Anjun Liu,Jie Chen ,Moshe Favelukis,Meng Guo, ,Meihong Yang,Chao Yang ,Tao Zhang,Min Wang,Hao-yue Quan

1 Qilu University of Technology (Shandong Academy of Sciences),Shandong Computer Science Center(National Supercomputer Center in Jinan),Jinan 250101,China

2 CAS Key Laboratory of Green Process and Engineering,Institute of Process Engineering,Chinese Academy of Sciences,Beijing 100190,China

3 School of Chemical Engineering,University of Chinese Academy of Sciences,Beijing 100049,China

4 Department of Chemical Engineering,Shenkar College of Engineering and Design,Ramat-Gan,5252626,Israel

5 Dynamic Machinery Institute of Inner Mongolia,Inner Mongolia,Hohhot 010010,China

Keywords:External mass transfer Nonlinear extensional flow Sphere Empirical correlation

ABSTRACT This work systematically simulates the external mass transfer from/to a spherical drop and solid particle suspended in a nonlinear uniaxial extensional creeping flow.The mass transfer problem is governed by three dimensionless parameters:the viscosity ratio (λ),the Peclet number (Pe),and the nonlinear intensity of the flow (E).The existing mass transfer theory,valid for very large Peclet numbers only,is expanded,by numerical simulations,to include a much larger range of Peclet numbers(1 ≤Pe ≤105).The simulation results show that the dimensionless mass transfer rate,expressed as the Sherwood number (Sh),agrees well with the theoretical results at the convection-dominated regime (Pe ＞103).Only when E ＞5/4,the simulated Sh for a solid sphere in the nonlinear uniaxial extensional flow is larger than theoretical results because the theory neglects the effect of the vortex formed outside the particle on the rate of mass transfer.Empirical correlations are proposed to predict the influence of the dimensionless governing parameters (λ,Pe,E) on the Sherwood number (Sh).The maximum deviations of all empirical correlations are less than 15% when compared to the numerical simulated results.

1.Introduction

Extensional flow exists widely in the process of high viscosity multiphase flows,such as food processing,production of polymer materials,spinning of fiber [1].The properties of stretching in one direction and squeezing in another direction will have a significant effect on the heat and mass transfer between the dispersed particles and the continuous phase,and then impact on the structural characteristics and physical properties of the product.In-depth understanding of particle size,distribution and transport phenomena in multiphase flows is essential to studying the coupled effect of transport,reaction and separation in chemical production processes.However,it is very difficult to directly study the transport laws in real production processes due to a number of influencing factors.In order to analyze the mass transfer process,it is usual to simplify the model to a single bubble,drop or solid particle immersed in a simple flow field [1–7].For example,Batchelor [8],Gupalo and Riazantsev[9]theoretically analyzed the relationship of Sh with Pe and viscosity ratio λ of a single spherical solid particle or droplet in the extensional flow field,respectively(Eqs.(1)–(2)in Table 1).

The traditional theoretical solution to the transfer rate(Sh)from a single droplet to the continuous flow depended on Pe for ideal cases,such as convection-dominated transfer and diffusiondominated transfer [10–13],but the theoretical solution has not been reported yet for a common case when convection and diffusion jointly controlled the mass transfer.With the development of numerical simulation technique,the numerical simulation research of single particle transport in the flow field becomes gradually popular.Using numerical simulation techniques to solve the transport behavior of a single droplet in the flow field and summarize it as empirical correlations has now become a popular research route in this field.

Table 1 summarizes existing relevant literature works.In order to supplement the theoretical analysis of external mass transfer when both convection and diffusion control,Kurdyumov and Polyanin[14]numerically simulated the external mass transfer Sh with respect to Pe of a solid spherical particle in a linear extensional flow field(Eq.(3)in Table 1).At the same time,they also examined the effect of two parameters,Pe,and viscosity ratio λ,when a single spherical droplet in the transition from diffusion-dominated to convection-dominated mass transfer (Eqs.(4)–(5) in Table 1).It was found that when the mass transfer from/to solid spherical particles is jointly controlled by convection and diffusion,Sh varies along with the 1/3 power of Pe.For the external mass transfer of spherical droplets,when the internal to external viscosity ratio λ is small,Sh is proportional to the 1/2 power of Pe;when the internal to the external viscosity ratio λ is large,it is again proportional to the 1/3 power of Pe.As the viscosity ratio λ increases,the flow velocity inside the droplet decreases,and the transfer behavior is similar to that of a solid sphere.

