Solution Using Solver

You can solve the same problem using the Solver option in Excel. 

Step 1 Under the Tools menu, click on Solver. Note If the choice Solver does not appear, choose Add-Ins and load Solver from the Analysis ToolPak or the original Excel program disk (or see your system administrator for help). 

Step 2 When the window opens, choose the option to make a cell equal to a value (or a maximum or minimum) by changing another cell. If you insert the appropriate cell locations, you will obtain the same answer as with Goal Seek. This time, however, it is much more accurate  

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We want values of pA"w, pAj, and pK2 that minimize the sum of squares of residuals in cell B12. Select SOLVER from the TOOLS menu. In the SOLVER window, Set Target Cell B12 Equal to Min By Changing Cells B9. B10. Bl 1. Then click Solve and SOLVER finds the best values in cells B9, BIO, and Bll to minimize the sum of squares of residuals in cell B12. Starting with 13.797, 2.35, and 9.78 in cells B9, B10, and Bll gives a sum of squares of residuals equal to 0.110 in cell B12. After SOLVER is executed, cells B9, B10, and Bll become 13.807, 2.312, and 9.625. The sum in cell B12 is reduced to 0.0048. When you use SOLVER to optimize several parameters at once, it is a good idea to try different starting values to see if the same solution is reached. Sometimes a local minimum can be reached that is not as low as might be reached elsewhere in parameter space. 

E. feU (a) Using the ion-pair equilibrium constant in Appendix J, with activity coefficients = 1, find the concentrations of species in 0.025 M MgS()4. Hydrolysis of the cation and anion near neutral pH is negligible. Only consider ion-pair formation. You can solve this problem exactly with a quadratic equation. Alternatively, if you use SOLVER, set Precision to le-6 (not le-16) in the SOLVER Options. If Precision is much smaller. SOLVER does not find a satisfactory solution. The success of SOLVER in this problem depends on how close your initial guess is to the correct answer. 

Rearrange Equation A to solve for lAg+] as a function of [H+] or use solver to find [Ag4 ] as a function of [H+], We will use the algebraic solution, which is easy for this exercise. Multiply both sides by [Ag+] and solve ... 

The spreadsheet uses Equation B to find [Ag+] in column C. pH is input in column A. To find the pH of unbuffered solution, we find the pH at which the net charge in column H is zero. We used solver to find that pH = 7.28 in cell A12 makes the net charge in cell HI2 equal to 0. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

Since the Solver operates by a search routine, it will find a solution most rapidly and efficiently if the initial estimates that you provide are close to the final values. Conversely, it may not be able to find a solution if the initial estimates are far from the final values. To ensure that the Solver has foimd a global minimum rather than a local minimum, it s a good idea to obtain a solution using different sets of initial estimates. 

Step 11 You can use Solver to set cell H27 to zero by varying cell H20, the vapor fraction. As Excel changes v, the values in rows 23 -27 change, and these in tiun cause changes in rows 14-18 because they depend upon the vapor mole fractions. Unfortunately, Solver cannot find a solution, but you can vary v yourself to find the solution. 

Equations (9.11) to (9.13) must be solved iteratively. An equation solver (or spreadsheet) provides a convenient way to do this. Eirst, a Lightnin A-320 impeller is selected. (A 45° PBT could also have been a reasonable first choice.) Choose D = 28 in. so that DIT 0.4. The height of liquid is 72 in. and the volume including the elliptical head is 1162 USG. A single impeller is selected, since the HIT is 1.0. Assume a speed determine the Re and the crossover point for transitional flow (Equation (9.15)) and the mixing time 699. The trial-and-error solution using a spreadsheet is shown in Table 9.3. 

IP problems are hard problems (technically called NP-hard, 0-1 IP problems are NP-complete). The solution time for the problem grows enormously with the size of the problem (number of variables, number of constraints). One can imagine the intractability in enumerating all solutions and picking the best one as the optimal solution. Excel solver can be used to solve small instances of the problem. 

The nonlinear regression Excel template used is set up identically to that of the linear regression template. The only difference is that now the estimated parameter values are not computed using a formula. Instead, they must be determined using Solver. Given the problem set-up, initial parameter estimates can be a bit of an issue, as the solution is sensitive to them. A recommended initial guess would be 0.5 for b and 2.5 for B. The macros are shown in Sect. 8.7.2.3 VB Macros. 

When ttie GRG engine is used. Solver has found atleasta local optimal solution. When Simplex LP is used, this means Solver has found a global optimal solution. 

There are two ways in which to use Solver for nonlinear equations. The direct way is to set up the nonlinear equations eis constraints with no objective function. The other way is to set up the spreadsheet to compute the sum of squares of residuals and use Solver to minimize this (without any constraints). The latter method is used in the following spreadsheet, where the feed consists only of component A with Qq = 1. The volumetric flow rate is 50 gmol/s, and the reactor volume is 100 L/s. The equations are rearranged in the form f(x) = 0 so that the left-hand sides are residuals whose value at a solution is zero (within tolerance). The initial guess for all concentrations is 0.5 gmol/L. 

First, solve this problem analytically by solving the constraint for xf and substituting this into the objective function. Then differentiate the objective function (the only remaining variable is x, set the derivative to zero, and find X2. Use the value for X2 to find the value(s) for x,. Next use Solver to find the solution(s). Use a starting point of [1, 1] and then [-1, -1] and see what solutions Solver finds from these starting points. 

Based upon this look at the errors achieved by the MATLAB funetions for these two test cases, eonsiderable confidenee ean be gained in the differential equation solvers developed in this chapter. The eode has been developed with the intent of easily estimating the accuracy of a solution using the h-2h teehnique. Because of the internal routines used in the MATLAB routines, it is not possible to readily evaluate the accuracy of the MATLAB eodes for general nonlinear differential equations. From the comparison in this section it can be expeeted that the codes developed in this work are comparable in accuraey to the MATLAB integration routines and with care in the selection of step distributions or with the use of the automatic adaptive selection algorithm to be more aecurate than the MATLAB routines with default parameters. 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

The highest level of integration would be to establish one large set of equations and to apply one solution process to both thermal and airflow-related variables. Nevertheless, a very sparse matrix must be solved, and one cannot use the reliable and well-proven solvers of the present codes anymore. Therefore, a separate solution process for thermal and airflow parameters respectively remains the most promising approach. This seems to be appropriate also for the coupling of computational fluid dynamics (CFD) with a thermal model. ... 

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