Mathcad Appendix

Before posing the problem for this computer project, we shall introduce another vei y useful piece of microcomputer software by repeating the integration of Eq. (l-36a) with Mathcad (Appendix A). Like other software of this kind, there is a short learning process before mathcad can be used with ease. Once one has entered the equation of interest, mathcad solves it with a click on the = sign. In the present example, the constants of (Eq. l-36a) are entered followed by the desired integral... 

The challenge now is to solve this equation. An analytical, closed-form solution to Eqn. (7-12) does not exist. However, differential equations like this can be solved numerically using programs such as Matlab, Maple, or Mathcad. Appendix 7-A shows how to solve Eqn. (7-12) using a spreadsheet, i.e., EXCEL. From Appendix 7-A, the value of Cb at r = 40 min is 0.22 mol/1. 

Here, since the measurements were done in an integral reactor, calculation must start with the Conversion vs. Temperature function. For an example see Appendix G. Calculation of kinetic constants starts with listed conversion values as vX and corresponding temperatures as vT in array forms. The Vectorize operator of Mathcad 6 tells the program to use the operators and functions with their scalar meanings, element by element. This way, operations that are usually illegal with vectors can be executed and a new vector formed. The v in these expressions indicates a vector. 

Multiple regression analysis can be executed by various programs. The one shown in the Appendix is from Mathcad 6 Plus, the regress method. Taking the log of the rates first and averaging later gives somewhat different result. 

Principles of Thermodynamics should be accessible to scientifically literate persons who are either learning the subject on their own or reviewing the material. At Emory University, this volume forms the basis of the first semester of a one-year sequence in physical chemistry. Problems and questions are included at the end of each chapter. Essentially, the questions test whether the students understand the material, and the problems test whether they can use the derived results. More difficult problems are indicated by an asterisk. Some problems, marked with an M, involve numerical calculations that are most easily performed with the use of a computer program such as Mathcad or Mathematica. A brief survey of some of these numerical methods is included in Appendix B, for cases in which the programs are unavailable or cumbersome to use. 

The equations to be solved are first the Peng-Robinson equation of Eqs. 6.7-1 to 6.7-4 for the initial molar volume or compressibility, and then the initial number of moles in the tank using Eq. d of Illustration 6.5-2. The results, using the Visual Basic computer program described in Appendix B.I-2, the DOS-based program PRl described in Appendix B.II-1, the MATHCAD worksheet described in Appendix B.ni, or the MATLAB program described in B.IV included... 

To use this equation, we first, at the specified value of T and P, solve the Peng-Robinson equation of stats for the liquid compressibility, Z, and use this value to compute f T, P). Of,eburse. other equations of state could be used in Eq. 7.4-8. The Peng-Robinson equation-of-state programs or MATHCAD worksheets described in Appendix B can be used for this calculation for the Peng-Robinson equation of state. 

Using the Peng-Robinson equation-of-state programs or MATHCAD worksheets described in Appendix B, we obtain the results in Table 7.5-1. The vapor pressure as a function of temperature is plotted in Fig. 7.5-3. The specific volumes and molar enthalpies and entropies of the coexisting phases have been added as the two-phase envelopes in Figs. 6.4-3, 6.4-4, and 6.4-5.B... 

Computer programs and a MATHCAD worksheet to do this calculation are discu.ssed in Appendix B. 

The critical properties for both carbon dioxide and toluene are given in Table 6.6-1. The binary interaction parameter for the COi-toluene mixture is. not given in Table 9,4-1. However, as the value for CO -benzene is 0.077 and that for COi-rt-heptane is 0.10, we estimate that the COi-toluene interaction parameter will be 0.09. Using this value and the bubble point pressure calculation in either the programs or the MATHCAD worksheet for the Peng-Robinson equation of state for mixtures (de.scribed in Appendix B and on the CD-ROM accompanying this book), the following values were obtained ... 

The activity coefficient model parameters in Problems 11.2-18 to 11.2-25 are easily determined using the MATHCAD worksheet ACTCOEFF on the CD-ROM accompanying this book and described in Appendix B.lll. 

