Diffusional interaction of drops in a liquid

The method of combining asymptotic expansions (with respect to a large Peclet number) is used to investigate the three-dimensional problem of steady-state convective diffusion to the surface of drops, around which flows a laminar stream of a viscous incompressible liquid whose velocity field is assumed to be known from the solution of the corresponding hydrodynamic problem. It is shown that for large Peclet numbers the heat and mass transfer between drops is completely determined by the mutual arrangement of special (starting or ending at the surface of a drop) lines of flow; under these circumstances, in the flow there are chains of drops which have no mutual diffusional effect on one another, and the total diffusional flow to a drop is determined by diffusion to particles located upstream in the same chain. For the case where the distance between the drops in the chain is much leas than P^1/2 (P is the Peclet number), formulas for the distribution of the concentration and the total diffusional flow to the surface of each drop are obtained. It is shown that the total diffusional flow to the surface of a drop approaches zero in inverse proportion to its order number in a chain, which generalizes [1], in which the axisymmetric case is considered. A solution of the diffusional case is obtained for the case where there are critical lines at the surface of the drop. The problem is solved to the end if the singular flow lines are not closed and depart to infinity. With the presence of a region of closed circulation behind the drops, the problem is reduced to an integral equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika, Zhidkosti i Gaza, No. 2, pp. 44–56, March–April, 1978.The author thanks Yu. P. Gupalo and Yu. S. Ryazantsev for their interest in the work.     

The extended Graetz problem with Dirichlet wall boundary conditions

The extension of the Graetz problem to include axial conduction has been of interest in view of its application to a number of low Peclet number heat or mass transfer situations. Past efforts in dealing with this problem have been plagued with uncertainties arising from expansion in terms of “eigenfunctions” and “eigenvalues” belonging to a nonselfadjoint operator. The uncertainties spring from a lack of basis for the assumptions that no complex eigenvalues exist and that the calculated eigenvectors originate from a complete set. Other methods have been entirely numerical. The present work produces an entirelyanalytical solution to the Graetz problem for the Dirichlet boundary condition based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. Physically, the decomposition views the convective diffusion process as a pair of stipulations on how the temperature (or concentration) and theaxial energy (or mass) flow through a partial tube cross-section vary with radial and axial distances. The solution obtained is simple, and readily computed. To whom correspondence may be addressed     

Numerical investigation of different modes of internal circulation in spherical drops: Fluid dynamics and mass/heat transfer

The results of detailed, three-dimensional numerical simulations of fixed spherical drops in a uniform flow are presented. The fluid dynamics outside and inside of the drops as well as the internal problem of mass (or heat) transfer are studied. Liquid drops in both a liquid and a gaseous ambient phase are considered. Special emphasis is put on the investigation of different modes of internal circulation.At low Reynolds numbers of the inner fluid, the flow field inside the drop resembles the well known Hill’s vortex solution. However, at higher internal Reynolds numbers, stable steady or quasi-steady alternative modes of internal circulation are found. As these modes are not cylindrical symmetric around the streamwise axis, the often applied assumption of a two-dimensional, axisymmetric flow field is not justified in these cases. Thus, major discrepancies to previous numerical studies are obtained. However, it is shown that experimental results support our findings.For liquid drops surrounded by a liquid, a major influence of the state of internal circulation on the drag is discovered, whereas the drag is nearly unaltered in the case of a liquid drop in gas.Concerning the internal problem of mass/heat transfer, the various internal flow modes show different characteristics. At low internal Peclet numbers, higher Sherwood numbers are reached for the Hill’s vortex-like cases, whereas at higher Peclet numbers, the transfer is faster for the alternative modes. For cases with a Hill’s vortex-like solution, asymptotic Sherwood numbers for very high Peclet numbers of around 20 are found, whereas no upper limit for cases with alternative modes can be determined. In the present study a maximum internal Sherwood number of 130 is reached, more than six times the maximum value for a case with a Hill’s vortex-like internal solution.     

