Liquid-Phase Enthalpies with the Yang-Yendall Equation of State The Yang-Yendall equation i s applied to the calculation of enthalpy departures of saturated liquid mixtures of methane and propane. A combination rule for equation-of-state constants i s not used. Instead enthalpy departures of mixtures are found with the help of an empirical rule which states that any residual property of a mixture i s the molal average of the residual properties of the components, all measured at the same pseudoreduced or reduced conditions. While the accuracy of the results i s limited by that of the Yang-Yendall equation, the procedure should be of interest in engineering applications.
T h e four-l)arameter Yang-Yeiidall equation of state
has been found useful for predicting vapor-liquid equilibria :md other thermodyiianiic properties iii the two-phase regioii (Yang aiid Yelidtill, 19ila,b). A unique feature of this equatioii is that the equation-of-state coiistaiits of a pure compoileiit are determined from its critical temperature, critical l)ressure, tlie iiornial boiling poiiit temperature, and the latent lieat of vaporization a t the iiormal boiling point. S o volumetric data are utilized i i i the coiistant fitting procedure. This tylie of equation c u i i be expected to predict eiithalpies aiid fugacities iii the two-l)hase region better than a correspoiidiiigly simple equatioii fitted with PT’T data. A i unwelcorne development iti the applicatiori of the TangYetjdall equation to mixtures was the discovery by its authors that 110 siiigle set, of cornbiiiatioii rules could be fouiid for the mixture parameters. Instead, it was found necessary to develop a separate combiliation rule for each given misture (Yang aiid Yendall, 19ila,b). This greatly limits t,he usefuliiess of the equation, since good mixture data are required t o develop a cornbiiiatioii rule for a mixture. Tang aiid Yelidall speculated on the need for biliary interactioii colistaiits to interpret mixture behavior but failed to show hoiv these could be iiitroduced into their mixture rules. The purpose of t’liis 1)aper is to shon. how t h e eyuatioiiof-state combination rules caii be circuniveiited in tlie case of the Yang-Yeiidall equatioii by making use of the Pseudocritical Method. Specific applicatioii is made to calculating the enthalpy departure from ideal gas, H a - M ,for saturated liquid mixtures of methane aiid propane. Equatioii-of-state constants for pure methane and for pure propane have been established by Yang and Yendall (197la,b). The enthalpy departure of a pure saturated liquid compoiieiit may be calculated with the equation derived from the work of Yaiig and Yendall
Ho - H
=
RT[1
- pl-m/(pl-m -
1)
Proceediiig to the calculatioii of tlie enthalpy departures of saturated liquid mixtures of methane and Iiropane, it, is first necessary to establish the pseudocritical coiistants of the mixtures. This can be done by any of several different procedures of which the siniplest is I i a ~ - ’ rule s (Kay, 1938). The method clioseii i i i this paper is that of I3arner and Quiiilan (1969) which includes tlie use of binary iiiteractioii paranieters. 13ariier a d (2iiiiilnli recommend a hitiar>- iiiteractioii parameter, K Z j= 1.07, for methaiie-prol)aiie mixtures. Their pseudocritical re1a t‘ioiisare ~
T,’
=
zizjTc,j
(3)
I-,’=
.l.,xjvc,j
(-1)
u’ =
Z,’
=
P,’
CziWz
0.891 - 0 . 0 8 ~ ’ =
RZ,’T,’ V,’
i5 )
(6) (7)
tern temperature niid pressure tlie pseudocritical rules of Harrier and Quiiilaii yield a pseudoreduced temperature aiid :i pjeutloreduced pressure. If the elithall)!- departure:, of pure iiiethxiie aiid pure propalie are fouiid a t the saiiie reduced coiitlitioiis of teniperature atid pressure ax thohe of the iiiixture. the>-ma\- be conibined accordiiig to t h e einpirical 1)riticil)le, that the molal :iverage of the residual 1)roperty values of tlie coinponetits gives tlie residual propert,y value for tlie niistiur (.Tofie, 1971)
I n this work the eiitliiilpy departure- oi pure iiiethatie a i d 1)ure propane \vere found ivitli eq 1 aiid 2 a t the bpec+ic temperatures aiid pressure. of the pure coml~ouiitls,calculated from tlie pseudoreduced \-dues us
+
n T * / ( n - 1)8”-1]
(2)
To test the performance of the Yang-Yendall equation iii predicting enthalpy departures of pure saturated liquid methane and propane, the experimental vapor pressures of these substances were substituted a t a series of temperatures into eq 1 and the correspondirig volumes were found by iteration. The enthalpy departures were found by substituting t h e volumes into eq 2. The results are compared with experimental data in Tables I and 11. Except ab or very close to the critical point, deviations of calculated enthalpy departures from experimental data are of the order of 2 or 3 13tu/ lh .
