CALCULATION OF ACID GAS DENSITY IN THE VAPOR, LIQUID, AND DENSE-PHASE REGIONS

Transcription

1 CALCULATION OF ACID GAS DENSITY IN THE VAPOR, LIQUID, AND DENSE-PHASE REGIONS Tm B. Boyle PanCanadan Petroleum Ltd Avenue SW Calgary, Alberta TP 1S John J. Carroll Gas Lquds Engneerng Ltd. #300, Avenue NE Calgary, Alberta T1Y 4T8 Acd gas njecton has quckly become the method of choce for the dsposal of unwanted acd gas (hydrogen sulfde and carbon doxde). Ths s especally true f the quantty of acd gas s small. One of the mportant parameters n the desgn of an njecton scheme s the densty of the acd gas mxture. Ths paper revews the avalable expermental data for the denstes of these mxtures and compares several methods for calculatng the densty. The frst element of the study wll be an nvestgaton of the densty predctons for pure hydrogen sulfde and carbon doxde where a substantal amount of data exsts. Next the bnary mxture wll be examned. For the bnary mxture only a lmted amount of data are avalable. In acd gas njecton the range of temperature of nterest s from 0 to 150 C and pressures from about atmospherc up to 30 MPa. Ths range of condtons ncludes the vapor, lqud, and dense-phase regons. It s therefore mportant that the model be able to predct the densty n all three regons. Ths study focuses on the ablty of the common cubc equatons of state (Soave-Redlch-Kwong [SRK], Peng-Robnson [PR], and Patel-Teja [PT]) to predct the denstes. Also, the effect of volume shftng on the SRK and PR equatons s examned. Of the sx equatons examned n ths study, only the SRK was deemed to be unsatsfactory for predctng denstes over the entre range of condtons. The errors for ths equaton were often larger than 10%. The volume-shfted SRK s margnally satsfactory. The volume-shft dramatcally mproves the densty n the lqud regon, but not n the supercrtcal regon. The other four equatons (PR, two volume shfted PR, and PT) exhbted errors less than 10% except n the near crtcal regon. Boyle and Carroll 1

2 INTRODUCTION Acd gas, a mxture of carbon doxde and hydrogen sulfde, s a byproduct of natural gas sweetenng processes. The acd gas mxture s toxc and envronmentally problematc so t must be dealt wth approprately. If there s a large quantty of acd gas, the hydrogen sulfde contaned theren s typcally converted to elemental sulfur usng a Claus process. In ths process the carbon doxde s usually released to the atmosphere. Acd gas njecton has become an economcal way to deal wth small quanttes of acd gas. Furthermore, acd gas njecton s a very low emsson process, typcally on a percentage bass, much lower than a Claus plant. In Western Canada there are about 5 such projects currently n operaton and several more are beng planned. In other parts of the world acd gas njecton s startng to be looked upon favorable as a dsposal method. It should be noted that acd gas njecton s not lmted to small projects. Sulfur prces are currently very low and the stockplng of elemental sulfur may not be an opton. Lmted space may force sulfur producers to fnd alternatve means for dealng wth the acd gas. Thus even larger producers are lookng to acd gas njecton as an opton to the producton of sulfur. The acd gas njecton process s smple n concept. The acd gas comes from the amne regenerator tower at low pressure (typcally less than 00 kpa) and at about 50 C, the temperature of the overhead condenser. In addton t s saturated water. The gas s compressed usng a multstage compressor. The pressure of the gas must be at least that requred for njecton nto a deep reservor. The pressure at the surface s usually substantally less than the reservor pressure. Ths s due to the hydrostatc head of the flud beng njected. Injecton pressures depend upon the njecton zone but they may be as large as 15 MPa. Reservor pressure can be as hgh as 30 MPa, and n extreme cases even hgher. One of the key propertes n the desgn of an acd gas njecton scheme s the densty of the flud (Carroll and Lu, 1997; Carroll and Maddocks, 1999). Accurate predctons of the densty are requred for the vapor, lqud, and supercrtcal (dense-phase) regons. Ths paper wll revew the avalable expermental data for acd gas denstes. Several popular methods for densty calculatons wll be examned. Densty predcton methods that are phase specfc wll not be examned n ths study. For example, there are several correlatons desgned for estmatng the denstes of lquds only (see Red et al, 1987). As was just noted, n acd gas njecton t s mportant to be able to predct the Boyle and Carroll

