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0.2 0.18 0.16 0.14 Berthelot 0.12 VW 0.1 0.08 RK 0.06 0.04 0.02 0 1 10 100 1000 TR Figure 11: The deviation function for Berthelot, RK,and VW gases, PR ®0 P = (RT/v) + (1/v2) (bRT – a/Tn). (63) Therefore, Pv/RT = Z (v,T) = (1 + b/v) –a/(vRT(n+1)) = 1 + B(T)/v, (64) Where B(T) = b – a/RT(n+1). As P ® 0, v ® ¥, and Z ® 1. Solving for v from Eq. (64), v = RT/2P (1 ± (1 + (4 P/RT)(b – a/RT n+1))1/2). As P ® 0, v » RT/2P (1 ± (1 + 2 P/RT (b–a/RTn+1))), only positive values of which are acceptable. Therefore, v = RT/P + (b – a/RTn+1), Since RT/P = v0, v = v0 + b – a/(RTn+1), for RK, VW and Berthelot (65) Therefore, at lower pressures, (v – v0)P®0 = b – a/RTn+1, where n ≥ 0. For example, if n = 0, the volume deviation function has a value equal to (b – a/RT), and is a function of tem-perature. In this case, the real gas volume never approaches the ideal gas volume even when T ® ¥. In dimensionless form vR – vo,R = a∗ – b∗/TR(n+1), for RK, VW and Berthelot fluids. Figure 12. Illustration of Pitzer factor estimation. where n = 0, 1, 1/2, a∗ = (27/64), (27/64), 0.4275, and b∗ = 0.125, 0.125, 0.08664, respectively, for the Van der Waals, Berthelot and RK equations. The difference between v¢ and v¢,R are illustrated with respect to TR for these equations as PR ® 0 in Figure 11. If a= b = 0 in any of the real gas equations of state, these equations are identical to the ideal gas state equation. 8. Three Parameter Equations of State If v = vc (i.e., along the critical isochore), employing the Van der Waals equation, P = RT/(vc – b) – a/vc2, which indicates that the pressure is linearly dependent on the temperature along that isochore. Likewise, the RK equation also indicates a linear expression of the form P = RT/(vc – b) – a/(T1/2vc (vc + b)). However, experiments yield a different relation for most gases. Simple fluids, such as argon, krypton and xenon, are exceptions. The compressibility factors calculated from either the VW or RK equations (that are two parameter equations) are also not in favorable agreement with experiments. One solution is to increase the number of parameters. a. Critical Compressibility Factor (Zc) Based Equations Clausius developed a three parameter equation of state which makes use of experi-mentally measured values of Zc to determine the three parameters, namely P = RT/( v – b) – a/(T ( v + c)2). (66) where the constants can be obtained from two inflection conditions and experimentally known value of ZC, critical compressibility factor. b. Pitzer Factor The polarity of a molecule is a measure of the distribution of its charge. If the charge it carries is evenly or symmetrically distributed, the molecule is non–polar. However, for some chemical species, such as water, octane, toluene, and freon, the charge is separated across the molecule, making it uneven or polar. The compressibility factors for nonsymmetric or polar fluids are found to be different from those determined using two parameter equations of state. Therefore, a third factor, called the Pitzer or acentric factor ω has been added so that the em- pirical values correspond with those obtained from experiments. This factor was developed as a measure of the structural difference between the molecule and a spherically symmetric gas (e.g., a simple fluid, such as argon) for which the force–distance relation is uniform around the molecule. In case of the saturation pressure, all simple fluids exhibit universal relations for PRsat with respect to TR (as illustrated in Figure 14). In Chapter 7 we can derive such a relation using a two parameter equation of state. For instance, when TR = 0.7, all simple fluids yield PRsat » 0.1, but polar fluids do not. The greater the polarity of a molecule, the larger will be its deviation from the behavior of simple fluids. Figure 14 could also be drawn for log10 Prsat vs. 1/TR as illustrated in Figure 12. The acentric factor ω is defined as ω = –1.0 – log10 (PRsat)TR=0.7 = –1 – 0.4343 ln (PRsat)TR=0.7. (67) Table A-1 lists experimental values of ω” for various substances. In case they are not listed, it is possible to use Eq. (68). i. Comments The vapor pressure of a fluid at TR = 0.7, and its critical properties are required in or-der to calculate ω. For simple fluids ω = 0. For non-spherical or polar fluids, a correction method can be developed. If the com-pressibility factor for a simple fluid is Z(0), for polar fluids Z ¹ Z(0) at the same values of TR and PR. We assume that the degree of polarity is proportional to ω. In general, the difference (Z – Z(0)) at any specified TR and PR increases as ω becomes larger (as illustrated by the line SAB in Figure 13). With these observations, we are able to establish the following relation, namely. (Z (ω,TR,PR)– Z(0) (PR, TR)) = ωZ(1) (TR, PR). (68) Evaluation of Z(ω,TR,PR) requires a knowledge of Z(1), w and Z(0) (PR, TR). c. Evaluation of Pitzer factor,ω i. Saturation Pressure Correlations The function ln(Psat) varies linearly with T–1, i.e., ln Psat = A – B T–1. (69) Using the condition T = Tc, P = Pc, if another boiling point Tref is known at a pressure Pref, then the two unknown parameters in Eq. (70) can be determined. Therefore, the saturation pressure at T = 0.7Tc can be ascertained and used in Eq. (69) to determine ω. ii. Empirical Relations Empirical relations are also available, e.g., ω = (ln PRsat – 5.92714 + 6.0964/TR,BP+1.28862 ln TR,BP – 0.16935 TR,BP)/ (15.2578 – 15.6875/TR,NBP + 0.43577 TR,NBP), (70) where PR denotes the reduced vapor pressure at normal boiling point (at P = 1 bar), and TR,NBP the reduced normal boiling point. PR2, TR2 PR1, TR1 Zref Z(1) (PR, TR) A S B wref Figure 13: An illustration of the variation in the compressibility factor with respect to the acentric factor. An alternative expression involves the critical compressibility factor, i.e., ω = 3.6375 – 12.5 Zc. (71) Another such relation has the form ω = 0.78125/Zc – 2.6646. (72) 9. Other Three Parameter Equations of State Other forms of the equation of state are also available. a. One Parameter Approximate Virial Equation For values of vR > 2 (i.e., at low to moderate pressures), Z = 1 + B1(TR) PR, (73) where B1(TR) = B(0)(TR) + ωB(1)(TR), B(0)(TR) = (0.083 TR–1) – 0.422 TR–2.6, and B(1)(TR) = 0.139 – 0.172 TR–5.2. b. Redlich–Kwong–Soave (RKS) Equation Soave modified the RK equation into the form P = RT/(v–b) – a a (ω,TR)/(v(v+b)), (74) where a = 0.42748 R2Tc2Pc–1, b = 0.08664 RTcPc–1, and a (ω,TR) = (1 + f(ω)(1 – TR0.5))2 , which is determined from vapor pressure correlations for pure hydrocarbons. Thus, f(ω) = (0.480 + 1.574 ω – 0.176 ω2). c. Peng–Robinson (PR) Equation The Peng–Robinson equation of state has the form P = (RT/(v – b)) – (a a(ω,TR)/((v + b(1 + 20.5))(v + b(1 – 20.5))), (75) where a = 0.45724 R2Tc2Pc–1, b = 0.07780 RTcPc–1 and a(ω,TR) = (1 + f(ω) (1 – TR0.5))2, f(ω) = 0.37464 + 1.54226 ω – 0.26992 ω2. Equation (75) can be employed to predict the variation of Psat with respect to T, and can be used to explicitly solve for T(P,v). 10. Generalized Equation of State Various equations of state (e.g., VW, RK, Berthelot, SRK, PR, and Clausius II) can be expressed in a general cubic form, namely, P = RT/(v–b) – aa (ω,TR)/(Tn(v+c)(v+d)). (76) In terms of reduced variables this expression assumes the form PR = TR/( v¢ – b´) – a´a (ω,TR)/(TRn ( v¢ + c´) ( v¢ + d´)), (77) where a´ = a/(Pc v¢2 Tcn), b´ = b/ v¢, c´ = c/ v¢, and d´ = d/ v¢. Tables are available for the pa-rameters a´ to d´. Using the relation Z = PR v¢ /TR, we can obtain a generalized expression for Z as a function of TR and PR, i.e., Z3 + Z2 ((c´ + d´ – b´) PR/TR – 1) + Z (a´a (ω,TR) PR/TR2+n – (1 + b´PR/TR) (c´ + d´)PR/TR + c´d´ PR2/TR 2) – (a´a (ω,TR) b´PR2/TR(3+n) + (1 + PR b´/TR) (c´d´PR2/TR2)) = 0. (78) Writing this relation in terms of v¢ , v¢ 3 PR + v¢ 2((c´ + d´ – b´)(PR/TR) –1) + v¢ ((c´d´ – b´c´ –b´d´)PR – (c´ + d´)TR + a´/TRn) – PR b´c´d´ – a´a (ω,TR)b´/TR – TRc´d´ = 0. (79) Using this equation along with the relation TR = PR v¢ /Z, the compressibility factor can be obtained as a function of PR and v¢ , i.e., Z(3+n) (a´a(w, TR) /( v¢ (2+n)PR(1+n))) (1–b´/ v¢ ) + Z3(1+(c´+d´–b´)/ v¢ –(b´/ v¢ 2)(c´+d´–d´/vR)) – Z2(1 + (c´ + d´)/ v¢ –c´d´PR/v¢ + d´/ v¢ 2) – Z(b´ c´/ v¢ ) – c´ = 0.. (80) where TR in a (w, TR) expression must be replaced by PR vR´/Z. Table 2 tabulates values of a, n, a´, b´, c´, and d´ for various equations of state. Table 2: Constants for the generalized real gas equation of state. a´ b´ c´ d´ n f(ω), H2O Berthelot 0.421875 0.125 0 0 1 Clausius II 0.421875 -0.02 0.145 0.375 1 PR 0.45724 0.0778 0.187826 -0.03223 0 PR with w 0.4572 0.0778 0.187826 -0.03223 0 0.873236 RK 0.42748 0.08664 0.08664 0 0.5 SRK 0.42748 0.08664 0.08664 0 0 1.000629 VW 0.421875 0.125 0 0 0 Note that Zc is required for Clausius II while ω is required for RKS, PR, f(ω) for H2O with ω = 0.344 ... - tailieumienphi.vn