1 FEBRUARY 2005 Javier Pérez PHASE DIAGRAM PREDICTION OF ACID GAS MIXTURES BY MOLECULAR SIMULATION Javier Pérez Pellitero †, P. Ungerer ‡, Allan D. Mackie † † † Departament d’Enginyeria Química,,Univ. Rovira i Virgili ‡ ‡ Institut Français du Pétrole

2 INTRODUCTION HISTOGRAM REWEIGHTING THE IDEA COLLECTING DATA REWEIGHTING DATA CALCULATING PHASE EQUILIBRIA FINITE SIZE SCALING GENERAL CONSIDERATIONS BINDER’S PARAMETER RESULTS CONCLUSIONS INDEX:

3 INTRODUCTION: Binary mixtures containing acid gases are of interest in the crude reinjection process of the petroleum industry. Gibbs ensemble technique is not useful in the critical region where industry demands predictions In this work histogram reweighting and finite size scaling techniques are presented to compute phase diagrams and to complete the phase envelope in the critical region. Phase diagrams and critical constants has been computed for the: Long range corrected Lennard-Jones fluid Binary Lennard-Jones mixture Potential r

4 N U H SIMULATIONS: 1. Monte Carlo simulations are performed in the Grand Canonical ensemble  μ i, V, T = constants 1. Information about the visited states in the system is recorded in histograms.

5 HISTOGRAM REWEIGHTING: THE IDEA Method to combine results taken at different state conditions Histograms are combined according to the method outlined by Ferrenberg and Swendsen. This method is based on the minimization of error in the microcanonical density of states The main idea: Combine simulation data at different temperatures to improve quality of all data via their mutual relation to  (U)

6 HISTOGRAM REWEIGHTING: COLLECTING AND REWEIGHTING DATA N U H H is the number of times the system visits a given state during a simulation Pure Compound Binary mixture H ( N,U) H (N 1,N 2,U) μ, V, T μ 1, μ 2, V, T

7 #U #N HISTOGRAM REWEIGHTING: COLLECTING AND REWEIGHTING DATA Different phase space regions are sampled with different simulations keeping “overlap” between them #N is the maximum number of particles allowed in the system and #U the maximum number of intervals (bins) for the energy. Farrenberg & Swendsen: error minimization in the calculation of  (U) Knowledge of the whole phase space of interest

8 HISTOGRAM REWEIGHTING: CALCULATING PHASE EQUILIBRIA Equilibrium is calculated tuning chemical potential until equal area (equal pressure) is obtained for both phases.

9 FINITE SIZE SCALING STUDY: CALCULATING THE CRITICAL PARAMETERS I For T>>T c and L = large, we find U L  3 For T<<Tc and L = large, we find U L  1

10 Lennard-Jones fluid L 1 = 7.9, L 2 =5.3 L 1 = 7.9 L 2 = 5.3 FINITE SIZE SCALING STUDY: CALCULATING THE CRITICAL PARAMETERS II For T  T c we can show that U L = universal number. This holds for large L. For small L there will be corrections in the form L -x with x > 0. This means:

11 RESULTS I: PURE FLUIDS: Cumulant intersection L 1 = 7.9, L 2 =5.3 1,3136This work 1,326Caillol (1998) 1,3120Potoff & Panagiotopoulos (1998) 1,3145Shi and Johnson (2001)

12 RESULTS II: LENNARD-JONES BINARY MIXTURES

13 RESULTS II: LENNARD-JONES BINARY MIXTURES

14 CONCLUSIONS: Histogram reweighting techniques combined with finite size scaling studies improve the behaviour of Gibbs ensemble near the critical region and allow for an accurately calculation of the critical constants. The results are in good agreement with Gibbs ensemble simulations in the region of appliance of the last technique Application of histogram reweighting techniques allows for the calculation of binary mixture pressure- temperature-composition phase diagrams from a relative small number of simulations.

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