3 densty for all flud phases. It s also mportant to have a well-behaved functon as the flud transverses the varous phase regons (Carroll and Lu, 1987). Therefore, the lqud densty correlatons are less useful for ths applcaton, even though they are of hgh accuracy. EXPERIMENTAL DENSITIES The amount of expermental data avalable for the denstes of carbon doxde and hydrogen sulfde s qute large. Therefore, nstead of revewng the data n ths paper, lterature revews wll be used. These complatons of the pure component propertes cover the entre range of pressure and temperature of nterest n ths study. Pure Carbon Doxde There s sgnfcantly more data avalable for carbon doxde than for hydrogen sulfde, partcularly for the transport propertes. One reason for ths s that carbon doxde s consderably easer to deal wth than hydrogen sulfde. In addton, carbon doxde has a much lower crtcal pont placng ths nterestng regon n a range more accessble to expermenters. The vcnty near the crtcal pont s attractve to researchers because of the nature of the physcal propertes n that regon the propertes change dramatcally wth small changes n ether the temperature or the pressure. The latest revew of the thermodynamc propertes of carbon doxde s that of Span and Wagner (1996). Ths nvestgaton s a through and crtcal revew of the avalable expermental data. The tables were generated usng a hghly accurate, but complex, equaton of state. The tabulaton of Span and Wagner (1996) wll be used here n order to compare model densty predctons. In addton, Angus et al. (1976) thoroughly revewed the thermodynamc propertes of CO and formulated several tables. Vukalovch and Altunn (1968) revewed both thermodynamc and transport propertes of CO. Although these data sets are useful, they are not used n ths study as they have been superceded by the newer tables of Span and Wagner (1996). Pure Hydrogen Sulfde Goodwn (1983) extensvely revewed the thermodynamc propertes of hydrogen sulfde. Usng an advanced equaton of state, a table of propertes was constructed over a wde range of Boyle and Carroll 3

4 pressures and temperatures. Hydrogen sulfde densty data from Goodwn (1983) wll be used n ths study. As was noted earler, much less data was used to buld the correlaton for hydrogen sulfde than was used for carbon doxde. Notwthstandng, the tables of Goodwn (1983) are probably the best currently avalable for the thermodynamc propertes of hydrogen sulfde. Bnary Mxtures In a study of the phase behavor and volumetrc propertes of sour gas mxtures, Robnson et al. (1960) [also see Macrygeorgos (1958)] reported the denstes for three mxtures of carbon doxde and hydrogen sulfde. These data were for mxtures contanng 17.75%, 0.35% and 60.5% hydrogen sulfde at 71.1 C (160 F) and pressures from 1.0 to 1.4 MPa (150 to 1800 psa). All of these data are n the gaseous regon (compressblty factors n the range 0.95 to 0.45). In a more thorough nvestgaton of the H S+CO bnary system, Kellerman et al. (1995) measured the denstes of four mxtures: 6.07%, 9.55%, 9.33%, and 49.99% hydrogen sulfde. Temperatures n ths study ranged from 3. to C (-9.7 to F) and pressures up to 0.0 MPa (900 psa) These measurements ncluded both lqud and vapor regons. The data from both of these sources wll be used n ths study. MODELLING ACID GAS DENSITY Equatons of state, and most notably cubc equatons of state, have become the models of choce n the process modelng busness. Ths s partcularly true for the smulaton of processes for the treatment of natural gas. Two popular cubc equatons are the Soave (197) modfcaton of the Redlch-Kwong equaton (SRK) and the equaton of state of Peng and Robnson (1976) (PR). The lterature s flled wth addtonal modfcatons of these equatons. In fact the modfcatons are often mplemented under the orgnal names n commercal software packages. Thus the users of the software should do so wth some cauton. The PR and the SRK are classfed as two-parameter cubc equatons and can show sgnfcant devatons n predcted lqud densty when compared to expermental data. Errors are typcally on the order of 5-10%, although larger errors can be expected n the regon near a crtcal pont. Errors specfc to acd gases wll be presented n more detal later n ths paper. Boyle and Carroll 4

