Nusselt number for mass transfer

Figure 7. Reduced Nusselt number for mass transfer to the substrate In a vertical reactor for varying Inlet flow rate and susceptor temperature.

The Sherwood number is also known as the Nusselt number for mass transfer. Notice that the diameter of the catalyst pellet is used in the Reynolds and Sherwood numbers as the characteristic length dimension of the system. For flow... 

Equal increments in i correspond to approximately equal decrements in In 4tAbuik-As illustrated in Table 23-8, ai, a2, and aj depend on the aspect ratio for rectangular channels. The asymptotic Nusselt number for mass transfer is given by cKi for constant transverse diffusional flux at the catalytic wall. The exponent m in equation (23-82) is either for plug flow or for viscous flow. 

Note that Nusselt number for mass transfer is usually known as Sherwood number. That is ... 

The Sherwood number is a nondimensional mass-transfer coefficient that is analogous to the Nusselt number for heat transfer. For the situation of A being dilute in B, the mass transfer at the stagnation surface is derived from the solution to the species equation by... 

The parameters (2 + 0.60 Re1/2 Sc1/3) and (2 -f 0.60 ReI/2Pr1/3) represent the Nusselt numbers for mass and heat transfer, respectively (87). Equation 3 derives from Froessling directly, while Spalding substituted thermal diffusivity for molecular diffusivity to establish the basis for Equation 4. 

Heat transfer data usually are correlated by use of dimensionless groups, such as the Nusselt number (analogous to the Sherwood number for mass transfer)... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

In this chapter, we return to forced convection heat and mass transfer problems when the Reynolds number is large enough that the velocity field takes the boundary-layer form. For this class of problems, we find that there must be a correlation between the dimensionless transport rate (i.e., the Nusselt number for heat transfer) and the independent dimensionless parameters, Reynolds number Re and either Prandtl number Pr or Schmidt number Sc of the form... 

Graetz number for mass transfer, mfDcLp Nusselt number, hD /k Peclet number, DpUo/D ... 

The appropriate dimensionless group characterizing film mass transfer is the Sherwood [number, defined by Sh = IR kjfD which is the analog of the Nusselt number for heat transfer. A simple analysis of heat conduction from... 

Furthermore, Figures 5.12 and 5.13 can also be used to show the dimensionless concentration as a function of dimensionless time and position for the case in which there is resistance to mass transfer at the interface between a solid and a fluid —Bab (9Ca/9z) = kc (Ca, — Caoo), where kc is the convective mass transfer coefficient (Section 4.4), Ca, is the concentration of species A at the interface in the fluid side, and Caoo is the concentration of species A in the fluid far away from the interface. Note that the constant concentration boundary condition referred to in the previous paragraph could be considered as a special case of the convective-type boundary condition for a Sherwood number, Sh (or Nusselt number for diffusion), equal to oo. Also, in Figures 5.12 and 5.13, the Biot (or Nusselt) number for heat transfer should be replaced by kcb/BAB) (l/ ) = (Sh/AT) = Sh, where K is the ratio of the equilibrium concentration in the solid to the... 

For mass transfer, which is considered in more detail in Chapter 10, an analogous relation (equation 10.233) applies, with the Sherwood number replacing the Nusselt number and the Schmidt number replacing the Prandtl number. 

Chakraborty S (2006) Analytical solutions of Nusselt number for thermally fully developed flow in microtubes under a combined action of electroosmotic forces and imposed gradients. Int J Heat Mass Transfer 49 810-813... 

In the previous chapters on temperature variations in reactors, we needed heat transfer coefficients h to calculate rates of heat transfer within the reactor. For a more detailed examination of temperature variations, we must use Nusselt number correlations similar to those developed here for mass transfer. 

Since the dimensionless equations and boundary conditions governing heat transfer and dilute-solution mass transfer are identical, the solutions to these equations in dimensionless form are also identical. Profiles of dimensionless concentration and temperature are therefore the same, while the dimensionless transfer rates, the Sherwood number (Sh = kL/ ) for mass transfer, and the Nusselt number (Nu = hL/K ) for heat transfer, are identical functions of Re, Sc or Pr, and dimensionless time. Most results in this book are given in terms of Sh and Sc the equivalent results for heat transfer may be found by simply replacing Sh by Nu and Sc by Pr. 

Thus, in Figure 17.36(c), if the Nusselt number is replaced by the Sherwood and the Prandtl by the Schmidt, the relation will be equally valid for mass transfer. 

This phenomenon is illustrated in Figure 11, which shows the average rate of mass transfer to the substrate in terms of the Nusselt number for various susceptor temperature and inlet gas flow rates. 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl, Sci, Eng, 17,336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Int, J, Heat Mass Transfer, 6, 483, 495 (1963)]. Lee [Int, J, Heat Mass Transfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. FuUy developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Npr < 30,10 < < 2 x 10, 1.01 <... 

The empirical correlations developed by Ranz and Marshall (23) for the Nusselt numbers for simultaneous heat and mass transfer during droplet vaporization are as follows ... 

The determination of heat transfer coefficients with the assistance of dimensionless numbers has already been explained in section 1.1.4. This method can also be used for mass transfer, and as an example we will take the mean Nusselt number Num = amL/ in forced flow, which can be represented by an expression of the form... 

Corresponding equations also hold for mass transfer. The Sherwood number Shx appears in place of the Nusselt number Nux, and in the same way the Prandtl number Pr number is replaced by the Schmidt number Sc. In the region 0.6 < Pr < 10 the agreement with the approximation equations (3.176) is excellent. 

As the previous illustrations showed, the heat and mass transfer coefficients for simple flows over a body, such as those over flat or slightly curved plates, can be calculated exactly using the boundary layer equations. In flows where detachment occurs, for example around cylinders, spheres or other bodies, the heat and mass transfer coefficients are very difficult if not impossible to calculate and so can only be determined by experiments. In terms of practical applications the calculated or measured results have been described by empirical correlations of the type Nu = f(Re,Pr), some of which have already been discussed. These are summarised in the following along with some of the more frequently used correlations. All the correlations are also valid for mass transfer. This merely requires the Nusselt to be replaced by the Sherwood number and the Prandtl by the Schmidt number. 

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