Dimensionless number Sherwood

The dimensionless numbers in tlris equation are the Reynolds, Schmidt and the Sherwood number, A/ sh. which is defined by this equation. Dy/g is the diffusion coefficient of the metal-transporting vapour species in the flowing gas. The Reynolds and Schmidt numbers are defined by tire equations... 

Because D is independently determined, and p is obtainable from initial conditions and thermod5mamic equilibrium, the problem of determining the convective dissolution rate now becomes the problem of estimating the boundary layer thickness. In fluid dynamics, the boundary layer thickness appears in a dimensionless number, the Sherwood number Sh ... 

Both methods yield dimensionless groups, which correspond to dimensionless numbers (1), e.g.. Re, Reynolds number Fr, Froude number Nu, Nusselt number Sh, Sherwood number Sc, Schmidt number etc. (2). The classical principle of similarity can then be expressed by an equation of the form ... 

Dijfusional dimensionless numbers The Peclet, Prandtl, Schmidt, Sherwood, and Nusselt number are the most common ones. 

The correlations in Table II are most often written in dimensionless numbers. The mass transfer coefficient k, which most frequently has dimensions of velocity, is incorporated into a Sherwood number Sh... 

Nsc Schmidt number (= p/pD), dimensionless Nsh Sherwood number (= kEdp/D around bubbles) (= ksdp/D around particles), dimensionless Nwe Weber number, pe (o2dx/a, dimensionless p pressure in reactor, N/m2... 

Guo et al.79 proposed a model based on a description of the mass transport by so-called similitude numbers. Similitude numbers are dimensionless numbers determined by factors influencing mass transfer. A standard description of the Sherwood number for mass transfer of solid... 

Reynolds number (-) gas constant (J K 1 mol 1) radial distance (m) modified Sherwood number (-) dimensionless number for particle bath concentration (-)... 

Considering these Biot numbers, we can observe that they are similar to the Nusselt and Sherwood numbers. The only difference between these dimensionless numbers is the transfer coefficient property characterizing the Biot numbers transfer kinetics for the external phase (a x heat transfer coefficient for the external phase, k ex- mass transfer coefficient for the external phase). We can conclude that the Biot number is an index of the transfer resistances of the contacting phases. 

C What is the physical significance of the Sherwood number How is it defined To ivhal dimensionless number does it correspond in he-al transfer What does a Sherwood number of 1 indicate for a plain fluid layer ... 

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... 

Various correlations exist for mass and heat transfer coefficients in terms of dimensionless numbers. Table 8.7 surveys the most appropriate ones for laboratory fixed-bed reactors [8,17-19]. The Sherwood number, Sh, and the Nusselt number, Nu, express the ratio of total mass transfer to diffusive mass transfer, and the ratio of total heat transfer to conductive heat transfer, respectively. Values of kf and h for gases in laboratory systems range from 0.1 to 10 mf " s ... 

Both these geometric parameters altered diffusion data measured as Sherwood dimensionless number or as diffusion coefficients maxima and minima in these parameters mirrored nodes and antinodes from the ultrasound. This involved relative motions between the various components of several centimeters since the wavelength of sound at 20 kHz is of this order, depending on the medium. These workers were using electrochemistry as a probe to monitor ultrasonic power, and a fuller account of this work is given in another chapter of this volume, but the effects of geometry upon behavior of the electrochemical probe are noteworthy. 

Kataoka et al. [100] foxmd that, when only the data for Re < 100 are considered, most experimental data published earlier fit the following equation using the Sherwood, Reynolds, and Schmidt dimensionless numbers... 

The characteristic Nerast parameter 5, the thickness of the film around the ion exchange particle, may be converted to the mass transfer coefficient and dimensionless numbers (Reynolds, Schmidt and Sherwood) that engineers normally employ. 

This equation represents the usual functional relationship between the dimensionless groups (Sherwood, Reynolds, and Schmidt numbers) plus the assumption that the effective packing element... 

Grashof number D -iPAPZ ) dimensionless number of sparger orifices or sites power number P JpO jN ), dimensionless bubble Reynolds number D fjiVdimensionless impeller Reynolds number (Dj Npjp, dimensionless Schmidt number (Pc/Pc ab) dimensionless Sherwood number (kiDy /DAB), dimensionless total pressure, N/m ... 

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. 

Reynolds number, dimensionless number of imxing stages on tray number of moles in stillpot Schmidt number, dimmsionless Sherwood number dimensionless... 

Dimensionless numbers used to calculate the transfer coefficients a-l Nu = — K Nusset number (ratio of total heat transfer over heat conduction alone) P-l Sh = — D Sherwood number or second Nusset number (ratio of total mass transfer over molecular mass transfer)... 

In addition to the dimensionless numbers, there are well-known others, such as the Sherwood (Sh), Reynolds (Re), Schmidt (Sc), Froude (Fr), Bodenstein (Bo), and Weber (We) numbers. On the basis of these types of dimensionless numbers, empirical correlations for a large number of bioreactors have been made (for example, Blanch, 1979 Schiigerl, 1980 Zlokarnik, 1979). The results of the experimental measurements of process engineering data are often presented in the form of a graph they have the form of the relationships given in Equs. 3.77a and 3.77b. For the volumetric mass transport coefficient (Ryu and Humphrey, 1972) (see Fig. 3.21)... 

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) ... 

Schmidt number Sherwood number dimensionless time dimensionless fluid temperature initial dimensionless temperature distribution dimensionless soUd phase temperature dimensionless volume-averaged solid temperature scale for temperature change, K dimensionless external temperature rise dimensionless inlet temperature maximum dimensionless temperature in the solid maximum dimensionless internal temperature rise... 

The interphase coefficients for mass (A ) and heat h) transfer are made dimensionless into Sherwood and Nusselt numbers, both defined as... 

The dimensionless number, sometimes called the Sherwood number, is taken at R = 100, by considering two values of the components of R, e.g., the thickness of the linen and the coefficient of convection h. With R = 100, the process of release is controlled by diffusion through the thickness of the linen, as the coefficient of convection is rather high. 

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. 

Like the convection heat transfer coefficient expressed as the Nusselt nmn-ber, the convection mass transfer coefficient is also expressed in the form a dimensionless number known as the Sherwood number, which is defined as... 

The design of heat and mass transfer operations in chemical engineering is based on the well-known correlations that use the dimensionless numbers Nu (Nusselt) for heat transfer and Sh (Sherwood) for mass transfer By balancing the acting forces, energies, and mass flows within the boundary layers of velocity, temperature, and concentration, the theoretical derivation of general relations for Nu and Sh is given in fundamental work [35]. 

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]. 

Sherwood number A dimensionless number, Sh, that represents the relationship between mass diffusivity and molecular diffusivity ... 

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