Plug flow reactors equations, initial conditions

Finally, the mathematical model of a plug flow reactor consists of a system of ordinary differential equations with initial conditions. 

Note that all the conditions are known at one time, t = 0. Thus it is possible to calculate the function on the right-hand side at f = 0 to obtain the derivative there. This makes the set of equations initial value problems. The equations are ordinary differential equations because there is only one independent variable. Any higher-order ordinary differential equation can be turned into a set of first-order ordinary differential equations they are initial value problems if all the conditions are known at the same value of the independent variable [Finlayson, 1980, 1997 (p. 3-54), 1990 (Vol. BI, p. 1-55)]. The methods for initial value problems are explained here for a single equation extension to multiple equations is straightforward. These methods are used when solving plug-flow reactors (Chapter 8) as well as time-dependent transport problems (Chapters 9-11). 

Equation 7.5.16 is the dimensionless, differential energy balance equation for cyhndrical tubular flow reactors, relating the temperature, 0, to the extents of the independent reactions, Z s, and P/Pq as functions of space time t. To design a plug-flow reactor, we have to solve design equations (Eq. 7.1.1), the energy balance equation (Eq. 7.5.16), and the momentum balance (Eq. 7.5.12), simultaneously subject to specified initial conditions. 

We solve the design equations simultaneously with the energy balance equation, subject to the initial condition that at t = 0, the extents of aU the independent reactions and the dimensionless temperature are specified. Note that we solve fiiese equations for a specified value of T t (or reactor volume). The reaction operating curves of plug-flow reactors with side injection are the final value of Z s and 9 for different values of Ttot-... 

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-... 

Ideal plug flow behavion The behavior of the reactor (residence time distribution) should be such that we can consider the fixed bed as an ideal plug flow reactor (PFR). If this condition is fidfiDed we can use the (relatively simple) equations valid for a PFR that correlate the conversion with the rate constant, residence time, and initial reactant concentration. For example, we can determine the rate constant for a reaction with order n by Eq. (4.10.26) if we have measured the conversion of reactant A at a given value of the residence time by ... 

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... 

The reactor model adopted for describing the lab-scale experimental setup is an isothermal homogeneous plug-flow model. It is composed of 2NP + 2 ordinary differential equations of the type of Equation 16.11 with the initial condition of Equation 16.12, NP + 3 algebraic equations of the type of Equation 16.13, and the catalytic sites balance (Equation 16.14) ... 

The experimental methods and the quantification of data from a TS-CST-SSR is fully delineated by the above treatment and its trivial but interesting extension to cases where the initial concentration of adsorbent in the sweeping stream is not zero. The same equations can be used to quantify commonly available TPD results as long as the reactor configuration and run conditions conform to the assumptions used in the derivations presented here. Since most TPD experiments are carried out in plug flow configurations one can take Vv=0, but the volume expansion factors remain necessary if one intends to calculate the correct quantities desorbed from the sample and/or to quantify the kinetics and thermodynamics of adsorption/desorption. 

The initial and boundary conditions that apply to this equation depend on whether one is dealing with a pulse or a step stimulus and the characteristics of the system at the tracer injection and monitoring stations. At each of these points the tubular reactor is characterized as closed or open, depending on whether or not plug flow into or out of the test section is assumed. A closed boundary is one at which there is plug flow outside the test section an open boundary is one at which the same dispersion parameter characterizes the flow conditions within and adjacent to the test section. There are then four different possible sets of boundary conditions on equation (11.1.29), depending on whether a completely open or completely closed vessel, a closed-open vessel, or an open-closed vessel is assumed. Different solutions will be obtained for different boundary conditions. Fortunately, for small values of the dispersion parameter, the numerical differences between the various solutions will be small. 

The dimensionless equations with the boundary and initial conditions governing the unsteady mass and heat treuisfer in the one-dimensional plug flow tubular reactor with high heat diffusivity ceui be written in the form... 

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