First-order reactions axial dispersion

Equations 8-148 and 8-149 give the fraction unreacted C /C o for a first order reaction in a closed axial dispersion system. The solution contains the two dimensionless parameters, Np and kf. The Peclet number controls the level of mixing in the system. If Np —> 0 (either small u or large [), diffusion becomes so important that the system acts as a perfect mixer. Therefore,... 

FIGURE 9.10 Relative error in the predicted conversion of a first-order reaction due to assuming piston flow rather than axial dispersion, kt versus Pe. 

Example 9.6 Compare the nonisothermal axial dispersion model with piston flow for a first-order reaction in turbulent pipeline flow with Re= 10,000. Pick the reaction parameters so that the reactor is at or near a region of thermal runaway. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

This example models the dynamic behaviour of an non-ideal isothermal tubular reactor in order to predict the variation of concentration, with respect to both axial distance along the reactor and flow time. Non-ideal flow in the reactor is represented by the axial dispersion flow model. The analysis is based on a simple, isothermal first-order reaction. 

The dispersion model of example DISRE is extended for the case of non-isother-mal reactions and to include the axial dispersion of heat from a first-order reaction. 

Another approach to evaluate the performance of a trickle-bed reactor (particularly a pilot-scale reactor) is to incorporate the RTD with intrinsic kinetics. Since the liquid holdup, catalyst wetting, or the degree of axial dispersion can all be obtained from the RTD, this approach is not exclusive of the ones described above. For a first-order reaction, if the residence-time distribution E(t) and the degree of conversion are known, they can both be related by an expression... 

Pavlica and Olson38 outlined a generalized axial dispersion model for the isothermal bubble-column reactor in which a pseudo-first-order reaction occurred in both the gas and liquid phases. The model considered axial mixing in both the gas and the liquid phases. Here, we review a model for the reactor in which a generalized (m, n)th-order reaction between a gaseous species A and a liquid species C is carried out in the liquid phase. There are many chlorination, nitration, sulfonation, alkylation, and hydrogenation reactions which can be... 

Consider (so that an analytical solution can be obtained) an isothermal, axially-dispersed PFR accomplishing a first-order reaction. The material balance for this reactor can be written as ... 

As illustrated above, dispersion models can be used to described reactor behavior over the entire range of mixing from PFR to CSTR. Additionally, the models are not confined to single-phase, isothermal conditions or first-order, reaction-rate functions. Thus, these models are very general and, as expected, have found widespread use. What must be kept in mind is that as far as reactor performance is normally concerned, radial dispersion is to be maximized while axial dispersion is minimized. 

Figure 5.19 F t) curves for CSTR and axial dispersion models demonstrating identical effects on conversion in a first-order reaction.

A certain first-order reaction is being carried out in a fixed-bed reactor in which the axial dispersion coefficient has been determined to be... 

Figure 6.12 Effect of axial dispersion on conversion for a first-order reaction in a...

Once the model parameters are determined and correlated, the flow model can be used to predict chemical reactor behavior. For example, with a first-order reaction occurring in a steady flow reactor with axial dispersion, the mass balance... 

Backmixing in a tubular reactor has a direct influence on the axial concentration profile. With decreasing axial dispersion time compared to the space time (decreasing Bo) the concentration profile flattens and finally a uniform concentration results Bo => 0). This is demonstrated in Figure 3.19 for an irreversible first order reaction at Dal = k x = "i. 

Another result, which is interesting for model simplification, concerns dispersion. In catalytic vapor phase reaction the effect of backmixing on reactor efficiency generally proves to be negligible except for cases of high conversion and short beds. For concurrent trickle-flow operation Hears [47] has developed on the basis of a perturbation solution of the one-dimensional plugflow dispersion (PD-)model, a criterion for negligible (< 5 %) influence of axial dispersion in case of first order reactions ... 

Axial gradients may arise as a result of reactant conversion along the catalyst bed, which may be important in integral reactors. The rule of thumb for minimizing axial dispersion effects is concerned with the reactor tube length to particle diameter ratio, which is reported as being at least 50 for first-order reactions and particle Reynolds numbers greater than 10 ... 

Nevertheless, the inclusion of axial dispersion may be interesting from the point of view of the numerical methods used to solve the conservation equations or in studies regarding the appearance of multiple steady-state solutions [141, 142], Petersen [81] presented an analysis for a ID reactor in terms of the dispersion factor E, which is the ratio between the length of a plug-flow reactor (no dispersion) and the one for a reactor with dispersion yielding the same conversion (Lm). Due to the coordinate transformations employed, F is a function of oP = kDAefu (isothermal first-order reaction). Asymptotic solutions for the dispersion ratio were obtained and are given by... 

Axial Dispersion of Heat According to Mears (1976), solution of the differential equations for the heat and mass balance (for a first-order reaction) lead to the following equation for the deviation of the axial temperature in a wall-cooled or heated fixed bed reactor from the corresponding value in an ideal plug-flow reactor ... 

Axial and Radial Dispersion of Mass and Heat The criterion for a negligible influence of axial dispersion of mass (first-order reaction) is ... 

Note that this equation has no physical significance It is only for first order reactions that these two models pr ict the same conversion for the same mean residence time. There is, however, an important physical difference between the two models in the cascade model there is no bacl xing from reactor number N to reactor number N-1, whereas in the model for plug flow with axial dispersion there is only one discontinuity, that is at the reactor entrance. 

FIGURE 4.33 Reactor size predicted by an axial dispersion model compared with the size predicted by a plug flow model. First-order reaction, — ta = aca-... 

Because the parameter of the axial dispersion model, as observed from numerous experimental studies (58), has been so extensively correlated with Peclet number, designers consider the model useful for scaleup and use it for reactor calculations. The model gives a nice analytical expression for prediction of conversion of a single, irreversible first-order reaction (E(s) in Table 1 with Da replacing s). The expressions for exit concentrations for a system of reversible first-order reactions with the same axial dispersion coefficient (turbulent flow) are much more complex and their evaluation is computationally demanding. 

The cell model ( ,2) is particularly well suited to describing the two dimensional temperature and concentration profiles in crossflow, including axial dispersion in these short passes (L/Dp<25 for the monoliths and catalyst used in this study). After neglecting the radial and interphase concentration and temperature gradients, we can write the modelling equations for a first order reaction with simultaneous... 

Let us assume that a first-order reaction A Bi considered again. If there is dispersion present, the diffusion term can be described by a second-order derivative in the axial direction Dd c Idz. This term has to be added to Eqn. (13.6), hence... 

Find a criterion for neglecting the axial dispersion of a first-order reaction in a tubular reactor. A first-order reaction in a tubular reactor without dispersion is described by... 

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