Using solver function solution

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 lines that follow the initial % MATLAB comment lines in fixedbedreact.m set up default values for the seven optional parameters. Then we prepare for the MATLAB IVP solver ode.. . that solves our problem by using the function dydt to evaluate the right-hand side of our IVP (4.22). Having solved (4.22) we plot two curves of the solution to the two joint DEs. 

Figure 4.8 Calculation of the hydrogen 2,1 energy using the Slater 2s orbital rendered orthogonal to the hydrogenic Is orbital with the best choice for the Slater exponent subject to two conditions. In the first diagram, the SOLVER based solution is determined by the requirement that the Virial theorem coefficient be 1.00. In the second diagram, the best values for the kinetic and potential energy terms have been determined for the choice of the Slater 2s exponent. Note, the substantial cancellation of mismatches of the r grad transformed Slater function compared to the variation of the transformed exact function.

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. 

The h-2h calculation needed for the Richardson type extrapolation is also the calculation needed to estimate the error in the solution of a boundary value problem as discussed in the previous section. Thus it seem appropriate to combine these into an eigenvalue solver and such a function has been coded as the function odebveveO which is also available with the require obebv statement. An example of using this function for a higher order eigenvalue and function of the same constant potential problem is shown in Listing 11.10. The difference from Listing... 

The Wegstein method is a secant method applied to g(x) = x - Fix). In Microsoft Excel, roots are found by using Goal Seek or Solver. Assign one cell to be x, put the equation for/(x) in another cell, and let Goal Seek or Solver find the value of x that makes the equation cell zero. In MATLAB, the process is similar except that a function (m-file) is defined and the command fzeroCf .xO) provides the solution x, starting from the initial guess xO. 

The Excel Solver. Microsoft Excel, beginning with version 3.0 in 1991, incorporates an NLP solver that operates on the values and formulas of a spreadsheet model. Versions 4.0 and later include an LP solver and mixed-integer programming (MIP) capability for both linear and nonlinear problems. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formula cells whose values are to be constrained (the constraints), and a formula cell designated as the optimization objective. The solver uses the spreadsheet interpreter to evaluate the constraint and objective functions, and approximates derivatives, using finite differences. The NLP solution engine for the Excel Solver is GRG2 (see Section 8.7). 

The absorbance of a mixture is the sum of absorbances of the individual components. At a minimum, you should be able to find the concentrations of two species in a mixture by writing and solving two simultaneous equations for absorbance at two wavelengths. This procedure is most accurate if the two absorption spectra have regions where they do not overlap very much. With a spreadsheet, you should be able to use matrix operations to solve n simultaneous Beer s law equations for n components in a solution, with measurements at n wavelengths. You should be able to use Excel SOLVER to decompose a spectrum into a sum of spectra of the components by minimizing the function (Aca c — Am)2. 

The code axialdisp4DaPerunbackw. m consists of a comments block, followed by the general default and initialization block, as well as the BVP initialization and solver block. The actual solution is found inside the central try. . . catch. . . end lines of code. This is followed by two blocks of plotting code and the coded DE in dydx with its boundary conditions in Rand, both expressed as vector valued functions. Finally four initial guess functions for the shape of the solution are given that are used inside the BVP solver for different parameter data. 

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 ... 

Takeuchi et al. 7 reported a membrane reactor as a reaction system that provides higher productivity and lower separation cost in chemical reaction processes. In this paper, packed bed catalytic membrane reactor with palladium membrane for SMR reaction has been discussed. The numerical model consists of a full set of partial differential equations derived from conservation of mass, momentum, heat, and chemical species, respectively, with chemical kinetics and appropriate boundary conditions for the problem. The solution of this system was obtained by computational fluid dynamics (CFD). To perform CFD calculations, a commercial solver FLUENT has been used, and the selective permeation through the membrane has been modeled by user-defined functions. The CFD simulation results exhibited the flow distribution in the reactor by inserting a membrane protection tube, in addition to the temperature and concentration distribution in the axial and radial directions in the reactor, as reported in the membrane reactor numerical simulation. On the basis of the simulation results, effects of the flow distribution, concentration polarization, and mass transfer in the packed bed have been evaluated to design a membrane reactor system. 

Equation (7-54) allows calculation of the residence time required to achieve a given conversion or effluent composition. In the case of a network of reactions, knowing the reaction rates as a function of volumetric concentrations allows solution of the set of often nonlinear algebraic material balance equations using an implicit solver such as the multi variable Newton-Raphson method to determine the CSTR effluent concentration as a function of the residence time. As for batch reactors, for a single reaction all compositions can be expressed in terms of a component conversion or volumetric concentration, and Eq. (7-54) then becomes a single nonlinear algebraic equation solved by the Newton-Raphson method (for more details on this method see the relevant section this handbook). 

There are important differences between the initial value formulation and the boundary value approach. Initial value solutions are based on interpolation forward in time one coordinate set after another. The boundary value approach is based on minimization of a target function of the whole trajectory. Minimization (and the study of a larger system) is more expensive in the boundary value formulation compared to initial value solver. However, the calculations of state to state trajectories and the abilities to use approximations (next section), make it a useful alternative for a large number of problems. 

If the time step is not small (such that At" > 2M/an initial value solver becomes unstable. A solution to the finite difference equation can be found in which A is imaginary, and the cosine function is replaced by a hyperbolic cosine. This solution is not only a poor approximation but it is also numerically unstable. Using initial value solvers the above solution leads to exponential (unbound) growth of the coordinate as a function of time = Aexp(A(i)j)). 

Solution methods for optimization problems that involve only continuous variables can be divided into two broad classes derivative-free methods (e.g., pattern search and stochastic search methods) and derivative-based methods (e.g., barrier function techniques and sequential quadratic programming). Because the optimization problems of concern in RTO are typically of reasonably large scale, must be solved on-line in relatively small amounts of time and derivative-free methods, and generally have much higher computational requirements than derivative-based methods, the solvers contained in most RTO systems use derivative-based techniques. Note that in these solvers the first derivatives are evaluated analytically and the second derivatives are approximated by various updating techniques (e.g., BFGS update). 

Solver, can provide parameters that fit a user-specified function to a set of data, but does not yield estimates of the precision of those parameters. Here we exploit the approach used in section 10.3 to compute the precision of the parameters obtained with Solver. We will make the usual assumptions that, in fitting a function F a ) to /Vexperimental data pairs x,y, all indeterminate uncertainties are restricted to y, and follow a single Gaussian distribution. Furthermore we will assume that Solver has already been used to find a solution ycaic based on a mathematical model expression of the type yta c = F (a(-), where at are the parameters Solver has adjusted. We can then use a second macro, called SolverAid, to estimate the standard deviations of those parameters a,. 

In this chapter, we have already discussed the unsteady operation of one type of reactor, the batch reactor. In this section, w C discuss two other aspects of unsteady operation startup of a CSTR and seniibatch reactors. First, the startup of a CSTR is examined to determine the time necessary to reach steady-state operation [see Figure 4-14(a)], and then semibaich reactors are discussed, in each of these cases, we are interested in predicting the concentration and conversion as a function of lime. Closed-form analytical solutions to the differential equations arising from the mole balance of the.se reaction types can be obtained only for zero- and first-order reactions. ODE solvers must be used for other reaction orders. 

---