Mass transfer enhancement through Marangoni instabilities during single drop formation

The impact of Marangoni convection on the extraction efficiency during the drop formation stage is investigated in the system toluene/acetone/water for different initial solute concentrations and different drop diameters. Both mass transfer directions of the solute have been considered. Marangoni instabilities are supposed to increase the internal mixing and thus enhance mass transfer coefficients. Experimental results show a strong dependency on the mass transfer direction. The amount of solute extracted is between 19% and 55%. The total transferred mass M_A increases with drop diameter and initial concentration. Present models from the literature which predict extraction efficiencies do not take into account interfacial effects like Marangoni convection. A correlation is proposed introducing an effective diffusivity which depends on the initial solute concentration. The diffusivity factor increases linearly with initial solute concentration and is more sensitive in the mass transfer direction c → d.     

Predicting Marangoni convection caused by transient gas diffusion in liquids

The onset of convection driven by surface tension during gas diffusion in a liquid is investigated. Gas diffusion at the gas-liquid interface results in the variation of concentration of the solute that may cause an increase in surface tension leading to Marangoni convection. The onset of convection for unsteady-state gas desorption can be predicted from the maximum transient Ma_t, which is here derived by analogy with its equivalent in thermal convection. It is a function of the transient Biot number (Bi_D) for interfacial gas diffusion, which depends strongly on the state of vapour-liquid equilibrium at the interface. The transient Marangoni numbers, critical times for stable mass diffusion and the critical sizes of convection cells have been formulated. The desorption of ethyl-ether from chloro-benzene in L.M. Blair’s [The onset of cellular convection in a fluid layer with time-dependent density gradients, PhD thesis, University of Illinois, Urbana, 1968] experiments is liquid phase-controlled, hence, the highly soluble system is characterized by Bi_D = 0. Therefore, his experiments that were initiated with a step-change in pressure cannot be analyzed by a step-function boundary that is characterized by Bi_D = ∞. The surface concentration may change very slowly, it has been approximated to be about 0.1% of the initial pressure change at the point of onset of convection. The average critical Marangoni number for this condition was estimated to be 53.3, which is fairly close to the theoretical value of 67 for an interface with a Biot number of 0. Therefore, the high value of 3100 calculated by I.F. Davenport and C.J. King [The initiation of natural convection caused by time-dependent profiles, Lawrence Berkeley Lab, Report NBR LBL-600, 1972] is wrong, who wrongly assumed a fixed surface-concentration boundary that is applicable only to a sparingly soluble solute. The critical sizes of convection cells predicted by theory are generally less than 1 mm for reported critical times of less than 20 s, they would be difficult to measure.     

Topological structure evolvement of flow and temperature fields in deformable drop Marangoni migration in microgravity

Using the level-set method and the continuum interface model, the axisymmetric thermocapillary migration of a deformable liquid drop immerged in an immiscible bulk liquid with a temperature gradient is simulated numerically with constant material properties of the two phases. Steady terminal state of the motion can always be reached. The dimensionless terminal migration velocity decreases monotonously with the increase of the Marangoni number. Good agreements with space experimental data and most of previous numerical studies in the literature are evident. The terminal topological structure of flow field, in which a recirculation identical to Hill’s vortex exists inside the drop, does not change with the Marangoni number. Only slight movement of the location of vortex center can be observed. On the contrary, bifurcations of the terminal topological structure of temperature field occur twice with increasing Marangoni number. At first, the uniform and straight layer-type structure of temperature field at infinitesimal Reynolds and Marangoni numbers wraps inside of the drop due to convective transport of heat as the Marangoni number increases, resulting in the emergence of an onion-type local cooler zone around the center of the drop beyond a lower critical Marangoni number. Expanding of this zone, particularly in the transverse direction, with the increasing of the Marangoni number leads to a cap- or even shell-type structure. The coldest point within the liquid drop locates on the axis. There is a middle critical Marangoni number, beyond which the coldest point will jump from the rear stagnation into the drop, though the topological structure of the temperature field does not change. The second bifurcation occurs at an upper critical Marangoni number, where the shell-type cooler zone inside drops ruptures from the central point and then a torus-type one emerges. The coldest point departs from the axis, and the so-called “cold-eye” appears in the meridian. It is also found that the inner and outer thermal boundary layers along the interface may exist both inside and outside the drop if Ma > 70. But the thickness decreases with the increasing Marangoni number more slowly than the prediction of potential flow at large Marangoni and Reynolds numbers. A velocity shear layer outside the drop is also introduced formally, of which modality may be affected by the convective transports of heat and/or momentum.     

