Mass Transfer at Low Reynolds Numbers

G. Strong Convection Effects in Heat and Mass Transfer at Low Reynolds Number... 

G. STRONG CONVECTION EFFECTS IN HEAT AND MASS TRANSFER AT LOW REYNOLDS NUMBER - AN INTRODUCTION... 

Bakhtiary-Davijany H, Dadgar F, Hayer F, Phan XK, Myrstad R, Venvik HJ, Pfeifer P, Flolmen A. Analysis of external and internal mass transfer at low Reynolds numbers in a multiple-slit packed bed microstructured reactor for synthesis of methanol from syngas. Industrial and Engineering Chemistry Research 2012 51 13574-13579. 

For the mass transfer at low Reynolds number flow (no turbulence), the conversation equation of a mass species (component substance) is known to be... 

Garner, F.H. and Keey, R.B. Chem, Eng. Sci. 9 (1958) 19. Mass transfer from single solid spheres — 1. Transfer at low Reynolds numbers. 

This classic equation, which combines well-known results from mass transfer and low-Reynolds-number hydrodynamics, is very useful to predict the effect of molecular size on diffusion coefficients. The assumptions that must be invoked to arrive at the Einstein diffusion equation and the Stokes-Einstein diffusion equation are numerous. A single spherical solid particle of species A experiences forced diffusion due to gravity in an infinite medium of fluid B, which is static. Concentration, thermal, and pressure diffusion are neglected with respect to forced diffusion. Hence, the diffusional mass flux of species A with respect to the mass-average velocity v is based on the last term in equation (25-88) ... 

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

Judy J, Maynes D, Webb BW (2002) Characterization of frictional pressure drop for liquid flows through micro-channels. Int J Heat Mass Transfer 45 3477-3489 Kandlikar SG, Joshi S, Tian S (2003) Effect of surface roughness on heat transfer and fluid flow characteristics at low Reynolds numbers in small diameter tubes. Heat Transfer Eng 24 4-16 Koo J, Kleinstreuer C (2004) Viscous dissipation effects in microtubes and microchannels. Int J Heat Mass Transfer 47 3159-3169... 

SK Friedlander. Mass and heat transfer to single spheres and cylinders at low Reynolds numbers. AIChE J 3 43-48, 1957. 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

Correlation 7.181 should be used with care at low Reynolds numbers. Typical values for gas-solid transfer are 1 m mi"2 s 1 for the mass transfer coefficient and 102 W m-2 K-1 for the heat transfer coefficient. 

The elimination or estimation of the axial dispersion contribution presents a more difficult problem. Established correlations for the axial dispersion coefficient are notoriously unreliable for small particles at low Reynolds number(17,18) and it has recently been shown that dispersion in a column packed with porous particles may be much greater than for inert non-porous particles under similar hydrodynamic conditions(19>20). one method which has proved useful is to make measurements over a range of velocities and plot (cj2/2y ) (L/v) vs l/v2. It follows from eqn. 6 that in the low Reynolds number region where Dj. is essentially constant, such a plot should be linear with slope Dj, and intercept equal to the mass transfer resistance term. Representative data for several systems are shown plotted in this way in figure 2(21). CF4 and iC io molecules are too large to penetrate the 4A zeolite and the intercepts correspond only to the external film and macropore diffusion resistance which varies little with temperature. 

Study of the Behavior of Heat and Mass Transfer Coefficients in Gas—Solid Fluidized Bed Systems at Low Reynolds Numbers... 

Correlations to estimate heat and mass transfer coefficients in gas-solid fluidized beds operating in the controversial low Reynolds numbers zone are proposed.The correlations incorporate the influence of particle diameter to bed length and particle diameter to bed diameter ratios and gas flowrate. Also, the experimental data are used to analyse the models proposed by Kato and Wen, and Nelson and Galloway in order to explain the behaviour of fluid bed systems operating at low Reynolds numbers. 