Mass transfer and dispersion around an active cylinder in cross flow and buried in a packed bed

The present work describes the mass transfer process between a moving fluid and a slightly soluble cylinder, with the axis perpendicular to flow direction, buried in a packed bed of small inert particles, with uniform voidage. Fluid flow in the packed bed around the cylinder was assumed to follow Darcy’s law and, at each point, dispersion of solute was assumed to be determined by radial and axial dispersion coefficients, in the cross-stream and streamwise directions, respectively. Numerical solutions of the differential equation describing solute mass conservation were undertaken to obtain the concentration field near the soluble surface and the mass transfer flux was integrated to give the Sherwood number as a function of the relevant parameters. Mathematical expressions are proposed that describes accurately the dependence found numerically between the value of the Sherwood number and the values of Peclet number, Schmidt number and the ratio between the diameter of cylinder and the diameter of inerts.     

Heat and mass transfer in a dispersive medium

Macroscopic equations for the conservation of heat (or the mass of a diffusing impurity) in a continuous medium containing distributed particles of a dispersed phase are formulated neglecting the effect of random fluctuations of the medium and particles by the transfer process. The problem of convective heat conduction or diffusion near an isolated particle is also formulated, the solution of which permits calculation of all the parameters entering into the indicated equations. This problem has been solved in the particular case of small Peclet numbers, which characterize heat and mass exchange in the vicinity of a single particle.     

Numerical analysis of transient laminar forced convection of nanofluids in circular ducts

In this study, forced convection heat transfer characteristics of nanofluids are investigated by numerical analysis of incompressible transient laminar flow in a circular duct under step change in wall temperature and wall heat flux. The thermal responses of the system are obtained by solving energy equation under both transient and steady-state conditions for hydro-dynamically fully-developed flow. In the analyses, temperature dependent thermo-physical properties are also considered. In the numerical analysis, Al₂O₃/water nanofluid is assumed as a homogenous single-phase fluid. For the effective thermal conductivity of nanofluids, Hamilton–Crosser model is used together with a model for Brownian motion in the analysis which takes the effects of temperature and the particle diameter into account. Temperature distributions across the tube for a step jump of wall temperature and also wall heat flux are obtained for various times during the transient calculations at a given location for a constant value of Peclet number and a particle diameter. Variations of thermal conductivity in turn, heat transfer enhancement is obtained at various times as a function of nanoparticle volume fractions, at a given nanoparticle diameter and Peclet number. The results are given under transient and steady-state conditions; steady-state conditions are obtained at larger times and enhancements are found by comparison to the base fluid heat transfer coefficient under the same conditions.     

Numerical study of evaporation-condensation in a vertical diffusion gap

The present work deals with mass transfer between a vertical falling film over a heated plate and a condensing film over a parallel cooled plate in a diffusion gap. This is typically the case of the distillation process in a diffusion still. The governing equations for mass, momentum, and energy are considered for the evaporating film, the diffusion gap, and the condensing film, together with the boundary and interfacial conditions. The local similar technique is used to solve the problem numerically and to get the velocity and temperature distributions in the gap. A comprehensive analysis of the effect of evaporating temperature, condensing temperature, and diffusion gap width over the diffusion mass flow rate and evaporation heat transfer coefficient are carried out. Performance charts of air and helium diffusion gaps are given. Additionally, the analytical results are experimentally validated.     

Cooling of a viscoelastic film during unsteady extrusion from an annular die

The unsteady extrusion of a viscoelastic film from an annular and axisymmetric die is examined. External, elastic, viscous and inertia forces deform the film, which is simultaneously cooled via forced convection to the ambient air. This moving boundary problem is solved by mapping the liquid/air interfaces onto fixed ones and by employing a regular perturbation expansion for all the dependent variables. The ratio of the film thickness to its inner radius at the exit of the die is used as the small parameter in the perturbation expansion. The fluid mechanical aspects of the process depend on the Stokes, Deborah, Reynolds, and Capillary numbers. The heat transfer in the film and to the environment gives rise to four additional dimensionless groups: the Peclet, Biot and Brinkman numbers and the activation energy, which determines the temperature dependence of fluid viscosity and elasticity. A variable heat transfer coefficient is also considered. For typical fluid properties and process conditions, the Peclet number is very large. In this case it is the ratio of the Biot to the Peclet number, the Stanton number, which arises in the energy conservation equation. It is shown that film cooling becomes important when the Stanton number and/or the activation energy are in the high-end of their typical values. In such cases, the cooling of the parison leads to a more uniform flow and shape for the film. The influence on the process of a variable heat transfer coefficient and the Brinkman number is small. Received: 7 April 1999/Accepted: 10 August 1999     