5 Recent attempts at mprovng lqud denstes have employed hgher-order equatons. For example, the model by Patel and Teja (198) (PT) has three parameters, and the model by Trebble et al. (TBS) has four parameters (Trebble and Bshno, 1986; Salm and Trebble, 1991). These hgher-order equatons rarely mprove the vapor-lqud equlbrum compared to the predctons from the smpler two-parameter models. Due to ther added complexty, they have yet to gan wde acceptance. Ths study focuses on the ablty of the PR, SRK and PT equatons of state to predct acd gas densty n the vapour, lqud and dense-phase regons. The pure component propertes and bnary nteracton parameters requred as nput to the varous equatons of state are gven n Appendx A. These values were taken from the lterature and no attempt was made to optmze them n order to mprove the densty predctons. Detals of the three equatons of state are summarzed n Appendx B. Densty from Equaton of State Models Equatons of state relate the temperature, pressure and specfc (or molar) volume of a flud. Cubc equatons of state model the pressure of a flud as a sum of an attractve and repulsve term. The equatons are therefore n the form P = f(t,v). Rearrangement of the equatons to a volume explct form produces a cubc (thrd-order) polynomal. The soluton to a cubc equaton result n one real or three real roots (two or all three of whch may be equal). For unsaturated fluds n the sub-crtcal regon only one of the roots, the most thermodynamcally stable, s physcally meanngful. For sub-crtcal saturated fluds two roots of equal thermodynamc stablty are physcally meanngful. It s worth notng that hgh temperature sngle-phase fluds may also produce three real roots, one or more beng negatve. Negatve roots, or specfc volume roots less than the co-volume of the equaton of state, b (see Appendx B), are neglected. Once the molar volume, v, has been calculated, the molar densty, ρ ~, s smply obtaned as follows: ~ 1 ρ = (1) v Boyle and Carroll 5

6 Furthermore, the mass densty, ρ, whch s the normal defnton of the densty, s gven by: M ρ = () v where M s the molar mass (molecular weght) of the flud. Volume-Shftng One method that has become qute popular to mprove the densty predcton form cubc equatons of state s called volume shftng. In ts smplest form, orgnally proposed by Peneloux et al. (198), volume shftng s a correcton to the calculated molar volume: v = v c (3) corrected EoS + where v EoS s the volume estmated from the equaton of state and c s the volume-shft parameter, whch n ths case s a constant. If the volume shft parameter s properly selected, then the corrected volume should be an mproved estmate of the true molar volume. Peneloux et al. (198) suggest fttng the saturated lqud densty at T r = 0.7 to obtan the volume-shft parameter. Alternatvely, the volume-shft parameter could be used as an adjustable parameter, whch s ft by mnmzng the error n the densty predcton. To apply ths method to mxtures, t s assumed that the c for the mxture, c mx, s the mole-fracton weghted average of the parameters for the pure components. NC c mx = x c (4) = 1 where x s the mole fracton of component and c s the volume shft parameter for component. Mathas et al. (1989) noted that the volume-shft method of Peneloux et al. (198) mproved the predcton of the lqud densty only up to reduced temperatures of about To Boyle and Carroll 6

