Reaction Sherwood number

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. 

Mass transfer in the continuous phase is less of a problem for liquid-liquid systems unless the drops are very small or the velocity difference between the phases is small. In gas-liquid systems, the resistance is always on the liquid side, unless the reaction is very fast and occurs at the interface. The Sherwood number for mass transfer in a system with dispersed bubbles tends to be almost constant and mass transfer is mainly a function of diffusivity, bubble size, and local gas holdup. 

Open-channel monoliths are better defined. The Sherwood (and Nusselt) number varies mainly in the axial direction due to the formation ofa hydrodynamic boundary layer and a concentration (temperature) boundary layer. Owing to the chemical reactions and heat formation on the surface, the local Sherwood (and Nusselt) numbers depend on the local reaction rate and the reaction rate upstream. A complicating factor is that the traditional Sherwood numbers are usually defined for constant concentration or constant flux on the surface, while, in reahty, the catalytic reaction on the surface exhibits different behavior. 

In Table 1.4, the characteristic time-scales for selected operations are listed. The rate constants for surface and volume reactions are denoted by and respectively. Furthermore, the Sherwood number Sh, a dimensionless mass-transfer coefficient and the analogue of the Nusselt number, appears in one of the expressions for the reaction time-scale. The last column highlights the dependence of z p on the channel diameter d. Apparently, the scale dependence of different operations varies from dy f to (d ). Owing to these different dependences, some op-... 

Note that, since k i depends on the tube diameter d, the dependence of the conversion on d is different for reaction- and mass-transfer-limited cases. If the flow is turbulent, the Sherwood number is given by the correlation = 0.023Rej Pr , which predicts a different dependence on the tube diameter. 

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. 

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). 

Equation 7.146 for the utilization factor corresponds to 7.107 for the case of heterogeneous catalysis with diffusional limitations. The analogy between 7.146 and 7.107 is complete when Shm = 1, i.e. when the reaction occurs simultaneously with diffusion throughout the complete liquid volume. The presence of a Sherwood number, besides the Hatta number, is needed to describe situations where a significant part of the reaction occurs in the bulk of the liquid, i.e. in series with the transport through the film. Such a situation is often encountered. Typical values for the Sherwood number are ... 

Fig. 7.15. Liquid utilization factor versus < l for an irreversible first-order reaction at different values of the modified Sherwood number.

If a transport parameter rc — CS/CL is defined, where Cs is the concentration of C at the catalyst surface, then Peterson134 showed that for gas-solid reactions t)c < rc, where c is the catalyst effectiveness factor for C. For three-phase slurry reactors, Reuther and Puri145 showed that rc could be less than t)C if the reaction order with respect to C is less than unity, the reaction occurs in the liquid phase, and the catalyst is finely divided. The effective diffusivity in the pores of the catalyst particle is considerably less if the pores are filled with liquid than if they are filled with gas. For finely divided catalyst, the Sherwood number for the liquid-solid mass-transfer coefficient based on catalyst particle diameter is two. 

The last three pi groups are well known in chemical engineering (113 is recognized as the Fourier reaction number (Fo ), 114 is the famous Biot diffusion number (Bi(j) and Hs is the Sherwood number (Sh)). 

Considerations along the above lines lead to analogous correlations for the Sherwood number for the description of mass transfer in a single channel. The application of the rather simple Nusselt and Sherwood number concept for monolith reactor modeling implies that the laminar flow through the channel can be approached as plug flow, but it is always limited to cases in which homogeneous gas-phase reactions are absent and catalytic reactions in the washcoat prevail. If not, a model description via distributed flow is necessary. 

Pg pressure inside the gas plug pi pressure in the liquid at the interface AP pressure drop q volumetric flow rate r radial coordinate reaction rate Tc radius at chaimel comer Ri principal radius of curvature Ri principal radius of curvature Re Reynolds number S selectivity Sc Schmidt number Sh Sherwood number SR slurry reactor STYv space lime yield t time... 

It was shown in [ 166,351 ] that Eq. (5.1.5) provides several valid initial terms of the asymptotic expansion of the Sherwood number as Pe —> 0 for any kinetics of the surface chemical reaction. (Specifically, one obtains three valid terms for the translational Stokes flow and four valid terms for an arbitrary shear flow.)... 

Figure 5.1. The Sherwood number against the rate constant of second-order surface chemical reaction 1, by formula (5.1.5) 2, for a solid sphere 3, for a circular cylinder and 4, for a spherical drop or bubble...

Formula (5.3.5) guarantees an exact asymptotic result in both limit cases fcv - 0 and kv - oo for any function /v(c). For a first-order volume reaction (/v = c), the approximate formula (5.3.5) is reduced to the exact result (5.3.4). The maximum error of formula (5.3.5) for a chemical volume reaction of the order n = 1 /2 (/v = fc) in the entire range of the dimensionless reaction rate constant fcv is 5% for a second-order volume reaction (/v = c2), the error of (5.3.5) is 7% [360], The mean Sherwood number decreases with the increase of the rate order n and increases with kv. 

Nonspherical particles. For nonspherical particles in a stagnant medium with the first-order volume chemical reaction taken into account, the mean Sherwood number can be calculated by using the approximate expression... 

Here Sho is the Sherwood number corresponding to mass transfer of a particle in a stagnant medium without the reaction. Each summand in (5.3.6) must be reduced to a dimensionless form on the basis of the same characteristic length. The value of Sho can be determined by the formula Sho = all/S , where a is the value chosen as the length scale and, is the surface area of the particle the shape factor II is shown in Table 4.2 for some nonspherical particles. 

Moderate Peclet Numbers. For spherical particles, drops, and bubbles (under limiting resistance of the continuous phase), in the case of a first-order volume reaction, the mean Sherwood number can be calculated [358] by the formula... 

Here Sho = Sho(Pe) is the Sherwood number in the absence of chemical reactions... 

For an arbitrary rate of volume chemical reaction, the mean Sherwood number at high Peclet numbers can be calculated according to the approximate formula... 

For an nth-order volume reaction, one can calculate the mean Sherwood number [360] by the approximate formula... 

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