Heat transfer Sherwood-numbers

The mass transfer coefficient is normally reported as a Sherwood number (Sh). Sherwood number is the mass transfer analog to the Nusselt number of heat transfer. Sherwood number is defined as follows ... 

The dimensionless quantity Sh is called the Sherwood number. The heat transfer factor a is defined bv... 

The correlation studies of heat and mass transfer in pellet beds have been investigated by many, usually in terms of the. /-factors (113-115). According to Chilton and Colburn the two. /-factors are equal in value to one half of the Fannings friction factor / used in the calculation of pressure drop. The. /-factors depend on the Reynolds number raised to a factor varying from —0.36 to —0.68, so that the Nusselt number depends on the Reynolds number raised to a factor varying from 0.64 to 0.32. In the range of the Reynolds number from 10 to 170 in the pellet bed, jd should vary from 0.5 to 0.1, which yields a Nusselt number from 4.4 to 16.1. The heat and mass transfer to wire meshes has received much less attention (110,116). The correlation available shows that the /-factor varies as (Re)-0-41, so that the Nusselt number varies as (Re)0-69. In the range of the Reynolds number from 20 to 420, the j-factor varies from 0.2 to 0.05, so that the Nusselt number varies from 3.6 to 18.6. The Sherwood number for CO is equal to 1.05 Nu, but the Sherwood number for benzene is 1.31 Nu. 

Experimental results for fixed packed beds are very sensitive to the structure of the bed which may be strongly influenced by its method of formation. GUPTA and Thodos157 have studied both heat transfer and mass transfer in fixed beds and have shown that the results for both processes may be correlated by similar equations based on. / -factors (see Section 10.8.1). Re-arrangement of the terms in the mass transfer equation, permits the results for the Sherwood number (Sh1) to be expressed as a function of the Reynolds (Re,) and Schmidt numbers (Sc) ... 

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. 

Correlations for heat transfer coefficient between a single sphere and surrounding gas have been proposed by many researchers (Table 5.2), for example, Whitaker,1584 and Ranz and Marshall,15051 among others. The correlation recommended by Whitaker is accurate to within 30% for the range of parameter values listed. All properties except jus should be evaluated at Tm. For freely falling liquid droplets, the Ranz-Marshall correlation 505 is often used. The correlations may be applied to mass transfer processes simply by replacing Nu and Pr with Sh and Sc, respectively, where Sh and Sc are the Sherwood number and Schmidt number, respectively. Modifications to the Ranz-Marshall correlation have been made by researchers to account... 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

The designer now needs to make some estimates of mass transfer. These properties are generally well known for commercially available adsorbents, so the job is not difficult. We need to re-introduce the adsorber cross-section area and the gas velocity in order to make the required estimates of the external film contribution to the overall mass transfer. For spherical beads or pellets we can generally employ Eq. (7.12) or (7.15) of Ruthven s text to obtain the Sherwood number. That correlation is the mass transfer analog to the Nusselt number formulation in heat transfer ... 

Now we need and h, the mass and heat transfer coefficients around a sphere. These come from Sherwood and Nusselt numbers, respectively, for flow around a sphere,... 

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. 

For flow parallel to a cylinder the rate of mass or heat transfer decreases with axial distance. Far from the leading end, the transfer at low Pe may be considered as transfer from a line source into a uniform stream and the local Sherwood number becomes... 

For Re > 10 there are a number of studies of the effect of walls on heat and mass transfer from solid particles in wind and water tunnels. In these studies it was customary to define a velocity ratio K based on the same Sherwood number in bounded and infinite fluids ... 

Here Kjj is obtained from Fig. 9.5. Equation (9-27) and the equations of Chapter 5 can be used to determine the decrease in Sh for a rigid sphere with fixed settling on the axis of a cylindrical tube. For example, for a settling sphere with 2 = 0.4 and = 200, Uj/Uj = 0.76 and UJUj = 0.85. Since the Sherwood number is roughly proportional to the square root of Re, the Sherwood number for the settling particle is reduced only 8%, while its terminal velocity is reduced 24%. As in creeping flow, the effect of container walls on mass and heat transfer is much smaller than on terminal velocity. 

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... 

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). 

Here Sh denotes the dimensionless Sherwood number for mass transfer and Nu is the dimensionless Nusselt8 number for heat transfer. 

For gas-solid fluidized beds, Wen and Fane (1982) suggested that the determination of the bed-to-surface mass transfer coefficient can be conducted by using the corresponding heat transfer correlations, replacing the Nusselt number with the Sherwood number, and replacing the Prandtl number by Sc(cpp)/(cpp)/(l — a). Few experimental results on bed-to-surface mass transfer are available, especially for gas-solid fluidized beds operated at relatively high gas velocities. 

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. 

There are many correlations available for heat and mass transfer to particles. These are all have the Nusselt number Nu = adjk (or the Sherwood number Sh = kgd/D) as a function of the Reynolds number Re = udjv and the Prandtl number Pr = vl (Al pcp) (or the Schmidt number Sc = v/D). 

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. 

All the statements given before for the Nusselt number have the same significance for the Sherwood number if we change the words couple heat transfer to couple mass transfer . We have to specify that, as far as the Sherwood number is... 

To this end, experimental heat and mass transfer coefficients were determined in a fluidized bed. Nusselt and Sherwood numbers were obtained in terms of Reynolds number and aspect ratios dp/L and dp/D. The results are also analyzed in terms of the Kato and Wen(5) and Nelson and Galloway(6) models. 

Which is less than 2300 and thus the flow is laminar. Therefore, based on the analogy between heat and mass transfer, the Nusselt and the Sherv/ood numbers in this case are Nu = Sh = 3.66, Using the definition of Sherwood number, itie mass transfer coefficient is determined to be... 

Unfortunately, no empirical mass transfer or heat transfer relations were found that consider the type of artificial surface roughness that is encountered in a BSR. The relations that are discussed in the first part of this section pertain to hydraulically smooth surfaces and are therefore expected to predict the lower limit of Sherwood numbers in a BSR. 

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