Heat transfer inside solid particles

This method was applied to produce carbides of titanium, zirconium, lead, and bismuth (Barcicki Myrdzik, 1974). Applications ofthe method are limited becanse of insufficient contact between solid particles in the presence of plasma flow if the solid mixtnre is not bricked or sintered. The efficiency of the process arranged in bricks or sintered solid mixture is limited by radiation heat losses and insufficient heat transfer inside of the brick, especially taking into account the decrease of density dne to CO formation (7-92). The most effective condensed-phase synthesis of carbides (7-92) is that from melt containing carbon compounds (Tumanov, 1981). A relevant example is the synthesis of carbides of uraninm and plutonium from a melt containing their nitrites and carbon compoimds (Coppinger Johnson, 1969). 

However, usually catalysts are porous materials to strongly increase the active reaction area per unit of reactor volume. Therefore, the intra-particle gradients, i.e., the mass and heat transfer inside the catalyst particle, have to be considered as well. Fluid phase equations are the same as Eqs. 4.10, 4.11, while solid phase equations have to be modified as follows ... 

The transfer of heat from one molecule to an adjaeent molecule while the particles remain in fixed positions relative to each other is conduction. For example, if a piece of pipe has a hot fluid on the inside and a eold fluid on the outside, heat is transferred through the wall of the pipe by conduc tion. This is illustrated in Figure 2-1. The molecules stay intact, relative to each other, but the heat is transferred from molecule to molecule by the process of conduetion. This type of heat transfer occurs in solids or, to a much lesser extent, within fluids that are relatively stagnant. 

Most of the actual reactions involve a three-phase process gas, liquid, and solid catalysts are present. Internal and external mass transfer limitations in porous catalyst layers play a central role in three-phase processes. The governing phenomena are well known since the days of Thiele [43] and Frank-Kamenetskii [44], but transport phenomena coupled to chemical reactions are not frequently used for complex organic systems, but simple - often too simple - tests based on the use of first-order Thiele modulus and Biot number are used. Instead, complete numerical simulations are preferable to reveal the role of mass and heat transfer at the phase boundaries and inside the porous catalyst particles. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

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. 

Equation (4.8) indicates that the one-dimensional transient temperature distribution inside a solid sphere without internal heat generation varies with Fo and Bi. On the basis of the equation, the temperature distribution in the solid can be considered uniform with an error of less than 5 percent when Bi < 0.1, which is the condition for most gas-solid flow systems. In transient heat transfer processes where the gas-solid contact time is very short, it also requires Fo > 0.1 [Gel Perin and Einstein, 1971] for the internal thermal resistance within the particles to be neglected. In the following, unless otherwise noted, it is assumed that the temperature inside a solid particle is uniform. 

The conventional macroscopic Fourier conduction model violates this non-local feature of microscale heat transfer, and alternative approaches are necessary for analysis. The most suitable model to date is the concept of phonon. The thermal energy in a uniform solid material can be jntetpreied as the vibrations of a regular lattice of closely bound atoms inside. These atoms exhibit collective modes of sound waves (phonons) wliich transports energy at tlie speed of sound in a material. Following quantum mechanical principles, phonons exhibit paiticle-like properties of bosons with zero spin (wave-particle duality). Phonons play an important role in many of the physical properties of solids, such as the thermal and the electrical conductivities. In insulating solids, phonons are also (he primary mechanism by which heal conduction takes place. 

There are three means of heat transfer that apply to drying processes. These are conduction, convection, and radiation. Conduction is the transfer of heat from one body to another part of the same body, or from one body to another body in direct physical contact with it. This transfer of heat must occur without significant displacement of particles of the body other than atomic or molecular vibrations. Conductive heat transfer is analogous to electrical flow and can be described by similar terms such as potential and resistance. Some examples of conduction would include heating of metal pipes by a hot liquid inside of them, or heat supplied to a solids bed via a metal shelf. 

Advantages of three-phase fluidized beds over trickle beds and other fixed bed systems are temperature uniformity, high heat transfer, ability to add and remove catalyst particles continuously, and limited mass transfer resistances (both external to the particles and bubbles, because of turbulence and limited bubble size, and inside the particles owing to relatively small particle diameters). Disadvantages include substantial axial dispersion (of gas, liquid, and particles), causing substantial deviations from plug flow, and lack of predictability because of the complex hydrodynamics. There are two major applications of gas-liquid-solid-fluidized beds biochemical processes and hydrocarbon processing. 

