Reactors differential

Catalytic differential reactors are frequently mentioned in textbooks as typical test reactors. In fact, a differential reactor used in catalysis is a special case of the fixed bed reactor, nothing else. The conversions are maintained low, which allows us to approximate the molar flow 

In principle, r, is directly obtained by means of a measurement of the molar flow difference, hi — hoi. Mathematically, the differential reactor model coincides with the model of a gradientless test reactor presented in the next section. 

To determine the kinetic parameters, we use the differential reactor. In this continuous flow system, the variation in concentration between the inlet and outlet of the reactor should be small and finite. The conversions should be around 5-10%. Under these conditions, the diffusive and mass transfer effects are avoided, assuring a kinetic regime for the determination of the kinetic parameters. Unlike the case of the batch system, the spatial time and consequently, the inlet flow and the mass or volume of the reactor are varied. Therefore, the reaction rate is directly determined. 

In a continuous system, the differential molar balance in a finite volume A V will be as follows  

The molar flow in the reactor inlet and outlet will be Fao and Fas, respectively. Therefore (Equation 4.13)  

If the reaction takes place at constant volume, we can approximate the rate for 

Note that when the reactor volume is small, the space velocities are high, but the space time is very small, and consequently, the conversions are very low. Thus, the rate practically corresponds to the initial rate, i.e.  

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Tubular Reactors. The tubular reactor is exceUent for obtaining data for fast thermal or catalytic reactions, especiaHy for gaseous feeds. With sufficient volume or catalyst, high conversions, as would take place in a large-scale unit, are obtained conversion represents the integral value of reaction over the length of the tube. Short tubes or pancake-shaped beds are used as differential reactors to obtain instantaneous reaction rates, which can be computed directly because composition changes can be treated as differential amounts. Initial reaction rates are obtained with a fresh feed. Reaction rates at... 

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 ... 

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. 

Various experimental methods to evaluate the kinetics of flow processes existed even in the last centuty. They developed gradually with the expansion of the petrochemical industry. In the 1940s, conversion versus residence time measurement in tubular reactors was the basic tool for rate evaluations. In the 1950s, differential reactor experiments became popular. Only in the 1960s did the use of Continuous-flow Stirred Tank Reactors (CSTRs) start to spread for kinetic studies. A large variety of CSTRs was used to study heterogeneous (contact) catalytic reactions. These included spinning basket CSTRs as well as many kinds of fixed bed reactors with external or internal recycle pumps (Jankowski 1978, Berty 1984.)... 

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... 

In Figure 3.6.1 the inner balance accounts for differences between just before and just after the catalyst bed. In essence this is a balance for a differential reactor and written for a reactant ... 

For the following calculations it is assumed that experiments are conducted in a good recycle reactor that is close to truly gradientless. Conceptually the same type of experiment could be conducted in a differential reactor but measurement errors make this practically impossible (see later discussion.) The close to gradientless conditions is a reasonable assumption in a good recycle reactor, yet it would be helpful to know just how close the conditions come to the ideal. 

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. 

Each experimental run gives the reaction rate at the composition of the exit fluid. Tubular reactors can be operated as differential reactors (i.e., at high throughputs and low conversions) or as integral reactors (i.e., at low throughputs and high conversions). Differential reactors give the rate as ... 

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 catalyst consists of 3-mm pellets that pack to a bulk density of 1350 kg/m and = 0.5. Mercury porosimetry has found 7 ore = 5nm. The feed mixture to a differential reactor consisted of 5mol% SO2 and 95mol% air. The following initial rate data were obtained at atmospheric pressure ... 

Steady-state reactors with ideal flow pattern. In an ideal isothermal tubular pZi/g-yZovv reactor (PFR) there is no axial mixing and there are no radial concentration or velocity gradients (see also Section 5.4.3). The tubular PFR can be operated as an integral reactor or as a differential reactor. The terms integral and differential concern the observed conversions and yields. The differential mode of reactor operation can be achieved by using a shallow bed of catalyst particles. The mass-balance equation (see Table 5.4-3) can then be replaced with finite differences ... 

Kinetic analysis of the data obtained in differential reactors is straightforward. One may assume that rates arc directly measured for average concentrations between the inlet and the outlet composition. Kinetic analysis of the data produced in integral reactors is more difficult, as balance equations can rarely be solved analytically. The kinetic analysis requires numerical integration of balance equations in combination with non-linear regression techniques and thus it requires the use of computers. 

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

Figure 5.4-18. Differential reactor with recycle of the reaction mixture.

It follows from the equation above that c, if Fv.rec Fvj This means that for a recycle stream much larger than the feed stream, the catalyst bed operates as a differential reactor, while the whole system gives an outlet concentration differing significantly from that of the feed. This significantly simplifies problems of chemical analysis. In practice, the recycle reactor operates differentially if the recycle ratio Fv.ret/Fv.f is larger than 25. The rate is then given by the overall rate ... 

Hint 1. The reactor can be treated as a differential reactor, since the conversions are so low. 

A material balance on this differential reactor volume yields the following result. 

For a differential reactor and the same residence time the ratio of these rates will be equal to the ratio of the conversion levels attained. 

For a differential reactor, the change in composition across the reactor will be very small, and the bulk fluid composition may be estimated from the inlet molal flow rates. Assuming that the inlet air is 79% nitrogen and 21% oxygen, the calculations below indicate the bulk fluid mole fractions and partial pressures of the various components of the reaction mixture. 

In a series of laboratory scale experiments, streams of oxygen and sulfur dioxide were fed at different rates to a differential reactor containing 2.372 g of catalyst. The data below were recorded under essentially isothermal conditions... 

Figure 6. Decomposition of N20 over NiO in a differential reactor. The steady-state surface oxygen coverage 6 was 0.58 (curve 1), 0.69 (curve 2), and 0.28 (curve...

A conmercial catalyst frcm Harshaw was used, a 3 1 mixture of molybdenum trioxide and ferric molybdate, as well as the two separate phases. Kinetic experiments were done previously in a differential reactor with external recycle using these same catalysts as well as several other preparations of molybdenun trioxide, including supported samples. Hie steady state kinetic experiments were done in the temperature range 180-300 C, and besides formaldehyde, the following products were observed, dimethylether, dimethoxymethane, methyl formate, and carbon-monoxide. Usually very little carbon dioxide was obtained, and under certain conditions, hydrogen and methane can be produced. 

In the case of a PFR, it is usually easier to vary Tin a controllable and measurable way if it is operated as a differential reactor rather than as an integral reactor. In the latter case, it may be difficult to eliminate an axial gradient in Tover the entire length of the reactor. 

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