Anjun Liu, Jie Chen*, Meng Guo,,*, Chengmin Chen, Meihong Yang, Chao Yang*

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 Jinan Key Laboratory of High Performance Industrial Software, Jinan Institute of Supercomputer Technology, Jinan 251013, China

Keywords:

ABSTRACT

1. Introduction

The mass transfer rate of a single particle can be modeled as an overall mass transfer coefficient and further integrated into a multiphase system [1–6], which can be simply decoupled the multiphase flow and interphase mass transfer problem. With development of supercomputers and the numerical computation methods of partial differential equations, the interior flow field[7–9] and transfer [10–13] can be solved closer to the analytical solution, which bring new development to mass transfer of multiphase.

We noticed that in the analysis of the heat/mass transfer between drops and a continuous phase, three categories of problems are usually considered [14]: the internal-controlled mass transfer processes [10,15–17], the external-controlled cases[4,8,9,18,19] and the conjugate-controlled cases [1,6,20–27]. For the external-controlled cases,the vortices outside[28]the droplet may affect mass transfer rate. Therefore, Liu et al. [18] researched vortexes and separation surfaces outside of particle influence mass transfer and summarized the corresponding correlations.Based on the Liu et al. [18] work, this work focuses on internal-controlled mass transfer processes.

The internal mass transfer rate is decided by the internal flow field of particle, which is produced by outside flow field. The simple shear and extensional flow fields are common flow fields in chemical industry, which produce different forms of vortex inside the droplet. For example, one internal circulation will be formed inside the droplet in the simple shear flow field [11,12]. When the internal and external viscosity ratio λ is far less than 1, two internal circulations will be formed. Frankel [29] and Acrivos [30]theoretically analyzed the relationship between the internal Nuinand Pe of the spherical droplet in the shear flow field.It was found that unlike the external mass transfer law of the spherical particle,the internal Nuinwould eventually reach a limit value of 4.5, and wouldn’t increase with the increase of Pe any longer.

For the simple extensional flow, two internal circulations will be formed inside the droplet.Zhang et al.[31]suggested the correlation for internal mass transfer rate of a spherical droplet in simple extensional flow via numerical simulation, that is,

when Pe=1,Shin=3.3;when Pe=10000,Shin=14.1.Under the special situations of diffusion-dominated mass transfer process or convection-dominated one, the simulation results agree with that of Oliver et al.[16,17].Oliver et al.[16,17]obtained the extreme values of Sh changing with Pe in a spherical droplet. When Pe →0,Nuin= 3.3; when Pe →∞, Nuin= 14.9. The internal mass transfer coefficient of a drop in the electric field is the same as that of a spherical drop in the simple extensional flow field because the internal circulation driven by a simple extensional field is similar to that driven by an electric field.

Through literature investigation,we have assembled the following list of five enhancement methods are changed internal flow field to improve the internal mass transfer rate:

(1) Increase the velocity of the external flow field. Juncu [14]gave the maximum Shin= 19.851 at Re = 1000 and exceeding the results from Oliver et al. [16,17] by approximately 30%.

(2) Change the particle shape. The drop in the extensional flow will be stretched to prolate or oblate. Based on the Zhang’s [31]correlation, Liu et al. [10] added the capillary number (Ca) to express the droplet deformation, and proposed the correlations(Eqs. (2) and (3)). Liu’s work [10] showed the inner mass transfer of peanut shape droplet and red-blood-cell shape one in the extensional flow. The internal mass transfer rate of red-blood-cell droplet is accelerated (the maximum Shin= 20.52), faster than that of the peanut shape droplets.

The correlation of prolate and peanut shape droplet in uniaxial extensional flow:

The correlation of oblate and red-blood shape droplet in uniaxial extensional flow:

(3) Ultrasound. The mass transfer coefficient is enhanced by 72.5%for the synthesis of AgInS2nanocrystals in ultrasound condition [32].