7 mprove the predcton over the entre range Mathas et al. (1989) proposed the followng extended correcton procedure. They begn wth the followng equaton: 0.41 v corrected = v EoS + s + f c (5) δ where s s a volume-shft parameter and t s a constant, and δ, the bulk modulus, s a dmensonless parameter whch s defned as follows: v P δ = RT v (6) T where R s the unversal gas constant, T s the absolute temperature, and P s the total pressure. Ths expresson can be evaluated from the equaton of state. Fnally the functon, f c, was chosen such that the volume shftng procedure calculated the true crtcal pont. For the PR equaton, ths functon s gven by the followng expresson: ( 3.946b s) f = v c + (7) c where b s the co-volume from the equaton of state. For mxtures they used the usual, smple mxng rule: NC s mx = x s (8) = 1 Others have proposed makng the volume shft parameter other functons of the temperature. Ths adds to the complexty of the model. In addton, a poorly constructed temperature-dependence can lead to thermodynamc consstency problems (for example, see Monnery et al., 1998). Boyle and Carroll 7

8 For the purposes of ths study only the correctons proposed by Peneloux et al. (198) and Mathas et al. (1989) wll be examned. The values of c and s for carbon doxde and hydrogen sulfde used n ths study are lsted n Appendx A. DISCUSSION A total of sx densty calculaton methods wll be examned n ths paper: 1. The orgnal SRK,. The SRK wth a Peneloux-type volume shft, 3. The orgnal Peng-Robnson equaton, 4. The PR equaton wth a Peneloux-type volume shft, 5. The PR equaton wth a Mathas-type volume shft, and 6. The PT equaton. A complete lst of parameters used n ths study s presented n Appendx A. The followng expressons wll be used as a measure of the accuracy of the varous equatons. The error for a gven pont, as a percentage, s defned as value() estmate() Error = 100% (9) value() where value s the value from the tabular data and the estmate s from the equaton the same condtons. The absolute error s: value() estmate() Abs. Error = 100% (10) value() The average error, AE, expressed as a percentage, s defned as: AE 1 NP NP = = 1 value() estmate() 100% value() (11) where NP s the number of ponts. The average error can have ether a postve or negatve values. However, the better the ft, the closer ths value s to zero. The absolute average error, AAE, s defned as: Boyle and Carroll 8

9 AAE 1 NP NP = = 1 value() estmate() value() 100% (1) The dfference between the AE and the AAE s that n the average error postve and negatve errors tend to cancel each other, whch makes the predcton look better than t may actually be. The average absolute error can only have a postve value, because of the absolute value functon. It s a better ndcaton of the goodness of ft than s the average error. A small AE and a relatvely large AAE usually ndcates a systematc devaton between the functon (values) and the predctons (estmates). Fnally the maxmum error, MaxE, s: value() estmate() MaxE = maxmum 100%, = 1,,..., NP (13) value() The maxmum error gves the largest devaton of the model from the data values. our purposes. There are other methods for estmatng the error of a model, but these are suffcent for Pure Components The errors for the varous densty predcton methods wll be dscussed n sx regons. The sx regons are detaled n Table 1. Tables 4 through 7 lst the errors for the sx equatons for predctng the densty of pure carbon doxde and Tables 8 through 13 are smlar lstngs for pure hydrogen sulfde. Fgures 1 through 6 dsplay a porton of the carbon doxde error data for each equaton over varous regons. Fgures 7 through 1 show smlar data for hydrogen sulfde. From these tables and fgures t can be observed that the densty predctons from the SRK equaton are unsatsfactory. Although the overall average error s only about 5% the maxmum errors often exceed 15% and not only n the near crtcal regon. The PR equaton s better at predctng the denstes of these components than s the SRK equaton, whch s as expected. For H S the PT equaton s a further mprovement over the PR Boyle and Carroll 9

