Tubular with axial dispersion

Dynamics of an Isothermal Tubular Reactor with Axial Dispersion... 

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. 

Chemical Kinetics, Tank and Tubular Reactor Fundamentals, Residence Time Distributions, Multiphase Reaction Systems, Basic Reactor Types, Batch Reactor Dynamics, Semi-batch Reactors, Control and Stability of Nonisotheimal Reactors. Complex Reactions with Feeding Strategies, Liquid Phase Tubular Reactors, Gas Phase Tubular Reactors, Axial Dispersion, Unsteady State Tubular Reactor Models... 

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). 

In the case of tubular reactor with axial dispersion it is possible to express the steady state activity profile as a fxmction of 6(x)... 

To illustrate the features of our proposed algorithm, we apply it to the case of a tubular reactor with axial dispersion, where an elementary first order irreversible exothermic reaction takes place A —[8]. The steady state problem is described two nonlinear partial differential equations (PDEs), which in dimensionless form are ... 

Ordinary differential equations govern systems that vary either with time or space, but not both. Examples are equations that govern the dynamics of a CSTR or the steady state of mbular reactors. Both the dynamics of a CSTR and the steady state of a plug-flow reactor are governed by first-order ordinary differential equations with prescribed initial conditions. The steady-state tubular reactors with axial dispersion are governed by a second-order differential equation with the boundary conditions spec-... 

Lynn ei ai (1970) obtain the optimal temperature profiles for a tubular reactor with axial dispersion (Lynn et ai, 1970). Weighted residual techniques were used to solve the state and adjoint differential equations which result from the application of Pontryagin s maximum principle to the optimal control problem. 

The nonlinearity of chemical processes received considerable attention in the chemical engineering literature. A large number of articles deal with stand-alone chemical reactors, as for example continuously stirred tank reactor (CSTR), tubular reactor with axial dispersion, and packed-bed reactor. The steady state and dynamic behaviour of these systems includes state multiplicity, isolated solutions, instability, sustained oscillations, and exotic phenomena as strange attractors and chaos. In all cases, the main source of nonlinearity is the positive feedback due to the recycle of heat, coupled with the dependence of the reaction rate versus temperature. 

Since the residence time varies between the channels, a tracer pulse at the inlet of the microreactor will be broadened similar to the case for a tubular reactor with axial dispersion. As a first approximation, the relative standard deviation in the residence time is twice the relative standard deviation in the channel diameter [9] ... 

Danckwarts model equation for tubular vessel with axial dispersion is... 

The first of these is the residence time distribntion (RTD) method. This technique allows us to classify the dispersion properties of a constituent in a chemical reactor, with reference to ideal behaviors of simple reactors. The RTD theory does not explicitly associate a RTD with a flow configuration inside the reactor. We examine this particular issue when the flow is turbulent, by considering successively the cases of a tubular reactor with axial dispersion and of a continuous stirred tank reactor (CSTR). Matching up the dimertsiorts of the reactor with the mean residence time and the velocity and length scales of turbulence allows the determirration of the hydrodynamic conditions associated with either type of reactor, for which the RTD laws are recovered, using the trrrbulent dispersion concepts introduced in Chapter 8. 

Modeling RTD via a turbulent diffusion approach cases of a tubular reactor with axial dispersion and of a CSTR... 

We recover the classic form of the RTD for a tubular reactor with axial dispersion ... 

The hypothesis [9.4], under whieh we have just recovered the RTD law for a tubular reactor with axial dispersion, can be restated in the form of a bounding of the turbulence intensity between two values comparing the dimensions of the reactor to the integral scale of turbulence ... 

We have determined in this section the hydrodynamic conditions on a turbulent flow that lead to the RTD laws of a CSTR or a tubular reactor with axial dispersion. The reverse analysis, for deriving turbulence characteristics of a flow from the measurement of a RTD, should be used with caution if it caimot be positively asserted that the flow is turbulent. A laminar flow (e.g. a Poiseuille flow) in a tubular reactor also produces an axial dispersion measured by a RTD, because the product injected on the pipe axis is carried faster than that injected near the walls. Clearly, it would be meaningless to derive turbulence characteristics from the measured RTD law." ... 

The methods listed in Sections 5.1.1-5.1.7 are illustrated by a tubular reactor with a first-order reaction and laminar flow. Models for species, heat, and momentum have been formulated and simplified. In addition to showing the methods, we discuss the assumptions in a traditional 1D lumped-parameter model for a tubular reactor with axial dispersion,... 

The assumptions in a lumped-parameter model are not always transparent. For example, in the ID model for a tubular reactor with axial dispersion (Equations (5.36) and (5.37), repeated here for convenience)... 

Determine the yield of a second-order reaction in an isothermal tubular reactor governed by the axial dispersion model with Pe = 16 and kt = 2. 

Fialova, M., Redlich, K. H., and Winkler, K., Axial dispersion of the liquid phase in vertical tubular contactors with static mixers, Collect. Czech. Chem. Commun., 51, 1925-1932 (1986). 

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 TUBULAR REACTOR WITH AXIAL MIXING THE DISPERSION MODEL... 

Note that setting one of the terms on the left side of the equation equal to zero yields either the batch reactor equation or the steady-state PFTR equation. However, in general we must solve the partial differential equation because the concentration is a function of both position and time in the reactor. We will consider transients in tubular reactors in more detail in Chapter 8 in connection with the effects of axial dispersion in altering the perfect plug-flow approximation. 

---