The Stefan outflow in a multicomponent vapor-gas atmosphere around a droplet and its role for cloud expansion

# The Stefan outflow in a multicomponent vapor-gas atmosphere around a droplet and its role for cloud expansion

A. E. Kuchma    A. K. Shchekin    D. S. Martyukova Department of Statistical Physics, Faculty of Physics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia
###### Abstract

A new comprehensive analysis of Stefan’s flow caused by a free growing droplet in vapor-gas atmosphere with several condensing components is presented. This analysis, based on the nonstationary heat and material balance and diffusion transport equations, shows the appearance of the Stefan inflow in the vicinity of the growing droplet and the outflow at large distances from the droplet as a consequence of nonisothermal condensation. For an ensemble of droplets in the atmospheric cloud, this flow provides an increase of the total volume of the cloud, which can be treated as cloud thermal expansion and leads to floating the cloud as a whole due to buoyancy. We have formulated the self-similar solutions of the nonstationary diffusion and heat conduction equations for a growing multicomponent droplet and have derived analytical expressions for the nonstationary velocity profile of Stefan’s flow and the expansion volume of the vapor-gas mixture around the growing droplet. To illustrate the approach, we computed these quantities in the case of droplet of stationary composition in air with several specific vapors (; ; ).

## Introduction

Growth and evaporation of small droplets are key phenomena in the physics of aerosols and clouds. The regularities of phase changes and latent heat release have a global influence on the Earth atmosphere and climate (Kessler (1969), Heintzenberg and Charlson (2009), Pruppacher and Klett (2010)). Cloud microphysical schemes are a central part of every model of the atmosphere. In numerical weather simulation, they are important for quantitative precipitation forecasts.

Our goal in this paper is to give a detailed analysis of one of mechanisms for thermal cloud expansion in the atmosphere. This mechanism is associated with the Stefan flow caused by nonisothermal growth of small droplets in the cloud. The Stefan flow of the vapor-gas mixture around a droplet compensates the diffusive flux of noncondensable gas molecules from the impenetrable surface of the growing droplet. This flow under ordinary atmospheric conditions is usually considered to be a small effect (Fuchs (1959); Clement (2008); Kuchma and Shchekin (2011, 2012)). However there are some atmospheric situations where the Stefan flow may be important. The particle scavenging efficiency of vapor-grown ice crystals falling from mixed clouds proves to be very high due to the Stefan flow which pushes airborne particles away from the surface of the supercooled droplets evaporating in the vicinity of an ice crystal (Vittori (1984)). The influence of the Stefan flow on the heat balance of a char particle should be taken into account in oxy-fuel combustion (temperature K), because, the presence of Stefan’s flow greatly weakens the heat transfer by conduction and accelerates the heat loss from the particle, which makes the particle temperature lower (Zhou et al. (2013)). Another effect, as follows from Kuchma et al. (2016), owes to nonstationary heating a vapor-gas mixture around a growing droplet (due to releasing the condensation heat) which produces the Stefan outflow at large distances from the droplet. For an ensemble of droplets in the cloud, such effect results in increase of the total volume of the cloud that can be treated in physics of clouds as cloud thermal expansion followed by the rise of the cloud as a whole due to buoyancy. In the case of several vapors, also the vertical structure of a cloud with droplets of different size and composition can change. Thus we have an example where a small effect on a level of a single droplet becomes an origin for strong mass and heat transport on the macroscopic scale of a cloud.

We will formulate below the theory of the Stefan flow in the case of multi-component nonisothermal and nonstationary growth of a free droplet with non-ideal composition. As a first step, we will derive in Sec. I the general expressions for the velocity of the Stefan flow based on the material and enthalpy balance equations at nonstationary diffusion. In Sec. II, we will consider the self-similar solutions of the nonstationary diffusion and heat conduction equations for a growing multicomponent droplet and obtain analytical expressions for the whole nonstationary velocity profile of Stefan’s flow and the expansion volume of the vapor-gas mixture around the growing droplet. Finally in Sec. III, we will illustrate the approach by computing velocity profile of Stefan’s flow and the expansion volume of the vapor-gas mixture around the growing droplet for droplets of stationary composition in air with several specific vapors (binary mixture of ethanol-water vapors; binary mixture of sulfuric acid-water vapors, and pure water vapor ).