10 equaton, agan as expected. However for CO the PT equaton does not mprove the densty predctons over the PR equaton. As a check that the mplementaton of the PT was correct ζ c was set to , whch reduces the PT equaton to the PR equaton. The calculated results obtaned for ths form of the PT equaton were dentcal to those from the PR. In all cases the volume-shftng results n an mprovement n the predcted lqud densty. However, overall, volume shftng does not always result n an mprovement. For example, for carbon doxde, the densty predctons wth the Peneloux-type volume shft of the PR equaton actually are worse than the orgnal PR equaton. The reason for ths s because the volumeshftng results n worse predctons of the vapor densty. Mxtures Table 14 shows the errors for predctng the data from Robnson et al. (1960). Snce these data are only for the vapor phase they were not dvded nto regons for analyss. In general the errors for these mxtures are relatvely small although a few ponts have larger errors. The data set of Kellerman et al. (1995) s suffcently large that t was examned n three regons and the defntons of these three regons are gven n Table 15. The errors for the sx equatons are then summarzed n Tables 16 through 1. Fgures 13, 14, and 15 are plots of the errors for the predctons from the sx equatons for the varous mxtures. In general the observatons for the pure components hold true for the bnary mxture data. For example, the densty predctons from the SRK equaton are unsatsfactory. The PR equaton predcts the mxture data wth acceptable accuracy the maxmum error s less than 10%. Although the volume-shft methods mprove the predctons for the lqud densty, overall the densty predctons are only margnally mproved. Fnally, the PT equaton s an mprovement over the PR equaton when you consder the overall errors. However, the maxmum error for the PT equaton s approxmately equal to that for the PR. CONCLUSIONS Based on the results presented here s far to conclude that the orgnal SRK equaton s not suffcently accurate for predctng the densty of acd gas. Although the average errors are acceptable, the maxmum errors n the varous regons exceed 10%. Boyle and Carroll 10

11 The Peneloux-type volume shftng mproves ths equaton sgnfcantly. For the pure components, only n the near crtcal regon do the errors exceed 10%. For the mxture data the maxmum errors are typcally less than 11%. The orgnal PR equaton s qute accurate for predctng the denstes of the pure components. Overall the average error for both the pure components s less than about 5% and for the mxtures s less than about 10%. Only n the near crtcal regon do the errors exceed 10%. Volume-shftng of the PR equatons also tends to mprove the densty predctons n the lqud regon. The predctons from the Peneloux-type volume shft have an overall error of about.5% and the errors from the Mathas-type are less than %. The PT equaton, although somewhat more complex, does not result n sgnfcant mprovement n the densty calculatons. The observatons for the pure components are bascally the same for the mxtures examned. The SRK s unsatsfactory for predctng the densty of these mxtures. The other fve equatons of state and modfcatons are all suffcently accurate for engneerng calculatons, except n the regon near a crtcal pont. REFERENCES Angus, S., B. Armstrong, and K.M. de Reuck, Internatonal Thermodynamc Tables of the Flud State Carbon Doxde, Pergamon Press, Oxford, UK (1976). Carroll, J.J. and D.W. Lu, Densty, phase behavor keys to acd gas njecton, Ol & Gas J., 95 (5), 63-7, (1997). Carroll, J.J. and J. Maddocks, Desgn consderatons for acd gas njecton, 49th Laurance Red Gas Condtonng Conference, Norman, OK, Feb. 1-4, (1999). Goodwn, R.D., Hydrogen Sulfde Provsonal Thermophyscal Propertes from 188 to 700 K at Pressure to 75 MPa, Report No. NBSIR , Natonal Bureau of Standards, Boulder, CO, (1983). Haar, L., J.S. Gallagher, and G.S. Kell, NBS/NRC Steam Tables, Hemsphere, Washngton, DC (1984). Kellerman, S.J., C.E. Stouffer, P.T. Eubank, J.C. Holste, K.R. Hall, B.E. Gammon, and K.N. Marsh, Thermodynamc Propertes of CO + H S Mxtures, GPA Research Report RR-141, Tulsa, OK, (1995). Boyle and Carroll 11

