Intraparticle heat transfer resistance

In the adiabatic case we set Kc = 0 in equation (7.196). The above equations with Kc = 0 also represent the nonporous catalyst pellet with external mass- and heat-transfer resistances and negligible intraparticle heat-transfer resistance but the parameters have a different physical meaning as explained earlier on p. 552. 

There are many cases where the external mass transfer resistance can be neglected while the external heat resistance in not negligible. In gas-solid systems, external heat transfer resistances are usually much higher than external mass transfer resistances especially for light components. Also there are cases where the intraparticle resistances are appreciable while the intraparticle heat transfer resistance is negligible due to the high thermal conductivity of the metal or metal oxides forming the bulk of the catalyst pellet. 

The importance of the intraparticle heat transfer resistance is evident for particles with relatively short contact time in the bed or for particles with large Biot numbers. Thus, for a shallow spouted bed, the overall heat transfer rate and thermal efficiency are controlled by the intraparticle temperature gradient. This gradient effect is most likely to be important when particles enter the lowest part of the spout and come in contact with the gas at high temperature, while it is negligible when the particles are slowly flowing through the annulus. Thus, in the annulus, unlike the spout, thermal equilibrium between gas and particles can usually be achieved even in a shallow bed, where the particle contact time is relatively short. 

Catalyst particles in three-phase fixed-bed reactors are usually completely filled with liquid. Then intraparticle temperature gradients are negligible due to the low effective diffusivities in the liquid phase, as pointed out by Satterfield [13] and Baldi [92]. However, if the limiting reactant and the solvent are volatile, vapor-phase reaction may occur in the gas-filled pores, causing significant intraparticle temperature gradients [109, 110]. In these conditions, intraparticle heat transfer resistance is necessary to describe the heat transfer. 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

The form of equation (21) is interesting. It shows that the uptake curve for a system controlled by heat transfer within the adsorbent mass has an equivalent mathematical form to that of the isothermal uptake by the Fickian diffusion model for mass transfer [26]. The isothermal model hag mass diffusivity (D/R ) instead of thermal diffusivity (a/R ) in the exponential terms of equation (21). According to equation (21), uptake will be proportional to at the early stages of the process which is usually accepted as evidence of intraparticle diffusion [27]. This study shows that such behavior may also be caused by heat transfer resistance inside the adsorbent mass. Equation (22) shows that the surface temperature of the adsorbent particle will remain at T at all t and the maximum temperature rise of the adsorbent is T at the center of the particle at t = 0. The magnitude of T depends on (n -n ), q, c and (3, and can be very small in a differential test. 

The Biot number Bib for heat transport. Analogous to Bim, this is defined as the ration of the internal to external heat transfer resistance (intraparticle heat conduction versus interphase heat transfer). 

From the above it follows that in most practical situations a model, that takes into account only an intraparticle mass and an interparticle heat transfer resistance will give good results. However, in experimental laboratory reactors, which usually operate at low gas flow rates, this may not be true. In the above criteria the heat and mass transfer coefficients for interparticle transport also have to be known. These were amply discussed in Section 4.2. 

The methods outlined by Satterfield94 for taking into account the effects of intraparticle mass- and heat-transfer resistances on the effective reaction rate are applicable to three-phase reactors and, therefore, they will not be repeated here. The importance of these resistances depends upon the nature of the reaction and... 

Note that in the limit of external diffusion control, the activation energy 0b,—>0, as can be shown when substituting Eq. (7-110) inEq. (7-108). For more details on how to represent the combined effect of external and intraparticle diffusion on effectiveness factor for more complex systems, see Luss, "Diffusion-Rection Interactions in Catalyst Pellets. Heat-Transfer Resistances A similar analysis regarding external and intraparticle heat-transfer limitations leads to temperature... 

The particle is isothermal, i.e. flat intraparticle temperature profile, with the temperature difference between the pellet surface and the bulk fluid concentrated across the external heat transfer resistance. 

The lumped parameter approximation for porous catalyst pellets represents, in the majority of cases, a gross simplification that needs to be justified thoroughly before being used in simulating industrial gas solid catalytic systems. Usually when intraparticle mass and heat transfer resistances are not large this approximation can be used in two ways ... 

The heart of the catalytic reactor is the catalyst pellet itself. This in quantitative terms involves the rates of mass and heat transfer between the bulk of the fluid and the surface of the catalyst pellet, the intraparticle mass and heat transfer within the tortuous structure of the pellet (in the case of the porous catalyst pellet) as well as the heart of the catalyst pellet behaviour itself which is the intrinsic rate of reaction (i.e. the rate of reaction in terms of the local concentration and temperature after these variables have been modified from the bulk conditions values through the mass and heat transfer resistances). The reaction kinetics include the effect of both the surface reaction and the chemisorption in an integrated form as explained earlier. 

Both radial and axial diffusion can be taken into account and the final equations to be solved are relatively simpler than those of the continuum model. Although, the equations of the model at steady state are algebraic equations, the dimensionality of the system increases considerably. McGuire and Lapidus (1965) used this model for the study of the stability of a packed bed reactor which included both interphase and intraparticle mass and heat transfer resistances. 

First, consideration is given to the simplest case that takes into account external mass and heat transfer resistances only. This situation is rigorous for non-porous catalyst pellets and approximate for porous catalyst pellets whenever the intraparticle resistances are much smaller than the external resistances. 

In the heterogeneous model for the high temperature shift converter, the effectiveness factor accounts for the external and the intraparticle mass and heat transfer resistances (e.g. Satterfield and Roberts, 1968 Hutchings and Carberry, 1966 Petersen et ai, 1970 Chu and Hougen, 1972) and is multiplied by the rate of reaction at bulk conditions to get the actual rate of reaction. 

The heterogeneous model developed in this section takes into account interphase as well as intraparticle mass and heat transfer resistances. The following classical simplifying assumptions are used for the modelling of the reactor. 

Particle Breakup, Inter- and Intraparticle Mass and Heat Transfer Resistance Models... 

Inter- and intraparticle mass and heat transfer resistances... 

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