(4) Breaking the vortex inside the droplet can greatly enhance the internal mass transfer. Christopher et al. [33] obtained the asymptotic Shin= 130, exceeding the value of the Hill’s [34],vortex-like solutions by more than a factor of six.

(5)Marangoni effect[35,36].Marangoni effect generated vigorous and fierce flow circulations in the droplet, which greatly promoted mass transfer rate by one order of magnitude.

All methods change the flow circulation state inside the droplet.The number and position of internal circulations can be taken as the key parameters. In order to study the mechanism for the enhancement of internal mass transfer by flow circulation, we must establish a parameter describe the number and position of internal circulations. Yang et al. [37] obtained the flow field with different number of vortices inside the droplet through experiment and numerical simulation. And, Favelukis [38] obtained a simple theoretical solution for the nonlinear extensional flow field. The flow circulations inside the droplet are controlled by nonlinear intensity(E). For example, when E =–1, there will be four internal circulations in the spherical droplet (Fig. 1). Therefore, we numerical simulated the mass transfer convection–diffusion equation(Section 2.2)based on Favelukis’theoretical flow field(Section 2.1)to elaborate the internal mass transfer of a spherical droplet in a nonlinear extensional flow field and discuss the effect of two more circulations on internal mass transfer (Section 3).

Fig. 1. The streamlines for a drop with E = –1 inside a droplet in nonlinear extensional flow (the variables in the colormap stand for the value of stream lines according to Ref. [38]).

2. Mathematical Formulation

2.1. Internal flow field of spherical droplet in nonlinear uniaxial extensional flow

Sherwood [39] proposed a nonlinear uniaxial extensional flow field and theoretically and experimentally studied the behavior of a slender drop in the flow field. Based on Sherwood [39] work,Favelukis [38] assumed that the droplet remained sphere in the uniaxial nonlinear flow field and drew the streamlines in spherical coordinates. And we changed the streamlines to the internal dimensional flow field of a spherical particle in the uniaxial nonlinear flow and given by

The constants in Eq. (4) are listed below:

where u is the velocity component,r and θ are the spherical coordinate components,λ is the interior-to-exterior viscosity ratio,E is the nonlinear intensity of the extensional flow.

The tangential surface velocity is

The flow field of internal is shown in Fig.2 according the Eqs.(4)and(5)and the surface velocity is drawn in Fig.3 according the Eq.(7). When E = 0, the flow field is simple uniaxial extensional flow(Fig. 2(c)). When E ≠0, the flow field is nonlinear uniaxial extensional flow.When E ≥–3/7,there are only two internal circulations in the droplet(Fig.2(a)–(d)).The interfacial velocity increases with the increase of E as shown in Fig. 3(a). The flow circulation moves up to the pole as the E gets bigger.

2.2. Governing equations of mass/heat transfer

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 not affected by the mass transfer. It follows that u from the theoretical solution (Eqs. (4) and (5)) 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.

For the case of internal mass transfer,the external resistance is assumed to be null.The solute concentration in the ambient fluid C is regarded as uniform and constant, and so does the interfacial solute concentration.The boundary conditions can be set as below:

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 [40].

For the study of internal resistance controlled mass transfer between the continuous phase and a single drop, the average internal mass transfer Sherwood number in an time interval of Δt = tend–tstartwas calculated as

For a certain molecular diffusion coefficient D, the Fourier number (Fo = τ/Pe) is proportional to the time of the mass transfer process. If the change in Shinduring the latest 1/5 of the computational time is less than 1%, the transport process would reach a quasisteady state. Shinapproached an asymptotic value Shin,∞which revealed the rate of a pseudo-steady mass transfer process.