12 Knapp, H., R. Dörng, L. Oellrch, U. Plöcker, and J.M. Prausntz, Vapor-Lqud Equlbra for Mxtures of Low Bolng Substances, DECHEMA Chemstry Data Seres Vol. VI, Frankfurt, Germany, (198). Macrygeorgos, C.A., Phase and Volumetrc Behavor n the Methane-Carbon Doxde-Hydrogen Sulfde System, M.Sc. Thess, Dept. Chemcal Engneerng, Unversty of Alberta, Edmonton, AB, (1958). Mathas, P.M., T. Naher, and E.M. Oh, A densty correcton for the Peng-Robnson equaton of state, Flud Phase Equl., 47, 77-87, (1989). Monnery, W.D., W.Y. Svrcek, and M.A. Satyro, Gaussan-lke volume shfts for the Peng- Robnson equaton of state., Ind. Eng. Chem. Res., 37, , (1998). Patel, N.C. and A.S. Teja, A new cubc equaton of state for fluds and flud mxtures, Chem. Eng. Sc., 37, , (198). Peneloux, A. E. Rausy, and R. Freze, A consstent correcton for Redlch-Kwong-Soave volumes, Flud Phase Equl., 8, 7-3, (198) Peng, D-Y. and D.B. Robnson, A new two-constant equaton of state, Ind. Eng. Chem. Fund., 15, 59-64, (1976). Red, R.C., Prausntz, J.M. and Polng, B.E., The Propertes of Gases & Lquds, 4th ed., McGraw-Hll, New York, NY, (1987). Robnson, D.B., C.A. Macrygeorgos, G.W. Gover, The volumetrc behavor of natural gases contanng hydrogen sulfde and carbon doxde Petrol. Trans. AIME, 19, 54-60, (1960). Salm, P.H. and Trebble, M.A., A modfed Trebble-Bshno equaton of state: thermodynamc consstency revsted, Flud Phase Equl., 65, 59-71, (1991). Span, R. and W. Wagner, A new equaton of state for carbon doxde coverng the flud regon from the trple-pont temperature to 1100 K at pressure up to 800 MPa J. Phys. Chem. Ref. Data, 5, , (1996). Soave, G., Equlbrum constants from a modfed Redlch-Kwong equaton of state, Chem. Eng. Sc., 7, , (197). Trebble, M.A. and Bshno P.R., Development of a new four-parameter equaton of state, Flud Phase Equl., 35, 1-18, (1986). Vukalovch, M.P. and V.V. Altunn, Thermophyscal Propertes of Carbon Doxde, Collet s Publshers Ltd. London, UK, (1968). translated from Russan. Boyle and Carroll 1

13 Table 1 Sx Regons for Flud Densty Study for the Pure Components Reduced Temperature Reduced Pressure 1. Lqud Regon T r < 0.95 P r > P r (sat). Vapor Regon T r > T r (sat) P r < P r (sat) or P r < Near Crtcal Regon 0.95 < T r < < P r < Supercrtcal Regon T r > 0.95 If 0.95 < T r < 1.05 then P r > 1.5 If T r > 1.05 then P r > Saturated Lqud T r = T r (sat), T r < 0.95 P r = P r (sat) 6. Saturated Vapor T r = T r (sat), T r < 0.95 P r = P r (sat) Table The Errors Assocated wth the Orgnal SRK Equaton for Carbon Doxde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated. Lqud Saturated Vapor Overall Table 3 The Errors Assocated wth the SRK-Peneloux Equaton for Carbon Doxde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Boyle and Carroll 13

14 Table 4 The Errors Assocated wth the Orgnal PR Equaton for Carbon Doxde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Table 5 The Errors Assocated wth the PR-Peneloux Equaton for Carbon Doxde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Table 6 The Errors Assocated wth the PR-Mathas Equaton for Carbon Doxde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Boyle and Carroll 14

15 Table 7 The Errors Assocated wth the Orgnal PT Equaton for Carbon Doxde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Table 8 The Errors Assocated wth the Orgnal SRK Equaton for Hydrogen Sulfde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Table 9 The Errors Assocated wth the SRK-Peneloux Equaton for Hydrogen Sulfde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Boyle and Carroll 15

