Computer Solutions

The solution proceeds as described below. AP/Az is calculated using Eq. 2.41 for a segment of length Az  

Here RisR° atz = 0. P must also be determined for each step and this is done for each segment using Eq. 2.39. Because neither Pq — Pl nor Q is known, one must use the equation for Xrz(R) in Table 2.5 to determine AP/Az over the last segment 

Approximation Fluid Newtonian Power Law Power Law Power Law 

Power-Law Parameters from the Ellis Model. Ellis model parameters for a polypropylene sample at 200 °C are = 1.24E-I-04 Pa-s, tiq = 6.90E-I-03 Pa, and a = 2.82. Estimate m and n in the power-law model using these values of the parameters in the Ellis model. 

3 Pressure Transducer Selection. It is desired to select pressure transducers to be mounted on the upper wall of the plates described in Problem 2A.2. The accuracy of the transducers depends on the range of pressure that must be measured. For the conditions described in Problem 2A.2, what is the maximum pressure that would have to be measured for pressure transducers mounted at the entrance, halfway down the channel, and at a distance of El from the exit  

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Symplectic integration methods replace the t-flow (pt,H by the symplectic transformation which retains Hamiltonian features of They poses a backward error interpretation property which means that the computed solutions are solving exactly or, at worst, approximately a nearby Hamiltonian problem which means that the points computed by means of symplectic integration, lay either exactly or at worst, approximately on the true trajectories [5]. 

Forsythe, G. E. and Meier, C. B., 1967. Computer Solution of Linear Algebraic Systems, Prentice Hall, Englewood Cliffs, NJ. 

A reahstic estimate of the temperature profile for theoretical plates can probably be obtained by the short-cut method developed on the basis of rigorous computer solutions for about 40 different hypothetical designs (108) which closely resemble those of Figure 27. 

Computer solutions entail setting up component equiUbrium and component mass and enthalpy balances around each theoretical stage and specifying the required design variables as well as solving the large number of simultaneous equations required. The expHcit solution to these equations remains too complex for present methods. Studies to solve the mathematical problem by algorithm or iterational methods have been successflil and, with a few exceptions, the most complex distillation problems can be solved. 

In principle, this set of equations can be solved for the various constants, a through Q, just as a and b were obtained previously. In practice, however, the actual numerical evaluation involves considerable computation in all but the simplest examples. Computer solution by matrix techniques designed specifically to handle this type of data correlation problem is usually required. 

Rail, L. B. Computational Solution of Nonlinear Operator Equations, Wiley, New York (1969) and Dover, New York (1981). 

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

This introduces the compositions Xi and y into the eqiiilibrium equations, but neither is explicit, because the are functions, not only of T and P, but of composition. Thus Eq. (4-304) represents N complex relationships connecting T, P, the Xi, and the y, suitable for computer solution. Given an appropriate equation of state, one or another of Eqs. (4-178) through (4-181) provides for expression of the i as functions of T, P, and composition. 

Complex Clieinical-Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy G in accord with Eq. (4-271). The treatment here is... 

Simultaneous computer solution of these eight equations, with RT = 8,314 J/mol and... 

One set of specifications that is particularly convenient for computer solutions is ... 

These results indicate that the 7 percent 1-C5 in D and the 3 percent n-C4 in B concentrations obtained in the original column can easily he obtained on the smaller column. Unfortunately, this disagrees somewhat with the answers obtained from a rigorous computer solution as shown in the following comparison ... 

FIG. 13-46 Comp arison of the assumed and calculated profiles from the first column iteration in Example 4 with the final computer solution,... 

A computer solution was obtained as follows. The only initial assumptions are a condenser outlet temperature of 65 F and a bottoms-prodiict temperature of 165 F, The bubble-point temperature of the feed is computed as 123,5 F, In the initiahzation procedure, the constants A and B in (13-106) for inner-loop calcu-... 

