Orr, F. M. - Theory of Gas Injection Processes Episode 3

3.4. FLASH CALCULATION 31 Table 3.2: Peng-Robinson Binary Interaction Parameters [99, 18] Component CH4 C2 C3 C4 C5 C6 C7 C8 C9 C10 C12 C16 C20 CH4 C2 0.010 0.020 0.010 0.020 0.010 0.025 0.010 0.025 0.010 0.035 0.010 0.035 0.010 0.035 0.010 0.035 0.010 0.035 0.010 0.035 0.010 C3 CO2 N2 0.100 0.120 0.010 0.150 0.120 0.134 0.120 0.010 0.130 0.120 0.010 0.125 0.120 0.010 0.119 0.120 0.010 0.100 0.120 0.010 0.112 0.120 0.010 0.100 0.120 0.010 0.102 0.120 0.010 0.095 0.120 0.010 0.105 0.120 0.010 0.093 0.120 in subsequent chapters. Interaction parameters for CO2 are those recommended by Deo et al. [18]. Otherwise Table 3.2 reports values of δij recommended by Peng and Robinson [99]. Eqs. 3.3.7–3.3.16 provide the description of volumetric behavior needed to complete the calcu-lation of the partial fugacity of a component in a mixture. Differentiation of Eq. 3.3.7 gives an expression for ∂ni . That expression is then substituted into Eq. 3.2.8, and the integration is performed. The result, after considerable algebraic manipulation, is ! ln xiij = lnφij = b i (z −1) −ln z 1 − m am 2 2bmRT bi − 2 nc xkj(aα)ik!ln m m k=1 1 + (√2 + 1)bm ! 1 −( 2 − 1)bm (3.3.17) 3.4 Flash Calculation The result of the thermodynamic analysis and the use of an equation of state to describe volumetric behavior is a set of nonlinear equations (Eqs. 3.1.30 in which each value of fij is given by an expression like Eq. 3.3.17) that must be solved for the compositions of the phases. The following procedure can be used: 1. Estimate composition, xij, of each of the phases present. 2. Solve Eq. 3.3.7 for the molar volume, V, of each phase. 3. Use Eq. 3.3.17 to calculate the partial fugacity, fij, for each component in each phase. 32 CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM 4. Check to see if component partial fugacities are equal in all of the phases (Eq. 3.1.30). If not, then adjust the estimates of phase compositions and return to step 2. While this procedure will give the compositions of phases that satisfy the requirement that component partial fugacities (or equivalently, chemical potentials) be equal at equilibrium, it should be noted that it is sometimes possible to find solutions for which the resulting phases are not stable [3]. In other words, the phase compositions make component chemical potentials equal but do not minimize free energy. For such situations, the stability of phases can be tested directly. See Baker et al. [3] or Michelsen [78, 80] for examples and details. Given the nonlinearity of Eq. 3.3.17, it is easy to see why flash calculations with van der Waals equation did not catch on in the 1880’s when van der Waals [122] performed the first phase equilibrium calculations for binary systems. In fact, equation-of state calculations of phase equilibrium did not come into widespread use until the late 1960’s when availability of computing resources made solution of the set of nonlinear equations reasonable. Step 1 requires that some sort of initial guess of phase compositions be made. For flash calcu-lations for two-phase systems, information about phase compositions is often expressed in terms of equilibrium ratios (also known as K-values), Ki = xi1, (3.4.1) i2 where xi1 and xi2 are the mole fractions of component i in phases 1 and 2, typically vapor and liquid. The Wilson equation [136], Ki = xi1 = ci exp 5.37(1 + ωi) 1 − ci , (3.4.2) is frequently used to estimate equilibrium K-values from which phase compositions can be estimated by the following manipulations. From the estimated or updated K-values, the phase compositions can be obtained from a mate-rial balance on each component. Consider one mole of a mixture in which the overall mole fraction of component i is zi. A material balance for component i gives zi = xi1L1 + xi2(1 − L1), i = 1,nc. (3.4.3) where L1 is the fraction of the one