Eq.(5) is a cubic polynomial.Given Pe,the only positive root is the required Sherwood number,which is not convenient to use.Therefore,Zhang et al.[15] studied external mass transfer process of a single particle in a simple extensional flow field,and summarized an empirical correlation with high accuracy.The relationship between Sh and Pe of a single droplet in the extensional flow is quite different from that of Kurdyumov and Polyanin [14].The empirical correlations of mass transfer are divided into three parts in the range of 1 ≤Pe ≤10,10 ＜Pe ≤103and 103＜Pe ≤105,which respectively represent that diffusion-dominated mass transfer(Eq.(6) in Table 1),diffusion-convection jointly dominated case(Eq.(7)),and convection-dominated case (Eq.(8) in Table 1).The correlations related to the external mass transfer process of spherical droplets are called as the Zhang-Yang-Mao [16] model,which is successfully used to predict the dissolution process of individual particles in cross channels.

For the external mass transfer from/to a slightly deformed droplet in the creeping extensional flow,Favelukis and Lavrenteva[17,18]gave the theoretical expressions of Sh versus Pe,Ca and viscosity ratio λ in uniaxial and biaxial extension flow fields(Eq.(9)in Table 1),where a new physical parameter,the capillary number(Ca),must be added to the theoretical solution.Recently,Favelukis[19,20] gave the theoretical expressions of external mass transfer around a compound drop in an extensional flow.

Table 1 A short summary of external mass transfer correlations of a single particle in extensional flow

So far,our discussion has been mainly limited to the transfer of mass transfer around spherical bubbles,drops,and solids in a linear extensional creeping flow.These two-phase systems,which appear in the form of foams,emulsions,and suspensions,are many times processed in the industry using rotating equipment.Yet,this type of machinery does not necessarily produce linear flows,such as simple(linear)shear flow or simple(linear)extensional flow,but rather more complicated motions such as nonlinear flows.Thus,despite the fact that the linear particle mass transfer literature is extensively present,very little is dedicated to the transfer of mass involving particles,but in a nonlinearflow.Hence,it is the purpose of this report to fill this gap by addressing the problem of mass transfer around a spherical drop or solid particle in a nonlinear extensional creeping flow.Nonlinear flow field is encountered in many industry processes,such as using nonlinear laminar flow to design a novel gas divider [21],learning dynamics of wormlike micellar solution in nonlinear shear and extensional flow[22],observing the properties of gellan gum in the nonlinear flow[23]and studying the structure and rheology of dual-associative protein hydrogels under nonlinear shear flow [24].For these nonlinear physical problems,the main models can be built on plate[25–27],channel[28,29] and particle[30,31].For the single particle in a nonlinear flow,Dimitrakopoulos[30]used the boundary element method to get the flow field of a single elastic dumbbell-shaped capsule in the nonlinear extensional flow,and observed the stress changes at the interface of a deformable capsule.In addition,fluid mechanics and mass transfer with slender drops in a nonlinear extensional flow field was studied by Favelukis[31–33].

Sherwood [34] put forward a nonlinear extensional flow to learn the tip streaming of a slender droplet.In order to explore small deformations of an initially spherical droplet in the nonlinear extensional creeping flow,Favelukis [35] presented the streamlines of a spherical droplet,and theoretically analyzed the external mass transfer from/to a single spherical particle in the nonlinear extensional flow field in the convection-dominated case [36].

However,in Ref.[36]the solution to the problem is for the large Peclet numbers(Pe ?1)regime only.The numerical work presented here,firstly verifies the theory in Ref.[36] and then explores the external mass transfer from/to a spherical particle in the uniaxial and biaxial nonlinear extensional flow in a wider range of 1 ≤Pe ≤105,-5 ≤E ≤10,with E being the nonlinear intensity of the flow.Beside the mass transport rate,the concentration field is also examined,from which the transport behavior can be understood more clearly.

2.Flow Fields of a Particle in Nonlinear Extensional Flow

A schematic diagram for a spherical particle,drop or bubble suspended in the uniaxial extensional flow is shown in Fig.1.Far away from the sphere,the undisturbed motion in spherical coordinates that was suggested by Sherwood [34] reads:

where u is the velocity component,r and θ are the spherical coordinate components.The nonlinear intensity of the extensional flow is E=Ba2/A.a is the drop radius.A is the strength of the extension rate,A ＞0 is the uniaxial flow and A ＜0 is the biaxial flow.B is related to the nonlinear term of the velocity flow.The streamlines of undisturbed flow are shown in Fig.2.

