FEMLAB program

P8-2.J Load the following Polymath/MATLAB/FEMLAB programs from the... 

Equation (14-52) was used in the FEMLAB program along with the rate law... 

It is suggested that one first use and play with the FEMLAB program for solutions to Examples 8-12 and 14-4, before making any changes. One should also review the web module for Chapter 8 on Axial and radial gradients in tubular reactors before running the program. 

Computational fluid dynamics (CFD) programs are more specialized, and most have been designed to solve sets of equations that are appropriate to specific industries. They can then include approximations and correlations for some features that would be difficult to solve for directly. Four major packages widely used are Fluent (http //www.fluent.com/), CFX (now part of ANSYS), Comsol Multiphysics (formerly FEMLAB) (http //www.comsol.com/), and ANSYS (http //www.ansys.com/). Of these, Comsol Multiphysics is particularly useful because it has a convenient graphical-user interface, permits easy mesh generation and refinement (including adaptive mesh refinement), allows the user to add phenomena and equations easily, permits solution by continuation methods (thus enhancing... 

This book demonstrates four computer programs Excel , MATLAB , Aspen Plus , and FEMLAB . You may have access to other programs created by other companies. While the exact details will not be the same, the steps you take will be similar. 

FEMLAB is treated in detail in Chapters 9-11 and Appendix D, but it can also be used to solve reactor problems. The advantage of FEMLAB is that you program with a GUI, so computer errors are less likely. It is still necessary to check your work, though. While the applications in this chapter are all one-dimensional (to compare with MATLAB solutions), it is easy to solve two-dimensional problems, as described in more detail in later chapters. We show here how to solve the same three problems already solved using MATLAB the simple exponential, Eq. (8.16) the isothermal flow reactor, Eqs. (8.21)—(8.22) and the nonisothermal reactor, Eqs. (8.24)-(8.26). 

One handy way to get all the solutions for several pressure drops (and avoid convergence problems) is to use the Parametric Nonlinear option in the Solver Parameters. Call the variable presdropx and set its value to 0 1 7. Then in the Physics/Subdomain Settings, where you put the Ap, replace it with lO presdropx. The program FEMLAB... 

Two methods are used in commercial computational fluid dynamics (CFD) codes the finite volume method and the finite element method. To a beginner, it probably makes little difference which method is used. The author has used finite element methods for fluid flow for over 30 years, and that is the method used in FEMLAB , which is the CFD program illustrated here. 

This chapter focuses on fluid flow, leaving the combination of fluid flow, heat transfer, and diffusion to Chapter 11. Examples of fluid flow include entry flow into a pipe, flow in a microfluidic T-sensor, turbulent flow in a pipe, time-dependent start-up of pipe flow, flow in an orifice, and flow in a serpentine mixer. The examples demonstrate many of the techniques that are useful in the program FEMLAB. 

Because the program FEMLAB includes all these terms, you merely need to define the problem you want to solve. 

The finite element method replaces a differential equation with a large set of algebraic equations. The details to make this switch are complicated, but fortunately, FEMLAB has done that for you. You still need to know how to use the program, because, after all, it is up to you to decide if you have solved the right problem, determined the accuracy, and derived useful properties from the solution. 

FEMLAB is a registered trademark of Comsol, Inc. The program name has recently been changed to Comsol... 

The examples are made with the Chemical Engineering addition to FEMLAB, version 3.1. Appendix F describes the finite element method in one dimension and two dimensions so you have some concept of the approximation going from a single differential equation to a set of algebraic equations. This appendix presents an overview of many of the choices provided by FEMLAB. Illustrations of how FEMLAB is used to solve problems are given in Chapters 9-11. Thus, you may wish to skim this appendix on a first reading, and then come back to it as you use the program to solve the examples. A more comprehensive account of FEMLAB is available in Zimmerman (2004). 

Figure D. 1 shows the opening screen of FEMLAB. Look at the options across the top. The New tab is what is shown. If you click the Model Library tab, you get a menu of solved examples. These are ideal to investigate the program. The User Models tab provides a place you can put your examples, if you wish. Settings lets you choose a white or black background.

This example will highlight the radial effects in a tubular reactor, which up until now have been neglected to simplify the calculations. Now, the effects of parameters such as inlet temperature and flow rate will be studied using the software program FEMLAB. Follow the step-by-step procedure in the Web Module on the CD-ROM. 

FEMLAB is a partial differential equation solver (PDE) available commercially from COMSOL. Inc, Included with this text is a special version of FEMLAB that has been prepared to solve problems intolving tubular reactors. Specifically, one can solve CRE problems with heat elTccts involving both axial and radial gradients in concentration and temperature simply by loading the FEMLAB CD on one s computer and running the program. One can also use it to solve isothermal CRE problem,s with reaction and diffusion. 

Figure 3.2.13 Variation of local Nusselt number along the circumference of a cylinder for cross flow of air (Pr=0.7) and Re = 2i comparison of experimental data (Eckert and Soehngen, 1952), a numerical solution (Sucker and Brauer, 1976), and results of a calculation by the finite element method [75 = 600°C, T = 500°C, X i, = 0.063 Wm (600°C),Vair=2.4x 10" m s (600°C), T , X T , program COMSOL PHYSICS, FEMLAB Company, Gottingen, Germany).

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