Mass transfer to a single particle

The Shenvoo In this section we consider two limiting cases of diffusion and reaction 

Schmidt numiters catalyst particle. In the first case the reaction is so rapid that the rate of are sed forced reactant to the surface limits the reaction rate. In the second 

Calculate the rriiass flux of reactant A to a single catalyst pellet 1 cm in diameter suspended in a large body of liquid. The reactant is present in dilute concentrations, and the reaction is considered to take place instantaneously at the external pellet surface (i.e., Cas = 0). The bulk concentration of the reactant is 1.0 M, and the free-system liquid velocity is 0.1 m/s. The kinematic viscosity is 0.5 centistoke (cS 1 centistoke = 10 mVs), and the liquid diffusivity of A is 10 ° m /s. 

Because reaction is assumed to occur instantaneously on the external surface of the pellet, Caj = 0. Also, Ca , is given as 1 mol/dm, The mass transfer coefficient for single spheres is calculated from the Frossling correlation  

Substituting M values above into Equation (11 0) gives us 

For mass transfer by molecular diffusion from a single sphere of diameter d to an infinite stationary medium, it can be shown that 

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The Mass Transfer Coefficient 771 Ma.ss Transfer Coefficient 773 Correlations for the Mass Traiurfer Coefficient Mass Transfer to a Single Particle 776 Ma.ss Transfer-Umited Reactions in Packed Beds 780 Robert the Worrier 783... 

Mass transfer to a single particle can be calculated from... 

To compare mass transfer in packed beds with transfer to a single particle, Sherwood numbers calculated from Eq. (21.62) are plotted in Fig. 21.5 along with the correlation for isolated spheres. The coefficients for packed beds are two to three times those for a single sphere at the same Reynolds number. Most of this difference is due to the higher actual mass velocity in the packed bed. The Reynolds number is based for convenience on the superficial velocity, but the average mass velocity is Gfe, and the local velocity at some points in the bed is even higher. Note that the dashed lines in Fig. 21.5 are not extended to low values of since it is unlikely that the coefficients for a packed bed would ever be lower than those for single particles. 

Because of the analogy between mass transfer by diffusion and heat transfer by conduction in a boundary layer, correlations for mass transfer and heat transfer to particles are similar. For mass transfer to a single isolated sphere,... 

Mass transfer between gas and particles affects gas-solids contact efficiencies in CFB risers. The mass transfer from a single particle to the suspension in CFB risers has been studied based on the sublimation of naphthalene spheres (Haider and Basu, 1988 Li et al., 1998), dehydration of 2-propanol (Masai et al., 1985), adsorption of CCI4, naphthalene, H2S, and NO (Kwauk et al., 1986 van der Ham et al., 1991, 1993 Vollert and Werther, 1994), and heat transfer between a heat pulse and suspension (Dry et al., 1987). For one-dimensional steady-state plug flow of the gas, a mass balance of the adsorbed species in a differential volume element of the reactor (Kwauk et al., 1986 Vollert and Werther, 1994) yields... 

A considerable amount of information has been reported regarding mass transfer between a single fluid phase and solid particles (such as those of spherical and cylindrical shape) forming a fixed bed. A recent review has been presented by Norman (N2). The applicability of such data to calculations regarding trickle-flow processes is, however, questionable, due to the fundamental difference between the liquid flow pattern of a fixed bed with trickle flow and that of a fixed bed in which the entire void volume is occupied by one fluid. 

In the limiting case of mass transfer from a single sphere resting in an infinite stagnant liquid, a simple film-theory analysis122 indicates that the liquid-solid mass-transfer coefficient R s is equal to 2D/JV, where D is the molecular diffusivity of the solute in the liquid phase and d is the particle diameter. In dimensionless form, the Sherwood number... 

In practice, the effects of heat transfer from the bulk of a fluid phase to the external surface of a solid catalyst particle are more important than the effects of mass transfer. An approach analogous to that of mass transfer, for a single reactant, leads to ... 