The mass transfer process between a spherical droplet and an extensional flow was applied here to test the independence of simulation results with respect to the grid fineness. The effect of grids(50(r) × 50(θ), 100(r) × 100(θ), 200(r) × 200(θ)) on the calculated Shin,∞with different Pe are listed in Table 1. A uniform grid of 100(r) × 100(θ) (Δr = 0.01) is sufficient for accurate results even at Pe = 100000. A constant time step Δτ = 10–3was employed from Pe = 1 to Pe = 10000.

The codes of this work is same as Ref. [18]. Only the boundary conditions are changed. By comparing the some results of Ref. [18]with the Ref. [38], the error is controlled within 0.5%.

3. Results and Discussion

The Pe in the simulation work is 1–100000, which cover the diffusion-dominated zones (Pe ≤ 10), the transitional zones(10 < Pe ≤1000) and convection-dominated zones (Pe > 1000). λ impacts on the internal velocity field of droplet. The flow field is inversely proportional to 1 + λ (Eqs. (4) and (5)). So λ only affects the fluid flow in the droplet as a whole and does not change the circulation shape. Here a fixed viscosity ratio (λ = 1) was considered.The internal mass transfer can be divided into two parts: E < –3/7 and E ≥–3/7 according to the internal flow field. When E ≥–3/7,there are only two internal circulations in the droplet (Fig. 2(a)–(d)).The flow circulation moves up to the pole as the E gets bigger.When E<–3/7,there are four vortices inside the droplet(Fig.2(e)–(h)). Therefore, the effect on mass transfer is discussed separately according to the number of vortices in the droplet.

3.1. E ≥–3/7

Fig.2. Evolutional flow fields with E inside a droplet in nonlinear extensional flow(the variables in the colormap stand for the value of stream lines according to Ref.[38]):(a)E = 5; (b) E = 1; (c) E = 0; (d) E = –3/7; (e) E = –4/7,α = 30°; (f) E = –6/7,α = 45°; (g) E = –12/7,α = 60°; (h) E = –5.

Fig.3. Local surface velocities around the droplet in one quadrant:(a)E ≥–3/7 and(b) E ≤–3/7.

Table 1 Verification of grid independence via Shin,∞with different Pe

The parameter E affects the flow field and thus the mass transfer.In order to reveal the effect of E on mass transfer,the condition of the convection-dominated(Pe>1000)must be considered carefully. Thus Fig. 4 shows the transient mass transfer rate and mean concentration change under condition of the Pe = 100000. At the very beginning, the mass transfer rate is largely determined by the surface velocity. The surface velocity becomes larger with the increase of E according to Eqs.(6)and(7).When Fo is smaller than 5×10–4,Shinenlarges with an increasing E.In particular,when E is bigger than 0,the surface velocity significantly speed up(as shown in Fig.3(a)),which leads to an obvious increase of Shin.As the mass transfer proceeds over time,the solute concentration gradient near the interface decreases and Shingradually reduces. The local peak of Shin, appearing during the mass transfer process, can be explained by the evolution of the internal circulation of concentration contours [10] (Fig. 5). The time of the crest of Shinvalue is around Fo = 10–3. The concentration decreases near the interface,especially near the equator that caused the concentration gradient normal to the interface increases sharply.So,it can be deduced that the mass transfer rate has a peak when the circulation of concentration contour is just formed. Then, the concentration gradient normal to the interface decreases gradually as time goes by,resulting in decreased Shinclose to the asymptotic value.

Fig.4. Transient transport behavior for the conjugate mass transfer between a drop and extensional flow about variation of Shin (a) and C-(b) versus Fo and E(Pe = 100000).

Fig. 6 shows that the stable Shin,∞varies with E and Pe. Under the condition of diffusion-dominated (Pe < 10), the mass transfer is not influenced by the flow field. The stable Shin,∞→3.3 and agrees with the theoretical solution.

Under the condition of the transitional zones(10 ≤Pe ≤1000),the Shin,∞increases with the increase of E when E ≥0.Because the flow field is more intense with the increase of E as shown in Figs.2 and 3.At the same Pe,the convection is stronger at larger E.When–3/7 < E < 0, the Shin,∞is smaller with decrease of E for the flow field becomes smaller.