The flow field of a single particle in the nonlinear extensional flow is a complicated theoretical solution.When the particle is sufficiently large(up to several mm in channels),the nonlinear terms of the imposed flow should not be neglected.So Sherwood [34]theoretically and experimentally studied the behavior of a slender drop in a nonlinear extensional flow field.Favelukis [35] assumed that the particle remained spherical in the nonlinear flow field and drew the streamlines in spherical coordinates.The external dimensional flow field of a spherical particle in the nonlinear flow is axisymmetric and given by

The dimensionless velocity is scaled by a characteristic length Aa.a is the drop radius.The nonlinear intensity of the extensional flow is E=Ba2/A.When B=0 or E=0,the linear flow is recovered.The constants in Eq.(11) are listed below:

Fig.1.A spherical particle,drop or bubble in a uniaxial extensional flow in spherical coordinates.

where λ is the interior-to-exterior viscosity ratio.

The tangential surface velocity is

The tangential velocity gradient at the surface:

Here,we give a short comment on the velocity field in order to present the mass transfer more clearly.More details can be found in Ref.[36].

When a single spherical droplet or bubble is placed in this nonlinear flow (E ≠0),the flow at the interface of the spherical particle will produce new hydrodynamic characteristics which deviate from that in the linear case (E=0).When E ＜0,separating surfaces will be formed outside the droplet (Fig.3(a)).According to the analysis of in Ref.[36],for bubbles and drops,the critical value is E=-3/7 (Fig.3(b)).As E continues to decrease,the angle of the separation surface α=arccosmoves towards 90°.When E ?1 or λ ?1,vortices(Fig.3(c)) are formed at the equator (θ=π/2) of droplet or the solid particle.As E is positive and the value increases,the size of the vortex becomes larger.When λ=0,no matter how large E is,there will be no vortex near the drop.

When λ →∞,the particle is considered as a solid sphere,and separation surfaces and vortexes will appear at E ≤-5/17 and E ≥5/4,respectively.When E ≤-5/17,the angles where the separation surfaces meet the surface of the drop are α=arccosas shown in Fig.4(a)–(b).When E ＞5/4,the angles that represent the location of two symmetrical vortex are from α=to θ=arccosas shown in Fig.4(c).The vortexes stick to surface,which are different from the vortexes detached to the spherical droplet as in Fig.3(c).

3.Governing Equations of Mass Transfer and Schemes of Numerical Simulation

For the numerical simulations,some simple assumptions should be adopted:(1) Both the dispersed (bubble and drop) and continuous phases are Newtonian and under creeping flow conditions;(2)The shape of the particle remains spherical;(3)The physical properties including mass diffusivity,fluid viscosity and density are constant,and affected by the mass transfer.It follows that u from the theoretical solution(Eqs.(10)-(12))can be inserted to the convection–diffusion equation and one needs to solve the convection–diffusion equation only in order to obtain the concentration field.

The dimensionless convection–diffusion equation of axisymmetric flow field in spherical coordinates is

where C is the dimensionless concentration,Pe=Aa2/D,D is the diffusion coefficient,and τ=At is the dimensionless time.

The boundary conditions can be set as below:

where the subscript s represents the drop surface.

Fig.2.Streamlines of undisturbed uniaxial flows.The values of the stream function for the circulations:(a) [-6,-3,-1,-0.25,-0.01],(b) [0.01,1,2,8],(c) [-2,-1,-0.5,-0.1,-0.01,0,0.01,0.25,2,8] and (d) [-0.25,-0.1,-0.01,0,0.01,1,5,20,40].

The Sherwood number Sh is calculated by

The convection–diffusion equation is coded in Fortran Language.The TVD Runge-Kutta scheme (third-order) is applied on the time term,the central difference discretion is applied on the diffusion term and F-WENO scheme(fifth-order)is used to deal with the convective term [37].

A schematic diagram was shown in Fig.5 to present a nonuniform grid.The grid settings should ensure the accuracy of simulation.At high Pe,a concentration boundary layer is formed near the sphere surface and it becomes thinner with increasing Pe.Thus,the grid near the sphere surface needs to be refined.Δr=0.0005 near the surface can satisfy the need of the thinner boundary layer when Pe=105.In other regions,the grid with exponential increment is used to complete the setting of calculation domain in the r direction.