mole of mixture that is phase 1. Elimination of xi1 from Eq. 3.4.3 gives xi2 = 1 + L1(Ki − 1), Similar elimination of xi2 using Eq. 3.4.2 gives xi1 = 1 + L1(Ki − 1), i = 1,nc. i = 1,nc. (3.4.4) (3.4.5) An equation for L1 is obtained by noting that Eqs. 3.4.4 and 3.4.5 each sum to unity, so that nc nc nc i=1 xi1 − i=1 xi2 = i=1 1 + L1(Ki −1) = 0. (3.4.6) 3.4. FLASH CALCULATION 33 Eq. 3.4.6, which is known as the Ratchford-Rice equation [103], can be solved for L1 by a Newton-Raphson iteration [80, p. 220]. Given the value of L1, Eqs. 3.4.4 and 3.4.5 give the phase compo-sitions consistent with the K-values. The equilibrium K-values defined in Eq. 3.4.2 are related to the equilibrium partial fugacity coefficients. The equilibrium relations, Eqs. 3.1.30, can be written using Eq. 3.1.28 as fi1 = φi1xi1P = φi2xi2P = fi2, i = 1,nc. (3.4.7) Rearrangement of Eq. 3.4.7 shows that the equilibrium K-value is just the ratio of partial fugacity coefficients, Ki = xi1 = φi2, i = 1,nc. (3.4.8) i2 i1 The form of Eq. 3.4.8 suggests a simple successive approximation scheme for updating K-values in step 4 of the flash calculation [98, p. 103][80, p. 245]. If the component partial fugacites are not equal at the kth iteration, then new K-values can be estimated from Kk+1 = Kk fi2, i = 1,nc. (3.4.9) i1 Eq. 3.4.9 modifies the K-values in the appropriate direction if fi2 = fi1. While iteration with Eq. 3.4.9 usually converges to solutions that satisfy the equilibrium relations even when the guess of initial phase compositions is relatively poor, convergence will be very slow for two-phase mixtures near a critical point (a pressure, temperature and composition for which the two phases become identical). For such situations, more sophisticated iterativeschemes can and should be used [79, 75]. Negative Flash The flash calculation can be performed whether a mixture forms one or two phases. Whitson and Michelsen [135] pointed out that Eq. 3.4.6 can be solved for L1 equally well when only one phase forms, a calculationthat is known as a negative flash. When the iterationfor L1 has converged for a single-phase system, the resulting value will be in the range L1 < 0 or L1 > 1. The phase compositions calculated with Eqs. 3.4.4 and 3.4.5 will be equilibrium compositions that can be combined to make the single-phase mixture. In other words, the single-phase mixture is a linear combination of the phase compositions, which means that the single-phase composition must lie on the extension of the line that connects the equilibrium compositions on a phase diagram. We will make repeated use of the negative flash to find that line, known as a tie line, for displacement calculations. Whitson and Michelsen showed that their negative flash calculation converges as long as L1 lies in the range 1 − Kmax < L1 < 1 −Kmin , (3.4.10) where Kmax and Kmin are the largest and smallest K-values. If the single-phase composition is far enough from the two-phase region that the condition 3.4.10 is not satisfied, a modified negative flash suggested by Wang [128] can be used. It is based on the 34 CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM idea that while L1 can vary over wide ranges, the equilibrium phase compositions, xij, are restricted to lie between zero and one. The mole fraction of phase 1, L1 can be eliminated by solving Eq. 3.4.4 written for component 1, L1 = (K1−1)x12. (3.4.11) Substitution of Eq. 3.4.11 into Eq. 3.4.4 gives an expression for the phase compositions in phase 2 in which the only unknown is x12, zix12 (K1 −1) i2 z1 (Ki − 1) + x12 (K1 −Ki) The revised version of Eq. 3.4.6 is nc nc zix12 (Ki − 1)(K1 −1) i=1 i 12 i=1 z1 (Ki −1) + x12 (K1 − Ki) Eq. 3.4.13 can be solved for x12 by a Newton-Raphson iteration. (3.4.12) (3.4.13) 3.5 Phase Diagrams It will be convenient for many of the flow problems considered here to represent the solutions as a collection of compositions on a phase diagram. Accordingly, we review briefly the terminology and properties of binary, ternary, and quaternary phase diagrams. 