Two problems of convective diffusion to the surfaces of poorly streamlined bodies

Problems of diffusion to particles of nonspherical shape at large Peclet numbers have been analyzed in many papers (see [1–7], for example). The solution of the problem of mass exchange of an ellipsoidal bubble at low Reynolds numbers was obtained in [1] while the solution at high Reynolds numbers was obtained in [2, 3]. In [4] an expression is given for the diffusional flux to the surface of a solid ellipsoidal particle over which a uniform Stokes stream flows. Generalization to the case of particles of arbitrary shape was done in [5, 6], while generalization to any number of critical lines on the surface of the body was done in [7, 8]. The two-dimensional problem of steady convective diffusion to the surface of a body of arbitrary shape is analyzed in the approximation of a diffusional boundary layer (ADBL). The simple analytical expressions obtained are more suitable for practical calculations than those in [5-8], since they allow one to determine at once, in the same coordinate system in which the field of flow over the particle was analyzed, the value of the diffusional flux to its surface (from the corresponding hydrodynamic characteristics). The plane problem of the diffusion to an elliptical cylinder in a uniform Stokes stream is solved. The problems of the diffusion to a plate of finite dimensions (in the plane case) and a disk (in the axisymmetric case) whose planes are normal to the direction of the incident stream are considered. It is shown that, in contrast to the results known earlier (see [4, 6-15], for example), where the total diffusional flux was proportional to the cube root of the Peclet number, here it is proportional to the one-fourth power.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 104–109, November–December, 1978.The authors thank Yu. P. Gupalo, Yu. S. Ryazantsev, and Yu. A. Sergeev for a useful discussion.     

The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli

In the present paper, the heat transfer characteristics in the thermal entrance region of concentric annuli have been analysed for laminar and turbulent internal flow. Axial heat conduction effects in the fluid have been taken into account. The present paper shows an exact analytical solution for the problem of a piecewise uniform wall heat flux. The obtained analytical solution for the extended Graetz problem is as simple and efficient to compute as the related solution of the parabolic problem. The obtained results show the effect of axial heat conduction in the fluid for a semi-infinite heated section as well as for a finite length of the heated section. It is shown, that for a finite length of the heated section, axial heat conduction effects might be important even for higher Peclet number.     

A novel non‐upwind,interconnected, multi‐grid,overlapping numerical procedure for problems involving fluid flow

A novel numerical procedure for heat, mass and momentum transfer in fluid flow is presented. The new scheme is passed on a non‐upwind, interconnected, multi‐grid, overlapping (NIMO) finite‐difference algorithm. In 2D flows, the NIMO algorithm solves finite‐difference equations for each dependent variable on four overlapping grids. The finite‐difference equations are formulated using the control‐volume approach, such that no interpolations are needed for computing the convective fluxes. For a particular dependent variable, four fields of values are produced. The NIMO numerical procedure is tested against the exact solution of two test problems. The first test problem is an oblique laminar 2D flow with a double step abrupt change in a passive scalar variable for infinite Peclet number. The second test problem is a rotating radial flow in an annular sector with a single step abrupt change in a passive scalar variable for infinite Peclet number. The NIMO scheme produced essentially the exact solution using different uniform and non‐uniform square and rectangular grids for 45 and 30° angle of inclination. All other schemes were unable to capture the exact solution, especially for the rectangular and non‐uniform grids. The NIMO scheme was also successful in predicting the exact solution for the rotating radial flow, using a uniform cylindrical‐polar coordinate grid. Copyright © 2008 John Wiley & Sons, Ltd.     