for each methane-propane composition studied. The eiithalpy departures of the pure components were tlieii combiiied iii accordaiice with ey 10 to yield the eiithallij- departure of the mixture. The results of these calculatioiis are compared in Table 111 with esperiiiieiital data aiid nit11 cal(uli1tioiis based on t h e extended Curl-Pitzer correlatioli (-%iiiericaii Petroleum Iiistitute, 1966). 111both sets of calculations Ixeiidoreduced coiiditiotis were derived from the pseudocriticals of 13ariier and Quiiilaii, eq 3-9 inclusive. I~ispectioiiof Table 111 shows that deviatioiis of enthalpy departures from esperimeiital \-allies are some\vhat larger when calculated with the Yaiig-Yetidall equatioii than if the Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
259
Table 1. Enthalpy Departures from Ideal Gas Saturated liquid Methane H o - H, Btu/lb Temp,
O F
- 270.67
- 243.67 -207.67 -171.67 -135.67 - 115.79
Pressure, psia
Calcd
Din (1961)
8.3 28.0 93.2 231.3 477.8 673.1
226 219 206 189 160 115
226 217 203 186 161 104
Table 111. Enthalpy Departures from Ideal Gas Saturated liquid Methane-Propane Mixtures Ho Mole fraction of propane
0.766
Table 11. Enthalpy Departures from Ideal Gas Saturated liquid Propane H o - H, Btu/lb Temp,
O F
-13.16 7.12 22.82 46.92 65.62 81.12 111.69 135.59 155.41
Pressure, psia
Calcd
Din ( 1 961 )
29.4 44.1 58.8 88.2 117.6 147.0 220.4 293.9 367.4
180 177 174 170 166 162 15‘4 147 139
180 175 172 167 163 160 152 145 137
Curl-Pitzer correlation is used. The difference in accuracy is attributable to the liniitntioiis of the Yang-Yeiidall equation rather than to the empirical rule on which eq 10 is based. I t lias been sliowii previously t,liat the 13arnei-Quiiilan eq 5 , this paper, used in applyiiig the Curl-Pit,zer correlation to nietliaiie-prol,aiie mistures, can he derived from tlie empirical principle of n-liicli eq 10 is an example (Joffe, 1 9 i l ) . On t’he other h i d , liquid enthalpy calculations with tlie YaiigYeiidall equation are better adapted to the computer than the Curl-Pitzer tables. For this reason t’he procedure illustrated in this paper merits consideration for applications where great accuracy is not iieeded. Nomenclature
H K ?n 71.
P
P
R
T
= enthalpy = = = = = = = =
T* = 1 : V
= =
5
=
Z
=
binary interaction constant a n equation-of-state constant an equation-of-state constant absolute pressure reduced pressure, as defined by Yaiig and Yendall universal gas constant absolute temperature reduced temperature, as defined by Yaiig and Ymdall characteristic temuerature molalvolume reduced volume, as defined by Yaiig and Yendall mole fraction compressibility factor = PV/RT
GREEKLETTER w
260
=
Pitzer’s acentric factor
Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
0,120
0.052
Temp, O F
Pressure, psia
Exptl
Yesavage, et al. (1969) -144 100 204 -100 200 194 -62 300 184 -24 400 176 12 500 167 47 600 158 ’76 700 150 102 800 140 -209 -150 -124 -101 -77 -49
Yesavage (1968) 100 211 300 188 500 171 700 159 900 145 1100 127
- H,
Btu/lb
Calculated This Curlwork Pitrer
197 191 178 171 163 155 147
197 190 183 175 166 158 150 142
208 188 177 166 151 130
206 186 176 166 144 115
185
13hirud and Powers (1969) -260 10 221 221 -200 102 198 204 -120 580 157 163 Average absolute deviation 4 1
222 203 164 3.6
SUBSCRIPTS c = crit’ical property i = pertaining to i t h component j = pertaining to j t h component ij = pertaining to i-j pair of components SUPERSCRIPTS o = pertaining t o ideal gas state ’ = pseudo property of mixture literature Cited American Petroleum Institute, Division of Refining, Kew York, N. Y., “Technical Data Book, Petroleum Refining,” 1966. Bamer, €1. E., Quinlan, C. R., Incl. Eng. Chem., Process Des. Develop. 8, 407 (1969). Bhirud, V. L., Powers, J. E., “Thermodynamic Properties of a 5 3Iole Per Cent Propane in Methane Mixture,” Report to Satural Gas Processors Association, Tulsa, Okla., Aug 1969. Din, F., Ed., “Thermodynamic Functions of Gases,” Butterworths, London, 1961. Joffe, J., IND.EXG.CHEM.,FUSDAM. 10, 532 (1971). Kay, W. B., Ind. Eng. Chem. 30, 459 (1938). Yang, C. L., Yendall, E. F., A.I.Ch.E. J . 17, 596 (1971a). Yang, C. L., lendall, E. F., A.I.Ch.E. J . 17, 602 (1971b). Yesavage, V. F., Ph.L). Thesis, University of Nichigan, 1968. Yesavage, V. F., Katz, L). L., Powers, J. E., J . Chem. Eng. Data 14, 137 (1969).
JOSEPH J O F F E Xewarb College of Engineering IYewarlc, N. J . 07102
RECEIVED for review August 16, 1972 ACCEPTED December 20, 1972