Under the condition of convection-dominated (Pe > 1000),When E ≥0, the stable Shin,∞has a tiny difference for different E.When –3/7 < E < 0, the stable Shin,∞has an apparent increase and become larger with an increase of |E|. The local Shin,∞varies with the θ.It can be find in Fig. 7 that the highest point of interfacial mass transfer is at the center of the circulation in the nonlinear uniaxial extensional flow field. The location of circulation is moving with E.The larger E is,the closer it approaches to the pole.Conversely,the circulation moves to the equator and the mass transfer area of spherical surface at the circulation become larger, that causes the increase of mass transfer rate.The stable concentration contours are shown in Fig. 8. When E ≥–3/7, there are only two flow circulations in the droplet under the condition of convection-dominated cases. The concentration circulations are consistent with the flow fields and move as the E changes.

Fig. 5. Evolutional concentration contours inside the droplet in nonlinear uniaxial extensional flow (E = 1, Pe = 100000): (a) Fo = 10–4; (b) Fo = 5 × 10–4; (c) Fo = 10–3.

Fig. 6. Variations of Shin,∞with E and Pe for nonlinear uniaxial extensional flow.

Fig.7. Local Shin,∞on the droplet surface in the case of Pe=100000 as a function of θ in uniaxial extensional flow.

Fig. 8. Stable concentration contours inside a droplet in the non-linear uniaxial extensional flow (Pe = 100000, Fo = 0.05): (a) E = 5; (b) E = 0; (c) E = –3/7.

3.2. E < –3/7

When E < –3/7, there are four circulations in the droplet. The positions of circulations change with the variation of E as shown in Fig. 2. Fig. 9 shows the transient development of mass transfer rate and the mean concentration under the condition of Pe = 100000. At the very beginning, the mass transfer rate is also largely determined by the surface velocity and the interfacial concentration gradient. The smaller E is, the faster the mass transfer rate is.As the mass transfer proceeds over time,the concentration gradient normal to the interface decreases gradually, leading to a decreasing Shinclose to the asymptotic value.

Fig. 10 shows the stable Shin,∞with E and Pe. Under the condition of convection-dominated (Pe < 10), the mass transfer is not influence by the flow field. The the stable Shin,∞→3.3 and agree with the theoretical solution. Under the condition of the transitional zones (10 ≤Pe ≤1000), the Shin,∞is increase with the decrease of E when E < –3/7. Because the flow field is larger with the decrease of E as show in Figs. 2 and 3.

Under the condition of convection-dominated (Pe > 1000),When –3/7 > E ≥–1, the stable Shin,∞increases with an decrease of E. When E < –1, the stable Shin,∞become smaller with decrease of E.When E=–5,Pe=100000,Shin,∞is 15.4.The Shin,∞is approximately equal to that at simple extensional flow.Because the circulation near the equator is become very small and can be neglected.The center position of big circulation is general coincide with that when E = 0 as shown in Fig. 2(c) and (h).

Fig.9. Transient transport behavior for the conjugate mass transfer between a drop and extensional flow about variation of Shin (a) and C-(b) versus Fo and E(Pe = 100000).

We noticed that at the case of E=–1,Shin,∞is largest,and even is twice of that at the case in simple extensional flow. It can be explained from interface mass transfer coefficient and stable concentration contours.

Fig. 10. Variation of Shin,∞with E and Pe for nonlinear uniaxial extensional flow.

Fig.11. Local Shin,∞on the droplet surface in the case of Pe=100000 as a function of θ in uniaxial extensional flow: (a) –1 ≤E ≤–3/7 and (b) –5 ≤E ≤–1.

Fig.12. Stable concentration contours inside the droplet in the non-linear uniaxial extensional flow(Pe=100000,Fo=0.05):(a)E=–4/7;(b)E=–6/7;(c)E=–0.9;(d)E=–1;(e) E = –1.1; (f) E = –12/7.

