The Differential Reactor

In a differential reactor the concentration change, i.e., the conversion increase, is kept so low that the effect of the concentration and temperature changes can be neglected. On the other hand the concentration change must be quantitatively known because, multiplied by the flow rate and divided by the catalyst quantity, it measures the reaction rate as  

This rate, measured the previous way, must be correlated with the temperature and concentration as in the following simple power law rate expression  

There are two contradictory requirements here. The first is to keep the difference between Ci and C as small as possible so that it can be neglected. The second is to analyze these two only very slightly different concentrations with such precision that the difference will be significantly greater than the measurement error. This second need is for calculation of the rate of reaction, as shown in the first equation of this section. 

This is an obviously difficult task, and it is rarely possible to satisfy both requirements reasonably and simultaneously. This difficulty is compounded by the need to use a preconverter to achieve the various conversion levels where the additional incremental increase in conversion can be measured. The alternative way to a preconverter is to feed the reactor various amounts of products in addition to the starting material. This does not ease the analysis difficulties. 

In a differential reactor the product stream differs from the feed only very slightly, so the addition of products to the feed stream can be avoided if most of the product stream is recycled. The feed can be made up mostly from the recycle stream with just enough starting materials added to replace that which was converted in the reaction and blown off in the discharge stream. This is the basis of loop or recycle reactors, as will be explained later. 

---

The differential reactor is the second from the left. To the right, various ways are shown to prepare feed for the differential reactor. These feeding methods finally lead to the recycle reactor concept. A basic misunderstanding about the differential reactor is widespread. This is the belief that a differential reactor is a short reactor fed with various large quantities of feed to generate various small conversions. In reality, such a system is a short integral reactor used to extrapolate to initial rates. This method is similar to that used in batch reactor experiments to estimate... 

The differential reactor is used to evaluate the reaction rate as a function of concentration for a heterogeneous system. It consists of a tube that contains a small amount of catalyst as shown schematically in Figure 4-17. The conversion of the reactants in the bed is extremely small due to the small amount of catalyst used, as is the change in reactant concentration through the bed. The result is that the reactant concentration through the reactor is constant and nearly equal to the... 

The differential reactor is simple to construct and inexpensive. However, during operation, care must be taken to ensure that the reactant gas or liquid does not bypass or channel through the packed catalyst, but instead flows uniformly across the catalyst. This reactor is a poor choice if the catalyst decays rapidly, since the rate of reaction parameters at the start of a run will be different from those at the end of the run. 

Steady-state temperatures along the length of a piston flow reactor are governed by an ordinary differential equation. Consider the differential reactor element shown in Figure 5.3. The energy balance is the same as Equation (5.14) except... 

The principle of the differential reactor with recycle is illustrated in Fig. 5.4-18. 

The differential reactor uses a thin catalyst bed in which only small changes of concentration and temperature occur (Fig. 3.3-3). The rate of reaction, r, can be obtained from the difference in concentration, Ac, over the catalyst bed or its thickness, Ax, the volumetric throughput, v, or the molar throughput, nges, and the quantity of catalyst, WK, using the material balance ... 

On the one hand, the differential reactor with recycle permits kinetic measurements of high accuracy. On the other hand, a transfer equipment is required to recycle a fraction of the reaction mixture. This can be difficult when the pressure is high. For this purpose, a jet loop reactor was developed which is equipped with an ejector to recycle the fluid. The design of the jet loop reactor is described in Chapter 4.3.4. 

In general, if heterogeneous catalytic reactions are to be conducted isothermally, the reactor design must provide for heat flow to or from the particles of catalyst so as to keep the thermal gradients small. Otherwise, temperatures within the catalyst bed will be non-uniform. The differential reactor and the various forms of the gradientless reactors are advantageous in this regard. 

The recycling reactor behaves similarly to the plug flow reactor, with one major difference, that the conversion range in the reactor is narrower than in the true plug flow reactor (Figure 8.10). This reactor type is also named the differential reactor. The performance equation is [1, 2, 6, 7]... 

