Equations tubular reactor

Another view is given in Figure 3.1.2 (Berty 1979), to understand the inner workings of recycle reactors. Here the recycle reactor is represented as an ideal, isothermal, plug-flow, tubular reactor with external recycle. This view justifies the frequently used name loop reactor. As is customary for the calculation of performance for tubular reactors, the rate equations are integrated from initial to final conditions within the inner balance limit. This calculation represents an implicit problem since the initial conditions depend on the result because of the recycle stream. Therefore, repeated trial and error calculations are needed for recycle... 

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

This model v/as used by Atwood et al (1989) to compare the performance of 12 m and 1.2 m long tubular reactors using the UCKRON test problem. Although it was obvious that axial conduction of matter and heat can be expected in the short tube and not in the long tube, the second derivative conduction terms were included in the model so that no difference can be blamed on differences in the models. The continuity equations for the compounds was presented as ... 

Figure 5 depicts the effect of calcination temperature on subsequent catalyst activity after reduction at 300°C (572°F). Activity was measured in laboratory tubular reactors operating at 1 atm with an inlet gas composition of 0.40% CO, 25% N2, and 74.6% H2, and an inlet temperature of 300°C. Conversion of CO is measured and catalyst activity is expressed as the activity coefficient k in the first order equation ... 

EQUATIONS FOR A PLUG FLOW POLYMER TUBULAR REACTOR WITH BRANCHING KINETICS... 

Chapter 2 developed a methodology for treating multiple and complex reactions in batch reactors. The methodology is now applied to piston flow reactors. Chapter 3 also generalizes the design equations for piston flow beyond the simple case of constant density and constant velocity. The key assumption of piston flow remains intact there must be complete mixing in the direction perpendicular to flow and no mixing in the direction of flow. The fluid density and reactor cross section are allowed to vary. The pressure drop in the reactor is calculated. Transpiration is briefly considered. Scaleup and scaledown techniques for tubular reactors are developed in some detail. 

The dAc/dz term is usually zero since tubular reactors with constant diameter are by far the most important application of Equation (3.7). For the exceptional case, we suppose that Afz) is known, say from the design drawings of the reactor. It must be a smooth (meaning differentiable) and slowly varying function of z or else the assumption of piston flow will run into hydrodynamic as well as mathematical difficulties. Abrupt changes in A. will create secondary flows that invalidate the assumptions of piston flow. 

The first of the relations in Equation (4.9) is valid for any flow system. The second applies specifically to a CSTR since p = pout- It is not true for a piston flow reactor. Recall Example 3.6 where determination of t in a gas-phase tubular reactor required integrating the local density down the length of the tube. 

The solution of Equations (5.23) or (5.24) is more straightforward when temperature and the component concentrations can be used directly as the dependent variables rather than enthalpy and the component fluxes. In any case, however, the initial values, Ti , Pi , Ui , bj ,... must be known at z = 0. Reaction rates and physical properties can then be calculated at = 0 so that the right-hand side of Equations (5.23) or (5.24) can be evaluated. This gives AT, and thus T z + Az), directly in the case of Equation (5.24) and imphcitly via the enthalpy in the case of Equation (5.23). The component equations are evaluated similarly to give a(z + Az), b(z + Az),... either directly or via the concentration fluxes as described in Section 3.1. The pressure equation is evaluated to give P(z + Az). The various auxiliary equations are used as necessary to determine quantities such as u and Ac at the new axial location. Thus, T,a,b,. .. and other necessary variables are determined at the next axial position along the tubular reactor. The axial position variable z can then be incremented and the entire procedure repeated to give temperatures and compositions at yet the next point. Thus, we march down the tube. 

Molecules must come into contact for a reaction to occur, and the mechanism for the contact is molecular motion. This is also the mechanism for diffusion. Diffusion is inherently important whenever reactions occur, but there are some reactor design problems where diffusion need not be explicitly considered, e.g., tubular reactors that satisfy the Merrill and Hamrin criterion. Equation (8.3). For other reactors, a detailed accounting for molecular diffusion may be critical to the design. 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

Based on the results of these researchers the tubular reactor in this study has been described by the axisymmetric model using effective diffusivities given by Equation 20. 

