Peclet number reactors

FIG. 23-10 Residence time distributions of pilot and commercial reactors. <3 = variance of the residence time distribution, n = number of stirred tanks with the same variance, Pe = Peclet number. 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

Water at room temperature is flowing through a 1.0-in i.d. tubular reactor at Re= 1000. What is the minimum tube length needed for the axial dispersion model to provide a reasonable estimate of reactor performance What is the Peclet number at this minimum tube length Why would anyone build such a reactor ... 

Via a passive scalar method [6] where or, denotes the volume fraction of the i-th phase, while T, represents the diffusivity coefiBcient of the tracer in the i-th phase. The transient form of the scalar transport equation was utilized to track the pulse of tracer through the computational domain. The exit age distribution was evaluated from the normalized concentration curve obtained via measurements at the reactor outlet at 1 second intervals. This was subsequently used to determine the mean residence time, tm and Peclet number, Pe [7]. 

Fig.4. Effect of thermal Peclet number on maximum reactor temperature and methanol conversion.

Simulation studies are also conducted for a dispersed PFR and a recycle reactor at 260 °C, 500 psig and feed with DCPD=0.32 mol/min, CPD=0.96mol/min and ethylene=3.2mol/min. Peclet number (Pe) or the recycle ratio is selected as a variable parameter for the dispersed PFR or for the recycle reactor, respectively. Conversion approaches to that of PFR over Pe=50 as can be seen in Fig.4. It is also worth mentioning that the reactor performance is improved with recycle if the residence time is low. 

When a number of competing reactions are involved in a process, and/or when the desired product is obtained at an intermediate stage of a reaction, it is important to keep the residence-time distribution in a reactor as narrow as possible. Usually, a broadening of the residence-time distribution results in a decrease in selectivity for the desired product. Hence, in addition to the pressure drop, the width of the residence-time distribution is an important figure characterizing the performance of a reactor. In order to estimate the axial dispersion in the fixed-bed reactor, the model of Doraiswamy and Sharma was used [117]. This model proposes a relationship between the dispersive Peclet number ... 

The figure shows the ratio of the widths of initially delta-like concentration tracers at the reactor exits as a function of the micro-channel Peclet number for different values of the porosity. Taking a value of = 0.4 as standard, it becomes apparent that the dispersion in the micro-channel reactor is smaller than that in the fixed-bed reactor in a Peclet number range from 3 to 100. Minimum dispersion is achieved at a Peclet number of about 14, where the tracer width in the micro-channel reactor is reduced by about 40% compared with its fixed-bed counterpart. Hence the conclusion may be drawn that micro-channel reactors bear the potential of a narrower residence time than fixed-bed reactors, where again it should be stressed that reactors with equivalent chemical conversion were chosen for the comparison. 

Figure 1.16 Ratio of the width of concentration tracers at the exit of the micro-channel reactor to the corresponding quantity in the fixed-bed reactor as a function of the micro-channel Peclet number udf /D, as obtained in [114. ...

The term Lu / D is known as the Peclet number, Pe, and its inverse as the dispersion number. The magnitude of the Peclet number defines the degree of axial mixing in the reactor. 

Study the effect of varying mass transfer and heat transfer diffusivities (D and X, respectively) and hence Peclet numbers (Pi and P2) on the resulting dimensionless concentration and temperature reactor profiles. 

Find the ratio of reactor volumes or residence times of reactions under dispersion conditions or plug flow over a range of Peclet numbers. Use the result of problem P5.08.08. 

Writing the model in dimensionless form, the degree of axial dispersion of the liquid phase will be found to depend on a dimensionless group vL/D or Peclet number. This is completely analogous to the case of the tubular reactor with axial dispersion (Section 4.3.6). 

The response of a plug flow with dispersion model to a pulse input, equation (6.42), is given in Figure 6.13 for various values of the Peclet number, Pe = UL/Dl. Equation (6.41) will be applied to the analysis of a pulse with dispersion in a reactor in Example 6.10. 

Effective thermal conductivities and heat transfer coefficients are given by De Wasch and Froment (1971) for the solid and gas phases in a heterogeneous packed bed model. Representative values for Peclet numbers in a packed bed reactor are given by Carberry (1976) and Mears (1976). Values for Peclet numbers from 0.5 to 200 were used throughout the simulations. 

Simulations show that the radial and axial temperature and bulk concentration profiles are effectively not influenced by these modeling differences. Figure 9 shows the radial concentration profiles at = 0.38 and at the reactor outlet. Even with very high Peclet numbers, the differences between the radial concentration profile across the relatively small bed and the assumed uniform profile are minimal. Under typical operating conditions with small Peclet numbers, there is no benefit to increasing the number of radial collocation points, especially in light of the increased dimensionality of the resulting system. 

The model discretization or the number of collocation points necessary for accurate representation of the profiles within the reactor bed has a major effect on the dimensionality and thus the solution time of the resulting model. As previously discussed, radial collocation with one interior collocation point generally adequately accounts for radial thermal gradients without increasing the dimensionality of the system. However, multipoint radial collocation may be necessary to describe radial concentration profiles. The analysis of Section VI,E shows that, even with very high radial mass Peclet numbers, the radial concentration is nearly uniform and that the axial bulk concentration and radial and axial temperatures are nearly unaffected by assuming uniform radial concentration. Thus model dimensionality can be kept to a minimum by also performing the radial concentration collocation with one interior collocation point. 

Concerning packed bubble bed reactors, the evaluation of the Peclet number of the liquid-phase is important in order to decide if we have to use a plug- or backmixed-flow model. The liquid-phase can be considered well mixed if (Ramachandran and Chaudhari, 1980)... 

Here, it has to be noted that for calculating the Peclet number in fixed beds, the actual velocity has to be used, i.e. the interstitial velocity, which influences the degree of mixing. In slurry bubble column reactors, the real velocity of the fluid is the bubble velocity, which is much higher than the gas superficial velocity. The mean bubble rise velocity for a batch liquid is (eq (3.201))... 

Concerning packed bubble bed reactors, the evaluation of the Peclet number of the liquid phase is important in order to decide if we have to use a plug- or backmixed-flow model. For the specified Reynolds number, the Peclet number for the liquid phase using the Stiegel-Shah correlation (eq. (3.422)) is 0.15, much lower than in the trickle bed, which was expected as the backmixing in the liquid phase in packed bubble bed reactors is relatively high. The liquid phase can be considered to be well mixed if (Ramachandran, and Chaudhari, 1980) (eq. (3.423))... 

Figure 17.1. Residence time distributions of some commercial and pilot fixed bed reactors. The variance, the equivalent number of CSTR stages, and the Peclet number are given for each.

The ratio, L/D, of length to diameter of a packed tube or vessel has been found to affect the coefficient of heat transfer. This is a dispersion phenomenon in which the Peclet number, uL/Ddisp, is involved, where D Sp is the dispersion coefficient. Some 5000 data points were examined by Schliinder (1978) from this point of view although the effect of L/D is quite pronounced, no dear pattern was deduced. Industrial reactors have LID above 50 or so Eqs. (6) and (7) of Table 17.18 are asymptotic values of the heat transfer coefficient for such situations. They are plotted in Figure 17.36(b). 

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