Sherwood number particle

Bo = - k- ax Dal = x/tT Dali = tm/tT pP — ud 1 e x Re = c AtJEj ° RT% Qr — -L. J Dm Sh = krt CX dVk<7 DnV T Dm Bodenstein number Capillary number first Damkohler number second Damkohler number axial Peclet number Reynolds number (channel) particle Reynolds number heat production potential Schmidt number Sherwood number (channel) Sherwood number (particle)... 

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

For a free-falling spherical particle of radius R moving with velocity u relative to a fluid of density p and viscosity p, and in which the molecular diffusion coefficient (for species A) is DA, the Ranz-Marshall correlation relates the Sherwood number (Sh), which incorporates kAg, to the Schmidt number (Sc) and the Reynolds number (Re) ... 

Sh particle Sherwood number, kigdpIDm (D, is difliisivity of species i)... 

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) ... 

Since the absolute thickness of the effective hydrodynamic boundary layer is very small, below a particular size range minimum, no hydrodynamic effects are perceived experimentally with varying agitation. This, however, does not mean, that there are no such influences Further, the mechanisms of mass transfer and dissolution may change for very small particles depending on a number of factors, such as the fluid viscosity, the Sherwood number (the ratio of mass diffusivity to molecular diffusivity), and the power input per unit mass of fluid. 

The values of Sherwood number fall below the theoretical minimum value of 2 for mass transfer to a spherical particle and this indicates that the assumption of piston flow of gases is not valid at low values of the Reynolds number. In order to obtain realistic values in this region, information on the axial dispersion coefficient is required. 

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 ... 

Combination of Eqs. (1-51) to (1-55) with either Eq. (1-56) or (1-58) yields an equation which may be solved to give concentration profiles from which mass transfer rates may be found. For a solid particle the average Sherwood number is... 

These equations have been solved for rigid (Nl) and circulating spheres (Jl, K6, W3, W4) in creeping flow. Since the dimensionless velocities within the particle are proportional to (1 + k) (see Eq. (3-8)), F is a function only of Tp and PCp/(l + k). In presenting the results, it is instructive to consider the instantaneous overall Sherwood number, Shp, as well as F. The driving force is taken as the difference between the concentration inside the interface, and the mixed mean particle concentration, Cp, giving... 

In some studies the surface area of the particle is measured and area-free Sherwood numbers are reported... 

For a rigid sphere k = oo) on the axis of a cylindrical tube, the Sherwood number is larger than in an unbounded fluid with the same particle/fluid velocity. The ratio of Sherwood numbers is approximated within 3% for 2 < 0.6 by... 

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. 

Rev = the particle Reynolds number Sh = the Sherwood number Sc = the Schmidt number us = the superficial velocity... 

Since the liquid is saturated with hydrogen, only the liquid-to-particle mass transfer coefficient and the intrinsic rate constant will be significant. In the case that the reaction is fast, the reaction rate will depend only on the liquid-sold mass transfer resistance. Since the particle are very small (10 micrometers), and the loading is moderate (0.8% mass), the Sherwood number will be that of lonely spheres, so Sh = 2. For this case we can take Sh = =4 [50], rather safely. 

It is obvious that the mass-transfer coefficient, ks, can be influenced by the diffusion coefficient, Di2, which is also included in the Sherwood number, Sh, as in the Schmidt number, Sc. Diffusion can be increased by shortening the diffusion length. For solid materials this is achieved by smaller particle sizes, which further leads to a higher specific interfacial area, as. However, there is a limit for reducing the particle size because if the particles are too fine, the problem of channelling arises, so an optimum has to be found. 

The transfer coefficient can be correlated in the form of a dimensionless Sherwood number Sh(= h0dp/D). The particle diameter dp is often taken to be the diameter of the sphere having the same area as the (irregular shaped) pellet. Thaller and Thodos(38> correlated the mass transfer coefficient in terms of the gas velocity u and the Schmidt number Sc(= p/pD) ... 

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). 

Although some controversy exists on the correlation of Sherwood Number applicable to fluidized beds, well-defined combustion experiments support the use of the Ranz and Marshall (35) or Frossling (36) correlation with an approximate correction of mf to allow for the obstruction to diffusion by the inert particles surrounding the burning char particles (37). Thus... 

Sherwood number residence time, s particle burnout time, s... 

Liquid-solid mass transfer is typically not limiting due to the small particle size resulting in large particle surface area/volume of reactor, unless the concentration of the particles is very low, and or larger particles are used. In the latter case, intraparticle mass-transfer limitations would also occur. Ramachandran and Chaudhari (Three-Phase Catalytic Reactors, Gordon and Breach, 1983) present several correlations for liquid-solid mass transfer, typically as a Sherwood number versus particle Reynolds and Schmidt numbers, e.g., the correlation of Levins and Glastonbury [Trans. Inst. Chem. Engrs. 50 132 (1972)] ... 

The capture efficiency or Sherwood number was shown to be a function of three dimensionless groups—the Peclet number, the aspect ratio (collector radius divided by par-dele radius), and the ratio of Hamaker s constant (indicating the intensity of London forces) to the thermal energy of the particles. Calculated values for the rate of deposition, expressed as Ihe Sherwood number, are plotted in Figure 6 as a function of the three dimensionless groups. 

For those cases in which gravitational forces are also included, values of the Sherwood number are summarized in Figure 8. The ratio G of the gravitational potential of a particle, located one particle radius above the collector, to the thermal energy was used to characterize the effect of gravity. For an aspect ratio of 100, gravitational forces were found to be negligible when G < I0-2. 

In this case Ijondon forces cause, adhesion of the particles coming in contact with tlic collector but do not aid in the transport hence A/kT is taken as zero. Because one of the subcases (Case lc) is pure diffusion, it seems mure reasonable to express the rate of deposition as the Sherwood number. 

Fig. 3- Cose 1. Sherwood numbers computed for the convective-diffusion of particles of finite sine to the surface of a spherical collector by neglecting interaction forces. The dashed line is the Levich-LighthilJ equation (19) which is valid when a diffusion boundary-layer exists and the particles are infinitesimal.

Fig. 5. Case 3. Sherwood numbers far the transpart of finite size particles through a stagnant fluid to n spherical cellectnr under the action of diffusion and London forces...

Reported values of A/kT nearly always are in the range 10 4 < A/kT < 10s. Thus Figure 5 snows that, even in the best of circumstances (R = 1), this variation in Hamaker s constant by a factor of 104, results in a change in the Sherwood number by only a factor of 7. Particle transport rates in stagnant fluids are not highly sensitive to small changes in Hamaker s constant. 

Fig. 6. Sherwood numbers computed for the transport of finite particles to a spherical collector under the combined action of convective-diffusion and London forces. Values of the aspect ratio are (a) ft - 104, (b) ft — 105, (c) ft =- 10s, and (d) ft = 10. For each aspect ratio, the value of A/kT was taken (upper curves to lower curves) as ID2, 1, 10 2 and 10 4. Dashed lines represent the Levich-Lighthill equation (19), while the dotted curves represent Sherwood numbers deduced from Figure 4 which ignores the transport from diffusion.

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... 

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

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). 

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