To represent these facts in an organized fashion, we start our deliberahons with a brief survey of the dimensionless groups pertinent to mass transfer operahons. One of these, the Biot number Bi, is singled out for a more detailed examination because it serves as an important criterion in mass transfer to and from particles. We next turn to transport coefficients that apply to systems in laminar flow and show how these coefficients are extracted from the solutions of the pertinent partial differenhal equahon (PDE) models. This is followed by an analysis of systems in turbulent flow where the approach of dimensional analysis is used. We describe the method and present the results obtained for some simple geometries, including flow in a pipe and aroimd spheres and cylinders. 

As shown in Example 22-3, for solid particles of the same size in BMF, the form of the reactor model resulting from equation 22.2-13 depends on the kinetics model used for a single particle. For the SCM, this, in turn, depends on particle shape and the relative magnitudes of gas-film mass transfer resistance, ash-layer diffusion resistance and surface reaction rate. In some cases, as illustrated for cylindrical particles in Example 22-3(a) and (b), the reactor model can be expressed in explicit analytical form additional results are given for spherical particles by Levenspiel(1972, pp. 384-5). In other f l cases, it is convenient or even necessary, as in Example 22-3(c), to use a numerical pro-... 

For heat and mass transfer through a stationary or streamline fluid to a single spherical particle, it has been shown in Volume 1, Chapter 9, that the heat and mass transfer coefficients reach limiting low values given by ... 

The double lines in Figure 3.44 represent the Sh number based on the mass transfer coefficient, in the case of a single-particle fall in water, for three different particle densities (Harriot, 1962). This value is considered to be the minimum mass-transfer coefficient in liquid-solid films in agitated vessels. Taking into account the fact that the actual Sh value in an agitated vessel is 1.5 -8 times its minimum value, it is apparent that the mass transfer coefficients are much higher in the case of agitated vessels. 

Results of these calculations for H mordenite are presented in Table IV. The macropore diffusion plays a role far from negligible even at high temperature and in some instances (e.g., low temperature and large particles) is the major contribution to the total mass-transfer resistance. No single step controls the overall mass-transfer process as no resistance has a relatively large enough contribution to dominate the process. In every... 

We first give a rather general mass-transfer model, which is useful for most processes of porous-solid extraction with dense gases. Two cases are possible [43] for a single particle loaded with solute. In (a), the solute is adsorbed over the internal surface of the particle, and is desorbed from the sites and diffuses out to the external surface, (b) The solute fills in the pore-cavities completely, and is dissolved from an inner core that moves progressively to the centre of the particle. 

The basic approach is to consider the problem in two parts. Firstly, the reaction of a single particle with a plentiful excess of the gaseous reactant is studied. A common technique is to suspend the particle from the arm of a thermobalance in a stream of gas at a carefully controlled temperature the course of the reaction is followed through the change in weight with time. From the results a suitable kinetic model may be developed for the progress of the reaction within a single particle. Included in this model will be a description of any mass transfer resistances associated with the reaction and of how the reaction is affected by concentration of the reactant present in the gas phase. 

Charge transfer occurs when particles collide with each other or with a solid wall. For monodispersed dilute suspensions of gas-solid flows, Cheng and Soo (1970) presented a simple model for the charge transfer in a single scattering collision between two elastic particles. They developed an electrostatic theory based on this mechanism, to illustrate the interrelationship between the charging current on a ball probe and the particle mass flux in a dilute gas-solid suspension. This electrostatic ball probe theory was modified to account for the multiple scattering effect in a dense particle suspension [Zhu and Soo, 1992]. 

This chapter describes the fundamental principles of heat and mass transfer in gas-solid flows. For most gas-solid flow situations, the temperature inside the solid particle can be approximated to be uniform. The theoretical basis and relevant restrictions of this approximation are briefly presented. The conductive heat transfer due to an elastic collision is introduced. A simple convective heat transfer model, based on the pseudocontinuum assumption for the gas-solid mixture, as well as the limitations of the model applications are discussed. The chapter also describes heat transfer due to radiation of the particulate phase. Specifically, thermal radiation from a single particle, radiation from a particle cloud with multiple scattering effects, and the basic governing equation for general multiparticle radiations are discussed. The discussion of gas phase radiation is, however, excluded because of its complexity, as it is affected by the type of gas components, concentrations, and gas temperatures. Interested readers may refer to Ozisik (1973) for the absorption (or emission) of radiation by gases. The last part of this chapter presents the fundamental principles of mass transfer in gas-solid flows. 