The local Shin,∞and the stable concentration contours varying with the θ are drawn in Figs.11 and 12.The two peaks of interfacial mass transfer are at the center of the two circulations in the nonlinear uniaxial extensional flow field.The positions of the two peak are moving with E.When–3/7>E ≥–1,the first mass transfer peak(0°–α°)increases with decrease of E.The mass transfer peak,when E=–1,is larger than other cases obviously.The largest second mass transfer peak (α°–90°) at the case of E = –0.9, not at E = –1. When E ≤–1, the largest first mass transfer peak (0°–α°) at the case of E = –1.1, not at E = –1. The second mass transfer peak (α°– 90°)is increase with decrease of E. Add the mass transfer of two circulations, the largest Shin,∞of total (0°–90°) is at the case of E = –1.

When E = –4/7, –6/7, –0.9, –1.1, –12/7, the mass in the first or second circulation has been dissipated and only two concentration circulations remains. Only when E = –1, the stable concentration contour has four circulations (Fig. 12(d)). So at the case of E = –1,the internal mass transfer rate is the largest, Shin,∞= 30.

3.3. Correlations

Based on the simulation results, empirical correlations of Eqs.(11)–(13) for non-linear uniaxial extensional flow around a deformable droplet are proposed to predict Shin,∞in the range of–5 ≤E ≤5,1 ≤Pe ≤10000. The fitting formula are modified with inclusion of E into to that for spherical droplet results [31]. As shown in Fig. 13, the empirical correlations match well with the simulation results.

with a maximum deviation of 9.8% in Fig. 13(a), and R-square is 0.9880.

with a maximum deviation of 10.2% in Fig. 13(b), and R-square is 0.9974.

Fig. 13. Comparisons of simulation and fitting results: (a) variation of Shin,∞for 0

with a maximum deviation of 6.4% in Fig. 13(c), and R-square is 0.9984.

4. Conclusions

The internal mass transfer of a single droplet in nonlinear uniaxial extensional flow is influenced by the number and positions of internal circulations.The simulation results show that when diffusion controls the interphase mass transfer (Pe →0), the mass transfer coefficient Shin,∞= 3.3,not influenced by the flow field.Under the condition of the transitional zones (10 ≤Pe ≤1000),the Shin,∞increases with the increase of |E|.

Under the condition of convection-dominated(Pe>1000)cases,when E ≥0, the mass transfer rate has slight changes with the increase of E. The Shin,∞becomes smaller and the concentration drops more slowly with the increase of E. The number of internal flow circulations inside a droplet is 2 and the limit value of mass transfer rate Shin,∞is around 15, no matter how larger E is. When 0 > E ≥–3/7, the mass transfer rate increases with the decrease of E. The Shin,∞becomes larger with decrease of E. But the mass transfer rate do not improve obviously. Thus, the condition E ≥–3/7 has a little influence on mass transfer rate, and is not a good strengthen mean of mass transfer. At the cases of E ≤–3/7,Shinfurther increased due to the internal circulations doubled.When E = –1, the peak of Shinis 30 and the internal concentration contours are four in the droplet.Therefore,the more the number of internal circulations is, the faster the mass transfer rate is.

Nomenclature

Superscripts and Subscripts

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 work was supported by the National Key Research and Development Program of China (2021YFC2902502); the National Natural Science Foundation of China (22078320, 22035007), the NSFC-EU project (31961133018); the Shandong Provincial Key Research and Development Program (2022CXGC020106); the Shandong Key Research and Development Program (International Cooperation Office) (2019GHZ018); the Shandong Province Postdoctoral Innovative Talents Support Plan (SDBX2020018) and the External Cooperation Program of BIC,Chinese Academy of Sciences(122111KYSB20190032). We would like to thank Professor M.Favelukis for helpful discussions regarding the application of velocity field.