The advantages of the differential reactor in kinetic studies in heterogeneous catalysts have been discussed in a previous volume of Advances... 

The rate of cracking as a function of pressure was studied in the pressure range of 0.4 to 1 atm. in order to test the applicability of the kinetic scheme and to determine the values of the constants k3B0 and G. Operation of the differential reactor at the required pressure was achieved by operating the gas outlet (21, Fig. 3) and the manometer outlet (23, Fig. 3) attached to a large reservoir kept at the required pressure. This made it possible to use the manometer system as before. The condensers were operated at the appropriate temperatures relative to the boiling points of cumene and benzene at the pressure used. 

A disadvantage of the differential reactor is the inaccuracy in the determination of conversion and selectivity due to the small concentration changes. The second difficulty in the treatment of experimental data is caused by possible flow nonuniformities. Since the average residence time is short and the fluid elements moving with different axial velocities do not mix, the simplified Equation 5.30 may not be valid. This is because the reactor operates as a segregated flow reactor rather than a plug flow or ideal mixed reactor, on which Equation 5.30 is based. 

CA, the conversion per cycle CAc approaches zero if the recycle ratio Rr is increased more and more. Thus, at sufficiently high recycle flow rates, the concentration and temperature gradients along the catalyst bed can be kept small in the same way as in the differential reactor. The overall conversion in the reactor can be set at any desirable, easily measurable level by regulating the net flow rate < . A further benefit of high recycle flow rates lies in the fact that, due to the high fluid velocities past the catalyst particles, any possible interface transport effects can be eliminated. The recycle reactors, therefore, are called gradient-free reactors. 

Knowing the inlet and exit compositions, the kinetics of the reaction can be elucidated with Equation 5.48 in the same way as for the differential reactor. 

Integral data (a) fits the conversion-space time graph and, (f>) varies in a particular way with reactant concentration. The differential reactor data have to fit only the reaction rate variation with initial concentration. 

Construction difficulty is the same as for the differential reactor, with a small amount of additional complexily added by the replacement of the addition of accurate pulses of reactant and direct measurement of the product by the chromatograph. 

It is of interest that similar ideas have been applied in the conception of a flow system for measuring the diffusion coefficients for gases in porous or microporous solids. Ruthven and Eic (20,21) use a zero-length column (ZLC) to suppress concentration gradients along the bed in the gas phase. As in the differential reactor described previously, a high gas flow rate is used so that the fixed bed acts as if it were very short. A preadsorbed adsorbate is removed by an inert gas stream. The diffusion inside the solid is very close to the classical solution for zero concentration on the surface, but the small concentration actually present in the gas leaving the bed (column) can be measured accurately. 

One advantage of the integral reactor is its ease of construction (see Figure 5-14). On ie other hand, while channeling or bypassing of some of the catalyst by the reactant stream may not be as fatal to diata interpretation in the case of this reactor as in that of the differential reactor, it may still be a problem. 

Modeling the differential reactor. The rate of reaction is calculated from the equation... 

A mole balance on the reactant A over the differential reactor volume... 

Since the HY at 573 K and 40 bar produces the higher amount of iso-butylbenzene, some other experimental runs have been performed under high pressure on this catalyst at 573, 613 and 643 K, to estimate the activation energy Ea for the three main reactions DEA, ISO and DIS. The runs were carried out at different space velocities to keep the conversion at a value < 10%. Samples of the effluent were collected every 15 minutes in the first three hours on stream. Initial conversion and selectivity were calculated by extrapolating the experimental values at zero time on stream. Initial reaction rates were calculated by the following relationship, typical for the differential reactor kinetic data ... 

The kinetic measurements were carried out in a 15 mm id. differential fixed bed reactor. Isothermality was ensured by inmersion of the differential reactor in an external fluidized bed at the desired reaction temperature. The temperature of the fluidized bed was controlled with a PlD controller. The experiments were performed under differential reactor conditions, at atmospheric pressure and at temperatures between 130 and 170 C. 

---