Models ideal plug flow

Non-ideal reactors are described by RTD functions between these two extremes and can be approximated by a network of ideal plug flow and continuously stirred reactors. In order to determine the RTD of a non-ideal reactor experimentally, a tracer is introduced into the feed stream. The tracer signal at the output then gives information about the RTD of the reactor. It is thus possible to develop a mathematical model of the system that gives information about flow patterns and mixing. 

The component mass balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

Except for the case of an ideal plug flow reactor, different fluid elements will take different lengths of time to flow through a chemical reactor. In order to be able to predict the behavior of a given piece of equipment as a chemical reactor, one must be able to determine how long different fluid elements remain in the reactor. One does this by measuring the response of the effluent stream to changes in the concentration of inert species in the feed stream—the so-called stimulus-response technique. In this section we will discuss the analytical form in which the distribution of residence times is cast, derive relationships of this type for various reactor models, and illustrate how experimental data are treated in order to determine the distribution function. 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

Explain carefully the dispersed plug-flow model for representing departure from ideal plug flow. What are the requirements and limitations of the tracer response technique for determining Dispersion Number from measurements of tracer concentration at only one location in the system Discuss the advantages of using two locations for tracer concentration measurements. 

The modeling of real immobilized-enzyme column reactors, mainly the fluidized-bed type, has been described (Emeiy and Cardoso, 1978 Allen, Charles and Coughlin, 1979 Kobayashi and Moo-Young, 1971) by mathematical models based on the dispersion concept (Levenspiel, 1972), by incorporation of an additional term to account for back-mixing in the ideal plug-flow reactor. This term describes the non-ideal effects in terms of a dispersion coefficient. 

Deviation from the ideal plug flow can be described by the dispersion model, which uses the axial eddy diffusivity (m s ) as an indicator of the degree of mixing in the flow direction. If the flow in a tube is plug flow, the axial dispersion is zero. On the other hand, if the fluid in a tube is perfectly mixed, the axial dispersion is infinity. For turbulent flow in a tube, the dimensionless Peclet number (Pe) deflned by the tube diameter (v dlE-Q is correlated as a function of the Reynolds number, as shown in Figure 10.3 [3] dz is the axial eddy diffusivity, d is the tube diameter, and v is the velocity of liquid averaged over the cross section of the flow channel. 

Trickle-bed reactors are widely used in the oil industry because of reliability of their operation and for the predictability of their large-scale performance from tests on a pilot-plant scale. Further advantages of trickle-bed reactors are as follows The flow pattern is close to plug flow and relatively high reaction conversions may be achieved in a single reactor. If warranted, departures from ideal plug flow can be treated by a dispersed plug-flow model with a dispersion coefficient for each of the liquid and gas phases. 

The deviation from ideal plug flow due to the axial mixing can be described by the dispersion model (Levenspiel, 1972). Let s look at the differential element with a thickness dx in a holding tube as shown in Figure 8.1. The basic material balance for the microorganisms suspended in the medium is... 

The reactive transport of contaminants in FePRBs has been modeled using several approaches [179,184,186,205-208]. The simplest approach treats the FePRB as an ideal plug-flow reactor (PFR), which is a steady-state flow reactor in which mixing (i.e., dispersion) and sorption are negligible. Removal rates (and therefore required wall widths, W) can be estimated based on first-order contaminant degradation and residence times calculated from the average linear groundwater velocity [Eq. (27)]. The usefulness of... 

In the literature many studies on LDPE tubular reactors are found (2-6).All these studies present models of the tubular reactor, able to predict the influence, on monomer conversion and temperature profiles, of selected variables such as initiator concentration and jacket temperature. With the exception of the models of Mullikin, that is an analog computer model of an idealized plug-flow reactor, and of Schoenemann and Thies, for which insufficient details are given, all the other models developed so far appear to have some limitations either in the basic hypotheses or in the fields of application. 

Remark This zeroth-order model is the ideal plug-flow model.) To obtain the averaged equation to order p, we write... 

