Solving Equations with Mathematica

The Poisson-Boltzmann equations (6.135) and (6.139) subject to boundary conditions (6.140)-(6.143) can be solved numerically with Mathematica to obtain the potential distribution y(x). The results of some calculations for the potential... 

SOLUTION This is the same equation that we solved with Mathematica in a previous example, so we know where the real roots are. We pretend that we know only that there is a real root near 1 and a real root near 4. We type in the value 1 in cell A1 and the value 4 in cell A2. We type the following formula in cell Bl =AlM-5 Al 3+4 A1 2-3 Al+2 and press the Return key. We then drag the cursor over cells B1 and B2 and fill down by holding down the Ctrl key and typing a letter d. We could also select the two cells and choose Fill Down in the Edit menu. We then select the Bl cell and choose Goal Seek from the Tools menu. A window appears with three blanks. The first says Set cell and should have Bl in the blank. The second blank... 

Solve the set of equations using Mathematica or by hand with the method of substitution ... 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions... 

Since equation 6.1-2 represents a set of linear equations, the path can be calculated from the stoichiometric number matrix and a particular net reaction by solving the set of linear equations (Alberty,1996a). In Mathematica this can be done with LinearSolve ... 

Ihese difficulties vanish if the system equations are simply collected and solved for all unknown variables. Several powerful equation-solving algorithms are available in commercial programs like Maple , Mathematica , Matlab , Mathcad , and E-Z Solve that make the equation-based approach competitive with the sequential modular approach. Many researchers in the field believe that as this trend continues, the former approach will replace the latter one as the standard method for flowsheet simulation. (Engineers are also working on simultaneous modular methods, which combine features of both sequential modular and equation-based approaches. We will not deal with these refinements here, however.)... 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

The numerical Method of Lines as implemented in the routine NDSolve of the Mathematica system deals with system (32) by employing the default fourth order finite difference discretization in the spatial variable Z, and creating a much larger coupled system of ordinnary equations for the transformed dimensionless temperature evaluated on the knots of the created mesh. This resulting system is internally solved (still inside NDSolve routine) with Gear s method for stiff ODE systems. Once numerical results have been obtained and automatically interpolated by NDSolve, one can apply the inverse expression (31.b) to obtain the full dimensionless temperature field. 

Finally, because the world is not an ideal gas, equations of state or activity coefficients frequently must be used to describe a real system of interest. This results in considerable mathematical complexity. Consequently, many problems are best solved using the computer software I have provided that is discussed in the previous appendix, or with MATHCAD, MATLAB, MATHEMATICA, POLYMATH, or equation-solving software and programs you develop on your own. 

The solution is represented by a column vector that is equal to the matrix product A C. In order for a matrix to possess an inverse, it must be nonsingular, which means that its determinant does not vanish. If the matrix is singular, the system of equations cannot be solved because it is either linearly independent or inconsistent. We have already discussed the inversion of a matrix in Chapter 9. The difficulty with carrying out this procedure by hand is that it is probably more work to invert an n by n matrix than to solve the set of equations by other means. However, with access to Mathematica, BASIC, or another computer language that automatically inverts matrices, you can solve such a set of equations very quickly. 

In Chapter 3, we introduced the use of Mathematica to solve a single algebraic equation, using the Solve statement and the NSolve statement. The Solve statement can also be used to solve simultaneous equations. The equations are typed inside curly brackets with commas between them, and the variables are listed inside curly brackets. To solve the equations ax + by = c gx + hy = k we type the input entry... 

Using Mathematica we have two primary choices on how to proceed with the solution to this equation—we can rearrange it into its separable components and then integrate both sides of the equation or we can solve it directly we will do the latter. 

The "seeds" for these rules are the solutions for the first CSTR, and then these exit concentrations become the inlet concentrations to the second CSTR, whose exit concentrations become the inlet concentrations for the third CSTR, and so it goes on through to n CSTRs. We can have Mathematica assemble the equations and the variables that we will need for the Solve routine. This is illustrated with n = 3 ... 

---