Equation-of-state measurements add to the scientific database, and contribute toward an understanding of the dynamic phenomena which control the outcome of shock events. Computer calculations simulating shock events are extremely important because many events of interest cannot be subjected to test in the laboratory. Computer solutions are based largely on equation-of-state models obtained from shock-wave experiments which can be done in the laboratory. Thus, one of the main practical purposes of prompt instrumentation is to provide experimental information for the construction of accurate equation-of-state models for computer calculations. 

We discuss, here, some examples of computational solutions to shock or impulsive loading problems. We consider, in turn, one-, two-, and three-dimensional simulations, and the role each typically plays in computational physics and mechanics investigations. 

The Smith-Brinkley Method can therefore be used to generate a hand base case beginning with either a heat and material balanced plant case, a rigorous computer solution of a plant case, or computer solution of a design case. Once the hand base case is established, alternate cases can be done by hand (or small computer having limited core) using the Smith-Brinkley Method. 

In Section 3.4, we consider the open gas turbine cycle in which fuel is supplied in a combustion chamber and the working fiuids before and after combustion are assumed to be separate semi-perfect gases, each with Cp(T), c (T), but with R = [Cp T) — Cv( )l constant. Some analytical work is presented, but recently the major emphasis has been on computer solutions using gas property tables results of such computations are presented in Section 3.5. 

Ab initio methods compute solutions to the Schrodinger equation using a series of rigorous mathematical approximations. These procedures are discussed in detail in Appendix A, The Theoretical Background. 

Sufficient information is given in this chapter to enable the more rigorous stepwise calculations. These calculations are ideally suited to digital computer solution and have been programmed by the author for a machine of the smaller type. 

A time dependent modulus is then calculated using the extreme fiber stress level for each of the materials at the initial stress value level using the loading-time curve developed. If the deflection at the desired life is excessive, the section is increased in size and the deflection recalculated. By iteration the second can be made such that the creep and load deflection is equal to the maximum allowed at the design life of the chair. This calculation can be programmed for a computer solution. 

The relatively complex form of Eq. (11-37) leads to some approximate solutions that were widely used before the modem computer solutions were available. One of these was based on the coalescence temperature. This is the point at which two separate resonances are no longer observable. It can usually be measured to 0.2 °C. The rate constant in either direction at this temperature is... 

There s another reason why the computed solution average temperature had decreasing accuracies in Tests 1, 2 and 3 respectively. The reason is that we started with increasingly viscous solutions, which caused the response time of the temperature measurement to increase rapidly. This response time becomes even more significant because as the solution viscosity increases there are significant rises in the reaction rates and temperatures. 

It is necessary to point out that calculations by these formulae may induce accumulation of rounding errors arising in arithmetic operations. As a result we actually solve the same problem but with perturbed coefficients A-i, Bi, Ci, Xj, Xj and right parts Fi, /Ij, /jj. If is sufficiently large, the growth of rounding errors may cause large deviations of the computational solution yi from the proper solution j/,-. 

We use computational solution of the steady Navier-Stokes equations in cylindrical coordinates to determine the optimal operating conditions.Fortunately in most CVD processes the active gases that lead to deposition are present in only trace amounts in a carrier gas. Since the active gases are present in such small amounts, their presence has a negligible effect on the flow of the carrier. Thus, for the purposes of determining the effects of buoyancy and confinement, the simulations can model the carrier gas alone (or with simplified chemical reaction models) - an enormous reduction in the problem size. This approach to CVD modeling has been used extensively by Jensen and his coworkers (cf. Houtman, et al.) ... 

In general, the Lewis-Matheson method has not been found to be an efficient procedure for computer solutions, other than for relatively straightforward problems. It is not suitable for problems involving multiple feeds, and side-streams, or where more than one column is needed. 

W. J. Irwin, Kinetics of Drug Decomposition, Basic Computer Solutions, Elsevier Science, Amsterdam, 1990. 

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