This chapter discusses four methods of gas phase ceramic powder synthesis by flames, fiunaces, lasers, and plasmas. In each case, the reaction thermodynamics and kinetics are similar, but the reactor design is different. To account for the particle size distribution produced in a gas phase synthesis reactor, the population balance must account for nudeation, atomistic growth (also called vapor condensation) and particle—particle segregation. These gas phase reactors are real life examples of idealized plug flow reactors that are modeled by the dispersion model for plve flow. To obtain narrow size distribution ceramic powders by gas phase synthesis, dispersion must be minimized because it leads to a broadening of the particle size distribution. Finally the gas must be quickly quenched or cooled to freeze the ceramic particles, which are often liquid at the reaction temperature, and thus prevent further aggregation. 

Some researchers use plug-flow reactors (PFRs), also known as packed bed reactors or column reactors (if run vertically) to model natural systems. In an ideal plug-flow or column reactor, fluid is pumped or drained through a packed bed of mineral grains and every fluid packet is assumed to have the same residence or contact time (Hill, 1977). The residence time equals the ratio of the pore volume of the reactor (Vo) divided by flow rate Q. With no volume change in the reaction, radial flow, or pooling of fluid in the reactor (Laidler, 1987), the outlet concentration varies from the inlet concentration according to ... 

If the piping only contributes to the dead time of the plant the delay can be described by a pipe model assuming an ideal plug-flow. 

In Fig. 6.15 two different models for parameter estimation are used and the resulting simulated concentration profiles are compared with the measurements. In one case ideal plug-flow (Eq. 6.116) and in the other axial dispersive flow (Eq. 6.117) is assumed for the pipe system, while both models use the C.S.T. model (Eq. 6.121) to describe the detector system. Figure 6.15 shows that the second model using axial dispersion provides an excellent fit for this set-up, while the other cannot predict the peak deformation. Because of the asymmetric shape a model without a tank would also be inappropriate. 

The main contribution from the work of Luo [95, 96] was a closure model for binary breakage of fluid particles in fully developed turbulence flows based on isotropic turbulence - and probability theories. The author(s) also claimed that this model contains no adjustable parameters, a better phrase may be no additional adjustable parameters as both the isotropic turbulence - and the probability theories involved contain adjustable parameters and distribution functions. Hagesaether et al [49, 50, 51, 52] continued the population balance model development of Luo within the framework of an idealized plug flow model, whereas Bertola et al [13] combined the extended population balance module with a 2D algebraic slip mixture model for the flow pattern. Bertola et al [13] studied the effect of the bubble size distribution on the flow fields in bubble columns. An extended k-e model was used describing turbulence of the mixture flow. Two sets of simulations were performed, i.e., both with and without the population balance involved. Four different superficial gas velocities, i.e., 2,4,6 and 8 (cm/s) were used, and the superficial liquid velocity was set to 1 (cm/s) in all the cases. The population balance contained six prescribed bubble classes with diameters set to = 0.0038 (m), d = 0.0048 (m), di = 0.0060 (m), di = 0.0076 (m), di = 0.0095 (m) and di = 0.0120 (m). 

The axial dispersion terms may be required to account for the mixing phenomena created by a non-ideal flow. However, the ideal plug flow model is often appropriate for packed bed reactors because the axial mixing is negligible compared to the convective flux for many processes. 

Recently, Agrawal et al. (2006) had modeled the LDPE reactor as an ideal plug flow reactor and presented all the model equations and parameters for use by researchers. The model equations include ordinary differential equations for overall and component mass balances, energy balance and momentum balance. The reactor model of Agrawal et al. (2006) is adopted, and cost expressions and economic objectives are... 

Consequently, we see that Equation (1-11) applies equally well to our model of tubular reactors of variable and constant cross-sectional area, although it is doubtful that one would find a reactor of the shape shown in Figure 1-11 unless it were designed by Pablo Picasso. The conclusion drawn from the application of the design equation to Picasso s reactor is an imponant one the degree of completion of a reaction achieved in an ideal plug-flow reactor tPFR) does not depend on its shape, only on its total volume. 

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