Using the bubble point pressure option in the Visual Basic program of Appendix B.1-3, the DOS-based program VLMU, or the MATHCAD worksheet PRBUBP, and the computed composition for the liquid richer in n-decane, we obtain the following results for the three-phase coexistence region ... 

Finally, because the world is not an ideal gas, equations of state or activity coefficients frequently must be used to describe a real system of interest. This results in considerable mathematical complexity. Consequently, many problems are best solved using the computer software I have provided that is discussed in the previous appendix, or with MATHCAD, MATLAB, MATHEMATICA, POLYMATH, or equation-solving software and programs you develop on your own. 

Calculate the binary diffusivities of ammonia in nitrogen (ID12) and ammonia in hydrogen (D13) using the Mathcad routine developed in Example 1.4 to implement the Wilke-Lee equation. Values of the Lennard-Jones parameters (a and e/k) are obtained from Appendix B. The data needed are ... 

The next step is to calculate the diffusivity of UF6 vapors in air at 314 K and 1 atm. The Lennard-Jones parameters for these gases are obtained from Appendix B. The Mathcad routine developed in Example 1.6 gives... 

The resulting pressure drop is too high therefore, we must increase the tower diameter to reduce the pressure drop. Appendix D presents a Mathcad computer program designed to iterate automatically until the pressure drop criterion is satisfied. Convergence is achieved, as shown in Appendix D, at a tower diameter of D = 0.738 m, at which AP/Z - 300 Pa/m of packed height. 

Next, enter the data presented above into the Mathcad sieve-tray design computer program of Appendix E. Since the dimensions of the tray are known, the fractional approach to flooding is adjusted until the tray design coincides with the tray dimensions determined in part (a) of this example. Convergence is achieved at a value of/= 0.431. This means that at the bottom of the distillation column the gas velocity is only 43.1% of the flooding velocity. Other important results obtained from the program are as follows ... 

If the equilibrium data are given in analytical form, a McCabe-Thiele computer program can be used. Appendix F is an example of a Mathcad programs to implement the McCabe-Thiele method (Hwalek, 2001). It generates the required VLE data from the Antoine equation for vapor pressure and the NRTL equation for liquid-phase activity coefficients. Appendix F-l is for column feed as saturated liquid Appendix F-2 is for column feed as saturated vapor. 

The slope of the equilibrium curve is m = 8.95. The gas-phase diffusivity is from the Wilke-Lee equation, DG = 0.177 cm2/s. The liquid-phase diffusivity is from the Hayduk-Minhas correlation for aqueous solutions, DL = 5.54 x 10-5 cm2/s. Take dQ = 4.5 mm on an equilateral-triangular pitch 12.0 mm between hole centers, punched in stainless steel sheet metal 2 mm thick. Use a weir height of 50 mm. Design for a 70% approach to the flood velocity. From the Mathcad program in Appendix E we get the following results for conditions at the bottom tray ... 

The Mathcad program in Appendix D is used to determine the tower diameter that satisfies the gas-pressure drop criterion at the bottom of the rectifying section. The result is D = 1.41 m, which corresponds to a 67% approach to flooding. The effective specific interfacial area is ah = 130.2 m2/m3. The mass-transfer coefficients are k - 3.705 mol/m2-s and k = 2.125 mol/m2-s. 

Using the data from Table 7.1, generate a cubic spline interpolation formula for the saturated raffinate curve, one for the saturated extract curve, and one for the tie-line data (see the Mathcad program in Appendix G-l). Equations in the form... 

It is desired to reduce the pyridine concentration of 5000 kg/h of an aqueous solution from 50 to 5 wt% in a single batch extraction with pure chlorobenzene. What amount of solvent is required Solve on right-triangular coordinates. Corroborate your results using the Mathcad program of Appendix G-l. 

Mathematics packages, such as Mathematica, Matlab or Mathcad and MathGV make it easy to fit these model distributions to sets of experimental data. It is often helpful to do so, since this allows the particle size distribution to be described by only two parameters. This also has its limitations, though. We shall come across one in Appendix 3. A. 

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