In general,the computational area is big enough when the range in the r direction is 40 times the particle radius.But for the nonlinear case,the velocity is proportionate to Er3in Eqs.(11) and (12).The dimensionless velocity gradient reaches 104when the radius is 40 times.Large velocity gradient needs very fine mesh which will demand too much computing resource.We define a variable Vmaxto represent the maximum velocity value in the whole computational domain.The Vmaxis selected to control the calculation parameters such as the mesh and the time step.

The grid of 200(r) × 40(θ),Δr=0.0005,Vmax(maximum velocity) ＞20 has enough precise to get the Sh values as shown in Table 2.The theoretical Sh is 206.1 for the case with E=1,λ=1,and Pe=105.The simulated Sh is 206.7 as listed in Table 2.So Vmax=20 guarantees the computation has enough accuracy.

A constant time step is employed for each run with the time step ranging between Δτ=10-7at Pe=1 and Δτ=5 × 10-5at Pe=104.The grid in θ is uniform and the space is π/40.

However,vortexes near the particle surface will form when E ?1 and separating surfaces will appear when E ?0.40 meshes in theta direction are difficult to accurately express information on the interface.Therefore,the 200 meshes are add to θ direction.The Sh will not be changed with the refined grids in the θ direction as listed in Table 3.The simulated Sh is 145.1 and the theoretical Sh is 144.8 in the case of E=10,λ=1,Pe=104.However,the concentration contour is crude with the 40 meshes in the θ direction as shown in Fig.6.Therefore,the 200 meshes are adopted for the simulation work.

Fig.3.Disturbed external streamlines for a drop/bubble with λ=1.The values of the stream function for the circulations:(a)[-0.3,-0.1,-0.01,-0.0001,0,0.0001,0.01,0.1,0.5,1,2,4,8],(b) [-0.9,-0.6,-0.3,-0.1,-0.01,0,0.01,0.1,0.3,0.6,0.9] and (c) [-50,-10,-1,-0.2,-0.01,0,0.001,0.007].

In order to verify the accuracy of the calculation,the simulated Sh is compared with that of the external mass transfer from/to a single spherical particle in nonlinear extensional flow with Pe=105of Favelukis[36]as listed in Table 4.The total error is controlled within 0.5%.The comparison cases include the Sh of three kinds of spherical particles in non-linear extensional flow field,i.e.,liquid drop (λ=1),bubble (λ=0) and solid sphere (λ=∞).The validation range is -5 ≤E ≤10.

Table 2 Effect of mesh parameters on computation of Sherwood number

Table 3 Effect of mesh parameters on computation of Sherwood number

3.1.Mass transfer outside a solid sphere

The external mass transfer from/to a solid sphere (λ=∞) in the nonlinear flow can be categorized into three parts:E ≤-5/17,-5/17 ＜E ＜5/4 and E ≥5/4 according to the external flow field.When E ＜ -5/17,the angle of the separation surface is atWhen-5/17 ≤E ≤5/4,the closed circulation or the separation surface are not present.When E ＞5/4,the two symmetrical vortexes are located from θ=arccos

Fig.4.Disturbed external streamlines for a particle with λ=∞.The values of the stream function for the circulations:(a)[-0.5,-0.2,-0.01,0,0.00001,0.001,0.1,0.5,2],(b)[-0.5,-0.2,-0.01,-0.00001,0,0.00001,0.001,0.05,0.2] and (c) [-160,-50,-10,-1,-0.2,-0.01,0,0.001,0.01,0.1,0.05].

3.1.1.Solid sphere with E ＜-5/17

When E ＜-5/17 and Pe ?1,the external mass transfer from/to a solid sphere in the nonlinear flow is given by Ref.[36]

Fig.5.Schematic diagram of the non-uniform grid setting outside a sphere.

The coefficient left to the bracketed term in Eq.(18)shows that the local Sh/Pe1/3on the interface is related to the angle θ for the case of convection-dominated mass transfer under different E.As shown in Fig.8,in the convection-dominated mass transfer process(Pe=103,Pe=104and Pe=105),the concentration gradient at the interface is directly proportional to Pe1/3.When E=-5,the angle of the separation surface α=1.06.The area with the slowest mass transfer rate is also at θ=1.06 (Fig.8),which coincides with the separation surface.With a decrease in Pe,the diffusion increases,and the region with the slowest mass transfer moves to the equator.When the mass transfer is completely controlled by diffusion,the local mass transfer Sh decreases with the increase of the angle(Pe=1).Therefore,as shown in Fig.9(b),the region with the lowest mass transfer rate θ=1.01 is slightly larger than the separation surface angle α=0.939 when Pe=100 and E=-1.5.

