Internal heat transport limitations

Internal heat transport limitations were considered by J.B. Anderson (Chemical Engineering Science 18 (1963) 147) and according to the following criterion such limitations do not occur if... 

Internal heat transport limitations are rather unlikely at laboratory scale given the small particle diameters that are generally used in order to avoid internal mass transport limitations. An appropriate criterion for the verification of the absence of internal heat transport limitations is derived based on assuming a parabolic temperature profile inside the catalyst pellet. In that case, it can be shown that the following relation holds between the observed reaction rate and... 

Solving equation 24 for y and substitution in equation 23 leads to the final criterion for the absence of internal heat transport limitations ... 

Whether the criteria for intra- or extraparticle heat transport limitations are more severe than the corresponding ones for mass transport depends on the absolute value of the products ys x ft, and yb x Pe, respectively. In some cases the former product exceeds unity, but more often not [8], so internal temperature gradients do not occur frequently. The latter product is a factor Bim/Bih larger, generally >10 (gases), which means that in this case the extraparticle temperature gradient occurs earlier than an extraparticle concentration gradient. 

Note that criterion 7.186 requires a knowledge of the apparent activation energy for the reaction. Criterion 7.186 holds whether internal diffusion limitations exist or not. When the criterion concerning external heat transfer is compared to criterion 7.167 concerning radial heat transport limitations through the bed, it can been seen that the latter are more critical unless ... 

The Diffusion Limit Model. In this model 34, 41 5) internal motion does not exist such that diffusion is the only heat transport mechanism. This represents the slowest internal heat transfer limit and is relevant for more viscous fuels or during the initial period when insuflBcient amount of internal motion has been generated. 

Internal Mass and Heat Transport Limitations. While external diffusion occurs in series with the reaction at the catalyst surface, internal diffusion occurs in parallel with the chemical reaction. As a consequence, the mathematical formulation of the problem is somewhat more complex. A mass balance for the reacting component A over an internal volume element in the catalyst can be written as... 

It is, however, possible to select fairly accurate A values because it varies within a rather narrow range experimental results reported vary between 0.1 and 0.4 Btu/h-ft-°F, excluding vacuum conditions [9]. Other order of magnitude values that have been reported as 4 x 10 J/s-cm-K [16] or 3 x 10 cal/s-cm-K [19] are also based on the latter results. Bearing in mind the lesser impact of internal heat transport compared with internal mass transfer limitations, relatively little attention has been paid to correlations of A inside pellets. 

GP 9] [R 16[ The extent of external transport limits was made in an approximate manner as for the internal transport limits (see above), as literature data on heat and mass transfer coefficients at low Peclet numbers are lacking [78]. Using a Pick s law analysis, negligible concentration differences from the bulk to the catalyst sur-... 

The intrinsic catalytic properties of enzymes are modified either during immobilization or after they were immobilized [25-27], In heterogeneous catalysis such as is carried out by immobilized enzymes, the rate of reaction is determined not simply by pH, temperature and substrate solution, but by the rates of proton, heat and substrate transport, through the support matrix to the immobilized enzyme. In order to estimate this last phenomenon, we have studied the internal mass transfer limitation both in hexane and in SC C02, with different enzymatic support sizes. 

Drying kinetics, which are affected by temperature, particle size, and structure, are limited by external heat and mass transfer to and from the particle surface in the early stages, but internal moisture transport is the main parameter at lower moisture. 

For a non-isothermal catalyst particle according to the Weisz-Hicks criterion no internal heat and mass transport limitations occur if... 

To conclude, an overall summary of calculations based on the above results indicates that the usual order of events as transport limitations occur is to begin with no limitations—chemical reaction controls throughout the pellet. Next, internal pore diffusion begins to have an effect, followed by external heat transfer... 

No limits are established for these radionuclides in Class B or C wastes. Practical considerations such as the effects of external radiation and internal heat generation on transportation, handling, and disposal will limit the concentrations for these wastes. These wastes shall be Class B unless the concentrations of other nuclides in Table 16.6 determine the waste to the Class C independent of these nudides. 

The general problem of diffusion-reaction for the overall effectiveness factor D is rather complicated. However, the physical and chemical rate processes prevailing under practical conditions promote isothermal particles and negligible external mass transfer limitations. In other words, the key transport limitations are external heat transfer and internal mass transfer. External temperature gradients can be significant even when external mass transfer resistances are negligibly small. 

The analogous Biot number for heat is defined as the ratio of internal to external heat transport resistances and is used together with (BOm to assess the relative importance of all heat and mass transport limitations ... 

Because of the enhanced internal moisture migration under vacuum, the rate of drying can be as rapid as that at a much higher temperature at atmospheric pressure. However, the higher specific volume of vapor associated with the reduced pressure is a severe limitation for heat transport by convection (Perre et al., 1995). 

In the previous chapter we discussed the polarization curve and all of the losses associated with the generation of current that result in decreased operating efficiency and generation of heat. At a fundamental level, all of these polarizations are a result of transport limitations. The ohmic polarization is a result of ion and electron transport losses, and the concentration and activation polarization is a result of mass transport limitations of the reactant to the catalyst surface and charged particles across the double layer, respectively. Even the crossover and internal short current loss from the expected Nemst potential is a result of transport. Optimization of the fuel cell design therefore must include an optimization of the (desired) modes of transport and minimization of the undesired modes of transport. In this chapter, the modes of transport relevant to fuel cells are described in greater detail. 

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