E-Z Solve

The E-Z Solve software also has a sweep feature that allows the user to perform sensitivity analyses and examine a variety of design outcomes for a specified range of parameter values. Consequently, it is also a powerful design and optimization tool. 

Many of the examples throughout the book are solved with the E-Z Solve software. In such cases, the computer file containing the program code and solution is cited. These file names are of the form exa-b.msp, where ex designates an example problem, a the chapter number, and b the example number within that chapter. These computer files are included with the software package, and can be readily viewed by anyone who has obtained the E-Z Solve software accompanying this text. Furthermore, these example files can be manipulated so that end-of-chapter problems can be solved using the software. 

This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9 (with n = 2), which correspond to the relationships developed for first-order and second-order kinetics, respectively. However, here we illustrate the use of nonlinear regression applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized integrated forms. The method employs user-defined functions within the E-Z Solve software. The rate constants estimated for the first-order and second-order cases are 0.0441 and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental data and the predictions from either the first- or second-order rate expression. This lack of sensitivity to reaction order is common when fA < 0.5 (here, /A = 0.28). 

Values of the rate constants kx and k2 can be obtained from experimental measurements of cA and cB at various times in a BR. The most sophisticated procedure is to use either equations 5.5-2 and -3 or equations 3.4-10 and 5.5-6 together in a nonlinear parameter-estimation treatment (as provided by the E-Z Solve software see Figure 3.11). A simpler procedure is first to obtain kx from equation 3.4-10, and second to obtain 2 fr°m and either of the coordinates of the maximum value of cB (tmax or cB max). These coordinates can be related to kx and k2, as shown in the following example. 

In the examples in Sections 7.1 and 7.2.1, explicit analytical expressions for rate laws are obtained from proposed mechanisms (except branched-chain mechanisms), with the aid of the SSH applied to reactive intermediates. In a particular case, a rate law obtained in this way can be used, if the Arrhenius parameters are known, to simulate or model the reaction in a specified reactor context. For example, it can be used to determine the concentration-(residence) time profiles for the various species in a BR or PFR, and hence the product distribution. It may be necessary to use a computer-implemented numerical procedure for integration of the resulting differential equations. The software package E-Z Solve can be used for this purpose. 

This problem is an extension of problems 7-10 and 7-11 on the dehydrogenation of ethane to produce ethylene. It can be treated as an open-ended, more realistic exercise in reaction mechanism investigation. The choice of reaction steps to include, and many aspects of elementary gas-phase reactions discussed in Chapter 6 (including energy transfer) are significant to this important industrial reaction. Solution of the problem requires access to a computer software package which can handle a moderately stiff set of simultaneous differential equations. E-Z Solve may be used for this purpose. 

The integral may be evaluated either numerically by means of E-Z Solve, or analytically by means of the substitution x = 1 + 2KBcAs + (KA - 2KB)cA. The latter results in... 

For given values of Ha and ,-, may be calculated by solving equations 9.2-51 and -52 numerically using the E-Z Solve software (or by trial). The results of such calculations are shown graphically in Figure 9.8, as a plot of E versus Ha with Et — 1 as a parameter. 

These can be solved numerically (e.g., with the E-Z Solve software), and the results used to obtain cE(t) and cP(t) from equations 10.2-2 and 10.2-12, respectively. 

To determine Km and Vmax, experimental data for cs versus t are compared with values of cs predicted by numerical integration of equation 10.3-3 estimates of Km and Vmax are subsequently adjusted until the sum of the squared residuals is minimized. The E-Z Solve software may be used for this purpose. This method also applies to other complex rate expressions, such as Langmuir-Hinshelwood rate laws (Chapter 8). 

This example may also be solved using the E-Z Solve software (see file exl2-2.msp). 12.3.2.2 Variable-Density System... 

Alternatively, the E-Z Solve software may be used to integrate simultaneously the material- and energy-balance expressions and solve the equation of state. 

If the batch reactor operation is both nonadiabatic and nonisothermal, the complete energy balance of equation 12.3-16 must be used together with the iiaterial balance of equation 2.2-4. These constitute a set of two simultaneous, nonlincmr, first-flijer ordinary differential equations with T and fA as dependent variables and I as Iidependent variable. The two boundary conditions are T = T0 and fA = fAo (usually 0) at I = 0. These two equations usually must be solved by a numerical procedure. (See problem 12-9, which may be solved using the E-Z Solve software.)... 

