Using the SRK equation ofstate (or any other cubic

Chapter 5, Problem 5.63

(choose chapter or problem)

Using the SRK equation ofstate (or any other cubic equation ofstate) to determine a specific volume from a specified temperature and pressure requires a trial-and-error calculation. Three computerbased approaches to solving this problem may be used: (1) spreadsheeting; (2) mathematical packages such as Mathcad, Mathematica, Maple, and E-Z Solve: and (3) programming languages such as Fortran and C++. The goal of this problem is to use each approach to determine VeL/mol) for CO2 at (i) 200 K and 6.8 atm; (ii) 250 K and 12.3 atm; (iii) 300 K and 6.8 atm; (iv) 300 K and 21.5 atm; and (v) 300 K and 50.0 atm. (a) Starting with Equation 5.3-7, derive the following equivalent expression for the SRK equation of state: f(V) = Pv3 - RTV2 + (aD: - b2P - bRT)V - aab = 0 (b) Write a spreadsheet program to take as inputs a species identifier (such as CO2), the critical temperature, critical pressure, and Pitzer acentric factor, and the temperatures and pressures for which Vis to be calculated, and to calculate Vusing Equations 5.3-9 to 5.3-13 for each of the specified conditions. The spreadsheet should have the following structure: PROBLEM 5.63-SRK EQUATION SPREADSHEET I I , I I Species CO2 i I Tc(K) I 304.2 I Pc(atm) 72.9 I I ; w 0.225 I i a * **** I i I i I b * **** i I I I * **** \ I I m I I I i , I ! T(K) P(atm) alpha V(ideal) I V(SRK) f(V) 2.4135 i i 200 6.8 1.3370 2.1125 *.**E-** I 250 12.3 * **** * **** I * **** *.**E-** i 300 6.8 * **** * **** I * **** *.**E-** i 300 21.5 * **** * **** I * **** *.**E-** ! I 300 50.0 * **** * **** i * **** *.**E-** I Single digits should appear in place of each asterisk shown on the table. Formulas should be entered into the row for 200 K and 6.8 atm and copied into the next four rows. The goalseek tool should be used to determine each V(SRK), starting with the ideal gas value and varying the cell value to make f(V) as close as possible to zero. *Computer problem. 233 (c) Use a root-finding procedure in a mathematical software package to determine Vfor each of the five conditions. (d) Write a program (in Fortran or another programming language) to determine Vfor each of the five conditions using Newton's rule (Appendix A.2d). The program should (i) Read in values of the species formula (C02), the critical temperature and pressure, and the Pitzer acentric factor. (ii) Calculate a, b, and m. (iii) Read values of T and P for which Vis to be calculated. Terminate if a negative value of T is input. (iv) Use the ideal gas equation of state to generate the initial value of V. (v) Calculate and print successive estimates of V using Equation A.2-2, stopping when the fractional change in V from one iteration to the next (E of Equation A.2-8) is less than 1.0 X 10-5 . Build in an upper limit of 15 iterations for each process condition; if convergence is not achieved within that limit, print an error message and quit. (vi) Go back to Step (iii). (e) Briefly summarize the advantages and disadvantages of the three problem-solving approaches.