Effectiveness factor reactor length effect

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

The effectiveness factor is defined as the ratio of the rate of reaction with diffusional resistances over the rate of reaction at bulk conditions. It follows that the effectiveness factor is not constant along the reactor length and therefore has to be calculated at each axial point along the reactor length. In the case of a two dimensional model, the effectiveness factor radial variations should also be calculated. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed bed reactor model must be described by a heterogeneous model, and pseudo-homogeneous models cannot be used except in a few very special cases as discussed earlier. 

Wilke equation gives better (as compared with the experimental results) temperature prediction than the Stefan-Maxwell equation, but the conversion of the latter almost coincides with the experimental results. The use of the Stefan-Maxwell equation gives a slightly lower effectiveness factor along the reactor length. From the results it seems that the simplified equation of Wilke is quite satisfactory in this case. However, since the Stefan-Maxwell equation is the more rigorous, and its use in this case does not add any extra complications, it is the recommended method. 

The results for Na versus reactor length using 25 collocation points for the pellet are shown in Figure AJ. Also shown are the simplified effectiveness factor calculations for this problem from Example 7.5. A magnified view is shown in Figure A.8. Notice the effectiveness factor approach gives a good approximation for the bed performance. It is not exact because the reaction is second order. ... 

Catalyst particle size is a very important factor in possible pressure variation. While catalyst effectiveness normally improved upon the use of smaller particles, it is often at the expense of higher pressure drops along the reactor length, so a design compromise is sometimes necessary. 

Constant catalyst effectiveness factor all along the reactor length... 

The results of mathematical modeling developed for closed architecture with a catalyst effectiveness factor of 0.6 clearly show the effect of the presence of the membrane on CO conversion profile along the catalyst bed. The most important effect is that the reactor, in this case, can overcome the thermodynamic limitations and the maximum CO conversion is higher than that obtained without the membrane. In addition, it is worth noting that the equilibrium CO conversion can be obtained by using a lower catalyst volume, or in other words at a shorter reactor length. 

The evolution of the various effectiveness factors is shown in Fig. 11.9.1.A-7. The effectiveness factor of reaction II shows a discontinuity at a reactor length of 3.4 m because the direction of reaction II changes from positive to negative on the catalyst surface, while there is no such tendency yet inside the catalyst particle. The rj2 value switches back from negative to positive values only at a reactor length of 9 m, meaning that the overall effect of the reaction inside the catalyst pellet reverses from the formation of CO2 and H2 out of CO and H2O to the formation of CO and H2O out of CO2 and H2. 

The patterns shown in the figure are typical of an adiabatic reactor for exothermic reactions. The effectiveness factor generally decreases with reactor length, but more rapidly than shown in the figure. Because of the reaction equilibrium, the decrease is much more moderate in this case. The conversion and temperature usually increase rapidly near the inlet, but these increases are moderate toward the outlet, reaching plateaus as the reactant is depleted. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Concrete nuclear reactor vessels, of the order of magnitude of 15-m (50-ft) inside diameter and length, have iimer linings of steel which confine the pressure. After fabrication of the liner, the tubes for the cables or wires are put in place and the concrete is poured. High-strength reinforcing steel is used. Because there are thousands of reinforcing tendons in the concrete vessel, there is a statistical factor of safety. The failure of 1 or even 10 tendons would have little effect on the overall structure. 

Since the cooling jacket has cocurrent flow, the model consists of the set of four coupled initial value differential equations (7.5) to (7.8). Note that the first three DEs (7.5) to (7.7) contain the variable catalyst effectiveness factor rj. Thus there are other equations to be solved at each point along the length 0 < / < Lt of the reactor tube, namely the equations for the catalyst pellet s effectiveness factor rj. 

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. 

Choose a reaction from the literature that is similar to the reaction of this reactor, such as catalytic hydrogenation, and perform the above computations using MATLAB to obtain the concentrations, temperatures, and effectiveness factor profiles along the length of the reactor. 

The effectiveness factors at each point along the length of the reactor are calculated for the key components methane and carbon dioxide by using either the dusty gas model or one of our simplified models (A) and (B). The dusty gas model gives rise to more complicated two point boundary value differential equations (BVPs) for the catalyst pellets. These are solved in MATLAB via bvp4c or bvp4cf singhouseqr. m as done in Chapter 5. 

The effectiveness factors for the reactions and the components are changing along the length of the reactor. The data above shows the effectiveness factors and rates at the exit. Here the effectiveness factor of a component is computed from the rate of its consumption at the exit. For components which are formed in a single reaction and are not involved in any other reactions, such as styrene, benzene, toluene and carbon dioxide, the effectiveness factor is determined by the effectiveness factor of their reaction, i.e., by the reactions (1) to (3) and (6). Note that for most of the reactions (except for reactions (5)... 

Although more information is needed to determine details concerning factors that favor inactive coke formation, relatively high levels of surface sulfides probably promote formation of such coke. On the other hand, metal oxides on the surface likely favor production of active coke. Sulfiding the reactor tube immediately upon completion of the decoking step would form metal sulfides. An aluminized surface, such as provided by the alonized Incoloy 800 reactor, also has been found to be an effective way to prevent the production of active coke. Quite possibly, the initial type of coke formed on the just-cleaned tube would have an important effect on the length of time a reactor tube could be used in a commercial plant before decoking would be required. 

Thus far, the overall effectiveness factor has been used in the mass and energy balances. Since 77 is a function of the local conditions, it must be computed along the length of the reactor. If there is an analytical expression for 17, for example for an isothermal, first-order reaction rate ... 

A membrane reactor preceded or followed bv a conventional reactor. Consider a typical commercial porous membrane currently available that exhibits a moderate to low gas separation factor and a high gas permeance even for the gas intended to be the retentate. Much of this less permeable gas in the feed stream is lost to the permeate (low pressure) side in the entrance section of the reactor due to its high partial pressure difference across the membrane layer. This leads to the undesirable effect of a low reactant conversion in that section. An effective way of reducing this reactant loss is to have a membrane enclosed section preceded by an impermeable reaction zone. To achieve a maximum total conversion, the impermeable length relative to the membrane length needs to be optimized. 

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