16 Table 10 The Errors Assocated wth the Orgnal PR Equaton for Hydrogen Sulfde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Table 11 The Errors Assocated wth the PR-Peneloux Equaton for Hydrogen Sulfde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Table 1 The Errors Assocated wth the PR-Mathas Equaton for Hydrogen Sulfde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Boyle and Carroll 16

17 Table 13 The Errors Assocated wth the Orgnal PT Equaton for Hydrogen Sulfde Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Near Crtcal Regon Supercrtcal Regon Saturated Lqud Saturated Vapor Overall Table 14 The Errors for The Mxture Data of Robnson et al. (1960) Equaton AE (%) AAE (%) MaxE(%) SRK SRK-Peneloux PR PR-Peneloux PR-Mathas PT Table 15 Three Regons for Flud Densty Study for the Mxtures Reduced Temperature Reduced Pressure 1. Lqud Regon T r < 1.0 P r > P r (bubble). Vapor Regon T r > T r (dew) P r < P r (dew) or P r < Supercrtcal Regon T r > 1.0 P r > 1.0 Boyle and Carroll 17

18 Table 16 The Errors Assocated wth the Orgnal SRK Equaton for the Mxture Data of Kellerman et al. (1995) Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Supercrtcal Regon Overall Table 17 The Errors Assocated wth the SRK-Peneloux Equaton for the Mxture Data of Kellerman et al. (1995) Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Supercrtcal Regon Overall Table 18 The Errors Assocated wth the Orgnal PR Equaton for the Mxture Data of Kellerman et al. (1995) Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Supercrtcal Regon Overall Boyle and Carroll 18

19 Table 19 The Errors Assocated wth the PR-Peneloux Equaton for the Mxture Data of Kellerman et al. (1995) Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Supercrtcal Regon Overall Table 0 The Errors Assocated wth the PR-Mathas Equaton for the Mxture Data of Kellerman et al. (1995) Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Supercrtcal Regon Overall Table 1 The Errors Assocated wth the Orgnal PT Equaton for the Mxture Data of Kellerman et al. (1995) Regon AE (%) AAE (%) MaxE(%) Lqud Regon Vapor Regon Supercrtcal Regon Overall Boyle and Carroll 19

20 APPENDIX A Component Propertes Equaton of state models requre the crtcal pont and accentrc factor as pure component nput parameters. The data used n ths study are as follows: Carbon Doxde (Span and Wagner, 1996) Crtcal Temperature (K) Crtcal Pressure (MPa) Accentrc Factor (-) (see below) Molecular Weght (g/mol) Hydrogen Sulfde (Goodwn, 1983) Crtcal Temperature (K) Crtcal Pressure (MPa) Accentrc Factor (-) (see below) Molecular Weght (g/mol) The acentrc factor of the pure components were derved from the vapor pressure data and the basc defnton of the acentrc factor: sat ω = log( Tr = 0.7) + 1 For carbon doxde ths nvolved extrapolatng the vapor pressure expresson. The trple pont of carbon doxde s K, whch s a reduced temperature of Accordng to the method of Peneloux et al. (198), the volume shft parameter should be obtaned by fttng the saturated lqud densty at T r = 0.7. In ths work, the shft parameters were obtaned by mnmzng the AAE for the saturated lqud densty over the followng temperature ranges: CO : < T r < H S: 0.73 < T r < Boyle and Carroll 0

21 The lower reduced temperature corresponds to 0 C, the lowest temperature of nterest n ths study. For the SRK the followng Peneloux-type volume shft parameters were used: c c CO H S = cm = cm 3 3 /mol /mol And for the PR equaton the followng parameters were used here: c CO = cm 3 /mol c H S = cm 3 /mol For the Mathas-type correcton to the PR equaton the s parameter was obtaned by fttng the saturated lqud densty at T r = 0.7 (or nearly so). The v c values were obtaned by mnmzng the AAE for the range of saturated lqud gven above. 3 s CO = cm /mol(from matchng the saturated lqud densty at T r = 0.71, CO trple pont) s 3 S.998 cm /mol (from matchng the saturated lqud densty at T r = ) H = 3 v c, CO = cm /mol(% larger than the expermental crtcal volume) v c, H S 3 = cm /mol (.5% larger than the expermental crtcal volume) For the PT equaton the followng parameters were used n ths study: CO : F = HS: F = ζ c = ζ c = 0.30 Boyle and Carroll 1

