Packed bed heat exchanger models

Solution of Packed Bed Heat-Exchanger Models by Orthogonal Collocation Using Piecewise Cubic Hermite Functions... 

Dixon, A.G, Solution of Packed-Bed Heat-Exchanger Models by Orthogonal Collocation using Piecewise Cubic Hermite Functions (MRC Technical Summary Report 2116, Mathematics Research Center, Univ. of Wisconsin-Madison, 1980). 

The RFSG has a pair of regenerative packed-bed heat exchangers at the air inlet and product gas outlet, with foe gas flow reversing typically every 20 minutes. The heat exchangers are modelled using a one-dimensional flnite diflerence technique, after Schmidt and Willmott [4]. 

Due to the Langrangian formulation applied to the solid phase, the use of an effective thermal conductivity as usually applied to porous media is not necessary. In a packed bed heat is transported between solid particles by radiation and conduction. For materials with low thermal conductivity, such as wood, conduction contributes only to a minor extent to the overall heat transport. Furthermore, heat transfer due to convection between the primary air flow through the porous bed and the solid has to be taken into account. Heal transfer due to radiation and conduction between the particles is modelled by the exchange of heat between a particle and its neighbours. The definition of the neighbours depends on the assembly of the particles on the flow field mesh. 

Show that the steady- state and dynamic models for a double-pipe, counter-current heat exchanger can have the same form as the model of a packed bed absorber. Discuss the assumptions inherent in both the heat exchanger and the absorber models which might lead to significant differences in the kinds of model equations used to describe each system. 

An individual reactor in a train can, of course, run away because of its own internal disturbances, but it can also send disturbances ahead to affect the stability of downstream reactors. If there is no intermediate heat exchange, the train acts like a single adiabatic reactor, subject only to disturbances in its feed. Within the stages there is backmixing, but there is none otherwise. A reactor series can also serve as a model for packed-bed reactors, if the upstream propagation implied by Equation (12) is not realistic. 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

A numerical model is presented to describe the thermal conversion of solid fuels in a packed bed. For wood particles it can be shown, that a discretization of the particle dimensions is necessary to resolve the influence of heat and mass transfer on the conversion of the solid. Therefore, the packed bed is described as a finite number of particles interacting with the surrounding gas phase by heat and mass transfer. Thus, the entire process of a packed bed is view as the sum of single particle processes in conjunction with the interaction of the gas flow in the void space of a packed bed. Within the present model, neighbour particles exchange heat due to conduction and radiation with each other. 

It is very well possible that the axial heat conduction in the wall has an influence on the temperature in the region close to the wall. A detailed description of the temperature in the bed as a function of the radial and axial positions can be obtained by extending the model to two dimensions. However, it is also possible to get more insights into the behaviour by adjusting the one-dimensional model. Instead of accounting for the heat capacity of the wall in the accumulation term, an extra energy balance for the wall is included in the model, in which heat is exchanged with the packed bed ... 

The present chapter provides an overview of several numerical techniques that can be used to solve model equations of ordinary and partial differential type, both of which are frequently encountered in multiphase catalytic reactor analysis and design. Brief theories of the ordinary differential equation solution methods are provided. The techniques and software involved in the numerical solution of partial differential equation sets, which allow accurate prediction of nonreactive and reactive transport phenomena in conventional and nonconventional geometries, are explained briefly. The chapter is concluded with two case studies that demonstrate the application of numerical solution techniques in modeling and simulation of hydrocar-bon-to-hydrogen conversions in catalytic packed-bed and heat-exchange integrated microchannel reactors. 

While lumped parameter models are normally used to describe processes, many important process units are inherently distributed parameter, that is, the output variables are functions of both time and position. Hence, their process models contain one or more partial differential equations. Pertinent examples include shell-and-tube heat exchangers, packed-bed reactors, packed columns, and long pipelines carrying compressible gases. In each of these cases, the output variables are a function of distance down the tube (pipe), height in the bed (column), or some other measure of location. In some cases, two or even three spatial variables may be considered for example, concentration and temperature in a tubular reactor may depend on both axial and radial positions, as well as time. 

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