The E-Z Solve software may also be used to solve Example 12-7 (see file exl2-7.msp). In this case, user-defined functions account for the addition of fiesh glucose, so that a single differential equation may be solved to desenbe the concentration-time profiles over the entire 30-dry period. This example file, with die user-defined functions, may be used as the basis for solution of a problem involving the nonlinear kinetics in equation (A), in place of the linear kinetics in (B) (see problem 12-17). 

This problem may also be solved by numerical integration using the E-Z Solve software (file exl4-3.msp). This simulation is well-suited to the investigation of the effect of initial conditions on the time for a specified approach to steady-state. To optimize... 

From equation 14.3-19, with MA defined by equation 4.3-4. From equation 13.52, with 3.4-10 or 14.3-20 and 13.4-2. dE is an exponential integral defined by E x) = y le y dy, where y is a dummy variable the integral must be evaluated numerically (e.g., using E-Z Solve) tabulated values also exist. 

From the Descartes rule of signs, since there is one change in the sign of the coefficients in (C), there is only one positive real root. (The same rule applied to - fA in (C) indicates that there may be two negative real roots for fA, but these are not allowable values.) Solution of (C) by trial or by means of the E-Z Solve software (file exl4-6.msp) gives... 

The E-Z Solve numerical solution requires combination of the material balance (14.3-5), the rate law, and the relation of q to q0 (accounting for the density change in the system). 

This equation may have three real, positive roots (as indicated by the Descartes rule of signs see Example 14-6). Three roots can be obtained by trial or by a suitable root-finding technique, for example, as provided by the E-Z Solve software (see note below) ... 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

The system of three equations based on equation 14.4-1 for /A1, and /A3 may also be solved simultaneously using the E-Z Solve software (file exl4-9.msp). The same values are obtained. 

The E-Z Solve software can be used to integrate numerically the differential equation resulting from the combination of the material-balance equation (15.2-17) and the rate law [equation (A)]—see file exl5-3.msp. The same results are obtained for V and t. 

To characterize the performance of a PFR subject to an axial gradient in temperature, the material and energy balances must be solved simultaneously. This may require numerical integration using a software package such as E-Z Solve. Example 15-4 illustrates the development of equations and the resulting profile for fA with respect to... 

This example can also be solved by numerical integration of equation (A) using the E-Z Solve software (file exl5-6.msp). For variable density, equation (B) is used to substitute for q. For constant density, q = qg. 

The integrals in equations (C) and (F) can also be evaluated numerically using the E-Z Solve software (file exl5-7.msp). The calculated value of t is 15.8 s, slightly less than that obtained using the trapezoidal approximation. 

The E-Z Solve software can be used to integrate equations 15.2-4 and 15.2-11 numerically, while simultaneously updating q, u, p, Re, and / at each step (file exl5-8.msp). The predicted conversion for isothermal, nonisobaric conditions is 0.247 the calculated pressure drop is 114 kPa. If the pressure drop is ignored (i.e., P = 400 kPa throughout the reactor), the resulting conversion is 0.274. Thus, for this case, it is important that the pressure drop be accounted for. 

Using the E-Z Solve software, we may integrate equation 16.2-12 to develop the axial profile for cA or /A equation 16.2-13 gives the value of the final (exit) point on this... 

This integral is related to the exponential integral (see Table 14.1). It cannot be solved in closed analytical form, but it can be evaluated numerically using the E-Z Solve software the upper limit may be set equal to 10f. 

For the purpose of numerical integration using E-Z Solve (file exl6.3-msp), the upper limit in the integral is assumed to be 10 i(x). 

The examples above illustrate that the determination of fA (or cA) for an LFR can be lU o done either analytically or numerically (using the E-Z Solve software) most situations... 

The conclusions illustrated in Table 17.1 are (1) for a given order, n, the ratio increases as /a increases, and (2) for a given /A, the ratio increases as order increases. In any case, for normal kinetics, ST > Vpp, since the CSTR operates entirely at the lowest value of CA, the exit value. (Levenspiel, 1972, p. 332, gives a more detailed graphical comparison for five values of n. This can also be obtained from the E-Z Solve software.)... 

Example 18-1 can also be solved by means of the E-Z Solve software (file exl8-l.msp). In this case, the design equations for each stage are written for species A and B in terms of cA, CB> Cc> together with cB = cc. The resulting nonlinear equation set is solved by the software. In this approach, there is no need to introduce /A, nor to relate the concentrations to /A, although /A can be calculated at the end, if desired. 

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