## I General regularities for a multicomponent Stefan flow

Consider a small free spherical droplet suspended in the ideal mixture of several condensable vapors and a non-condensable carrier gas. The condensable components are assumed to be miscible in the droplet at any ratios, and the solution in the droplet may be non-ideal. The droplet is tiny enough to provide that the specific times of establishing internal thermal and chemical equilibrium in the droplet are small in comparison with the specific time of changing the droplet radius in time. As a consequence, the droplet is internally uniform in composition and temperature. The droplet can grow or evaporate depending on the ratio between the actual concentrations of vapors far away from the droplet and the concentrations of the vapors at saturation with the solution in the droplet. We will take into account releasing or absorbing the latent heat at condensation or evaporation. As a result, the temperatures of the droplet and surrounding vapor-gas mixture may differ. The regimes of vapor and heat transfer to or from the droplet are diffusive. The total pressure in the vapor-gas mixture is considered to be constant. Gravity is neglected.

We set the origin of the coordinate system in the center of the spherical droplet with radius changing in time . In view of the spherical symmetry, the local vapor concentration of the condensable component , i.e., the local number of molecules of th component per unit volume, and the local temperature in the vapor-gas mixture at any time moment depend only on distance from the center of the droplet to the observation point. The corresponding concentration of the non-condensable carrier gas will be denoted by . Thereby the total local volume concentration of molecules in the vapor-gas mixture is determined as

 n(r,t)=ng(r,t)+k∑i=1ni(r,t), (I.1)

The boundary conditions for the vapor concentrations and temperature can be written in the form

 ni(r,t)−−−→r→∞ni0,ni(r,t)∣∣r=R(t)=ni∞({x},Td),i=1,2…k, (I.2)
 n(r,t)−−−→r→∞n0=ng0+k∑i=1ni0, (I.3)
 T(r,t)−−−→r→∞T0,T(r,t)∣∣r=R(t)=Td, (I.4)

where and are the bulk concentration of th component and carrier gas, respectively, is the temperature far away from droplet, is the equilibrium concentration of th component over a planar surface of the liquid solution at droplet temperature (we neglect the effect of the curvature of droplet surface on the th vapor equilibrium pressure and concentration) and composition . The molar concentration of condensible components in the droplet (the total number of droplet components coincides with the number of the vapor components in the vapor-gas mixture) is defined by

 xi(t)=Ni(t)N(t),i=1,2…k. (I.5)

Here is the number of molecules of th component in the droplet, and is the total number of molecules in the droplet.

### i.1 Velocity of a multicomponent Stefan flow

The total number of all molecules inside an imaginary sphere around droplet with radius , which expands with the velocity of the hydrodynamic molecular flow of the vaporgas mixture (i.e., the velocity of the Stefan flow), does not depend on time and is distributed between the droplet and the vapor-gas phase as

 N(t)+4πR1(t)∫R(t)drr2n(r,t)=const, (I.6)

Differentiating both sides of eq.(I.6) with respect to time and setting , , and allows us to express the Stefan velocity in the form

 u(r,t)=−˙N(t)4πr2n(r,t)+R2(t)˙R(t)r2n(R(t),t)n(r,t)−−1r2n(r,t)r∫R(t)dr1r21∂n(r,t)∂t. (I.7)

At diffusive regime of droplet growth, the droplet composition and the droplet temperature become independent of time after some transient stage (Kulmala et al. (1993), Kuchma et al. (2011a, b),Kuchma et al. (2013)). Because the time of this transient stage is small, we will neglect this stage and consider that the droplet composition and the temperature of droplet do not depend on time, i.e. () and , respectively. Recognizing that the pressure is constant in the ideal vapor-gas mixture, we have the following relations between total local volume concentration of molecules and local temperature in the vapor-gas mixture

 n(r,t)T(r,t)=n0T0, (I.8)
 ∂n(r,t)∂t=−n0T0T2(r,t)∂T(r,t)∂t. (I.9)