Study of melt convection and interface shape during sapphire crystal growth by Czochralski method

Global simulation is performed to predict electromagnetic field, heat transfer, melt flow, and interface shape during the different growth stages of sapphire crystal by RF-heated Czochralski (Cz) process. Melt flow in the Cz-sapphire growth system includes Marangoni convection due to the variation of surface tension at the free surface, natural convection due to the complex thermal boundary conditions on the crucible walls, and forced convection due to the crystal rotation. As the crystal grow longer, the effects of the convections on the melt flow, temperature distribution and interface shape varies due to the drop of melt, which can be characterized by Grashof number (Gr) with the melt height as the characteristic length. It is found that as melt height reduces, natural convection gets weaker, and the interface becomes flatter. To examine systematically the effects of the convections on the crystal growth process, three dimensionless parameters, N_σ, Gr/Re² and N_σ2, are further introduced. N_σ denotes the influence of surface tension in the boundary layer as compared with buoyancy in the melt, Gr/Re² is the ratio of convective forces to the normalizing convective velocities with viscosity, and N_σ2 represents the relative strength of Marangoni flow to forced convection. The effects of the parameters on the melt flow and heat transfer are examined, and their relationships to the interface convexities are obtained. The understanding of melt convection and interface shape is not only important for the optimization of a stable sapphire crystal growth, but also critical for an accurate modeling of the growth process.     

Effect of surface tension on convection in a binary fluid layer under a zero gravity environment

The Marangoni flows in a horizontal layer of a binary mixture with an undeformable free upper surface are studied analytically and numerically. The system is heated and cooled by constant heat fluxes. The surface tension is assumed to vary linearly with temperature and solute concentration. Both double diffusive convection and Soret induced convection, in a zero gravity level, are considered. The governing parameters of the problem are the thermal Marangoni number Ma_T, the solutal Marangoni number Ma_S, the Prandtl and the Lewis numbers Pr and Le, the aspect ratio A and the parameter a defining the mechanism responsible for the occurrence of the solutal gradients (double diffusion or Soret effect). An approximate analytical solution, based on the parallel flow approximation, is proposed. Bifurcation diagrams are presented for the cases in which the solutal Marangoni effect acts in the same direction or competes with the thermal Marangoni effect. The stability of the parallel flow solution is studied numerically and the threshold for Hopf bifurcation determined. The validity of the analytical model is tested against the results obtained by solving numerically the full governing equations.     

Taylor dispersion coefficients for laminar,longitudinal flow past arrays of circular tubes

The problem of determining shell-side Taylor dispersion coefficients for a shell-and-tube configuration is examined in detail for both ordered as well as disordered arrangement of tubes. The latter is modeled by randomly placing N tubes within a unit cell of a periodic array. It is shown that shell-side Taylor dispersion coefficient D_T is expressed by D_T = D_M(1 + λPe²) and the coefficient λ is divergent with N, where D_M is the molecular diffusivity of solute on the shell side and Pe is the Peclet number given by aU/D_M with a and U being the radius of tube and the mean fluid velocity on the shell side, respectively. The coefficient λ depends on the spatial average and the fluid velocity weighted average of the concentration of solute on the shell side. The behavior of the coefficient λ with N arises due to logarithmically divergent nature of concentration disturbances caused by each tube in the plane normal to the axes of the tubes. An effective-medium theory is developed for determining conditionally-averaged velocity and concentration fields and hence the shell-side Taylor dispersion coefficients. Its predictions are compared with the results of rigorous numerical computations. The present study also presents formulas for determining the shell-side Taylor dispersion coefficients for square and hexagonal arrays of tubes with cell theory approximations.     

