By Richard G. Rice, Duong D. Do

This moment version of the go-to reference combines the classical research and smooth functions of utilized arithmetic for chemical engineers. The ebook introduces conventional recommendations for fixing traditional differential equations (ODEs), including new fabric on approximate resolution equipment similar to perturbation innovations and straight forward numerical recommendations. it's also analytical tips on how to take care of vital periods of finite-difference equations. The final part discusses numerical resolution recommendations and partial differential equations (PDEs). The reader will then be built to use arithmetic within the formula of difficulties in chemical engineering. just like the first variation, there are lots of examples supplied as homework and labored examples

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**Extra resources for Applied Mathematics And Modeling For Chemical Engineers, Second Edition, 2nd Edition**

**Example text**

Walas, S. M. Modeling with Differential Equations in Chemical Engineering. Butterworth-Heinemann, Boston (1991). 1 GEOMETRIC BASIS AND FUNCTIONALITY In Chapter 1, we took the point of view that elementary differential balances gave rise to certain differential equations, and for first order systems, we could represent these in general by dy ¼ f ðx; yÞ dx ð2:1Þ where f(x,y) is some arbitrary, but known, function ((x2 þ y2)1=2, xy2, etc). Our strategy was to find suitable solutions to the ordinary differential equation (ODE) so that we could compute y for any given x, such as y ¼ gðx; cÞ Suppose, we have a family of curves represented by the function f ðx; y; lÞ ¼ 0 ð2:3Þ where l is an arbitrary parameter.

Finite difference) processes, is Riccati’s equation k2 A À! B À! C hence, the original equation is now linear in v 35 ð2:60Þ QðuÞ ¼ 0; RðuÞ ¼ k1 k1 CA0 exp À u k2 k2 to see that dCB þ C 2B ¼ RðuÞ du ð2:63Þ 36 SOLUTION TECHNIQUES FOR MODELS YIELDING ORDINARY DIFFERENTIAL EQUATIONS If we make the Riccati transformation CB ¼ 1 du u du we finally obtain d 2 uðuÞ k1 k1 À C exp À u uðuÞ ¼ 0 A0 k2 k2 du2 ð2:64Þ CASE (c) We have thus transformed a nonlinear first-order equation to a solvable, linear second-order equation.

To start the solution of Eq. 12, we put forth the proposition that there exists an elementary, separable solution In the following section, we shall exploit this property of exactness in a formal way to effect solutions to certain classes of problems. 2 CLASSIFICATION OF ODE The broad classification of equations with which engineers must contend is as follows: (a) Linear equations (b) Nonlinear equations Much is known about linear equations, and in principle all such equations can be solved by well-known methods.