Mass convection Sherwood number

A theoretical analysis of the numerical results like those shown in Fig. 22 is greatly simplified by introducing the dimensionless mass-transfer Sherwood number Sh = k a/D and dimensionless parameters characterizing the role of various forces in particle adsorption phenomena. Except for the previously defined Pe number describing the role of convection, one often defines the external force parameter Ex by... 

Free convection heat transfer as a source of forced convection mass transfer. It has been demonstrated on numerous occasions that the Chilton-Colburn analogy appearing in Table 2.3 is applicable for converting a forced-convection Nusselt number to a forced-convection Sherwood number as a means of converting the imbedded HTC into its equivalent MTC. In the present situation, the thermal buoyant forces provide the momentum source, which in effect provides the forced-convective flow that drives the mass transfer process. In addition, Grj Gta and Sc > Pr. For this case the alternative equation is... 

The convective mass transfer coefficient hm can be obtained from correlations similar to those of heat transfer, i.e. Equation (1.12). The Nusselt number has the counterpart Sherwood number, Sh = hml/Di, and the counterpart of the Prandtl number is the Schmidt number, Sc = p/pD. Since Pr k Sc k 0.7 for combustion gases, the Lewis number, Le = Pr/Sc = k/pDcp is approximately 1, and it can be shown that hm = hc/cp. This is a convenient way to compute the mass transfer coefficient from heat transfer results. It comes from the Reynolds analogy, which shows the equivalence of heat transfer with its corresponding mass transfer configuration for Le = 1. Fire involves both simultaneous heat and mass transfer, and therefore these relationships are important to have a complete understanding of the subject. 

Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration c,. Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion-advection mass transport [48,49], The relative contribution of advection to total transport is characterised by the nondimensional Peclet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh - 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

So far we have considered an infinite value of the gas-to-particle heat and mass transfer coefficients. One may encounter, however, an imperfect access of heat and mass by convection to the outer geometrical surface of a catalyst. Stated in other terms, the surface conditions differ from those in the bulk flow because external temperature and concentration gradients are established. In consequence, the multiple steady-state phenomena as well as oscillatory activity depend also on the Sherwood and Nusselt numbers. The magnitudes of the Nusselt and Sherwood numbers for some strongly exothermic reactions are reported in Table III (77). We may infer from this table that the range of Sh/Nu is roughly Sh/Nu (1.0, 104). 

This relationship indicates that the Sherwood number takes the value 2 for Ap 0, as is the case in natural convection. A generalized correlation for mass and heat transfer coefficients is recommended by Calderbank and Moo-Young (1961). This correlation relates the mass-transfer coefficient to the power per unit volume and Schmidt number. The relationship is mainly applicable to low-viscosity liquids. 

In natural convection mass transfer, the analogy betsveen the Nusselt and Sherwood numbers still holds, and thus Sh =y(Gr, Sc). But the Grashof number in this case should be determined directly from... 

The counterparts of the Prandtl and Nusselt numbers in mass convection are the Schmidt mmtber Sc and the Sherwood number Sb, defined as... 

The calculation of diffusion fluxes and the mean Sherwood number is usually carried out in three steps. First, the problem of convective mass transfer is solved and the concentration field is determined. Second, the normal derivative dC /dC) =0 on the surface is evaluated. Finally, one applies formulas (3.1.25)—... 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

A wide class of nonlinear problems of convective mass and heat exchange is considered taking into account the dependence of the transfer coefficients on concentration (temperature). The results are presented in the form of simple unified formulas for the Sherwood number. 

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... 

This parameter, termed the Sherwood number, is equal to the dimensionless mass fraction (i.e., concentration) gradient at the surface, and it provides a measure of the convection mass transfer occurring at the surface. Following the same argument as before (but now for Eq. 1.77), we have... 

Each of these is a ratio of a convective transfer rate to the corresponding diffusion rate of transfer. Dimensionless analysis indicates that, for fixed geometry and constant properties, the Nusselt number and the Sherwood number depend on the Reynolds number (forced convection), Rayleigh number (natural convection), flow characteristics, Prandtl number (heat transfer), and Schmidt number (mass transfer). 

Investigators have studied the mass transfer from single spheres and have correlated the Sherwood number by direct addition of terms representing transfer by purely molecular diffusion and transfer by forced convection in the form... 

As an incompressible fluid of infinite extent approaches and flows past either a spherical solid pellet or a gas bubble, a mobile component undergoes inteiphase mass transfer via convection and diffusion from the sphere to the fluid phase. The overall objective is to calculate the mass transfer coefficient and the Sherwood number at any point along the interface (i.e., the local transfer coefficients), as well as surface-averaged transfer coefficients. The results are applicable in the laminar flow regime (1) when the sphere is stationary and the fluid moves,... 

If one divides the average mass transfer coefficient A c. average by the simplest mass transfer coefficient in the absence of convective transport, then the resulting dimensionless ratio is identified as the average Sherwood number. Hence,... 

Now, it is necessary to discuss the mass transfer coefficient for component j in the boundary layer on the vapor side of the gas-liquid interface, fc ,gas, with units of mol/(area-time). The final expression for gas is based on results from the steady-state film theory of interphase mass transfer across a flat interface. The only mass transfer mechanism accounted for in this extremely simple derivation is one-dimensional diffusion perpendicular to the gas-liquid interface. There is essentially no chemical reaction in the gas-phase boundary layer, and convection normal to the interface is neglected. This problem corresponds to a Sherwood number (i.e., Sh) of 1 or 2, depending on characteristic length scale that is used to define Sh. Remember that the Sherwood number is a dimensionless mass transfer coefficient for interphase transport. In other words, Sh is a ratio of the actual mass transfer coefficient divided by the simplest mass transfer coefficient when the only important mass transfer mechanism is one-dimensional diffusion normal to the interface. For each component j in the gas mixture. 

In Equations 10.1 through 10.3, the coefficients of convective heat and mass transfer, h and ho, are evaluated in terms of the appropriate Nusselt and Sherwood numbers for evaporating droplet [22]. 

Abstract Evaporation of multi-component liquid droplets is reviewed, and modeling approaches of various degrees of sophistication are discussed. First, the evaporation of a single droplet is considered from a general point of view by means of the conservation equations for mass, species and energy of the liquid and gas phases. Subsequently, additional assumptions and simplifications are discussed which lead to simpler evaporation models suitable for use in CFD spray calculations. In particular, the heat and mass transfer for forced and non-forced convection is expressed in terms of the Nusselt and Sherwood numbers. Finally, an evaporation model for sprays that is widely used in today s CFD codes is presented. 

Sherwood number (Sh) A dimensionless measure of the ratio of convective mass transfer to molecular mass transfer. If the mass transfer coefficient k is defined in terms of the film theory, then Sh is a measure of the ratio of hydraulic diameter to the thickness of the boundary layer. See Section 6.5. 

Moreover, the mixing in the liquid-liquid system can be characterised by dimensionless numbers, such as, Sherwood number (Sh), which is the ratio of convective mass transfer to the molecular diffusion, and Schmidt niunber (Sc), which is the ratio of the viscous diffusion rate to the molecular diffusion. In addition to these, the Fourier number (Fo) can also give an idea about the dynamics of diffusive transport process. 

The Sherwood number is similar to the Nusselt number but for mass transfer. It represents the ratio of convective to diffusive mass transfer ... 

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. 

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