2n+1 +Nn],δn = −12(2n + 1 + s)+12√(2n+ 1)2 − 2s+ s2, αn = −δn,βn = 1− δn − s, γn = 2(n+ 1),  = N0N0+y , s =ω1+α , Fn(αn, βn; γn; ) is hypergeometric function.4. CONCLUSIONSAssuming a weak temperature inhomogeneity in the angular part (the thermal conductivity coef-ficient of a particle is much larger in magnitude than the thermal conductivity coefficient of a gas),which is the case for most physical applications, a general solution of the stationary diffusion equationhas been found with a power-law dependence of the molecular transfer coefficients (viscosity, thermalconductivity, diffusion) and the density of a binary non-isothermal gas mixture on temperature.LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 43 No. 6 20221140 MALAI et al.REFERENCES1. A. V. Vasilyeva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations (Vyssh.Shkola, Moscow, 1990) [in Russian].2. N. B. Vargaftik, Tables on the Thermophysical Properties of Liquids and Gases (Nauka, Moscow, 1972)[in Russian].3. E. Kamke, Differentialgleichungen Losungsmethoden und Losungen (Akademische Verlagsgesellschaft,Leipzig, 1957).4. V. G. Levich, Physico-Chemical Hydrodynamics (Fizmatlit, Moscow, 1959) [in Russian].5. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow,1986; Pergamon, New York, 1987).6. Yu. P. Gupalo and Yu. S. Ryazantsev, “On mass and heat transfer of a spherical particle in a laminar flow of aviscous liquid,” Appl. Math. Mech. 35, 255–265 (1971).7. N. V. Malai, D. N. Efimtseva, E. R. Shchukin, and Z. L. Shulimanova, “Diffusiophotesis and photophoresis ofheat large nonvolatile aerosol spherical particles,” J. Appl. Mech. Tech. Phys. 61, 426–435 (2020).8. E. R. Shchukin, N. V. Malai, Z. L. Shulimanova, and L. A. Uvarova, “Diffuse vaporization (sublimation) of alarge aerosol particle under precipitous changes in the ambient temperature,” High Temp. 53, 531–537 (2015).9. A. Acrivos and T. D. Taylor, “Heat and mass transfer from single spheres in Stokes flow,” Phys. Fluids 5,378–394 (1962).LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 43 No. 6 2022

Boundary value problem of mass and heat transfer of an evaporating spherical droplet

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Boundary value problem of mass and heat transfer of an evaporating spherical droplet

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