Mass transfer correlation, dimensionless

The polar velocity component is linearized within a thin mass transfer boundary layer on the liquid side of the solid-liquid interface to facilitate the development of dimensionless mass transfer correlations. 

Dimensionless Mass Transfer Correlation for Solid-Liquid Interfaces... 

Effect of Flow Regime on the Dimensionless Mass Transfer Correlation. For creeping flow of an incompressible Newtonian fluid around a stationary solid sphere, the tangential velocity gradient at the interface [i.e., g 9) = sin6>] is independent of (he Reynolds number. This is reasonable because contributions from accumulation and convective momentum transport on the left side of the equation of motion are neglected to obtain creeping flow solutions in the limit where Re 0. Under these conditions. 

The dimensionless mass transfer correlation, given by (12-1), reveals the complete dependence of the surface-averaged Sherwood number on the Reynolds and Schmidt nnmbers ... 

In this problem, we explore the dimensionless mass transfer correlation between the effectiveness factor and the intrapellet Damkohler number for one-dimensional diffusion and Langmuir-Hinshelwood surface-catalyzed chemical reactions within the internal pores of flat-slab catalysts under isothermal conditions. Perform simulations for vs. A which correspond to the following chemical reaction that occurs within the internal pores of catalysts that have rectangular symmetry. 

The mass transfer coefficient describes the effect of mass transfer resistance of the reactants flowing from the gas phase to the surface of the individual particles in the bed. The mass transfer coefficient can be obtained from a correlation for the Sherwood number (or dimensionless mass transfer coefficient) given by Eq. (7) ... 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

In order to characterize mass transfer in the boundary layers, it is necessary to determine the respective mass transfer coefficients. These coefficients depend on the properties of the solutions and on the hydrodynamic conditions of the system. Such coefficient can either be obtained by experiments or be estimated with the help of empirical correlations of dimensionless numbers. The majority of the correlations referred to in the literamre for various hydrodynamic conditions have the same general form. These include Sherwood number Sh), which contains the mass transfer coefficient, as a function of the Reynolds number Re) and Schmidt number (5c) [89-91]. General mass transfer correlation can be written as... 

Mass transfer correlations have been derived that are used for membranes of various configurations [16]. For flow through lubes, the following are used (see Appendices A and B for details on dimensionless numbers and mass transfer correlations, respectively). 

Gudmundsson [1981] has quoted a correlation for the dimensionless mass transfer coefficient K in terms of the particle Schmidt number for particles in the diffiision regime in turbulent pipe flow. 

The primary focus of this chapter is to analyze the dimensionless equation of motion in the laminar flow regime and predict the Reynolds number dependence of the tangential velocity gradient at a spherical fluid-solid interface. This information is required to obtain the complete dependence of the dimensionless mass transfer coefficient (i.e., Sherwood number) on the Reynolds and Schmidt numbers. For easy reference, the appropriate correlation for mass transfer around a solid sphere in the laminar flow regime, given by equation (11-120), is included here ... 

I-power dependence of the dimensionless mass transfer coefficient on Re reveals fbat the flow regime is laminar. Turbulent mass transfer across high-shear no-slip interfaces also scales as Shaverage Sc, but the exponent of Re in this correlation is somewhere between 0.8 and 1. AU of these dimensionless scaling laws for interphase mass transfer are summarized in Table 12-1 for solid-liquid and gas-Uquid interfaces. 

The scale-up problems arise from the fact that all STR gas-liquid mass transfer correlations are empirical. They are, for the most part, unable to account for hydrodynamic or liquid property changes with scale and time. Extensive attempts have been made in using nondimensional groups, especially toward solving gas-liquid processes involving non-Newtonian liquids. These correlations tend to be more complicated and require numerous static, but only few dynamic, inputs. One of the simplest correlations is presented by Ogut and Hatch (1988), which involves four dimensionless groups and requires six inputs. One of the more complicated forms. 

The use of dimensionless groups characterizes the second class of mass transfer correlations. For instance, Nakanoh and Yoshida [61] correlated their gas/liquid mass transfer results as ... 

