Temperature and concentration gradient

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

As mentioned above batch crystallizers are usually simple vessels provided with some means of mechanical agitation or particulate fluidization. These have the effect of reducing temperature and concentration gradients, and maintain crystals in suspension. Baffles may be added to improve mixing and heat exchange or vacuum systems may be added, as appropriate. Various design combinations are available and some are illustrated in Figure 7.1. 

The stirred batch reactors are easy to operate and their configurations avoid temperature and concentration gradient (Table 5). These reactors are useful for hydrolysis reactions proceeding very slowly. After the end of the batch reaction, separation of the powdered enzyme support and the product from the reaction mixture can be accomplished by a simple centrifugation and/or filtration. Roffler et al. [114] investigated two-phase biocatalysis and described stirred-tank reactor coupled to a settler for extraction of product with direct solvent addition. This basic experimental setup can lead to a rather stable emulsion that needs a long settling time. 

Now let us consider the possibility that there will be a significant temperature difference between the bulk fluid and the external surface of the catalyst pellet. Equation 12.5.6 indicates that the temperature and concentration gradients external to the particle are related as follows ... 

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

The existence of radial temperature and concentration gradients means that, in principle, one should include terms to account for these effects in the mass and heat conservation equations describing the reactor. Furthermore, there should be a distinction between the porous catalyst particles (within which reaction occurs) and the bulk gas phase. Thus, conservation equations should be written for the catalyst particles as well as for the bulk gas phase and coupled by boundary condition statements to the effect that the mass and heat fluxes at the periphery of particles are balanced by mass and heat transfer between catalyst particles and bulk gas phase. A full account of these principles may be found in a number of texts [23, 35, 36]. 

A significant step in the numerical solution of packed bed reactor models was taken with the introduction of the method of orthogonal collocation to this class of problems (Finlayson, 1971). Although Finlayson showed the method to be much faster and more accurate than that based on finite differences and to be easily applicable to two-dimensional models with both radial temperature and concentration gradients, the finite difference technique remained the generally accepted procedure for packed bed reactor model solution until about 1977, when the analysis by Jutan et al. (1977) of a complex butane hydrogenolysis reactor demonstrated the real potential of the collocation procedure. 

Clearly, in the absence of a radial temperature or velocity gradient, no radial mass transfer can exist unless, of course, a reaction occurs at the bed wall. When a system is adiabatic, a radial temperature and concentration gradient cannot exist unless a severe radial velocity variation is encountered (Carberry, 1976). Radial variations in fluid velocity can be due to the nature of flow, e.g. in laminar flow, and in the case of radial variations in void fraction. In general, an average radial velocity independent of radial position can be assumed, except from pathological cases such as in very low Reynolds numbers (laminar flow), where a parabolic profile might be anticipated. 

It has already been mentioned that the experiments of this sort realize a practically nongradient course of reaction. Specifically, large-scale nongradient conditions exist in the reactor, viz, the composition of the gas mixture and its temperature in the space between catalyst grains are virtually identical in the whole volume of the reactor. But this does not preclude considerable temperature and concentration gradients directed from the surface to the center of the catalyst grain. These gradients can be eliminated,... 

Analysis of flows in which more than one driving force exists has been limited to several idealized cases of thermosolutal convection driven by vertical and constant temperature and concentration gradients. A discussion of thermosolutal convection is presented by Brown (5). 

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). 

Temperature and concentration gradients are absent inside an ideal stirred reactor. Mixing can be achieved by forced or free-convective movement of a material. For such a reactor the energy balance equation can be written as ... 

The driving forces of heat and mass transfer are temperature and concentration gradients, respectively. To a considerable extent, they are limited by the characteristics of the specific processes involved, such as stocks, heat sources, and equipment materials, etc. In most cases only a limited increasing magnitude is permitted. 

Table 2 lists most of the available experimental criteria for intraparticle heat and mass transfer. These criteria apply to single reactions only, where it is additionally supposed that the kinetics may be described by a simple nth order power rate law. The most general of the criteria, 5 and 8 in Table 2, ensure the absence of any net effects (combined) of intraparticle temperature and concentration gradients on the observable reaction rate. However, these criteria do not guarantee that this may not be due to a compensation of heat and mass transfer effects (this point has been discussed in the previous section). In fact, this happens when y/J n [12]. 

Figure 11. Temperature and concentration gradients in and around a catalyst particle for exothermal and endothermal reactions.

In this equation, the concentration C of reactive species is expressed as a mass fraction, u denotes the local velocity of the fluid and results both from the movement of the triple line and from convection caused by temperature and concentration gradients in the liquid. 

When we can control the temperature and concentration gradients, the coupling coefficients between the chemical reaction and the flows of mass and heat may be determined experimentally by using Eqs. (9.116)—(9.118)... 

Numerous studies on the kinetics and mechanisms of CVD reactions have been made. These studies provide useful information such as activation energy and limiting steps of deposition reactions which are important for the understanding of deposition processes. The main problem in the CVD kinetics studies is the complexity of the deposition process. The difficulty arises not only from the various steps of the CVD process but also from the temperature and concentration gradient, geometric effects, and gas flow patterns in the reaction zones. Exact kinetic analysis is therefore usually not possible as the kinetic data are reactor dependent. There are several possible rate-limiting factors but mass transport and surface kinetics control are the most... 

Because pressure drop measurements are much faster and cheaper than mass transfer or heat transfer measurements, it is tempting to try to relate the Sherwood and Nusselt numbers to the friction factor. A relation that has proved successful for smooth circular tubes is obtained from a plausible assumption that is known as the film layer model. The assumption is that for turbulent flow the lateral velocity, temperature, and concentration gradients are located in thin films at the wall of the channel the thickness of the films is indicated with 8/, 87, and 8., respectively. According to the film model, the lateral velocity gradient at the channel surface equals (m)/8/, the lateral temperature gradient equals (T/, - rj/87 and the lateral concentration gradient equals (c. /, - C , )/8,.. From these assumptions, and the theoretical knowledge that 8//8r Pr and 8//8e Sc (for... 

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