The governing heat transfer modes in gas-solid flow systems include gas-particle heat transfer, particle-particle heat transfer, and suspension-surface heat transfer by conduction, convection, and/or radiation. The basic heat and mass transfer modes of a single particle in a gas medium are introduced in Chapter 4. This chapter deals with the modeling approaches in describing the heat and mass transfer processes in gas-solid flows. In multiparticle systems, as in the fluidization systems with spherical or nearly spherical particles, the conductive heat transfer due to particle collisions is usually negligible. Hence, this chapter is mainly concerned with the heat and mass transfer from suspension to the wall, from suspension to an immersed surface, and from gas to solids for multiparticle systems. The heat and mass transfer mechanisms due to particle convection and gas convection are illustrated. In addition, heat transfer due to radiation is discussed. 

To study the chemistry of highly concentrated particles in bulk solution one must avoid mass transfer limitations and the effects of container surfaces. Both of these problems are eliminated by directly using aerosol particles. Two approaches have been used to study aerosol chemistry (1) aerosol reactors in which the evolution of a suspension of particles is followed, and (2) experiments in which the changes occurring in a single particle can be followed. 

We further mention that at low values of the Reynolds number (that is at very low fluid velocities or for very small particles) for flow through packed beds the Sherwood number for the mass transfer can become lower than Sh = 2, found for a single particle stagnant relative to the fluid [5]. We refer to the relevant papers. For the practice of catalytic reactors this is not of interest at too low velocities the danger of particle runaway (see Section 4.3) becomes too large and this should be avoided, for very small particles suspension or fluid bed reactors have to be applied instead of packed beds. For small particles in large packed beds the pressure drop become prohibitive. Only for fluid bed reactors, like in catalytic cracking, may Sh approach a value of 2. 

In order to assess transport mechanisms due to convection various correlation for heat and mass transfer coefficients in a packed bed have been derived. For the present application the transfer coefficient in the bed is related to the transfer coefficient of a single particle in a gas flow according to [15]. Due to the outflow of the gases during pyrolysis and char conversion the calculated transfer coefficient is decreased, thus Stefan correction is included to calculate the transfer coefficient at a finite flow over the boundary. 

Heat and mass transfer coefficients in a fluidised bed lie between the values for a packed bed and those for a single particle. The fundamental pattern of the Nusselt or Sherwood number as functions of the Reynolds number is illustrated in Fig. 3.40. In this the Nusselt number Nu = adP/A or the Sherwood number Sh = l3dP/rj and the Reynolds number Re = wmdP/v are all formed with the particle diameter, which for non-spherical particles is the same as the equivalent sphere diameter according to (3.273). In the Reynolds number wm is the mean velocity in the imaginary empty packing. 

The mass transfer coefficient to a single solid spherical particle immersed in a liquid flowing with velocity rsL past the particle can be calculated from ... 

Another interesting example is that of gas bubbles dispersed in a continuous liquid phase with which mass is exchanged. Also for this case the rate of change of the internal coordinates due to mass transfer is written starting from a simple mass balance for a single bubble. Following the standard notation for gas-liquid systems, the single-particle mass balance becomes... 

Given T, the expression for is closed, thereby fixing the mass-transfer rate. The discussion above is applicable to single-component droplets. In many applications, the liquid/gas phase will contain multiple chemical species, for which additional internal coordinates will be necessary in order to describe the physics of evaporation (Sazhin, 2006). In the context of a single-particle model for a multicomponent droplet, the simplest mesoscale model must include the particle mass Mp, the component mass fractions Yp and Yf, and the temperatures Tp and Tf. 

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