Software Maple

Maple V, Release 2, Waterloo Maple Software, University of Waterloo, Ontario, Canada. 

The system (4.197) through (4.199) with the boundary conditions (4.202), (4.203), (4.204), and (4.209) has been solved numerically using Maple software. The physical and operational parameters are listed in Table 4.1. Note that the boundary conditions (4.202) and (4.204) contain both the cell current density jo and the potential loss rjox,o- However, these values are related by the polarization curve, which a priori is unknown. Thus, iterations are required to find ijox,o, providing the prescribed cell current jo. 

The system of Equations 5.228 and 5.229 was solved numerically using Maple software. Figure 5.36a shows the shape of normalized methanol concentration and membrane potential . Equation 5.219 prescribes rapid decrease of Cmi from unity to zero at X = 5 cm (Figure 5.36a). As can be seen, this decrease induces the drop of membrane potential by 450 mV (from —0.25 V in the MR-domain down to —0.7 V in the MD-domain, Figure 5.36a). 

Clearly for larger matrices the situation becomes quite complicated and, while the principle ought to be understood, computer programs are available for the numerical as well as for the symbolic evaluation of large determinants. I find the programs MapleVl for Windows (Waterloo Maple Software, 160 Columbia Street West, Waterloo, Ontario, Canada N2L... 

Calculations (accurately to 99%) of ti and t2 parameters from Eq. (27) using Maple software (Table 2) and the subsequent non-linear approximation of the results with the Origin program make it possible to quantitatively estimate the values from the following equations ... 

Finally, the apparent foam modulus can be obtained through combination and numerical integration of Equ.(l-3,5-7). For this. Maple 9.5 (Waterloo Maple Software) was used and the results are discussed next. 

Figure 2. Transition state complex in the ethanol + 2-pentanol 8, 2 reaction activated by the proton at the chaimel intersection of H21SM-5 [14]. The zeolite pore structure is represented as a wire-frame section of the intersecting channels produced by the MAPLE V software package. The zeolite proton that activates the 2-pentanol molecule is marked with. ...

Any software that includes matrix reduction can be used similariy. For example, wilh Maple (Watedoo Maple, Inc.), the first three steps in Example 1-3 are initiated by (1) with (linalg) (2) transpose (array ([list of species as in (1)])) (3) rref ( ). In many cases, the matrix reduction can be done conveniently by hand manipulation. 5Chemical reaction stoichiometry is described more fully on a Web site located athttp //www.chemical-stoichiometry.net. The site includes a tutorial and a Java applet to implement the matrix reduction method used in the examples here. 

In practice, the solution of polynomial equations is problematic if no simple roots are found by trial and error. In such circumstances the graphical method may be used or, in the cases of a quadratic or cubic equation, there exist algebraic formulae for determining the roots. Alternatively, computer algebra software (such as Maple or Mathematica, for example) can be used to solve such equations... 

The integrals in Eqs. (B) and (C) must be evaluated through the exponential integral, E(x), a special function whose values are tabulated in handbooks and are also found from such software packages as MAPLE . The necessary equations, as found from MAPLE , are ... 

In this section we analyze processes involving the second-order A + B —> 2P reaction. Such processes have been studied, among others, by Luyben andTyreus [10]. It has been noticed [11] that certain control structures lead to state multiplicity and instability. Here, we use dimensionless models to derive general feasibility and stability conditions. The reader is encouraged to check carefully the balance equations, writing them first in the dimensional form, and then deriving the dimensionless versions. To solve these equations, software such as Maple or the symbolic toolbox of Matlab can be used. 

As an example of a symbolic calculation. Fig. 5 displays a Mathcad solution of one 4X4 block of the secular determinant of a Hiickel molecular orbital calculation done in Exp. 41 for orf/ro-benzosemiquinone (compare with Table 41-2). Mathcad is a software package for numerical analysis but also makes use of a subset of the symbohc routines of Maple. The algebraic expansion of the determinant is generated and solved with two... 

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. 

Roots of equations of state are most easily found with a software package such as Mathcad or Maple , in winch iteration is an integral part of the equation-solving routine. Starting values or bounds may be reqnired, and must be appropriate to the particular root of interest. A Mathcad programfor solving Ex. 3.8 is given in App. D.2. 

Dewpoint and bubblepoint calculations are readily made witlr software packages such as Mathcad and Maple , in wliich iteration is an integral part of an equation-solving routine. Mathcad programs for solution of Ex. 10.3, parts ( aitlirough(d), are given in App. D,2. 

Expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived by Kirkwood and BufP in compact matrix forms (see Appendix 1). The derivation of explicit expressions for the above quantities in multicomponent mixtures required an enormous number of algebraic transformations, which could be carried out by using a special algebraic software (Maple 8 was used in the present paper). A full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibilities in a quaternary mixture were derived. However, our main interest in this paper is related to the derivatives of the activity coefficient with respect to the mole fractions (all of the expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility can be obtained from the authors at request), namely, the derivatives of the form (9 In where Xg, Xg,... 

Combining Eqs. (C4)-(C11) with Eqs. (C2) and (C3) allows one to obtain the following Kirkwood-Buff integrals of ternary mixtures for an infinitely dilute solute (of course, the numerous algebraic transformations necessary could be carried out by using an algebraic software such as MATH-EMATICA or maple). 

However, the derivation of analytical expressions for the KB Is in ternary and multicomponent mixtures is algebraically extensive and requires the use of an algebraic software, such as Mathematica or Maple. A procedure was developed by us before and used to obtain expressions for Gn and G23 in water (1)—protein (2)—cosolvent (3) for infinite protein dilution. The expressions have the forms... 

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