Using eqs.(I.8) and (I.9) and the relationship for the droplet volume ,

 Vd(t)=4π3R3=k∑i=1vil({x},Td)xiN≡vl({x},Td)N, (I.10)

where is the partial volume per molecule of th component and is the mean molecular volume in the droplet with total composition and temperature , allows us to express the velocity of the Stefan flow from (I.7) as

 u(r,t)=˙RR2r2T(r,t)Td(1−Tdvl({x},Td)n0T0++TdR2˙Rr∫R(t)dr1r211T2(r1,t)∂T(r1,t)∂t). (I.11)

Here we took into account that the composition and the droplet temperature are fixed in time. Eq. (I.11) is the general expression for the velocity of the Stefan flow in the case of nonisothermal multicomponent growth of a droplet with non-ideal solution at stationary composition and temperature.

It strictly follows from eq.(I.11) at and that

 u(r,t)−−−−→r→R(t)−˙RTdvl({x},Td)n0T0<0. (I.12)

Thereby, the Stefan flow is directed to the droplet in the droplet vicinity. As is seen from eq.(I.11) at and at isothermal condensation or stationary heat conduction when , the velocity of the Stefan flow is directed to the growing droplet at any

 u(r,t)=−˙RR2r2T(r,t)vl({x},T0)n0T0<0. (I.13)

### i.2 The hydrodynamic flow rate far from the droplet

In the case of non-stationary heat conduction, the integral term in brackets in eq.(I.11) is able to change the sign of the velocity at large . To check its role, we consider below the balance of the total enthalpy of the system consisting of droplet and vapor-gas mixture. Let us express the total enthalpy of the droplet and surrounding vapor-gas atmosphere within the sphere of radius as

 H=hl({x},Td)N++4πR2(t)∫R(t)drr2(hg(T(r,t))ng(r,t)+k∑i=1hi(T(r,t))ni(r,t)), (I.14)

where

 hl({x},Td)≡k∑i=1hil({x},Td)xi (I.15)

is the mean enthalpy per molecule in the droplet solution, is the partial enthalpy of th component in droplet solution, and are the enthalpies per molecule of carrier gas and of th component of vapor, respectively. If we take the radius as the radius of the sphere which is located outside the diffusion layer (this is possible only for nonstationary diffusion and heat conduction) and which boundary shifts with the velocity of the hydrodynamic molecular flow of the vapor-gas mixture, i.e. , the enthalpy will conserve in time.

Differentiating both sides of eq.(I.14) with respect to time at fixed and (we recall that the transition stage of the droplet growth is considered to be fast comparatively with the time of observation) and setting allow us to express the Stefan velocity outside the diffusion shell around the droplet in the form

 4π(hg(T0)ng0+k∑i=1hi(T0)ni0)r2u(r,t)==−hl({x},Td)˙N++4πR2(t)˙R(t)(hg(Td)ng(R(t),t)++k∑i=1hi(Td)ni(R(t),t))−−4πr∫R(t)dr1r21∂∂t(hg(T(r1,t))ng(r1,t)++k∑i=1hi(T(r1,t))ni(r1,t)). (I.16)

With the help of eq.(I.1), it is convenient to write

 hg(T(r,t))ng(r,t)+k∑i=1hi(T(r,t))ni(r,t)==hg(T(r,t))n(r,t)+k∑i=1(hi(T(r,t))−hg(T(r,t)))ni(r,t). (I.17)

In view of eq.(I.8) we have

 hg(T(r,t))n(r,t)=cgT(r,t)n(r,t)=cgT0n0, (I.18)

where is the heat capacity at constant pressure per carrier gas molecule. Substituting eq.(I.18) into eq.(I.17) and neglecting the dependence of on temperature and differentiating with respect to time yields

 ∂∂t[hg(T(r,t))ng(r,t)+k∑i=1hi(T(r,t))ni(r,t)]==∂∂tk∑i=1(hi(T(r,t))−hg(T(r,t)))ni(r,t). (I.19)