3.1.2.Solid sphere with -5/17 ≤E ≤5/4

For the range of -5/17 ≤E ≤5/4,the correlation of external mass transfer Sh of a solid sphere in the nonlinear extensional flow was not presented in Ref.[36]because the analytical result is cumbersome.Therefore,it is necessary to fit an empirical correlation of the mass transfer rate when 1 ≤Pe ≤105and-5/17 ≤E ≤5/4 for convenient use.

Table 5 shows the results of the mass transfer rate (Sh) outside the solid sphere as a function of Pe in the range of-5/17 ≤E ≤5/4.When E=-5/17,Sh is at the minimum when compared to other E cases in the whole range of Pe.This conclusion is consistent with the mass transfer conclusion of Favelukis [36] in the non-linear flow field when the mass transfer is convection-dominated.When E=0 and Pe →∞,the simulated Sh agrees with the analytical result of Sh=0.968Pe1/3as listed in Table 5.

The local concentration gradient(-?C/?r)/Pe1/3at surface varies with θ as shown in Fig.10.When E=0,(-?C/?r)/Pe1/3coincides well with Zhang’s data [15].When E=-5/17,a minimum value of mass transfer rate appears in θ=0.2 due to separation surfaces appearing at the critical E=-5/17.The separation surfaces with zero velocity appear at α=0,but the area with the slowest mass transfer rate is near θ=0.2 when Pe=105to 100 and E=-5/17 as shown in Fig.11.Under the combined influence of diffusion and convection,the location of the minimum mass transfer rate moves forward a little to the equator.The concentration contours show the trails of the solute concentration outside the particle,which is different from that in linear flow field(for a linear concentration contour [15]).When E ＞0,the local mass transfer rate increases to a maximum value,and then decreases at equator.

3.1.3.Solid sphere with E ＞5/4

When E ＞5/4,there are two vortices near the equator(Fig.4(c)),which have significant effect on the rate of mass transfer.Favelukis[36]analyzed that when E ＞5/4 under convection-dominated mass transfer,the empirical correlation formula is Eq.(19),and the calculation integral domain is from 0 to α.It is assumed that the mass transfer near the flow vortex is mainly diffusion-dominated on mass transfer,and its mass transfer Sh can be neglected by comparing with other positions of thin concentration boundary layer region without vortex.However,the simulated Sh of the external mass transfer from/to a single spherical particle in nonlinear extensional flow is between Eq.(18) and Eq.(19) when E=2 and E=10,Pe ＞103as listed in Table 6,that is,Sh at the vortex effected on mass transfer cannot be neglected.

Table 4 Comparison of the literature data with this simulation data for a spherical particle in nonlinear extensional flow (Pe=105)

At the same time,there is a certain concentration gradient near the equator as shown in Fig.12.The concentration contour does not agree with the flow field as shown in Fig.4(c).The existence of velocity vortex does not lead to concentration vortex for the outside of droplet is not the concentration source.

The local concentration gradient (-?C/?r)/Pe1/3is sharply declining when α ＜θ ＜π-α as shown in Fig.13.Eq.(18)(integral domain from 0 to π/2)and Eq.(19)(neglecting the mass transfer at vortex)cannot describe the mass transfer situation with E ＞5/4,so new empirical correlations are needed to be fitted.When Pe ≤103,the data of in Ref.[36] for calculating convective controlled mass transfer is no longer applicable.Therefore,for the external mass transfer from/to a single solid sphere under the joint control of convection and diffusion,a new empirical correlation is needed to explain the relationship between Sh and Pe:

Fig.6.Different concentration contours for three different grids when Pe=100.

3.1.4.Correlations for a solid sphere

The analytical correlations put forward by Favelukis [36] for a spherical particle in the nonlinear extensional flow are only in an integral form.And at the range of -5/17 ≤E ≤5/4,Favelukis [36]only gave the curve of Sh and E which is inconvenient to use.Moreover,when E ＞5/4,the vortex effect on the mass transfer should not be neglected.So,a correlation between Sh and Pe should be given in the range of 1 ≤Pe ≤105which contains the diffusion-dominated,convection-dominated and the Pe in the transitional zone.

As empirical correlations of mass transfer from/to a deformed droplet in extensional flow field are available [17,18,38],correlations are proposed based on Eq.(3) by adding the parameter E in the range of 1 ≤Pe ≤105.According to the discussion above,the correlations are divided in two parts.

For -5 ≤E ≤-5/17,the correlation is

Fig.7.Variation of Sh with Pe and E for mass transfer from a solid sphere in a nonlinear uniaxial extensional flow.

