shown to be small, and their effect upon transients is imperceptible 1n the cases treated.-ivThese studies show that mathematical models which adequately simulate chemical reactors in unsteady state must incorporate the following phenomena: the thermal capacity of the packing, finite (as distinguished from infinite) rates of heat transfer between fluid and packing, and coupling between concentration and temperature, Both of the one-dimensional models studied include these phenomena and yet are simple enough to be useful in further studies of reactor dynamics and in studies of reactor control.
L o c a l l y linearized equations for dynamic disturbances from a steady state have been solved analytically for a n adiabatic chemical reactor with chemically inert packing and a homogeneous chemical reaction. T h e solutions, in the form of transfer functions, include the effects of heat capacity of the packing and heat transfer between the packing and the fluid, as well as the coupling effects of chemical reaction and the associated h e a t generation.Several limiting cases o f the transfer functions are discussed and the tronsfer functions are simplified by on approximation of the temperature dependence of the reaction rate. Frequency responses and step responses of the reactor are calculated from the transfer functions.T h e first mathematical analyses of thermal dynamics in a packed bed (heat regenerator) were made by Anzelius (3) and Nusselt ( 1 4 ) and independently by Schumann ( 1 7 ) . Their results were confirmed experimentally by Furnas (11) and extended mathematically by Brinkley ( 6 ) to a packed catalytic reactor in which the rate of heat generation i n the packing i s a linear function of packing temperature.Amundson (2) expanded Brinkley's solution to account for radial gradients within the bed.T h e s e analyses do not include, however, an important feature of many chemical reactors: the interaction of temperature and reactant concentration. For most chemical reactions, the reaction rate depends upon both temperature and reactant concentration, and the rate of heat generation does also. T h e s e dependences are usually nonlinear.This temperature and concentration interaction has, however, been included in a study of the dynamics of unpacked reactors by Bilous and Amundson (5). Furthermore, Douglas and Eagleton (10) have given analytic solutions for the dynamics of adiabatic unpacked reactors. T h e s e results for unpacked reactors cannot b e readily extended, however, t o packed reactors, because it h a s been shown that the exchange of heat between the fluid and the packing h a s important effects upon the dynamics of packed reactors In the present paper, an approximate analytic solution to the dynamics of an adiabatic packed reactor i s presented. This solution shows the dynamic effects of packing heat (19,20,9).J . E. Crider is with Shell Development Company, Houston, Texas.capacity, heat transfer between the packing and the fluid, and interaction of concentration and temperature through the chemical reaction and associated heat generation. This analytic solution applies only to small disturbances from steady s t a t e operating conditions.The reactor considered here is not a catalytic reactor, however; it i s a reactor that contains chemically inert packing and a homogeneous chemical reaction occurring in the fluid phase, which i s here a liquid. Such reactors, which have recently been studied experimentally (20, 1 9 ,
Expressions are derived for an effective wall heat transfer coefficient useful in one-dimensional representations of heat transport in packed beds. These expressions are obtained with two mathematical models of a cylindrical packed bed: a partial differential model and a finite stage model. These expressions relate the effective wall heat transfer coefficient, which is a local coefficient, to the actual wall heat transfer coefficient and the bed diameter (and the radial Peclet number in the partial differential model) i n regions of the bed where similar temperature profiles obtain. These relations involve a single dominant root of an equation characteristic of the heat balance equations of each model.From these relations, expressions for an effective thermal resistance of the bed are obtained. For each model, this bed resistance is found to be an approximately linear function of the bed diameter and to be rather insensitive to the actual wall heat transfer coefficient. For each model, an approximate bed resistance is found that is not dependent upon the actual wall heat transfer coefficient and with which the effective wall heat transfer coefficient can be estimated with an error of less than 7%.
The sphere can be covered by any of an infinite number of tiling sets of equilateral spherical quadrilaterals (diamonds). Five of these tiling sets have practical use for texture mapping application. Points on the sphere can be described by intersections of geodesics, which provide coordinate values in a new coordinate system, defined for each tiling set. Each of the diamonds can be subdivided by a grid of coordinate geodesics to pixel level and so can be directly mapped to and from a texture array. The diamonds can also be subdivided into spherical quadrilaterals that can be approximated by pairs of triangles for fast rendering in a graphics system. Because coordinates in the new system are readily converted to and from Cartesian coordinates, diamonds can be used easily in interactive graphics and ray-tracing applications.
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