Under steady-state conditions, variations with respect to time are eliminated and the steady-state model can now be formulated in terms of the one remaining independent variable, length or distance. In many cases, the model equations now result as simultaneous first-order differential equations, for which solution is straightforward. Simulation examples of this type are the steady-state tubular reactor models TUBE and TUBED, TUBTANK, ANHYD, BENZHYD and NITRO. 

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. 

With respect to reaction rates, an element of fluid will behave in the ideal tubular reactor, in the same way, as it does in a well-mixed batch reactor. The similarity between the ideal tubular and batch reactors can be understood by comparing the model equations. 

Equating the time of passage through the tubular reactor to that of the time required for the batch reaction, gives the equivalent ideal-flow tubular reactor design equation as... 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

The tubular reactor, steady-state design equation is of interest here. The dimensional and dimensionless forms are compared for an nth-order reaction. 

At steady-state conditions, the mass balance design equations for the ideal tubular reactor apply. These equations may be expressed as... 

Two-phase mass transfer and heat transfer without phase change are analogous, and the results of mass-transfer studies can be used to help clarify the heat-transfer problems. Cichy et al. (C5) have formulated basic design equations for isothermal gas-liquid tubular reactors. The authors arranged the common visually defined flow patterns into five basic flow regimes, each... 

Consider the segment of tubular reactor shown in Figure 8.3. Since the fluid composition varies with longitudinal position, we must write our material balance for a reactant species over a different element of reactor (dVR). Moreover, since plug flow reactors are operated at steady state except during start-up and shut-down procedures, the relations of major interest are those in which the accumulation term is missing from equation 8.0.1. Thus... 

The summation involves the effluent molal flow rates. This equation and equation 10.4.2 must be solved simultaneously in order to determine the tubular reactor size and to determine the manner in which the heat transfer requirements are to be met. For either isothermal or adiabatic operation one of the three terms in equation 10.4.7 will drop out, and the analysis will be much simpler than in the general case. In the illustrations which follow two examples are treated in detail to indicate the types of situations that one may encounter in practice and to indicate in more detail the nature of the design calculations. 

Example Consider the equation for convection, diffusion, and reaction in a tubular reactor. 

In order to implement the PDF equations into a LES context, a filtered version of the PDF equation is required, usually denoted as filtered density function (FDF). Although the LES filtering operation implies that SGS modeling has to be taken into account in order to capture micromixing effects, the reaction term remains closed in the FDF formulation. Van Vliet et al. (2001) showed that the sensitivity to the Damkohler number of the yield of competitive parallel reactions in isotropic homogeneous turbulence is qualitatively well predicted by FDF/LES. They applied the method for calculating the selectivity for a set of competing reactions in a tubular reactor at Re = 4,000. 

In estimating the length of the tubes, the mass of catalyst, W, is calculated from the design equation for a tubular reactor as ... 

Ethyl formate is to be produced from ethanol and formic acid in a continuous flow tubular reactor operated at a constant temperature of 303 K (30°C). The reactants will be fed to the reactor in the proportions 1 mole HCOOH 5 moles C2H5OH at a combined flowrate of 0.0002 m3/s (0.72 m3/h). The reaction will be catalysed by a small amount of sulphuric acid. At the temperature, mole ratio, and catalyst concentration to be used, the rate equation determined from small-scale batch experiments has been found to be ... 

A mixture of 5 mol butanol per mol acetic acid flows to a tubular reactor at the rate of 45.4 kg/h or 10 liters/h at 100 C. It also contains 0.032 wt% sulfuric acid as catalyst. Inlet concentration of the acid is 1.744 gmol/liter. The rate equation (Leyes Ofchmer, 7nd Eng Chem 37 968, 1945) is... 

Differential momentum equation, 11 741 Differential plug-flow tubular reactors, 25 269... 

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