TK Solver

Using the above equations, model simulations of the various Hg species in the Scheldt estuary were carried out using the TK-Solver program.115 A conditional stability constant of 1019 was estimated for Hg-humic acid interactions in the Scheldt. 

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. 

A computer program may be written to solve this problem for residence times as a function of the recycle ratio, 7 . The above outline of the solution procedure may be programmed using MATHCAD, TK SOLVER, FORTRAN, or any other method. The printouts below show the saturation temperature, Ts, and reciprocal rates, 1/Rate, for recycle ratios of 7 = 0, 1, and 5, and for 7 = 1 with only the fresh air rate increased, using Eq. 8. When there is no recycle, use Eq. (la) instead of Eq. (1). The residence times for the four cases are ... 

The procedure outlines in Steps 1 through 5 employing Eq. (1) through (15) will result in satisfactory fluid bed dryer design. The first step is to assume T2 and then other quantities can be evaluated in order. A computer program using Fortran, MATHCAD, TK SOLVER, or other such programs may be used to solve this problem. Partial results for the first interval are ... 

Figure 2.20 TK Solver Plus lets you view any two of its nine worksheets onscreen at once. Shown here are the Variable Sheet (top) and the Rule Sheet (bottom). After known values are entered into the Variable Sheet, the key is pressed to start the calculations. Once the solution is found, the values of the unknown variables appear in the output column spaces in the Variable Sheet.

TK Solver has two methods of solving equations. (1) The Direct Solver is just what it sounds like values are substituted for variables, and both sides of an expres-... 

A number of iterative methods exist, as described in Appendix L. TK Solver uses a modified Newton-Raphson iterative procedure (see Sec. L.2), which is satis-fectory for a wide variety of problems. 

Before closing this section it should be mentioned that most of the methods discussed above can be solved using commercially available programs such as symbolic computation (Mathematica, Maple, or Mathcad 3.0), equation solvers (TK Solver Plus or Mathcad 3.0), spreadsheets (e.g., Lotus 123, Quattro Pro, Microsoft Excel, or Wingz), simulators (Extend or Stella), or Microsoft FORTRAN 5.0. For example, one can introduce the idea of molecular mechanics on model systems using a spreadsheet. Actual applications of molecular mechanics are much better carried out on a software package designed for that purpose, however. 

Nonetheless, this numerical problem can be accomplished on a spreadsheet, Stella, TK Solver Plus, Mathcad, etc. 

TK Solver Plus, Universal Technical Systems, Inc., 1220 Rock Street, Rockford, Illinois 61101. (TK Solver, Release 2.0, Fall, 1992.)... 

Note that using eqs. (10-3) and (10-4) require an iterative calculation since the value of the Reynolds number determines the flow regime and thus which equation to use. On the other hand, to evaluate the Reynolds number, one needs the value of Vt. Such problems are easily solved with an equation solver such as TK Solver software from Universal Technical Systems, Inc. or Microsoft Excel software. 

See Table 11-5 for the calculations. The relevant equations, which are solved using TK Solver or a similar program, are... 

Penney, R. W., H. X. Vo, and G. K. Patterson (1997). Implementation of the Bourne engulfment model using TK-Solver, presented at the AIChE Annual Meeting, Los Angeles. 

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