Let us first consider the particular case when , , i.e. the enthalpies per molecule of carrier gas and all vapor components are equal. As follows from eqs.(I.19) and (I.3), eq.(I.16) can be rewritten in this particular case as

 hg(T0)n0r2u(r,t)=−R2(t)˙R(t)hl({x},Td)v−1l({x},Td)++R2(t)˙R(t)[hg(Td)ng(R(t),t)+k∑i=1hi(Td)ni(R(t),t)], (I.20)

where we used eqs.(I.10). Introducing the mean evaporation heat per molecule in the liquid with composition and temperature as

 q({x},Td)≡k∑i=1[hi(Td)−hil({x},Td)]xi==k∑i=1qil({x},Td)xi, (I.21)

where is the partial evaporation heat of th component, we find from eq.(I.20)

 hg(T0)n0r2u(r,t)=R2(t)˙R(t)q({x},Td)v−1l({x},Td)++R2(t)˙R(t)[hg(Td)ng(R(t),t)++k∑i=1hi(Td)(ni(R(t),t)−xiv−1l({x},Td))]. (I.22)

Recognizing equalities , , (I.1) and (I.18) allows us to rewrite eq.(I.22) as

 u(r,t)=˙R(t)R2(t)r2[1++1n0vl({x},Td)TdT0(q({x},Td)hg(Td)−1)]. (I.23)

As follows from eq.(I.23) at with and ,

 u(r,t)>0, (I.24)

i.e., the velocity of the Stefan flow is directed from the growing droplet at any point outside the diffusion shell around the droplet.

Let us now consider the case when vapors are only small admixtures to the carrier gas, i.e., when the strong inequality

 k∑i=1ni(r,t)≪n, (I.25)

fulfills, the relative deviations of the temperature of the vapor-gas medium due to release of the condensation heat are also small,

 |T(r,t)−T0|≪T0. (I.26)

Inequalities (I.25) and (I.26) together with eq.(I.19) and (I.10) allow us to rewrite eq.(I.16) with a sufficient accuracy as

 4πhg(T0)ng0r2u(r,t)≈−hl({x},Td)˙N++4πR2(t)˙R(t)hg(T0)ng0−−4πk∑i=1(hi(T0)−hg(T0))r∫R(t)dr1r21∂ni(r,t)∂t. (I.27)

There is no material sources in the vapor-gas atmosphere, and we can write a continuity equation for each vapor,

 ∂ni(r,t)∂t=−∇→ji(r,t)=−∇[→Ji(r,t)+ni(r,t)→u(r,t)], (I.28)

where and are the densities of the total molecular flux and, respectively, the diffusion molecular flux of th component of vapor. The integral term in eq.(I.27) can be rewritten with the help of eq.(I.28) as

 4πr∫R(t)dr1r21∂ni(r,t)∂t=4πk∑i=1(ji(R,t)R2−ji(r,t)r2)==4πk∑i=1(ji(R,t)R2−ni0u(r,t)r2). (I.29)

Substituting eq.(I.29) in eq.(I.27), recognizing smallness of the shift of the surface of droplet with the rate , using equalities

 ˙Ni=−4πR2(ji(R,t)−ni(R,t)˙R)≈−4πR2ji(R,t), (I.30)
 4πR2k∑i=1ji(R,t)=−˙N, (I.31)

eqs.(I.5) and (I.21) yield

 4πhg(T0)ng0r2u(r,t)≈q({x},Td)˙N++4πR2(t)˙R(t)hg(T0)ng0−hg(T0)˙N, (I.32)

where we took into account that . Replacing by , we find from (I.32)

 u(r,t)=˙R(t)R2(t)r2[1++1ng0vl({x},Td)(q({x},Td)hg(T0)−1)]. (I.33)

According to procedure of derivation, eq.(I.33) is applicable outside the diffusion shell around the droplet where . Under conditions (I.25) and (I.26), eq.(I.23) reduces to eq.(I.33). Thus eq.(I.24) is still valid in the same domain, and we should observe the Stefan outflow with positive . Taking into account eq.(I.12), one can expect that velocity should change its sign at some distance () within the diffusion shell.