Vermischung von zwei Fluiden durch Diffusion in einer Potentialströmung

The problem may be formulated as follows: “Subject to investigation is a plane flow in a long straight channel or duct of constant width. A fluid enters the channel at one of its ends. Another fluid is supplied by two sources located on opposit sides of the duct. The flow being a potential one, the process of mixing the fluids is mere diffusion. The direction of diffusion considered is perpendicular to the direction of flow. The mixture leaves the channel at its other end. The aim is to determine the steady state concentration field inside the duct. “ The solution given bases ona generalisation according to the principles of conformity. This involves the introduction of “confrom models.?     

Stress diffusion and high order viscoelastic effects in the 3D flow past a sedimenting sphere subject to orthogonal shear

The effect of polymer stress diffusion in the unbounded flow past a sedimenting, freely rotating, rigid sphere subject to shear in a plane perpendicular to the direction of sedimentation is investigated analytically. Steady state, creeping, incompressible, and isothermal flow is assumed. For viscoelastic fluids following the Oldroyd-B constitutive model, three-dimensional results for the velocity vector, pressure, and viscoelastic extra-stress tensor are derived by including an artificial diffusion term in the constitutive equation and using regular perturbation theory with the small parameter being the Deborah number. The analytical solution reveals that the influence of the stress diffusion term on the results may be significant (and sometimes unexpected) and strongly depends on the magnitude of the dimensionless diffusion coefficient. For instance, it is shown that the critical Deborah number, below which a physical solution arises, decreases with the increase in the diffusion coefficient. Also, comparison against simulation results from the literature shows excellent agreement up to shear Weissenberg number (defined as the product of the imposed shear rate with the single relaxation time of the fluid) approximately equal to unity.     

The theoretical study of the efficiency of diffusion deposition of nanoaerosols in the extended range of the Peclet numbers

The efficiencies of the diffusion deposition of nanoaerosols for a single fiber for the models of aerosol filter and wire mesh screen are studied numerically in the extended range of the Peclet number Pe. The rectangular periodic cell model for fluid flow and convective-diffusive transport of small aerosol particles is used. Most of the previous theoretical and experimental studies of single fiber diffusion deposition efficiency were for the case of Pe > 1. The array with uniform square or chess grid of fibers and of a row of circular cylindrical fibers are considered as the filter and wire mesh screen models. The flow and particles transport equations are solved numerically using the Boundary Element Method.The obtained numerical data are used to derive the approximate formulas for the deposition efficiency in the entire range of the Peclet number for the various porosities of the filter medium or distances between fibers in a wire mesh screen. The derived dependencies take into account nonlinearity of the deposition efficiency at the low Peclet numbers. The obtained analytical dependencies compare well with the numerical and experimental data.     

FLOW-INDUCED VIBRATION OF FREE EDGES OF THIN FILMS

During manufacturing processes of thin materials such as paper, photographic film, and magnetic film, which are handled as continuous sheets and subjected to drying air-flows, the interaction of the air with the web can cause the free edges to vibrate violently. This phenomenon is related to the waving motion of a flag in the wind, except that the thin films under consideration are under tension in the direction of the air-flow or at right angles to it. A travelling-wave analysis was done based on incompressible potential-flow theory; the critical flow speed, wave speed, wavelength, and flutter frequency were predicted. A closed-form solution of the critical flow speed is suggested. Experiments were carried out with stationary thin films mounted in a wind tunnel where the direction of tension was perpendicular to the flow direction. It was shown that the analysis, which assumes that the film is infinitely long in the flow direction, could successfully predict the critical flow speed above which violent edge vibrations occur.     

