Diffusion heat transfer resistance

A significant feature of physical adsorption is that the rate of the phenomenon is generally too high and consequently, the overall rate is controlled by mass (or heat transfer) resistance, rather than by the intrinsic sorption kinetics (Ruthven, 1984). Thus, sorption is viewed and termed in this book as a diffusion-controlled process. The same holds for ion exchange. 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

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. 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

Surface reaction with diffusion and heat transfer resistance In fast exothermic reactions, in addition to grad c, also grad T (TG Ts) is present in the boundary layer between the gas bulk phase and the catalyst surface. For the outer effectiveness factor qext this means that... 

Experimental Measurements of Reaction Kinetics. The reaction expressions discussed in the following model the intrinsic reaction on the catalyst surface, free of mass-transfer restrictions. Experimental measurements, usually made with very fine particles, are described by theoretically deduced formulas, the validity of which is tested experimentally by their possibility for extrapolation to other reaction conditions. Commonly the isothermal integral reactor is used with catalyst crushed to a size of 0.5-1.5 mm to avoid pore diffusion restriction and heat-transfer resistance in the catalyst particles. To exclude maldistribution effects and back mixing, a high ratio of... 

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... 

A preferred embodiment of this concept is the catalytic plate reactor, which consists of catalytically coated metal plates so that exothermic and endothermic reactions take place in alternate channels. In addition to minimizing the heat transfer resistances, this reactor facilitates mass transfer to the catalytic surface by reducing the diffusion length. Catalytic plate reactors can find applications in steam reforming, dehydrogenation, and hydrocarbon cracking which are strongly endothermic processes. In recent years those reactors have received considerable attention as steam reformers for fuel cell applications. " ... 

In many catalytic systems multiple reactions occur, so that selectivity becomes important. In Sec. 2-10 point and overall selectivities were evaluated for homogeneous well-mixed systems of parallel and consecutive reactions. In Sec. 10-5 we saw that external diffusion and heat-transfer resistances affect the selectivity. Here we shall examiineHEieHnfiuence of intrapellet res ahces on selectivity. Systems with first-order kinetics at isothermal conditions are analyzed analytically in Sec. 11-12 for parallel and consecutive reactions. Results for other kinetics, or for nonisothermal conditions, can be developed in a similar way but require numerical solution. ... 

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. 

Regardless of the way in which progress of the sorption is followed, all uptake rate measurements are subject to the intrusion of heat transfer resistance, and such effects are more severe for strongly adsorbed and rapidly diffusing species. To circumvent this problem Grenier et al. [27] introduced... 

In the frequency response method, first applied to the study of zeolitic diffusion by Yasuda [29] and further developed by Rees and coworkers [2,30-33], the volume of a system containing a widely dispersed sample of adsorbent, under a known pressure of sorbate, is subjected to a periodic (usually sinusoidal) perturbation. If there is no mass transfer or if mass transfer is infinitely rapid so that gas-solid mass-transfer equilibrium is always maintained, the pressure in the system should follow the volume perturbation with no phase difference. The effect of a finite resistance to mass transfer is to cause a phase shift so that the pressure response lags behind the volume perturbation. Measuring the in-phase and out-of-phase responses over a range of frequencies yields the characteristic frequency response spectrum, which may be matched to the spectrum derived from the theoretical model in order to determine the time constant of the mass-transfer process. As with other methods the response may be influenced by heat-transfer resistance, so to obtain reliable results, it is essential to carry out sufficient experimental checks to eliminate such effects or to allow for them in the theoretical model. The form of the frequency response spectrum depends on the nature of the dominant mass-transfer resistance and can therefore be helpful in distinguishing between diffusion-controlled and surface-resistance-controlled processes. 

Some simulation results of the first- and second-order FRFs for the nonisothermal micropore diffusion model with variable diffusion coefficient are given in Figure 11.16. They correspond to literature data for adsorption of CO2 on silicalite-1 [34], Ps= 10 kPa and Tg = 298 K, and to moderate heat transfer resistances [57], The functions H2,pp(co, —co), ff2,Tx(<. —co), and //2,px( , —co), which are identically equal to zero, are not shown. In Rgurc 11.16a we also give the FRFs corresponding to isothermal case (the parameter very large). Notice that for that case the Fp set of FRFs describes the system completely. 

The macropore diffusion of nc adsorbates is described by the Maxwell-Stefan equation as learnt in Chapter 8 (Section 8.8). The micropore diffusion in crystal is activated and is described by eq. (10.6-11), and the adsorption process at the micropore mouth is assumed to be very fast compared to diffusion so that local equilibrium is established at the mouth. Adsorption and desorption of adsorbates are associated with heat release which in turn causes a rise or drop in temperature of the pellet. We shall assume that the thermal conductivity of the pellet is large such that the pellet temperature is uniform and all the heat transfer resistance is located at the thin film surrounding the pellet. How large the pellet temperature will change during the course of adsorption depends on the interplay between the rate of adsorption, the heat of adsorption and the rate of heat dissipation to the surrounding. But the rate of adsorption at any given time depends on the temperature. Thus the mass and heat balances are coupled and therefore their balance equations must be solved simultaneously for the proper description of concentration and temperature evolution. 

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