Mass convection concentration boundary layer

The value of the concentration modulus depends on the convective velocity and the mass-transfer coefficient of the concentration boundary layer (D/ i) that means that on the membrane structure and the hydrodynamic conditions. If the retention coefficient is equal to 1, then c /ch = exp(Pe). The larger convective velocity (or smaller diffusion coefficient) causes higher concentration polarization on the membrane interface. 

When the concentration boundary layer is sufficiently thin the mass transport problem can be solved under the approximation that the solution velocity within the concentration boundary layer varies linearly with distance away from the surface. This is called the L6v que approximation (8, 9] and is satisfactory under conditions where convection is efficient compared with diffusion. More accurate treatments of mass transfer taking account of the full velocity profile can be obtained numerically [10, 11] but the Ldveque approximation has been shown to be valid for most practical electrodes and solution velocities. Using the L vSque approximation, the local value of the concentration boundary layer thickness, 8k, (determined by equating the calculated flux to the flux that would be obtained according to a Nernstian diffusion layer approximation that is with a linear variation of concentration across the boundary layer) is given by equation (10.6) [12]. 

Consider the flow of air over the free surface of a water body such as a lake under isothermal conditions. If the air is not saturated, the concentration of water vapor will vtsry from a maximum at the water surface where the air is always saturated to the free steam value far from the surface. In heat convection, we defined the region in which temperature gradients exist as the thermal boundary layer. Similarly, in mass convection, we define the region of the fluid in which concentration gradients exist as the conceniration boundary layer, as shown in Figure 14 -38. In external flow, the thickness of the concentration boundary layer S,. for a. species A at a. specified location on the surface is defined as the normal distance y from the surface at which... 

The equations (3.109), (3.117) or (3.118) and (3.120) for the velocity, thermal and concentration boundary layers show some noticeable similarities. On the left hand side they contain convective terms , which describe the momentum, heat or mass exchange by convection, whilst on the right hand side a diffusive term for the momentum, heat and mass exchange exists. In addition to this the energy equation for multicomponent mixtures (3.118) and the component continuity equation (3.25) also contain terms for the influence of chemical reactions. The remaining expressions for pressure drop in the momentum equation and mass transport in the energy equation for multicomponent mixtures cannot be compared with each other because they describe two completely different physical phenomena. 

Order-of-magnitude analysis indicates that diffusion is neghgible relative to convective mass transfer in the primary flow direction within the concentration boundary layer at large values of the Peclet number. Typically, liquid-phase Schmidt numbers are at least 10 because momentum diffusivities (i.e., i/p) are on the order of 10 cm /s and the Stokes-Einstein equation predicts diffusion coefficients on the order of 10 cm /s. Hence, the Peclet number should be large for liquids even under slow-flow conditions. Now, the partial differential mass balance for Cji,(r,0) is simplified for axisynunetric flow (i.e., = 0), angu-... 

Mass transfer coefficient (fe) A measure of the solute s mobility due to forced or natural convection in the system. Analogous to a heat transfer coefficient, it is measured as the ratio of the mass flux to the driving force. In membrane processes the driving force is the difference in solute concentration at the membrane surface and at some arbitrarily defined point in the bulk fluid. When lasing the film theory to model mass transfer, k is also defined as D/S, where D is solute diffusivity and d is the thickness of the concentration boundary layer. 

In Section 3.IOC an exact solution was obtained for the hydrodynamic boundary layer for isothermal laminar flow past a plate and in Section 5.7A an extension of the Blasius solution was also used to derive an expression for convective heat transfer. In an analogous manner we use the Blasius solution for convective mass transfer for the same geometry and laminar flow. In Fig. 7.9-1 the concentration boundary layer is shown where the concentration of the fluid approaching the plate is and in the fluid adjacent to the surface. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

Karabelas et al." reviewed many of the published correlations for fixed-bed coefficients and proposed different correlations to be used depending on the flow regime that is, at low Reynolds number the effects of molecular diffusion and natural convection must be considered. Kato et al. reviewed mass transfer coefficients in fixed and fluid beds and observed considerable deviations from established correlations in both the literature and their own data for Re < 10. In some cases it appeared that the limiting Sherwood number could be less than 2 for gas-particle transfer. They suggested that for small Re and Sc i the concentration boundary layers of the individual particles in a fixed bed would overlap considerably. They proposed two correlations for different flow regimes which also inclutted a particle diameter to bed height term. 

Analogous to velocity and thermal boundary layers is the concentration boundary layer. That is, it determines convection mass transfer, similar to the velocity boundary layer determining wall friction or the thermal boundary layer determining convection heat transfer. 

Physically, the Sherwood Number is define as the ratio of convection to diffusion mass transfer through the concentration boundary layer at the cathode surface [2,5,7]. Thus, Sh can be derived by using eqs. (4.3) and (4.5)... 

The convection mass transport of species i may also take place if there exists a bulk fluid motion. The convection mass transfer is analogous to convection heat transfer and occurs between a moving mixture of fluid species and an exposed solid surface. Like hydrodynamic and thermal boundary layers, a concentration boundary layer forms over the surface if the free stream concentration of a species i, differs from species concentration at the surface, Qs, in an external flow over a solid surface as demonstrated in Figure 6.13. 