Kato and Wen (5) found, for the case of packed beds,that there was a dependency of the Sherwood and Nusselt numbers with the ratio dp/L. They proposed that the fall of the heat and mass transfer coefficients at low Reynolds numbers is due to an overlapping of the boundary layers surrounding the particles which produces a reduction of the available effective area for transfer of mass and heat. Nelson and Galloway W proposed a new model in terms of the Frossling number, to explain the fall of the heat and mass transfer coefficients in the zone of low Reynolds numbers. 

This overlapping will in fact reduce the available area for heat and mass transfer. During the present work, some boundary layer thicknesses were estimated for the experimental conditions of this work. As a result, the boundary layers only overlap for Reynolds numbers below 0.826. For the case of Reynolds numbers of 1.74 and 3.05 using the particle diameter of 0.035 cm., the boundary layers do not overlap.Table III shows some of the values obtained.Clearly, this effect cannot explain completely the low heat and mass transfer coefficients at low Reynolds numbers. 

Discussion. The data and correlations at pressures of 10 atm to 200 atm are compared to literature values at low pressure (1 atm and 25 C) (20-22). Figure 4 shows that above the critical pressure mass transfer coefficients are less dependent on Re than below the critical pressure. At low pressure, the density gradient across the boundary layer is much smaller. Therefore, the effect of natural convection on mass transfer is very slight. Near the critical pressure, however, the effect of natural convection on mass transfer rates becomes important at low Reynolds number due to very large density gradients across the boundary layer. 

Schmidt give data in tree convection for wires and Satterfield and Cortez give data in forced convection for gauzes. The latter conclude that the data are better correlated according to the Reynolds number based on wire diameter (A Re.d) rather than that based on hydraulic radius. Values found were similar to values for infinite cylinders. From their work the mass transfer coefficient at low Reynolds numbers (<10 ) is proportional to Values of mass... 

Problem 9-25. Mass Transfer From a Gas Bubble in a Bioreactor. Gas bubbles are injected through a sparger into a bioreactor to oxygenate a liquid growth medium. The liquid is sufficiently viscous that the bubbles flow at low Reynolds number. You have been asked to model the mass transfer of oxygen in the reactor. 

Spherical bubble. The problem of mass transfer to a spherical bubble in an axisymmetric shear flow at low Reynolds numbers was solved numerically in the entire range of Peclet numbers in [251], The results for the mean Sherwood number can be approximated by formula (4.7.10), where the corresponding value from the second row in Table 4.4 at (3 = 0 must be substituted into the right-hand side. Thus, we obtain the formula... 

Mass transfer inside a drop is described by Eq. (4.12.1) and the first two conditions (4.12.2). The fluid velocity field v = (vr,vg) at low Reynolds numbers is given by the Hadamard-Rybczynski stream function and, in the dimensionless variables, has the form... 

Let us consider mass and heat transfer for a monodisperse system of spherical particles of radius a with volume density of the solid phase. We use the fluid velocity field obtained at low Reynolds numbers from the Happel cell model (see Section 2.9) to find the mean Sherwood number [74,76 ... 

The results predicted by formula (6.9.8) were compared with the experimental data of [92, 292] for the liquid phase mass transfer coefficients from the absorption of nitrogen bubbles in aqueous solutions of carboxymetil cellulose and carbopol at low Reynolds numbers. The maximum error of the formula is about 5% for 0.7 n < 1.0. 

Axial dispersion or backmixing31 36 can make a major contribution to the mass transfer rate in liquid-phase adsorption and cannot be ignored. This is especially true at low Reynolds numbers. 

Answer For boundary layer mass transfer across solid-liquid interfaces, = I and y =. In the creeping flow regime, z = - This problem is analogous to one where the solid sphere is stationary and a hquid flows past the submerged object at low Reynolds numbers. 

Equation 12.5.b-8 is the same as Eq. I2.5.b-4 except for the term in the latter with D 2, representing direct effective particle-to-particle transport. For several types of situations, this term may be of definite importance mass transfer in highly porous solids at low Reynolds numbers (Wakao [63]) and the heat transfer situation discussed by Littman and Barile [61], which is analogous to the model for radial heat transfer proposed by De Wasch and Froment [62]. 

Feng, Z.G. and Michaelides, E.E., Mass and heat transfer from fluid spheres at low Reynolds numbers. Powder TechnoL, 112, 63-69, 2000. 

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