The Influence of the Magnitude of Gravitational Acceleration on Marangoni Convection About an Isolated Bubble under a Heated Wall

Thermocapillary or Marangoni convection is the liquid motion caused by surface tension variation in the presence of a temperature gradient along a gas–liquid or vapor–liquid interface. This work numerically investigates the effect of the magnitude of gravitational acceleration on the flow and temperature fields resulting from the presence of a hemispherical air bubble of constant radius of 1.0 mm, situated on a heated wall immersed in a liquid silicone oil layer of constant depth of 5.0 mm. The model is oriented such that the Marangoni and gravitational forces act to oppose one another. To elucidate the effect of gravity on Marangoni flow and heat transfer, the simulations were carried out for a silicone oil of Prandtl number 83, at a Marangoni number of 915. The gravity levels tested were 0g, 0.01g, 0.1g, 0.25g, 0.5g, 0.75g, and 1g, where g represents the earth gravitational acceleration of 9.81 m/s ² . The influence of the magnitude of gravitational acceleration on the velocity profile along the bubble interface and on the location of maximum velocity was analyzed. It was found that the gravity level affects the velocity profile by influencing the interfacial temperature gradient, but that the location of maximum velocity was almost independent of gravity level. The increase in heat flux on the wall to which the bubble is attached was calculated and it has been determined that local heat transfer enhancement of up to nearly 1.7 times that of the conduction only case can be achieved for the parameter range tested. Furthermore, local enhancement was observed to occur up to a distance of seven bubble radii for the zero-gravity case, but increased gravity levels cause a reduction in the effective radius of enhancement. The influence of the Marangoni flow on the heat transfer for the opposite cooled wall has also been analyzed.     

Influence of interface materials on the thermal impedance of electronic packages

The thermal properties of different interface materials (grease, ceramic, silicone and mica) have been investigated using the thermal impedance Z_th(jω) represented in a Nyquist plot. It will be shown that an interface material not only influences the thermal resistance R_th = Z_th(0) but the entire shape of the thermal impedance Nyquist plot. From the wind tunnel experiments the specific contribution of the interface materials to the overall thermal resistance could be determined.     

Experimental study on Marangoni effect induced by heat and mass transfer

Surface tension gradients along a fluid–fluid interface provoke strong convective activity, such effect is called Marangoni effect. The Marangoni effect might be induced by heat and mass transfer. In both cases, instabilities develop at the interface. The importance of Marangoni effect has led to many investigations over several years [1], [2], [3], [4], [5], [6] [L. Scriven and C. Sternling, Nature, 187, pp. 186–188 (1960), H. Lu, Y. Yang and J. Maa, Ind. Eng. Chem. Res., 35, pp. 1921–1928 (1996), V. Kaminsky, A. Vyaz'min, N. Kulov and V. Dil'man, Chem. Eng. Sci., 53, No. 19 (1998), P. Lyford, H. Pratt, F. Greiser and D. Shallcross, Can. J. Chem. Eng., 76 (1998a), P. Lyford, H. Pratt, F. Greiser and D. Shallcross, Can. J. Chem. Eng., 76 (1998b), H. Lu, Y. Yang and J. Maa, Ind. Eng. Chem. Res., 36, pp. 474–482 (1997)]. The aim of this work is to study experimentally the Marangoni effect induced by heat and mass transfer, and to determine the Marangoni number (Ma) in both cases. Four aliphatic alcohol were studied, from C1 to C4. In the experiments, interfacial instabilities were observed and Ma was calculated for different temperatures and pressures. It was observed that Ma decreases as the carbon number atoms (n) increase, this behavior can be explained through the changes of some properties (surface tension, diffusivity) as heat and mass transfer occurred. In heat transfer experiments, the highest temperature gradients were reached for methanol, the Bond number (Bd) was also calculated and it was found that the natural convection predominated. In mass transfer experiments, CO₂ was used as gas phase, and it was observed that as pressure increases the Ma increases, it might be explained by the decrease of CO₂ diffusion in hydrocarbons as pressure increases. In conclusion, the Marangoni effect was observed for aliphatic alcohol, the influence of temperature and pressure was also observed, and finally, the Marangoni number decreases as n increases.     