Convection mass transfer coefficients are often used as convective boundary conditions for gas diffusion in a stationary media. However, while applying mass transfer correlations to describe mass species transport from the electrode-gas diffusion layer to gas flow stream in the channel, it is assumed that species mass transport rate at the wall is small and does not alter the hydrod5mamic, thermal, and concentration boundary layers like in boundary layers with wall suction or blowing. 

In free-convection mass transfer at electrodes, as well as in forced convection, the concentration (diffusion) boundary layer (5d extends only over a very small part of the hydrodynamic boundary layer <5h. In laminar free convection, the ratio of the thicknesses is... 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

At the RDE, various approximate analytical treatments have been presented by dropping the highest order convective term [237], neglecting convection completely [238], and by assuming a linear concentration profile within a time-dependent mass transfer boundary layer [239]. The last of these gives... 

The salt flux through the membrane is given by the product of the permeate volume flux. /,. and the permeate salt concentration c,p. For dilute liquids the permeate volume flux is within 1 or 2% of the volume flux on the feed side of the membrane because the densities of the two solutions are almost equal. This means that, at steady state, the net salt flux at any point within the boundary layer must also be equal to the permeate salt flux Jvcip. In the boundary layer this net salt flux is also equal to the convective salt flux towards the membrane Jvc, minus the diffusive salt flux away from the membrane expressed by Fick s law (Didcildx). So, from simple mass balance, transport of salt at any point within the boundary layer can be described by the equation... 

Although the assumption of a quasistationary distribution of the concentration of component A within the diffusion boundary layer seems to be very rough, nevertheless under conditions of sufficiently intensive convection the dissolution kinetics of solids in liquids is well described by equations (5.1) and (5.8)-(5.10) (see Refs 301, 303, 304, 306-308). Clearly, these equations are generally applicable at a low solubility of the solid in the liquid phase (about 10-100 kg m or up to 5 mass %). Note that they may also describe fairly well the dissolution process in systems of much higher solubility. An example is the Al-Ni binary system in which the solubility of nickel in aluminium amounts to 10 mass % even at a relatively low temperature of 700°C (in comparison with the melting point of aluminium, 660°C).308... 

When an electrochemical process takes place at the electrode surface, a concentration gradient develops near the surface, resulting in diffusion as an additional mode of mass transport. The liquid layer in which the transport by diffusion is comparable to the convectional motion is called the diffusion boundary layer, and its approximate thickness S is given by Eq. (86), corresponding to approximately 5% of Sq... 

Mass transfer in the feed and strip solutions is limited by the extent of concentration polarization. On the feed side of the membrane, concentration polarization refers to an increase in the concentration of solutes at and near the feed-membrane interface because of evaporation of water into the membrane pores (Fig. 1). The resulting solute concentration gradient between the membrane-feed interface, where the concentration is greatest, and the bulk solution induces diffusive transport of rejected solutes back through the concentration polarization boundary layer into the bulk stream. Bulk solution is simultaneously transported to the membrane wall by convection. When equilibrium has been established under a given set of operating conditions (stream flow rate, temperature, fluid dynamics imposed by membrane module design), the rate of back diffusion is equal to the rate at which the solutes are carried to the membrane surface by convective flow. ... 

In Chap. 9, we considered the solution of this equation in the limit Re 1, where the velocity distribution could be approximated by means of solutions of the creeping-flow equations. When Pe 1, we found that the fluid was heated (or cooled) significantly in only a very thin thermal boundary layer of 0(Pe l/3) in thickness, immediately adjacent to the surface of a no-slip body, or () Pe l/2) in thickness if the surface were a slip surface with finite interfacial velocities. We may recall that the governing convection di ffusion equation for mass transfer of a single solute in a solvent takes the same form as (111) except that 6 now stands for a dimensionless solute concentration, and the Peclet number is now the product of Reynolds number and Schmidt number,... 

In the diffusion boundary layer d, in the mass transfer equation one can neglect molecular tangential diffusion transfer compared with the diffusion in the radial direction the convective terms are retained (but somewhat simplified by linearization near the interface). The concentration distribution in that region was obtained earlier in Section 4.6. 

Let us investigate steady-state convective diffusion on the surface of a flat plate in a longitudinal translational flow of a viscous incompressible fluid at high Reynolds numbers (the Blasius flow). We assume that mass transfer is accompanied by a volume reaction. In the diffusion boundary layer approximation, the concentration distribution is described by the equation... 

In the inner problems of the convective mass transfer for kv - 0(1) as Pe —> oo, the concentration is leveled out along each streamline. The mean Sherwood number, by virtue of the estimate (5.4.8), is bounded above uniformly with respect to the Peclet number Sh < const kw. This means that the inner diffusion boundary layer cannot be formed by increasing the circulation intensity alone (i.e., by increasing the fluid velocity, which corresponds to Pe - oo) for moderate values of kv. This property of the mean Sherwood number is typical of all inner problems. For outer problems of mass transfer, the behavior of this quantity is essentially different here a thin diffusion boundary layer is usually... 

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