Transport Processes in One Dimension

Chemical engineering processes involve the transport and transfer of momentum, energy, and mass. Momentum transfer is another word for fluid flow, and most chemical processes involve pumps and compressors, and perhaps centrifuges and cyclone separators. Energy transfer is used to heat reacting streams, cool products, and run distillation columns. Mass transfer involves the separation of a mixture of chemicals into separate streams, possibly nearly pure streams of one component. These subjects were unified in 1960 in the first edition of the classic book. Transport Phenomena (Bird et al., 2002). This chapter shows how to solve transport problems that are one-dimensional that is, the solution is a function of one spatial dimension. Chapters 10 and 11 treat two- and three-dimensional problems. The one-dimensional problems lead to differential equations, which are solved using the computer. 

Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial locations At one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus methods have been developed especially for boundary value problems. There is a baseball analogy that illustrates the difference. If you are at bat and there is no one on base, you can hit and run as fast as you can. This is like an initial value problem. If you are at bat and someone is on first base, you can hit, but the way you run is influenced by what the runner ahead of you does. This is like a boundary value problem. 

This chapter illustrates heat transfer, diffusion, diffusion with reaction, and flow in pipes, considering steady and unsteady processes. These problems are all solved using FEMLAB , which is easy to use for these applications. Details of the finite element 

Introduction to Chemical Engineering Computing, by Bruce A. Finlayson Copyright 2006 John Wiley Sons, Inc. 

---

In many separation processes (chromatography, countercurrent distribution, field-flow fractionation, extraction, etc.), the transport of components, in one dimension at least, occurs almost to the point of reaching equilibrium. Thus equilibrium concentrations often constitute a good approximation to the actual distribution of components found within such systems. Equilibrium concepts are especially crucial in these cases in predicting separation behavior and efficacy. 

The one-dimensional model is by no means descriptive of everything that goes on in the reactor, because it provides calculated temperatures, concentrations, pressures, and so on only in one dimension — lengthwise, down the axis of the tube. Actually, transport processes and diffusion cause variations and gradients not only axially but also radially within tubes and within individual catalyst pellets. Furthermore, the reactor may not actually operate at steady-state, and so time might also be included as a variable. All of these factors can be described quite easily by partial differential equations in as many as four dimensions (tube length, tube radius, pellet radius, and time). 

The flux of chemicals to and from surfaces depends on the magnitude of forces causing molecular movement and on the dimensionality of the system. Molecules in solution are transported by the mean motion of water, the advection process (Fischer et al., 1979). Molecules also move relative to the water by diffusion, in response to concentration gradients. For ions, electrostatic forces that contribute to movement are also experienced in regions of changing electrical potential. These three forces are incorporated in the following equation for flux (in one dimension) of a migrating chemical species (Newman, 1973, p. 301) ... 

Within this book, two different approaches of geochemical models are presented. This paper is an attempt to model the transport and reaction of all major redox components observed. At current stage, it is possible to model the interaction of a larger number of participating redox-processes in one space dimension only. In Massmann and... 

A mathematical model for DEFC was proposed by Pramanik and Basu describing different overpotentials [191]. The assumptions of their model are (i) the anode compartment considered as a well-mixed reactor, (ii) 1 bar pressure maintained both at the anode and cathode compartments, (iii) the transport processes are modelled in one dimension. The model accounts for Butler-Volmer-based descriptions of the ethanol electrooxidation mechanisms, diffusive reactants transport and ohmic losses at the electrode, current collector and electrode-current collector interfaces. The experiment data on current-voltage characteristics is predicted by the model with reasonable agreement and the influence of ethanol concentration and temperature on the performance of DEFC is studied by the authors (Fig. 8.19). 

While there are many possible types of flux, a single common principle underpins all transport phenomena. This is the idea that a driving force must be present to cause the transport process to occur. If there is no driving force, there is no reason to move This applies equally well to the transport of matter as it does to the transport of heat or charge. The governing equation for transport can be generalized (in one dimension) as... 

A driving force is required in order for any type of transport process to occur. The governing equation for transport can be generalized (in one dimension) as... 

Eor 1-D sampling, one dimension in space dominates the lot completely as all increments extend fully over the other two physical dimensions, and there is a natural linear order between increments. Conveyor belts and process pipelines transporting a material flux are the two most important examples of 1-D lots. 

It follows from Equation 6.12 that the current depends on the surface concentrations of O and R, i.e. on the potential of the working electrode, but the current is, for obvious reasons, also dependent on the transport of O and R to and from the electrode surface. It is intuitively understood that the transport of a substrate to the electrode surface, and of intermediates and products away from the electrode surface, has to be effective in order to achieve a high rate of conversion. In this sense, an electrochemical reaction is similar to any other chemical surface process. In a typical laboratory electrolysis cell, the necessary transport is accomplished by magnetic stirring. How exactly the fluid flow achieved by stirring and the diffusion in and out of the stationary layer close to the electrode surface may be described in mathematical terms is usually of no concern the mass transport just has to be effective. The situation is quite different when an electrochemical method is to be used for kinetics and mechanism studies. Kinetics and mechanism studies are, as a rule, based on the comparison of experimental results with theoretical predictions based on a given set of rate laws and, for this reason, it is of the utmost importance that the mass transport is well defined and calculable. Since the intention here is simply to introduce the different contributions to mass transport in electrochemistry, rather than to present a full mathematical account of the transport phenomena met in various electrochemical methods, we shall consider transport in only one dimension, the x-coordinate, normal to a planar electrode surface (see also Chapter 5). 

---