FEMLAB

Packages to solve boundary value problems are available on the Internet. On the NIST web page http //gams.nist.gov/, choose problem decision tree and then differential and integral equations and then ordinary differential equations and multipoint boundary value problems. On the Netlibweb site http //www.netlib.org/, search on boundary value problem. Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. 

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... 

The Navier-Stokes equations are solved first to determine the velocity field throughout the reactor, as described by Armaou and Christofides [4], and subsequently by Brass and Lee [5] using FEMLAB. Then, the species mass balances are solved to determine the concentrations of SiFL (1), SiH2 (2), SiH3 (3), and H (4), throughout the reactor. Finally, the deposition rate of silicon is ... 

Arana et al. have performed extensive modeling and thermal characterization experiments on their reactor design. They modeled their design consisting of two suspended SiN - tubes linked with slabs of silicon using two-dimensional computation fluid dynamics and a heat transfer model (Femlab, Comsol Inc.). The heat of reaction of the steam reforming or... 

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. 

Two and Three Dimensions Appendix A Hints when Using Excel Appendix B Hints when Using MATLAB Appendix C Hints when Using Aspen Plus Appendix D Hints when Using FEMLAB Appendix E Parameter Estimation Appendix F Mathematical Methods... 

Note that the dependent variable, c, is a function of only one independent variable, z, and that the initial value is specified. For reactors, you start at the inlet and integrate down the reactor using either MATLAB or FEMLAB. 

Chapter 9 then solves transport problems in one space dimension (ID) using FEMLAB. If you consider heat transfer through a slab, one side of the slab is kept at one temperature, To, and the other side of the slab is maintained at another temperature, Tj,. The governing equation is... 

The differential equation, (1.2), is an ordinary differential equation because there is only one independent variable, x In this case, equations in one space dimension are boundary value problems, because the conditions are provided at two different locations. While it is also possible to solve this problem using Excel and MATLAB, it is much simpler to use FEMLAB. Transient heat transfer in one space dimension is governed by... 

Chapters 10 and 11 use FEMLAB to solve fluid flow, heat transfer, and mass transfer... 

USING FEMLAB TO SOLVE ORDINARY DIFFERENTIAL EQUATIONS... 

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). 

Thus, we must choose D = 0 and/= — 10 y. (For more details about FEMLAB, see Appendix D.)... 

Step 1 Open FEMLAB and choose ID (in the top menu). Chemical Engineering Module, then Mass Transfer and convective diffusion (the PDE/Classical/convective diffusion also works). Click OK. 

Figure 8.6. Solution to problem posed by Eqs. (8.21)—(8.22) using FEMLAB.

Note to experienced users Sometimes FEMLAB has difficulty with plug flow reactor problems, when MATLAB did not, because FEMLAB is solving them as boundary value problems, whereas MATLAB solves them as initial value problems. They are actually initial value problems you fool FEMLAB by setting the diffusion coefficient to zero, and the thermal conductivity to zero. The other difficulty is that the kinetic expression makes huge changes when the temperature changes. Anytime you have to iterate to find... 

The equations are differential equations for the molar flow rates. Fa, Fr, and Fc, but the rates of reaction are expressed as concentration, Ca- It is possible to derive the concentration of each chemical from the molar flow rates, and this section illustrates how to do that in MATLAB and FEMLAB. In a gas phase, the total concentration is governed by the perfect gas law, although more complicated equations of state are possible ... 

The same considerations apply when using FEMLAB. The variables being solved for are now Fa, Fb, and Fc. In Options/Expressions/Subdomain Expressions you define equations for new variables ca, cb, and cc representing Eq. (8.42), and then use the same reaction rate expression (in terms of ca). The rest of the solution proceeds as before ... 

Solve for the concentration distribution in a plug flow reactor governed by the following equations. (1) Use MATLAB (2) use FEMLAB ... 

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