## Ii A self-similar approach to the Stefan flow problem

### ii.1 The local temperature and vapor concentrations in the vapor-gas mixture

To describe all the details in the behavior of the Stefan flow velocity analytically at any distance from the droplet center, we will analyze the case of nonstationary self-similar regime of multicomponent nonisothermal droplet growth at fixed composition and temperature . Let be the self-similar variable determined as

 z≡r/R(t), (II.1)

then self-similar vapor densities and temperature profile should be taken as

 ni(r,t)=ni(z)(i=1,2…k),T(r,t)=T(z). (II.2)

We will assume below that inequalities (I.25) and (I.26) fulfill. These inequalities can be rewritten in the self-similar variables in the form

 k∑i=1ni(z)≪n, (II.3)
 T(z)−T0T0≪1. (II.4)

Correspondingly, eq.(I.11) for the Stefan velocity profile transforms under conditions (II.3), (II.4), and self-similar variables as

 u(z)≈−˙R(t)z2⎛⎜⎝1n0vl({x},Td)+1T0z∫1dz1z31dT(z1,t)dz1⎞⎟⎠. (II.5)

To find the velocity with the help of eq.(II.5), it is sufficient to derive expressions for rate and non-stationary temperature profile in the vapor-gas atmosphere in the lowest approximation with respect the small parameter . In this case, we can use the equations of diffusion and heat transfer obtained with neglecting the Stefan flow. Then nonstationary equations for the diffusion to the droplet and the heat transfer from the droplet to the medium can be written in the following form (Grinin et al. (2008, 2011); Kuchma et al. (2015))

 ∂ni(r,t)∂t=Dir2∂∂r[r2∂ni(r,t)∂r]−−˙RR2(t)r2∂ni(r,t)∂r,i=1,2…k, (II.6)
 ∂T(r,t)∂t=χr2∂∂r[r2∂T(r,t)∂r]−˙RR2(t)r2∂T(r,t)∂r, (II.7)

where is the diffusivity of th component and is the thermal diffusivity of the vapor-gas medium. The boundary conditions at the droplet surface at for eqs.(II.6) and (II.7) are the equations of material and heat balance at the droplet surface

 ˙N=ddt(4πR33vl({x},Td))=k∑i=1˙Ni==(4πr2k∑i=1Di∂ni(r,t)∂r)∣∣∣r=R, (II.8)
 (κ∂T(r,t)∂r)∣∣∣r=R=−(k∑i=1qil({x},Td)Di∂ni(r,t)∂r)∣∣∣r=R. (II.9)

Here is the thermal conductivity of the vapor-gas mixture. Equations (II.6) and (II.7) and the boundary conditions (II.8), (II.9) rewritten in the self-similar variables have the form

 d2ni(z)dz2+[2z+R˙RDi(z−1z2)]dni(z)dz=0,i=1,2…k, (II.10)
 d2T(z)dz2+[2z+R˙Rχ(z−1z2)]dT(z)dz=0 (II.11)

and

 R˙R=vl({x},Td)k∑i=1Didni(z)dz∣∣∣z=1, (II.12)
 κdT(z)dz∣∣∣z=1=−k∑i=1qil({x},Td)Didni(z)dz∣∣∣z=1. (II.13)

Note, as follows from eq.(II.12), the rate (entering eqs.(II.10) and (II.11)) is independent of , but generally depends on droplet composition and temperature. This fact is important for existence of the self-similar solution. It allows simultaneous solving of ordinary differential equations eq.(II.10) and eq.(II.11) together with conditions (I.2) and (I.4). As a result we have

 ni(z)=ni∞({x},Td)+(ni0−ni∞({x},Td))××z∫1dyy2exp[−R˙R2Di(y2+2y−3)]∞∫1dyy2exp[−R˙R2Di(y2+2y−3)],i=1,2…k, (II.14)
 T(z)=Td+(T0−Td)z∫1dyy2exp[−R˙R2χ(y2+2y−3)]∞∫1dyy2exp[−R˙R2χ(y2+2y−3)]. (II.15)

Here we took into account the conditions (I.2) and (I.4).