Analysis of first and second laws of thermodynamics between two isothermal cylinders with relative rotation in the presence of MHD flow

In this paper, an analysis of the first and second laws of thermodynamics is presented to show the effects of MHD flow on the distributions of velocity, temperature and entropy generation between two concentric rotating cylinders. The flow inside the gap is assumed to be steady state and laminar for an incompressible, viscous, and Newtonian fluid. The walls of the cylinders are kept at different constant temperatures. Governing equations in cylindrical coordinates are simplified and analytically solved to obtain the local and average (overall) entropy generation rate. Due to the nature of the problem, the velocity distribution in the annulus becomes as the modified Bessel functions I₁(MR) and K₁(MR). Therefore, to obtain the temperature field, the expansions of the modified Bessel functions I₁(MR) and K₁(MR), with 3 terms, are employed in the energy equation. The results are presented for various values of Hartmann number (M), radius ratio (Π), group parameter (Ω/Br), and Brinkman number (Br).     

Skin friction and heat transfer in power-law fluid laminar boundary layer along a moving surface

Analytical and numerical solutions are presented for momentum and energy laminar boundary layer along a moving plate in power-law fluids utilizing a similarity transformation and shooting technique. The results indicate that for a given power-law exponent n(0<n?1) or velocity ratio parameter ξ, the skin friction σ decreases with the increasing in ξ or n. The shear force decreases with the increasing in dimensionless tangential velocity t. When Prandtl number N_Pr=1, the dimensionless temperature w(t) is a linear function of t, and the viscous boundary layer is similar to that of thermal boundary layer. In particular, w(t)=t if ξ=0, i.e., the velocity distribution in viscous boundary layer has the same pattern as the temperature distribution in the thermal boundary and δ=δ_T. For N_Pr?1, the increase of viscous diffusion is larger than that of thermal diffusion with the increasing in N_Pr, and δ_T(t)<δ(t). The thermal diffusion ratio increases with the increasing in n(0<n?1) and ξ.     

Numerical investigation of the interfacial characteristics during Bridgman growth of compound crystals

The effects of the Bridgman process parameters and the double diffusive convection in the melt on the melt/solid interface shape and the solute distribution were numerically investigated for the II–VI semiconductor material HgCdTe. The results show that the melt/solid interface is concave at the center and the solute concentration at the interface is not uniform. A clockwise eddy forms near the interface in the melt when double diffusive convection is considered. With this eddy flow, the solute near the interface is well mixed so the solute distribution at the interface is very different from the case of no flow, but there are no significant changes in the interface shape or the thermal field. However, the growth parameters such as Bi, St, U and A significantly affect the melt/solid interface shape, position and the solute distribution at the interface.     

Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube

The present article address the nanofluid flow with the interaction of shape factor and heat transfer in a vertical tube with temperature-dependent viscosity. Flow study has been done in a flexible tube with low Reynolds number (Re<<0 i.e and long wavelength (δ<<0 i.e assumption. Mathematica software is employed to evaluate the exact solutions of velocity profile, temperature profile, axial velocity profile, pressure gradient and stream function. The influence of heat source/sink parameter (β), Grashof number (G_r) and the viscosity parameter (α) and nanoparticle volume fraction (?) on velocity, temperature, pressure gradient, pressure rise and wall shear stress distributions is presented graphically. Three types of shape factor i.e cylinder platelets and bricks are discussed. Streamline plots are also computed to illustrate bolus dynamics and trapping phenomena which characterize peristaltic propulsion. It is seen that with an increment in Grash of number, G_r, nanofluid velocity is significantly increases i.e. flow acceleration is induced across the tube diameter. Once again the copper-methanol nanofluid in shape of platelets achieves the best acceleration.     