Mixed convection adjacent to inclined flat surfaces embedded in a porous medium

An analysis is performed to study the flow and heat transfer characteristics of laminar mixed convection boundary layer flows from inclined (including horizontal and vertical) surfaces embedded in a saturated porous medium with constant aiding external flows and uniform surface temperature. Both the streamwise and normal components of the buoyancy forces are retained in the momentum equations. Nondimensionalization of the boundary layer equations results in the following three governing parameter: (1)Gr/Re, the ratio of the Grashof number to the Reynolds number; (2)Pe _x =Re _x Pr, the Peclet number; (3) φ, the angle of inclination from the horizontal. The resulting nonsimilar equations are solved by an efficient implicit finite-difference scheme. Numerical results are presented for flows with different values ofGr/Re in the range of 0 to 50, over a wide range of the Peclet numbersPe _x, and various values of φ ranging from 0 to 90 degrees. It is found that the local surface heat transfer rate increases with increasing the local Peclet number. In addition, as the plate is tilted from the horizontal to the vertical orientation, the local Nusselt number increases for a given Peclet number and the effect of the buoyancy force on the surface heat transfer rate increases.     

Thermal analysis of an infinite tube during quenching

 This paper deals with a numerical solution of the two-dimensional convection–diffusion equation in an infinite domain, arising out of quenching of an infinite tube. On the wetted side, upstream of the quench front, a constant heat transfer coefficient is assumed. The downstream of the quench front as well as the inside surface of the tube are assumed to be adiabatic. The solution gives the quench front temperature as a function of various model parameters such as Peclet number, Biot number and the radius ratio. The solution has been found to be in good agreement with the available analytical solutions and thus validates the numerical procedure suggested. Received on 10 July 2000     

Microchannel membrane separation applied to confined thin film desorption

The concept of a confined thin film to enhance the desorption process is based on a reduced mass diffusion resistance. A wide thin film is formed into a microchannel by using a porous membrane as one wall of the channel enabling vapor extraction along the flow. Heat added to the channel results in vapor generation and subsequent extraction through the membrane. This experimental study investigates the performance of vapor extraction as a function of confined thin film thickness, pressure difference across the membrane and inlet concentration to the microchannel. In addition, heat added to the system was varied and results are presented in terms of the wall superheat temperature relative to the inlet saturated conditions of the binary fluid. The test section was equipped with a transparent window to observe bubble formation and vapor extraction. Results show that the performance, measured by the vapor release rate, increases for reduced channel thickness, for increased pressure difference across the membrane, and for lower inlet concentration. Results show that lower wall superheat correspond to higher heat transfer coefficients. Trends of Nusselt number and Sherwood number versus both channel Reynolds number and the product of the Reynolds number and Schmidt number are presented. Bubble formation in the channel does not degrade overall performance provided a critical heat flux condition does not occur.     

Wiener–Hopf solution of rewetting of an infinite cylinder with internal heat generation

The two-dimensional quasi-steady conduction equation governing conduction controlled rewetting of an infinite cylinder with heat generation has been solved by Wiener–Hopf technique. The analytical solution yields the quench front temperature as a function of various model parameters such as Peclet number, Biot number and dimensionless heat generation rate. Also, the dry out heat generation rate is obtained by setting the Peclet number equal to zero, which gives the maximum permissible heat generation so as to prevent the dry out of the coolant.     

Simultaneous heat and mass transfer inside a vertical tube in evaporating a heated falling alcohols liquid film into a stream of dry air

A numerical study of the evaporation in mixed convection of a pure alcohol liquid film: ethanol and methanol was investigated. It is a turbulent liquid film falling on the internal face of a vertical tube. A laminar flow of dry air enters the vertical tube at constant temperature in the downward direction. The wall of the tube is subjected to a constant and uniform heat flux. The model solves the coupled parabolic governing equations in both phases including turbulent liquid film together with the boundary and interfacial conditions. The systems of equations obtained by using an implicit finite difference method are solved by TDMA method. A Van Driest model is adopted to simulate the turbulent liquid film flow. The influence of the inlet liquid flow, Reynolds number in the gas flow and the wall heat flux on the intensity of heat and mass transfers are examined. A comparison between the results obtained for studied alcohols and water in the same conditions is made.