Effectiveness factor endothermic reactions

The suitability of a cycle for hydrogen production depends upon the overall thermal efficiency and operational feasibility. A highly endothermic reaction step is required in a cycle to achieve effective heat-to-chemical energy conversion. For efficient mass and momentum transfer a fluid based system is preferred [96] and, ultimately, for large-scale hydrogen production other factors such as environmental effects and cost effectiveness must also be considered. 

Figure 7.9 shows the temperature and concentration profiles caused by transfer limitations in the case of an endothermic reaction. For an exothermic reaction the temperature at the external surface, T,-, increases with increasing transfer limitations. Hence the external effectiveness factor, Tie, becomes greater than 1 as soon as the decrease of the reactant concentration is compensated by the increase of the temperature. 

Since the conversion rate depends on a and e, the effectiveness factor will be determined by three parameters, namely a, e and a Thiele modulus. This is illustrated in Figure 6.4 [18]. For values of a larger than zero (exothermic reaction) an increase in the effectiveness factor is found, since the temperature inside the catalyst pellet is higher than the surface temperature. For endothermic reactions (a < 0) a decrease of the effectiveness factor is observed. 

When gum formation proceeds, the minimum temperature in the catalyst bed decreases with time. This could be explained by a shift in the reaction mechanism so more endothermic reaction steps are prevailing. The decrease in the bed temperature speeds up the deactivation by gum formation. This aspect of gum formation is also seen on the temperature profiles in Figure 9. Calculations with a heterogenous reactor model have shown that the decreasing minimum catalyst bed temperature could also be explained by a change of the effectiveness factors for the reactions. The radial poisoning profiles in the catalyst pellets influence the complex interaction between pore diffusion and reaction rates and this results in a shift in the overall balance between endothermic and exothermic reactions. 

Since F C ) s F C s) as discussed above, the effectiveness factor is always less than or equal to unity for an endothermic reaction. For an exothermic reaction, the opposite situation can occur. The temperature of the interior of the particle can exceed the surface temperature, T > T, which leads to ... 

In an exothermic reaction the temperature increases as the conyersion increases. At low conversions the rising temperature increases the rate more than it is reduced by the fall in reactants concentration. Normally the conversion will be greater than for isothermal operation. However, undesirable side reactions and other factors may limit the permissible temperatures. In these cases successful design depends on effective removal of the heat of reaction to prevent excessive temperatures (hot spots). In general, the same methods are employed as for adding energy in endothermic reactions. 

For an endothermic reaction there is a decrease in temperature and rate into the pellet. Hence 17 is always less than unity. Since the rate decreases with drop in temperature, the effect of heat-transfer resistance is diminished. Therefore the curves for various are closer together for the endothermic case. In fact, the decrease in rate going into the pellet for endothermic reactions means that mass transfer is of little importance. It has been shown that in many endothermic cases it is satisfactory to use a thermal effectiveness factor. Such thermal 17 neglects intrapellet mass transport that is, ri is obtained by solution of Eq. (11-72), taking C = Q. 

For more complex reaction networks, these component effectiveness factors are the correct numbers to indicate the effect of diffusion on the yield and selectivity for the different components. For this reason it is the components effectiveness factors formulation which should be used in connection with complex reaction networks. It will be shown that in these cases not only rj> I are possible but also ij <0 are possible for some intermediate components. This phenomenon will be discussed in sections 5.1.9 and 5.2.2, in connection with the highly endothermic steam reforming reaction, and the highly exothermic partial oxidation of o-Xylene to phthalic anhydride. 