### ii.2 Stationary composition and temperature of the droplet

The stationary rate of droplet growth and the droplet temperature satisfy two coupled integral equations following from eqs.(II.12) and (II.13)

 R˙R=vl({x}k∑i=1Di(ni0−ni∞({x},Td))∞∫1dyy2exp[−R˙R2Di(y2+2y−3)], (II.16)
 Td−T0=−∞∫1dyy2exp[−R˙R2Di(y2+2y−3)]××k∑i=1qil({x},Td)Di(ni0−ni∞({x},Td))κ∞∫1dyy2exp[−R˙R2Di(y2+2y−3)]. (II.17)

Finally we need an equation for the stationary droplet composition which is established independently of initial composition of the growing droplet. As follows from definition (I.5), the condition of a fixed composition in the droplet can be written as

 ˙xi=˙Ni(t)−xi˙N(t)N(t)=0,i=1,2…k, (II.18)

In view of eqs.(II.8) and (II.14),

 ˙Ni(t)=4πRDi(ni0−ni∞({x},Td))∞∫1dyy2exp[−R˙R2Di(y2+2y−3)],i=1,2…k. (II.19)

Taking into account eqs.(II.8) and (II.19), we find from eq.(II.18) equations for the composition in the growing droplet

 1xiDi(ni0−ni∞({x},Td))∞∫1dyy2exp[−R˙R2Di(y2+2y−3)]==k∑m=1Dm(nm0−nm∞({x},Td))∞∫1dyy2exp[−R˙R2Dm(y2+2y−3)],i=1,2…k. (II.20)

If we know the explicit thermodynamical expressions for , and at , eqs. (II.16), (II.17), (II.20) together with eqs.(II.5) and (II.15) form a complete set of equations for the problem of finding the profile of the Stefan flow velocity around a growing multicomponent droplet.

To simplify the Stefan velocity profile problem at the characteristic values of the parameters in the case of growing droplet in the vapor-gas atmosphere, the following approximations can be used,

 ∞∫1dyy2exp[−R˙R2χ(y2+2y−3)]≈1,∞∫1dyy2exp[−R˙R2Di(y2+2y−3)]≈1. (II.21)

Substituting eq.(II.21) into eq.(II.15) for the local temperature profile in a vapor-gas mixture gives

 T(z)≈Td−(Td−T0)z∫1dyy2exp(−R˙R2χy2), (II.22)

where the stationary droplet growth rate satisfies according to eq.(II.16) to the formula

 R˙R=vl({x},Td)k∑i=1Di(ni0−ni∞({x},Td)). (II.23)

Substituting eq.(II.21) into eqs.(II.17) and (II.20) for the stationary droplet temperature and droplet composition gives

 κ(Td−T0)=k∑i=1qilDi(ni0−ni∞({x},Td)), (II.24)
 Di(ni0−ni∞({x},Td))xi=D1(n10−n1∞({x},Td))x1,i=1,2…k,k∑i=1xi=1. (II.25)

### ii.3 The Stefan velocity profile

Differentiating eq.(II.22) with respect to variable yields

 dT(z1)dz1≈−Td−T0z21exp(−R˙R2χz21). (II.26)

Substituting eqs.(II.26) into eq.(II.5) for the velocity of the Stefan flow and using equality , eqs.(II.1) and (II.24) allow one to obtain

 u(z)≈˙R(t)z21vl({x},Td)n0[q({x},Td)cgT0××(1−exp(−R˙R2χz2))−1]. (II.27)

The distance from the center of the droplet, where the Stefan flow velocity changes its sign, corresponds to the value at which . As follows from eqs.(II.1) and (II.27)

 r2u=0R2(t)=−2χR˙Rln(1−cgT0q({x},Td)). (II.28)

Outside the non-stationary diffusion shell around the droplet, where , eq.(II.27) for the Stefan flow velocity reduces to

 u(z)≈˙R(t)z21vl({x},Td)n0(q({x},Td)cgT0−1), (II.29)

or, in spherical coordinates, to

 u(r,t)≈˙R(t)R2(t)r21vl({x},Td)n0(q({x},Td)cgT0−1). (II.30)

Eq.(II.30) coincides with a general asymptotic form given by eq.(I.33) at .

In the vicinity of the droplet, where , eq.(II.27) reduces to

 u(r,t)≈−˙Rn0vl({x},Td), (II.31)

which coincides with another general asymptotic form given by eq.(I.12) at .

Let us consider now the expansion of the vapor-gas medium around the growing droplet. The vapor-gas mixture expands in response to Stefan’s outflow. The corresponding increase of the volume of the vapor-gas mixture per a droplet at a given velocity profile (outside the non-stationary diffusion shell) can be estimated as

 ΔV(t)=4πt