Engineering Flow Models

The modelling example of the previous section shows that to simplify the general mathematical model of the studied process, the real flow in the filter unit has been considered in terms of its own simplified model. Indeed, it is difficult to understand why we have used a flow model, when in fact, for the flow characterization, we already have the Navier-Stokes equations and their expression for the computational fluid dynamics. To answer this question some precisions about the general aspects of the computational fluids dynamics have to be given. 

Concermng our problem of the modelling of the flow process, even if the CFD seems to be the most complete approach, the use of flow models for its characterization is sustained by the following statements  

For the majority of the specific apparatus, the flows present a turbulent comportment and, for such flow, a numerical solution is covered by high uncertainty because some hypotheses have to be accepted a priori [3.25] in all the studied cases, the real apparatus has a complicated geometry that imposes very complex and frequently uncertain umvocity conditions in the real CFD-based flow computation. 

---

Chemical Engineering Flow Models 75 Table 3.3 The transfer function and model equation for some flow models... 

Chemical Engineering Flow Models 89 Table 3.5 Evolution of the NaCI concentration at the reactor s exit. 

Chemical Engineering Flow Models 93 Table 3.9 Evolution of the sulfur dioxide concentration at the exit of the reactor. 

Chemical Engineering Flow Models 95 Table 3.11 MathCAD computation of Pe,, D parameters. 

The engineer is offered a large variety of flow-modeling methods, whose complexity ranges from simple order-of-magnitude analysis to direct numerical simulation. Up to now, the methods of choice have ordinarily been experimental and semi-theoretical, such as cold flow simulations and tracer studies. 

The first step is to define the objectives of the flow model, and to identify those flow aspects that are relevant for the performance of the reactor. Then, the engineer must identify and quantify the various times and space scales involved, as well as the geometry of the system. These actions allow the problem to be represented by a mathematical model. Creating this model accurately is the most crucial task in the flow modeling project. 

It must be reemphasized that the value of a flow model s reeom-mendations depends on how well the model represents the real proeess situation. The reaetor and the proeess streams must be deseribed aeeurately, as must the relationship between the fluid dynamies and the proeess performanee. Often, proeess engineers are tempted to rely on eommereial CFD programs for the fluid dynamies equations. However, any eommereial program may have partieular limitations for simulating eomplex proeess equipment. On the other hand, almost all... 

Yang, W. C., Revay, D., Anderson, R. G. Chelen, E. J., Keaims, D. L., and Cicero, D. C., Fluidization Phenomena in a Large-Scale Cold-Flow Model, Fluidization, (D. Kunii, and R. Toei, eds.), Engineering Foundation, New York, p.77 (1984b)... 

Ranade, V. V., Computational Flow Modeling for Chemical Reactor Engineering, Volume 5 of Process Systems Engineering (G. Stephanopoulos and J. Perkins, Eds.), Academic Press, San Diego (CA, USA) (2002). 

Some early spray models were based on the combination of a discrete droplet model with a multidimensional gas flow model for the prediction of turbulent combustion of liquid fuels in steady flow combustors and in direct injection engines. In an improved spray model,[438] the full Reynolds-averaged Navier-Stokes equations were... 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

Mann, R., P. Mavros, and J. C. Middleton (1981). A structured stochastic flow model for interpreting flow follower data from a stirred vessel. Transactions of the Institution of Chemical Engineers 59, 127. 

Wen, C.Y., Flow regimes and flow models for fluidised bed reactors, in Recent advances in the engineering analysis of chemically reacting systems (ed. L. K. Doraiswamy), Wiley Eastern Limited, India, 256-287 (1984)... 

Gottschalk, F., Scholz, R.W. and Nowack, B. (2010) Probabilistic material flow modeling for assessing the environmental exposure to compounds methodology and an application to engineered nano- 2 particles. Environ. Model. Software, 25, 320-332. 

In this section we have presented the first example of two-point boundary value problems that occur in chemical/biological engineering. The axial dispersion model for tubular reactors is a generalization of the plug flow model for tubular reactors which removes some of the limiting assumptions of plug flow. Our model includes additional axial diffusion terms that are based on the simple physics laws of Fick for mass and of Fourier for heat dispersion. 

Massmann, J. W. (1989). Applying Groundwater Flow Models in Vapor Extraction System DesignJournal oJEnvironmental Engineering, 115(1), 129-149. 

---