Inlet conditions differential reactors

Initial conditions for the system of differential equations shown before are given by the values of state variables known at the inlet of the reactor ... 

These are first-order ordinary differential equations that have two initial conditions at the inlet to the reactor,... 

To determine the kinetic parameters, we use the differential reactor. In this continuous flow system, the variation in concentration between the inlet and outlet of the reactor should be small and finite. The conversions should be around 5-10%. Under these conditions, the diffusive and mass transfer effects are avoided, assuring a kinetic regime for the determination of the kinetic parameters. Unlike the case of the batch system, the spatial time and consequently, the inlet flow and the mass or volume of the reactor are varied. Therefore, the reaction rate is directly determined. 

Perfectly mixed stirred tank reactors have no spatial variations in composition or physical properties within the reactor or in the exit from it. Everything inside the system is uniform except at the very entrance. Molecules experience a step change in environment immediately upon entering. A perfectly mixed CSTR has only two environments one at the inlet and one inside the reactor and at the outlet. These environments are specifled by a set of compositions and operating conditions that have only two values either bi ,..., Ti or Uout, bout, , Pout, Tout- When the reactor is at a steady state, the inlet and outlet properties are related by algebraic equations. The piston flow reactors and real flow reactors show a more gradual change from inlet to outlet, and the inlet and outlet properties are related by differential equations. 

If we are determining the response of the series of stirred tank reactors to a step change in inlet tracer concentration from 0 to Cq at time zero, the initial condition for this differential equation is... 

Figure 12.2 Application of the nonreflecting bonndary conditions (left part) and standard von Nenmann boundary conditions (right part) in the problem on pressure disturbance propagation in a flow reactor with open left and right boundaries. Time instants (a) 10 /rs, (b) 20 ps, (c) 30 /rs. Flow velocity at the inlet 40 m/s, po = 0.1 MPa, To = 300 K, fco = 9 J/kg, lo = 2 mm. Initial pressure differential Ap/po = 0.5. The size of the computational domain is 3.3 x 2 cm...

Knowing all the conditions at the reactor inlet, integrate down the length of the reactor, using the ordinary differential equations given in Eqs. (5.3)—(5.5), until... 

Third, there may be a concentration gradient of reactants and products along the length of the catalyst bed. If the structure of the catalyst depends upon the composition of the gas phase, then an average of the various structures will be measured. There is little discussion of this topic in the literature of XAFS spectroscopy of working catalysts. An extreme example of structural variations within a sample is discussed in Section 6, where there is a discussion of XAFS spatially resolved spectra recorded to allow direct observation of the axial distribution of phases present. If the XAFS data are not measured with spatial resolution, then it is recommended that XAFS data be measured under differential conversion conditions. However, if the aim of the experiment is to relate the catalyst structure directly to that in some industrial catalytic processes, then differential conversion conditions will only reflect the structure of the catalyst at the inlet of the bed. To learn about the structure of the catalyst near the outlet of the bed, the reaction has to be conducted at high conversions. If it is anticipated that this operation will lead to variations in the catalyst structure along the bed, then the feed to the micro-reactor should be one that mimics the concentration of reactants toward the downstream end of the bed (i.e., products should be added to the reactants). 

In the differential flow reactor, the residence time is short so that the conversion remains small, usually a few per cent. This can be achieved in short beds and/or with high feed flow rates. Since the conversion is small all pellets operate approximately under the same conditions and variation of the volumetric flow rate can be neglected. The observed or apparent conversion rate follows directly from the measured inlet and outlet concentrations via a material balance over the bed ... 

This set of hyperbolic partial differential equations for the gasifier dynamic model represents an open or split boundary-value problem. Starting with the initial conditions within the reactor, we can use some type of marching procedure to solve the equations directly and to move the solution forward in time based on the specified boundary conditions for the inlet gas and inlet solids streams. 

Measure the incremental conversion of ethanol per mass of catalyst and calculate the initial reactant product conversion rate with units of moles per area per time as a function of total pressure at the reactor inlet. One calculates this initial rate of conversion of ethanol to products via a differential material balance, unique to gas-phase packed catalytic tubular reactors that operate under plug-flow conditions at high-mass-transfer Peclet numbers. Since axial dispersion in the packed bed is insignificant. 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

These coupled partial differential equations can be solved for the temperature and composition at any point in the catalyst bed by using numerical procedures to solve the corresponding difference equations. As boundary conditions, one needs to know the temperature and composition profile across the tube diameter at the reactor inlet. In addition, the solution must satisfy the requirements of cylindrical symmetry that... 

In this kind of models second-order differential terms appear in mass, energy, and momentum balances and two boundary conditions on the axial coordinate have to be imposed. The set of differential equations together with the boundary conditions represents a boundary value problem (BVP), since the constraints are imposed at the inlet and at the outlet of the tubular reactor. 

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