Fig.8.Local concentration gradient (-?C/?r)/Pe1/3 on a solid sphere surface as a function of θ when E=-5 and λ=∞.

The maximum deviation is 14.74% and the average relative deviation is 3.9% as shown in Fig.14.

3.2.Mass transfer outside a liquid sphere

3.2.1.Comparison between simulated values and theoretical data

For the mass transfer process of a spherical gas bubble (λ=0)and a liquid drop (0 ＜λ ≤100) in the nonlinear extensional flow field,the simulated Sh with 0 ≤λ ≤100,1 ≤Pe ≤104and-5 ≤E ≤10 are listed in the Appendix tables,and compared with the theoretical Sh values in Ref.[36] (Fig.15).Convectiondominated mass transfer correlations of a single droplet in the nonlinear extensional flow can be found in Ref.[36].The mass transfer rate in range of -1.04 ＜E ＜0 (bubble and drop) and-0.490 ＜E ＜0 (solid sphere) is lower than that of the linear case and larger than that of linear case outside these ranges.The conclusion in Ref.[36] also applies to diffusion-dominated cases and the ones in the transitional zone as listed in the Appendix tables.When Pe=104,the simulated Sh agrees with Favelukis’ theoretical solution as shown in Fig.15(a).When Pe=103,small deviations occur between simulated Sh and theoretical solution Sh as shown in Fig.15(b).When Pe=100 and 10,the theory deviates from simulations as shown in Fig.15(c)–(d).Thus the influence of nonlinear E on mass transfer needs to be explored in the diffusion-dominated zones and in the transitional zones.

In the Appendix Table A5 (λ=10) and Table A6 (λ=100),we compare the simulated Sh with theorical solution of droplet and the comparison is not so good.However,when we compare our results with the theorical solution of a solid,we obtain better results.There is a nice explanation on this subject in the work of Lochiel and Calderbank[13]as shown in the supplementary material.For a drop with λ=100 and Pe=103–106,we should use the Sh～Pe1/3(Eq.(18)) formula,rather than the Sh～Pe1/2formula.

Table 5 Variation of Sh with Pe for mass transfer from a solid sphere in a nonlinear extensional flow in the range of -5/17 ＜E ＜5/4

Table 6 Sh of a solid sphere in the range of E ＞5/4 compared with theoretical data

The external mass transfer from/to a bubble(λ=0)or a droplet(0 ＜λ ≤100) in nonlinear flow can be divided into two parts:E ≤-3/7 and E ＞-3/7 according to the external flow field.Favelukis[36]has given correlations applicable to Pe ＞103.In addition,the whole range of 1 ≤Pe ≤105effecting on mass transfer is discussed as follows.

Fig.9.Concentration contours around a solid sphere at E=-1.5 and λ=∞.

Fig.10.Local concentration gradient (-?C/?r)/Pe1/3 on solid sphere surface as a function of θ when Pe=105 for -5/17 ≤E ≤5/4.

3.2.2.Bubble and drop with E ≤-3/7

When E ≤-3/7,Pe ＞1000,the separating surface by touches the surface of the drop at α ＝and theory for the convective dominated regime suggests:

Sh is directly proportional to Pe to the power of 1/2 and is obtained by the integration of Eq.23.

where vθ0is the tangential surface velocity defined in Eq.(13).Eq.(23) can quantitatively explain how the local Sh varies with θ.

Fig.16 shows the variation of (-?C/?r)/Pe1/2with θ.The lowest mass transfer rate is at the separation point.With the decrease of Pe,the local Sh smoothly varies with θ at the surface.The concentration contours are similar to the streamlines as shown in Fig.17 and Fig.3(a)–(b).

3.2.3.Bubble and drop with E ＞-3/7

An analytical correlation between Sh and E,λ,Pe ＞1000 is described by Eq.(24) which is calculated using Eq.(25) which expresses the convection-dominated regime.This analytical equation suggested for no vortex.when there is vortex,mass transfer may be lower especially in uniaxial.

Fig.11.Concentration contours around a solid sphere at E=-5/17 and λ=∞.

Fig.12.Concentration contours around a solid sphere at E=10,λ=∞,and Pe=100.

Fig.13.Local concentration gradient (-?C/?r)/Pe1/3 on solid sphere surface as a function of θ when E=10 and λ=∞.

Fig.14.Comparison of simulation results and correlations (1 ≤ Pe ≤ 105,-5 ≤E ≤10).