Presenting the process model as a mass transfer correlation is also conunon. This requires an understanding of the process s physical properties, namely, the density and viscosity of the SC-CO2 and the mass diffusion of the solute in SC-CO2. Dimensionless numbers, namely, Reynolds (Re) (Equation 5.16), which is related to fluid flow Schmidt (Sc) (Equation 5.17), which is related to mass diffusivity Grashof (Gr) (Equation 5.18), which is related to mass transfer via buoyancy forces due to difference in density difference between saturated SC-CO2 with solute and pure SC-CO2 and Sherwood (Sh) (Equation 5.19), which is related to mass transfer, are important in these correlations. In supercritical extraction, natural convection is not significant (Shi et al., 2007) and in this case, Shp is related only to Re and Sc, as shown in Equation 5.19. 

Heat and mass transfer coefficients are usually reported as correlations in terms of dimensionless numbers. The exact definition of these dimensionless numbers implies a specific physical system. These numbers are expressed in terms of the characteristic scales. Correlations for mass transfer are conveniently divided into those for fluid-fluid interfaces and those for fluid-solid interfaces. Many of the correlations have the same general form. That is, the Sherwood or Stanton numbers containing the mass transfer coefficient are often expressed as a power function of the Schmidt number, the Reynolds number, and the Grashof number. The formulation of the correlations can be based on dimensional analysis and/or theoretical reasoning. In most cases, however, pure curve fitting of experimental data is used. The correlations are therefore usually problem dependent and can not be used for other systems than the one for which the curve fitting has been performed without validation. A large list of mass transfer correlations with references is presented by Perry [95]. 

The characteristics of the common dimensionless groups frequently used in mass transfer correlations are given in Table 8.3-1. Sherwood and Stanton numbers involve the mass transfer coefficient itself. The Schmidt, Lewis, and Prandtl numbers... 

Some common correlations of heat transfer coefficients are reported in Table 20.4-3. These all refer to heat transfer across a solid-fluid interface because other situations either are rare or are described in different terms. Like the mass transfer correlations in Section 8.3, these are best presented in terms of dimensionless groups. The two most... 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

From an analysis of the electrochemical mass-transfer process in well-supported solutions (N8a), it becomes evident that the use of the molecular diffusivity, for example, of CuS04, is not appropriate in investigations of mass transfer by the limiting-current method if use is made of the copper deposition reaction in acidified solution. To correlate the results in terms of the dimensionless numbers, Sc, Gr, and Sh, the diffusivity of the reacting ion must be used. 

Dimensional analysis of the variables characteristic of mass transfer under flow conditions suggests that the following dimensionless groups are appropriate for correlating mass transfer data. 

Many of the results and correlations in heat and mass transfer are expressed in terms of dimensionless groups such as the Nusselt, Reynolds and Prandtl numbers. The definitions of those dimensionless groups referred to in this chapter are given in Appendix 2. 

The second method uses dimensionless numbers to predict scale-up parameters. The use of dimensionless numbers simplifies design calculations by reducing the number of variables to consider. The dimensionless number approach has been used with good success in heat transfer calculations and to some extent in gas dispersion (mass transfer) for mixer scale-up. Usually, the primary independent variable in a dimensionless number correlation is Reynolds number ... 

The mass transfer between phases is, of course, the very basis for most of the diffusional operations of chemical engineering. A considerable amount of experimental and empirical work has been done in connection with interphase mass transfer because of its practical importance an excellent and complete survey of this subject may be found in the text book of Sherwood and Pigford (S9, Chap. Ill), where dimensionless correlations for mass transfer coefficients in systems of various shapes are assembled. 

As mentioned in Chapter 2, close analogies exist between the film coefficients of heat transfer and those of mass transfer. Indeed, the same type of dimensionless equations can often be used to correlate the film coefficients of heat and mass transfer. 

Liquid-phase mass transfer data [13-15] were correlated by the following dimensionless equation [13]. 

If there is no mass transfer resistance within the catalyst particle, then Ef is unity. However, it will then decrease from unity with increasing mass transfer resistance within the particles. The degree of decrease in f is correlated with a dimensionless parameter known as the Thiele modulus [2], which involves the relative magnitudes ofthe reaction rate and the molecular diffusion rate within catalyst particles. The Thiele moduli for several reaction mechanisms and shapes of catalyst particles have been derived theoretically. 

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