A Very Robust and Easy to Implement Interval Approach to Solve VLE, LLE and VLLE Flash Problems

A very robust and easy to implement interval approach to solve VLE, LLE and VLLE flash problems

Hamid Mehdizadeh, Hallvard F. Svendsen, Tore Haug-Warberg

Norwegian University of Science and Technology

 

Flash calculation problems are one of the most frequently occurring problems in chemical engineering and lots of effort has been put into solving its challenges. Since the early work of Rachford and Rice[1] (RR) different methods have been suggested over the decades[2-6]. But, still, this problem is subject of interest where research effort is focused[7]. Different aspects of flash calculation procedures are studied like robustness[6], speed[8] and number of phases[9].

Equi-fugacity is the necessary condition for the phase equilibrium, while a minimum of Gibbs free energy can be considered a sufficient condition. A successive substitution (SS) approach used in the RR method decreases the Gibbs free energy in each iteration, with some exceptions[10]. SS method consists of two loops. In the internal loop the phase fraction is calculated and the external loop iterates on Ki (distribution coefficients). The external loop could always be convergent using damping factor[10], but there are problems associated with solving the internal loop as the vapor fractions could obtain negative values [11, 12]. Some other difficulties are discussed in Veeranna et al.[13].

To get rid of difficulties in the calculation of phase fraction in the inner loop an Interval-Newton method is used. Based on the this method, all roots of a function in a pre-specified interval could be calculated with mathematical proof, with some exceptions[14].

Interval calculation introduced by R.E. Moore in 60's, was first developed to estimate rounding error[14]. This method is able to predict the range of change of f(x) when x changes in the specified interval of [a,b]. For this work, a program based on the INTLAB[15] package in MATLAB environment is used.

The function that should be solved for 2 phase flash is:

                                                                                                            

Where b is the molar fraction of the reference phase. As the size of b is (NP-1), so the size of the problem that is to be solved by interval analysis will not be too big and could be done with a small computational effort. In the outer loop both equations of state and activity coefficient models could be used. This feature makes the model flexible and easy to use in different approaches.

Heidemann[10] describes that with the SS method in each iteration the Gibbs free energy will decrease, but there will be some cases with instability. To prevent instability and also to speed up the convergence, an adaptive damping factor is used.

The prepared model showed good capabilities in prediction of VLE, LLE and VLLE. For VLE, different hydrocarbon and non-hydrocarbon mixtures, in a wide range of temperatures and pressures were examined and good prediction was achieved. Binary interaction parameters were calculated from a group contribution approach developed by Jaubert[16] and his coworkers in a series of papers and further expanded by Ghadrdan [17] to new groups. Three different mixtures (nC10-CH4, CO-nC8, H2-nC6) were validated using this method. For the first mixture the model is predictive up to 24MPa which was the limit of available experimental data. Considering the big difference between the size of molecules and also the very high pressure reflects the performance of the model in reasonable computation time. All results were calculated without any additional data. Distribution values were estimated from the Watson equation. For VLLE cases, a mixture that is considered in CO2 flooding and EOR research was examined. For LLE, some different mixtures were investigated using the extended UNIQUAC framework as presented by Thomson.[18], but for non-electrolyte systems.  

Acknowledgements: Financial support from Aker Clean Carbon and the Research Council of Norway through the CLIMIT program SOLVit project is greatly acknowledged

 

References:

1.         Rachford, H.H. and J.D. Rice, Procedure for Use of Electronic Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium. 1952.

2.         Boston, J.F. and H.I. Britt, A radically different formulation and solution of the single-stage flash problem. Computers & Chemical Engineering, 1978. 2(2-3): p. 109-122.

3.         Michelsen, M.L., The isothermal flash problem. Part I. Stability. Fluid Phase Equilibria, 1982. 9(1): p. 1-19.

4.         Michelsen, M.L., The isothermal flash problem. Part II. Phase-split calculation. Fluid Phase Equilibria, 1982. 9(1): p. 21-40.

5.         Parekh, V.S. and P.M. Mathias, Efficient flash calculations for chemical process design -- extension of the Boston-Britt "Inside-out" flash algorithm to extreme conditions and new flash types. Computers & Chemical Engineering, 1998. 22(10): p. 1371-1380.

6.         Li, Y. and R.T. Johns, A Rapid and Robust Method To Replace Rachford-Rice in Flash Calculations, in SPE Reservoir Simulation Symposium. 2007: Houston, Texas, U.S.A.

7.         Nichita, D.V. and A. Graciaa, A new reduction method for phase equilibrium calculations. Fluid Phase Equilibria, 2011. 302(1-2): p. 226-233.

8.         Michelsen, M.L., Speeding up the two-phase PT-flash, with applications for calculation of miscible displacement. Fluid Phase Equilibria, 1998. 143(1-2): p. 1-12.

9.         Fraces, C., D.V. Voskov, and H.A. Tchelepi, A New Method for Thermodynamic Equilibrium Computation of Systems With an Arbitrary Number of Phases, in SPE Reservoir Simulation Symposium. 2009: The Woodlands, Texas.

10.       Heidemann, R.A. and M.L. Michelsen, Instability of Successive Substitution. Industrial & Engineering Chemistry Research, 1995. 34(3): p. 958-966.

11.       Michelsen, M.L., Calculation of multiphase equilibrium. Computers & Chemical Engineering, 1994. 18(7): p. 545-550.

12.       Leibovici, C.F. and D.V. Nichita, A new look at multiphase Rachford-Rice equations for negative flashes. Fluid Phase Equilibria, 2008. 267(2): p. 127-132.

13.       Veeranna, D., et al., An algorithm for flash calculations using an equation of state. Computers & Chemical Engineering, 1987. 11(5): p. 489-496.

14.       Hansen, E.R. and G.W. Walster, Global optimization using interval analysis. 2004, New York: Marcel Dekker. xvii, 489 s.

15.       Rump, S.M., INTLAB - INTerval LABoratory, in Developments in Reliable Computing, T. Csendes, Editor. 1999, Kluwer Academic Publishers: Dordrecht. p. 77-104.

16.       Privat, R., J.-N. Jaubert, and F. Mutelet, Addition of the sulfhydryl group (-SH) to the PPR78 model (predictive 1978, Peng-Robinson EOS with temperature dependent kij calculated through a group contribution method). The Journal of Chemical Thermodynamics, 2008. 40(9): p. 1331-1341.

17.       M. Ghadrdan, 2011, Personal communication

18.       Thomsen, K., P. Rasmussen, and R. Gani, Correlation and prediction of thermal properties and phase behaviour for a class of aqueous electrolyte systems. Chemical Engineering Science, 1996. 51(14): p. 3675-3683.