Equivalent Reactor

Because the equivalent reactor is defined by having identical total production to the two parallel isothermal PFRs, we can obtain the following equation  

Substitute the relationship between the molar flow rate and conversion 

Vg reactor volume of Rg Tg reactor time of Rg CONVg conversion of Rg FAin.T- total food Ct t IVg = 1 - Exp(-k Tg) molar flow Aout,e 

With molar flow rate, conversion and SV, we can calculate reactor volume to conduct reactor design. The idea of equivalent reactor provides us a convenient way to understand the performance of a complex reactor system, namely, two parallel PFRs. 

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World annual uranium requirements in 1993 were estimated at about 58,382 t natural uranium equivalent. Reactor-related requirements are expected to rise about 1015 t/yr on the average, reaching 75,700 t U total requirements in the year 2010. The cumulative aggregate world uranium requirements for the period 1993—2010 are estimated to be about 1.185 X 10 t U metal (29). 

A 15 1 (11 1 working volume) Applikon stirred tank, or equivalent, reactor is used as a medium reservoir. 

A question arises how to reduce this difference The solution consists of using a series of CSTRs instead a single PFR. Since the reaction rate is higher in each intermediate volume, finally a much higher productivity than in a single equivalent reactor is obtained (Fig. 8.6). At limit, an infinite series of CSTRs behaves as a single PFR of the same volume. 

The equivalent reactor volume concept, introduced by Hougen and Watson [1] allows for a second way of dealing with nonisothermal data it first reduces the data to isothermality and determines the temperature dependence of the rate parameters in the second stage only. The equivalent reactor volume has been defined as that volume, which, at the reference temperature T, and the reference total pressure p,i, would give the same conversion as the actual reactor, with its temperature and pressure profiles. It follows that... 

In a kinetic study the activation energy is generally not known a priori, or only with insufficient accuracy. The use of the equivalent reactor volume concept therefore leads to a trial-and-error procedure a value of is guessed and with this value and the measured temperature profile Vp is calculated by graphical or numerical integration. Then, for the rate model chosen, the kinetic constant is derived. This procedure is carried out at several temperature levels and from the temperature dependence of the rate coefficient, expressed by Arrhenius formula, a value of is obtained. If this value is not in accordance with that used in the calculation of Vp the whole procedure has to be repeated with a better approximation for . 

We see how the curves do not extrapolate through the origin. This results from the fact that not all of the volume accounted for is at the reference temperature considered. The equivalent reactor volume concept will be used to reduce the data to isothermal conditions. 

With this value of and the temperature profiles the equivalent reactor volumes may be obtained as shown in Fig. 3. The curve x versus is shown in Fig. 4. The curves now extrapolate through the origin. With such a diagram the derivation of a rate equation may now be undertaken. 

Figure 3 Acetone cracking. Calculation of equivalent reactor volume from Froment, et al. [5, 6]).

Perhaps the most important problem is the lack of understanding of bubble behavior in beds of different sizes and particle properties. An approximate but useful way of addressing this problem is to operate a pilot plant reactor of a certain size and design the larger reactor so as to simulate the bubble behavior of the pilot reactor. For this purpose, we define an equivalent reactor diameter given by the hydraulic diameter (see Volk et al., 1962)... 

The feeding rate policies for the fed-batch reactors may then be found from the critical DSR a policies, whereas the batch cycle times are found from the equivalent reactor residence times under continuous operation. 

Acetone cracking. Temperature profiles for calculation of equivalent reactor volume. From Froment et al. [1961a, b]. 

These reactors can be modelled in three ways (i) by direct experimental simulation, (ii) by equivalent reactor volume concept, and (iii) by conservation equations. Since the flow rates used in... 

If the temperature variations as a function of axial location can be predicted, the "equivalent reactor volume" concept (Hougen and Watson, 1947 Froment et al., 1961) can be applied to convert the data to a pseudo-isothermal basis. According to Hougen and Watson (1947), the equivalent reactor volume is defined as that volume which, at a constant reference temperature, would give the same conversion as the actual non-isothermal tubular reactor. The reference temperature has frequently been taken as the arithmetic mean of the process gas temperatures in the last 40% of the reactor (Van Damme et al., 1975). 

For example, if the reactor volume is 0.2 and the inlet volumetric flow rate is 0.01 mVs, it would take the upstream equivalent reactor volume (V = 0.2 m ) shown by the da.shed lines a lime t equal to... 

Construct an equivalent reactor to represent the two parallel reactor series ... 

This section demonstrates the concept of equivalent reactor. Considering a system with two parallel isothermal plug flow reactors (PFRs), where a first-order liquid-phase reaction takes place (see Figure 6.14), the relationship between conversion and residence time of each PFR is [33] ... 

Figure 6.16 HCR rate constant versus equivalent reactor volume.

Following reconciliation of the equivalent reactor model is using real operating data to build preliminary models for real H P H CR reactors. We apply the reaction activities from the equivalent reactor model into the preliminary reactor models. It is necessary to fine-tune the preliminary reactor models. From the Aspen Simulation Workbook, we create a MS Excel spreadsheet (Figure 6.18) to make it feasible to simultaneously fine-tune reactor models of the two parallel series. In the HP HCR model, we only fine-tune HCR selectivity from 4.5 to 3.9 and the resulting model agrees with real operation and production well. The development equivalent reactor model reduces time and makes it achievable to develop the HP HCR model of two parallel reactor series. 

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