Exact analytical solution of a two diode circuit model for organic solar cells showing S-shape using Lambert W-functions

Organic solar cells (OSCs) often show a kink, also called S-shape, in the current–voltage (I–V) characteristics, that has been attributed to different physical phenomena such as poor quality of cathode-active layer interface or unbalance charge carrier mobilities. This non-ideal behaviour can be electrically modelled including a second diode, in reverse bias, together with an extra shunt resistance (R_P2) in the traditional solar cell equivalent circuit. In this paper, we solve without approximations the transcendental equation system derived from this modified circuit. We have obtained an analytical expression for I–V curves decoupling the voltage drop in each diode using Lambert W function. This expression has been fitted to experimental data in order to obtain circuital parameters. Simulations varying saturation current of reverse diode (I₀₂) and R_P2 have been performed in order to study the dependence of S-shape with these parameters.     

Matched asymptotic solution for the solute boundary layer in a converging axisymmetric stagnation point flow

A novel boundary-layer solution is obtained by the method of matched asymptotic expansions for the solute distribution at a solidification front represented by a disk of finite radius R₀ immersed in an axisymmetric converging stagnation point flow. The detailed analysis reveals a complex internal structure of the boundary layer consisting of eight subregions. The development of the boundary layer starts from the rim region where the concentration, according to the obtained similarity solution, varies with the radius r along the solidification front as ∼ln^1/3(R₀/r). At intermediate radii, where the corresponding concentration is found to vary as ∼ln(R₀/r), the boundary layer has an inner diffusion sublayer adjacent to the solidification front, an inner core region, and an outer diffusion sublayer which separates the former from the outer uniformly mixed region. The inner core, where the solute transport is dominated by convection, is characterized by a logarithmically decreasing axial concentration distribution. The logarithmic increase of concentration along the radius is limited by the radial diffusion becoming effective in the vicinity of the symmetry axis at distances comparable to the characteristic thickness of the solute boundary layer.     

Numerical modeling of methane pyrolysis in a bubble column of molten catalysts for clean hydrogen production

Methane pyrolysis using molten catalysts in a bubble column reactor (BCR) has recently been proposed to produce hydrogen with separable carbon particles as byproducts. In this study, a numerical model of the BCR of molten catalysts for methane pyrolysis was developed and validated using experimental data. Based on a non-isothermal 1-D simplification, continuous liquid and discrete bubble phases were considered by incorporating submodels for bubble behaviors, catalytic and homogeneous reactions, heat/mass transfer, and a submerged orifice for methane supply. The initial bubble diameter was predicted using the correlation derived from measurements. When applied to experiments with Ni₍₂₇₎Bi₍₇₃₎ and mixtures of KCl–MnCl₂, the model accurately reproduced the methane conversion at different temperatures and column heights. Furthermore, detailed information on the key phenomena was acquired, including the profiles of the bubble diameter, rise velocity, reaction rates, temperature, and gas composition. A sensitivity analysis confirmed that the uncertainties regarding the physical properties of molten catalysts had a negligible impact. A comparison of the performances of Ni₍₂₇₎Bi₍₇₃₎ and KCl₍₅₀₎MnCl₂₍₅₀₎ under the same reaction conditions revealed a favorable influence of the catalyst density on methane conversion because of the increased pressure. The proposed model would be useful in reactor optimization and scale-up with high hydrogen productivity.     