3.5.1 Binary Systems Fig. 3.1 is a typical phase diagram for a binary mixture at some fixed temperature above the critical temperature of component 1. At pressure, P, liquid phase (phase 2) with mole fraction x12 of component 1 is in equilibrium with vapor phase (phase 1) containing mole fraction x11 of component 1. Those equilibrium compositions are connected by a tie line, along which a tie line material balance like Eq. 3.4.3 applies. There is one tie line in Fig. 3.1 for each pressure. A mixture with an overall molefraction z1 of component 1 between x12 and x11 forms two phases. Mixtures with z1 < x12 are all liquid, while those with z1 > x11 form only vapor. For a given overall composition, the mole fraction of phase 1 present is easily determined by rearrangement of Eq. 3.4.3 to be z1 − x12 x11 −x12 (3.5.1) Eq. 3.5.1 is a lever rule, which states that the mole fraction vapor is proportional to the distance from the overall composition to the liquid composition locus divided by the length of the tie line. A similar statement applies to systems with any number of components. Eq. 3.5.1 indicates that L1 ≤ 0 for mixtures that form only liquid, and L1 ≥ 1 for mixtures that are all vapor. At the top of the two-phase region in Fig. 3.1 is a critical point, at which the liquid and vapor phases, as well as all phase properties, are identical. The critical point can be thought of as a tie line with zero length. Because phase compositions are equal at a critical point, Eq. 3.4.2 indicates that all K-values must be equal to one for a critical mixture. 3.5. PHASE DIAGRAMS 35 2000 a 1800 Critical Point 1600 1400 1200 Liquid Phase Tie Line 1000 800 Two Phase Region 600 400 Vapor Phase 200 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction Component 1 Figure 3.1: Pressure-composition phase diagram for a two-component mixture. A phase diagram like Fig. 3.1 can be calculated with an equation of state. Fig. 3.2 shows such a diagram calulated with the Peng-Robinson equation for a two-component system, CO2/decane (C10) system. Also shown in Fig. 3.2 are experimental data of Reamer and Sage [104] along with the phase diagram calculated with van der Waals equation of state. Critical properties used for CO2 and C10 are given in Table 3.1, and the binary interaction parameter used in the Peng-Robinson flash calculations was δ12 = 0.102 (Table 3.2). Fig. 3.2 shows that the Peng-Robinson phase compositions agree much better with the experimental observations than do the van der Waals predictions. They should, of course, because the value of δ12 was chosen to minimize the disagreement between calculated phase compositions and measured data. Even so, with the average value of δij chosen by matching experimental data at several temperatures [18], there is some disagreement between experiment and calculation near the critical point. 3.5.2 Ternary Systems A typical vapor/liquid phase diagram for a three-component system is shown in Fig. 3.3, which displays phase behavior information at fixed pressure and temperature. Because the mole fractions at any composition point in the diagram always sum to one, it is useful to plot equilibrium phase compositions on an equilateral triangle. On such a diagram, the three mole (or volume or mass) fractions are read from the perpendicular distances from the composition point to the three sides. The corners of the diagram represent 100% of the component with which the corner is labeled, and the opposite side represents the zero fraction. The sides of the ternary diagram represent binary mixtures of the two components that lie on that side. For gas/oil systems, the component at the top corner of the diagram is usually the lightest component, and the heavies component is usually ... - tailieumienphi.vn