For liquid-phase catalytic or enzymatic reactions, catalysts or enzymes are used as homogeneous solutes in the liquid, or as solids particles suspended in the liquid phase. In the latter case (i) the particles per se may be catalysts (ii) the catalysts or enzymes are uniformly distributed within inert particles or (in) the catalysts or enzymes exist at the surface of pores, inside the particles. In such heterogeneous catalytic or enzymatic systems, a variety of factors which include the mass transfer of reactants and products, heat effects accompanying the reactions, and/or some surface phenomena, may affect the apparent reaction rates. For example, in situation (iii) above, the reactants must move to the catalytic reaction sites within catalyst particles by various mechanisms of diffusion through the pores. In general, the apparent rates of reactions with catalyst or enzymatic particles are lower than the intrinsic reaction rates this is due to the various mass transfer resistances, as will be discussed below. 

As discussed in chapter 5, diffusion through catalyst pores represents a resistance to mass and heat transfer, which gives rise to concentration and temperature gradients within the catalyst pellet. This causes the rate of reaction in the solid phase to be different from that if the bulk phase conditions prevail inside the particle, and the rate of reaction should be integrated along the radius of the pellet to get the actual rate of reaction. 

Very often the rates of chemical transformations are affected by the rates of other processes, such as heat and mass transfer. The process should be treated as a part of kinetics. The gas/liquid mass transfer in multiphase heterogeneous and homogeneous catalytic reactions could be treated in a similar way. The mathematical framework for modelling diffusion inside solid catalyst particles of supported metal catalysts or immolisided enzymes does not differ that much, but proper care should be taken of the reaction kinetics. 

The thermobalance is an apparatus capable of continuously weighing a coal sample which is undergoing reaction in a gaseous environment of desired composition at a constant pressure. The temperature can be kept constant or varied (10°F/min is the maximum rate for the apparatus used at IGT). The nature of gas-solid contact with the apparatus used in this study is shown in Figure 1. The coal sample is contained in the annular space of a wire mesh basket bounded on the inside by a hollow, stainless steel tube and on the outside by a wire mesh screen. To facilitate mass and heat transfer between the bed and its environment, the thickness of the bed is only 2-3 particle diameters when using —20+40 US sieve-size particles. Gas flow rates used with this system are sufficiently... 

This polymerization process can be separated into three different levels as proposed by Ray [22]. First this is the microscale level, modeling all processes at the surface and inside the growing polymer particle. The next level is the mesoscale level, describing all mass and heat transfer processes inside the three-phase slurry containing gas bubbles, hydrocarbon diluent with the dissolved aluminumalkyl compound, and the solid growing polymer particles loaded with the active sites. Finally, there is the macroscale level comprising the polymerization vessel as a whole, with sensors to control this slurry polymerization process. These three levels are shown in Fig. 4. 

These equations have to be solved simultaneously with the Reynolds averaged Navier-Stokes equations and the Ergun-equation given in Section 11.7.5 to account for the radial void profile and generate both axial and radial flow velocity components, Ur and Uz. In these equations it is assumed that the heat transfer through the fluid and solid phase occur in parallel [de Wasch and Froment, 1971 Dixon and Cresswell, 1979 Dixon, 1985]. and AV can be calculated from the correlations of de Wasch and Froment [1972] and Zehner and Schliinder [1972], The internal diffusion limitations appear in those equations by means of the effectiveness factor r , which is obtained by numerical integration of the diffusion and reaction equations inside the particles, as discussed in Chapter 3. 

Of course the spectrum of quantities, which need to be measured in a fluidized bed, is much wider. These include, for example, local solids volume concentrations, solids velocities and solids mass flows, the vertical and the horizontal distribution of solids inside the system or the lateral distribution of the fluidizing gas. In response to these needs a number of more sophisticated measurement techniques were proposed. For example, suction probes were developed to measure local solids and mass flow, heat transfer probes were proposed for detection of de-fluidized zones and solids flow inside fluidized-bed reactors. Other techniques include capacitance probes, optical probes, or y-ray densitometry - a detailed review was given recently by Werther [1]. Cody et al. 2 reported the use of an acoustic probe to measure particle velocity at the wall of fluidized beds. 

The principles of homogeneous reaction kinetics and the equations derived there remain valid for the kinetics of heterogeneous catalytic reactions, provided that the concentrations and temperatures substituted in the equations are really those prevailing at the point of reaction. The formation of a surface complex is an essential feature of reactions catalyzed by solids and the kinetic equation must account for this. In addition, transport processes may influence the overall rate heat and mass transfer between the fluid and the solid or inside the porous solid, > that the conditions over the local reation site do not correspond to those in the bulk fluid around the catalyst particle. Figure 2.1-1 shows the seven steps involved when a molecule moves into the catalyst, reacts, and the product moves back to the bulk fluid stream. To simplify the notation the index s, referring to concentrations inside the solid, will be dropped in this chapter. 

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