Pattern formation of drops in thermocapillary migration

The behavior of a drop cloud in thermocapillary motion in zero gravity is examined for both mono-dispersed and poly-dispersed cases. Numerical simulations of the thermocapillary motion of two- and three-dimensional fully deformable light drops are presented. The Navier–Stokes equations coupled with the energy conservation equation are solved by a front-tracking/finite-difference method. The material properties of the drop fluid and the ambient fluid are different, and the interfacial tension depends on the temperature. At moderate Reynolds (Re) and Marangoni (Ma) numbers, the results show that drops form layers nearly perpendicular to the temperature gradient.     

An asymptotic,large time solution of the convection Stefan problem with surface radiation

An asymptotic, large time solution has been obtained for the convection Stefan problem with surface radiation. The moving boundary problem has been reformulated as a fixed boundary problem where Lagrange-Burmann expansions are used to complete the variable transformation. An asymptotic solution of the problem is obtained by requiring that the asymptotic expansions assumed for the interface position X(t) and wall temperature u_w(t) for large times are consistent with the resulting interfacial Lagrange-Bürmann expansions. It is found that the asymptotic expansions admit Neumann's solution as the leading terms and that logarithmic terms start intervening at the third-order terms of the expansions for nonzero Stefan number.     

Critical heat flux on flow boiling of methanol–water mixtures in a diverging microchannel with artificial cavities

This study constitutes an experimental investigation into the convective boiling heat transfer and critical heat flux (CHF) of methanol–water mixtures in a diverging microchannel with artificial cavities. Flow visualization shows that bubbles are generally nucleated at both the artificial cavities and side walls of the channel. This confirms the proper functioning of such artificial cavities. Consequently, the wall superheat of the onset nucleate boiling is significantly reduced. Experimental results show that the boiling heat transfer and CHF are significantly influenced by the molar fraction (x_m) as well as the mass flux. The CHF increases with an increase in mass flux at the same molar fraction. On the other hand, the CHF increases slightly from x_m = 0 to 0.3, and then decreases rapidly from x_m = 0.3 to 1 at the same mass flux. The maximum CHF is reached at x_m = 0.3, particularly for a mass flux of 175 kg/m² s, due to the Marangoni effect. Flow visualization confirms that the Marangoni effect helps a region with a liquid film breakup persist to a higher heat flux, and therefore a higher CHF. Moreover, a new empirical correlation involving the Marangoni effect for the CHF on the flow boiling of methanol–water mixtures is developed. The present correlation prediction shows excellent agreement with the experimental data, and further confirms that the present correlation may predict the Marangoni effect on the CHF for the convective boiling heat transfer of binary mixtures.     

Onset of Marangoni instability of a two-component evaporating droplet

The temperature and solute concentration reductions across a thin boundary layer near the free surface of an evaporating droplet may induce cellular flow motion in the droplet because of Marangoni instability. The present study is aimed at investigating theoretically the onset of Marangoni instability due to the evaporation of a two-component evaporating droplet.

With the quasi-steady approximation which means that the surrounding gas motion is asymptotically steady, the size change of the droplet is negligible, and the temperature and concentration distributions of the droplet are temporarily frozen at each specified instant of interest, the onset condition for Marangoni instability is obtained through the linear stability analysis.

By assuming the surface tension is a monotonically decreasing function of both temperature and concentration of the higher-volatility substance, the thermocapillary and diffuso-capillary effects augment each other. Therefore, the theoretical analysis predicts a linear relation, with a negative slope, between the onset thermal Marangoni number, Ma_T, and the onset solute Marangoni number, Ma_S. Moreover, when liquid Lewis number Le_l>1, the critical wave number, l_c, may possess different values depending on the variation of the thermocapillary effect and diffuso-capillary effect. In addition, Le_l has a stronger effect on the critical solute Marangoni number Ma_S,C, than on the critical thermal Marangoni number Ma_T,C. That is, as Le_l decreases, Ma_T,C decreases mildly while Ma_S,C increases drastically.