The variation of (-?C/?r)/Pe1/2with θ is shown in Fig.18.The local Sh/Pe1/2along θ approaches a fixed curve when Pe ＞103.When Pe ≤103,the convective mass transfer weakens,and the local Sh/Pe1/2increases with the decrease of Pe.

Fig.19 shows the concentration contour at the case of E=10,λ=1 and Pe=100.The concentration contour is not consisted with the streamline as shown in Fig.3(c) as the same with that of solid sphere.

3.2.4.Correlations for a liquid sphere

The Sh data in this simulation work agrees with the correlations of convection-dominated mass transfer(Pe ＞103)which were proposed by Favelukis [36].Therefore,only new correlations in the diffusion-dominated case(1 ≤Pe ≤10)and in the transitional zone(10 ＜Pe ≤103)are required.

For the external mass transfer from/to a spherical droplet or bubble in the simple extensional flow,Zhang et al.[15] has given some correlations.Based on Zhang et al.[15]work,for the nonlinear flow field,the correlations are divided in four parts.

(1) When 1 ≤Pe ≤10 and -5 ≤E ≤-3/7,we propose

with a maximum deviation of 15% in Fig.20

Fig.15.Variation of Sh with E for mass transfer from a liquid drop in a nonlinear uniaxial extensional flow in case of Pe=105,103,100,10.

Fig.16.Local surface concentration gradient (-?C/?r)/Pe1/2 on a spherical droplet surface as a function of θ when E=-5 and λ=1.

(2) When 1 ≤Pe ≤10 and -3/7 ≤E ≤10,the correlation suggested is

with a maximum deviation of 15% in Fig.21.

(3) When 10 ≤Pe ≤1000 and -5 ≤E ≤-3/7,we suggest the following:

with a maximum deviation of 14% in Fig.22.

(4) When 10 ≤Pe ≤1000 and -3/7 ≤E ≤10,the correlation is

with a maximum deviation of 15% in Fig.23.

The Eqs.(28) and (29) show that:when λ is not so large,Sh～Pe1/2.For a solid with λ=∞Sh～Pe1/3.Thus for a drop with very large λ,the power over the Peclet number will be somewhere between 1/3 to 1/2.This can be seen in Eq.(34) of Favelukis [36].That is,when a drop with very large λ,the surface velocity gradient times the concentration boundary layer thickness (δ) is much greater than the surface velocityand the drop behaves like a solid particle with Sh～Pe1/3.

Fig.17.Concentration contours around a spherical droplet at E=-5 (left),-3/7 (right),λ=1,and Pe=100.

Fig.18.Local concentration gradient(-?C/?r)/Pe1/2 on a spherical droplet surface as a function of θ when E=10 and λ=1.

Fig.19.Concentration contours around a spherical droplet at E=10,λ=1 and Pe=100.

Fig.20.Comparison between simulation results and correlation.(1 ≤Pe ≤10,-5 ≤E ≤-3/7).

Fig.21.Comparison between simulation results and correlation.(1 ≤Pe ≤10,-3/7 ≤E ≤10).

Fig.22.Comparison between simulation results and correlation.(10 ≤Pe ≤1000,-5 ≤E ≤-3/7).

Fig.23.Comparison between simulation results and correlation.(10 ≤Pe ≤1000,-3/7 ≤E ≤10).

4.Conclusions

This work deals with the external mass transfer from/to a spherical particle(bubble,drop or solid sphere)in a nonlinear uniaxial extensional creeping flow.The flow outside the drop reveals a different picture than the linear case,where vortexes and separating surfaces may be present.Three dimensionless parameters describe the mass transfer problem:the drop to external fluid viscosity ratio(λ),the Peclet number(Pe),and the nonlinear intensity of the flow (E).Based on the convection-dominated external mass transfer theoretical results,this work expands the current knowledge by numerical simulating the external mass transfer to the diffusion-dominated and to the convection–diffusion regimes:1 ≤Pe ≤105and -5 ≤E ≤10.

For the convection-dominated regime(Pe ＞103),the simulated Sherwood number (Sh) agrees with the theoretical results,except for the case of a solid sphere at E ＞5/4,where the simulated Sh is larger than the theoretical predictions.The simulations indicate that the effect of a vortex formed outside a solid particle,caused by the nonlinear flow,on the rate of mass transfer cannot be neglected.