Thermal effects are often the key concern in reactor scaleup. The generation of heat is proportional to the volume of the reactor. Note the factor of V in Equation 5.31. For a scaleup that maintains geomedic similarity, the surface area increases only as Sooner or later, temperature can no longer be controlled by external heat transfer, and the reactor will approach adiabatic operation. There are relatively few reactions where the full adiabatic temperature change can be tolerated. Endothermic reactions will have poor yields. Exothermic reactions will have thermal runaways giving undesired byproducts. It is the reactor designer s job to avoid limitations of scale or at least to understand them so that a desired product will result. There are many options. The best process and the best equipment at the laboratory scale are rarely the best for scaleup. Put another way, a process that is less than perfect at a small scale may be better for scaleup precisely because it is scaleable. 

One of the most interesting features is that for > 0 (exothermic), there are regions where > 1. This behavior is based on the physical reasoning that with sufficient temperature rise caused by heat transfer limitations, the increase in the rate constant, Ic,., more than offsets the decrease in reactant concentration, C so that the internal rate is actually larger than that at surface conditions of C/ and T, , leading to an effectiveness factor greater than unity. The converse is, of course, true for endothermic reactions. 

The generation of heat inside a pellet due to reaction and its transport through the pellet can greatly affect the reaction rate. For endothermic reactions, there is a fall in temperature within the pellet. As a result, the rate falls, thus augmenting the retarding effect of mass diffusion. On the other hand, for exothermic reactions, there is a rise in temperature within the pellet. This leads to an increase in rate which can more than offset the decrease due to lowered concentration. Thus the effectiveness factor can actually be greater than one. 

We might expect that the maximum hydrogen primary kinetic isotope effect for a reaction in which a C-H bond is broken would be h/ d = 7, with smaller values for reactions that are either endothermic or exothermic. Indeed, the usual range for primary kinetic isotope effects is about 5—8. However, a ratio of A h/A d of 25 was seen in one case, and a value of 13,000 was reported in one unusual situation. Obviously, our analysis of the possible magnitude of h/ d has been oversimplified. Two factors were specifically ignored above ... 

Figure 1 shows the dependence effectiveness factor tj = roi/ri on Thiele modulus for the selected values of parameter p and for = 2, v = 2 (sphere), yt = 7, yi = 12, 5 = 1 (endothermic reactions) and xa(1) = 1. One can find that for Peffectiveness factor rj assumes in the certain ranges of Thiele modulus values much higher than unity. This means that in the cases discussed the internal diffusion, in contrast to the classical isothermal or endothermic catalytic reactions, may considerably increase the rate of the heterogeneous autocatalytic reactions. 

For endothermic reactions, fi<0, and the p curves at each value clearly indicate that the internal effectiveness factor ri will always be less than unity, since both the temperature and the reactant concentration decline toward the center of the particle. In this case, the impact of heat transfer decreases, but the effect of mass transfer becomes almost negligible. An approximate solution can be obtained by ignoring the concentration profile and solving the differential energy balance (Eq. 2.69) by assuming that the reactant concentration is equal to Cas within the particle ... 

Figure 4.5.15 External effectiveness factor as a function ofthe ratio ofthe (measurable) effective reaction rate to the maximum rate (complete control by external mass transfer) fora constant Arrhenius number of 20. For a Prater number jSex < 0, the reaction is endothermic, for /Sex > 0 exothermic, and for /Sex = 0 we have isothermal conditions. Arrows and dashed line indicate ignition, as explained in the text.

Theoretical work has been done on the effectiveness factor for nonisothermal particles. Fig. 3.13.1-1 shows the results of the computations by Weisz and Hicks [1962] for y = EIRT = 20. For > 0.1, that is, for sufficiently exothermic reactions, the effectiveness factor can exceed the value of 1. In such a case the temperature rise, which increases the value of the rate constant, would more than offset the decrease in reactant concentration Cas, so that fA averaged over the particle exceeds that at surface conditions. The converse is true for endothermic reactions. 

For endothermic reactions, P < 0 applies AH, is now a positive number. The effectiveness factor is now below the value seen in the isothermal case, due once again to the dependence of fc, on temperature. The effect is shown in the plots of Figure 9.8. 

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