For the diffusion-dominated and the convection–diffusion regimes(1 ≤Pe ≤103),empirical correlations for the external mass transfer from/to a single spherical bubble or droplet in the nonlinear extensional flow are suggested in order to predict the Sh in the range of-5 ≤E ≤10.In addition,the empirical correlations for the external mass transfer from/to a single solid sphere in the larger range of Pe(1 ≤Pe ≤105)are also presented.The correlations suggested in this work represent a maximum deviation of less than 15% when compared to the simulated data.The six empirical correlations,basically include the external mass transfer from/to a single particle in the range of 1 ≤Pe ≤105and -5 ≤E ≤-10.

In general,the nonlinear flow field increases the mass transfer efficiency.The only difference is that -3/7 ≤E ≤0 for droplet and -5/17 ≤E ≤0 for solid sphere,the mass transfer rate lower than the linear flow.If we want to reduce the mass transfer rate of the spherical particles,this range of E can be achieved.The mass transfer rate at the vortex and separation surface decreases locally,but the overall mass transfer rate increases.Through the comparison of the concentration contour,it is found that the concentration separation layer can appear with the velocity separating surfaces.Under the influence of the vortex outside the particle,the solute transfers out of the particle,and the concentration tail forms near the equator in uniaxial nonlinear extensional flow.

Mass transfer studies around bubbles,drops,and solid particles is a topic of basic scientific interest with many industrial applications.These two-phase systems are processed in mixing devices undergoing shear,extensional,but in general much more complicated nonlinear flows.While almost all of the models presented in the literature are restricted to linear flows,very little is dedicated to nonlinear flows,which can describe better(than the linear flow) the actual motion produced in industrial machinery.While mass transfer models around bubbles,drops,and solid particles in linear flows are relatively easy to obtain,this is not the case when dealing with nonlinear flows which require complicated numerical simulations.The external nonlinear motion in this report includes separation surfaces and vortexes which are absent in the linear flow.These nonlinear effects require numerical techniques,especially if all three mass transfer regimes(diffusion dominated,diffusion-convection dominated,and convection dominated) are be explored,as it was done in this report.Lastly,one need to remember that the nonlinear flow also influences the internal motion and,under certain conditions,the number of internal circulations may be doubled.Thus,future research may focus on internal mass transfer in nonlinear flows by combining theoretical and numerical techniques.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This paper is dedicated to Prof.Jiayong Chen at Institute of Process Engineering,Chinese Academy of Sciences.The authors are very grateful to Prof.Jiayong Chen for his support and helpful insight.This work was supported by the National Key Research and Development Program(2021YFC2902502),the National Natural Science Foundation of China(21938009,91934301,22078320),the Major Scientific and Technological Innovation Projects in Shandong Province(2019JZZY010302),the Shandong Key Research and Development Program (International Cooperation Office)(2019GHZ018),the Shandong Province Postdoctoral Innovative Talents Support Plan (SDBX2020018),the External Cooperation Program of BIC,Chinese Academy of Sciences(122111KYSB20190032),Chemistry and Chemical Engineering Guangdong Laboratory (1922006),and GHfund B(202107021062).

Supplementary Material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cjche.2021.11.017.

Nomenclature

A the strength of the extension rate,s-1

a sphere radius,m

B the nonlinear term of the velocity flow,m-2·s-1

C concentration,dimensionless

c concentration,kg·m-3

D diffusivity,m2·s-1

E nonlinear intensity of flow (E=Ba2/A),dimensionless

Pe Peclet number (Aa2/D),dimensionless

r radial coordinate,dimensionless

S droplet surface area,dimensionless

Sh Sherwood number,defined in Eq.(17),dimensionless

u velocity,dimensionless

Vmaxmax velocity,dimensionless

z cylindrical component,dimensionless

θ spherical polar angle,rad

φ spherical azimuthal angle,rad

ψ stream function,m3·s-1

Subscripts

r spherical radial component,dimensionless

θ spherical polar component,dimensionless

φ spherical azimuthal component,dimensionless

∞ far from the droplet or finite large time

Superscript

s droplet interface

Appendix

Appendix Tables A1–A6.

Table A1 Various Sh vs.Pe and E for a spherical drop with λ=0 compared with theoretical data

Table A2 Various Sh vs.Pe and E for a spherical drop with λ=0.01 compared with theoretical data

Table A4 Various Sh vs.Pe and E for a spherical drop with λ=1 compared with theoretical data

Table A5 Various Sh vs.Pe and E for a spherical drop with λ=10 (no theoretical solution)

Table A6 Various Sh vs.Pe and E for a spherical drop with λ=100 compared with theoretical data