22 whch were taken from the paper of Patel and Teja (198). The above values result n smaller errors n the densty than the values from the generalzed correlatons. Mxtures Equaton of state models requre bnary nteracton parameters for mult-component nput parameters. The bnary nteracton coeffcents for H S CO used n ths study are as follows: for SRK for PR for PT The values for the SRK and PR equatons are from Knapp et al. (198). These values are those gven n the popular book by Red et al. (1987). The nteracton parameter for the PT equaton s an estmate based on the values for the other two equatons. Boyle and Carroll

23 APPENDIX B Equaton of State Summary The materal that follows provdes the detals of the three equatons of state used n ths study. Only the equatons are provded, no dscusson or dervaton. Soave-Redlch-Kwong Equaton of State R T a P = v b v (v + b) Z 3 Z + (A B B ) Z (A B) = 0 where: a c (R T = P c c ) a = a c α α 0.5 = 1 + m (1 T 0.5 r ) m a = = ω N N j 0.5 ω x x (a a ) (1 k ) j j j a P A = (R T) b R T = P c c b N = x b b P B = R T Boyle and Carroll 3

24 Boyle and Carroll 4 Peng-Robnson Equaton of State b) (v b b) (v v a b v T R P + + = 0 ) B B B (A Z ) B 3 B (A Z B) (1 Z 3 3 = + where c c c P ) T (R a = a c a α = ) T (1 m r = α ù ù m + = = N j 0.5 j j N j ) k (1 ) a (a x x a T) (R P a A = c c P T R b = = N x b b T R P b B =

25 The Patel-Teja Equaton of State P = R T a v b v (v + b) + c (v b) Z 3 + (C 1) Z + ( B C B B C + A) Z + (B C + B C A B) = 0 where a c = Ω a (R T P c c ) a = a c α α 0.5 = 1 + F (1 T 0.5 r ) a = N N j 0.5 x x (a a ) (1 k ) j j j a P A = (R T) b = Ω b (R T P c c ) b N = x b B = b P R T c = Ω c (R T P c c ) c N = x c c P C = R T Boyle and Carroll 5

26 Ω a = ζc + 3 (1 ζc ) Ωb + Ωb 3 3 ζ 3 c and Ω b s the smallest real root of the followng equaton: 3 Ωb + ( 3 ζc ) Ωb + 3 ζc Ωb ζc = 0 Ω c = 3 1 ζ c 3 The parameters F and ζ c can be optmzed from a set of data or they can be obtaned from the followng generalzed equatons: F = ω ω ζc = ω ω Boyle and Carroll 6

27 Fgure 1. Densty Errors Assocated wth the Orgnal SRK Equaton for Carbon Doxde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.711 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 7

28 Fgure. Densty Errors Assocated wth the SRK-Peneloux Equaton for Carbon Doxde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.711 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 8

29 Fgure 3. Densty Errors Assocated wth the Orgnal PR Equaton for Carbon Doxde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.711 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 9

30 Fgure 4. Densty Errors Assocated wth the PR-Peneloux Equaton for Carbon Doxde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.711 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 30

31 Fgure 5. Densty Errors Assocated wth the PR-Mathas Equaton for Carbon Doxde % Error 0.0 Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.711 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 31

32 Fgure 6. Densty Errors Assocated wth the Orgnal PT Equaton for Carbon Doxde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.711 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 3

33 Fgure 7. Densty Errors Assocated wth the Orgnal SRK Equaton for Hydrogen Sulfde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.31 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 33

34 Fgure 8. Densty Errors Assocated wth the SRK-Peneloux Equaton for Hydrogen Sulfde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.31 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 34

35 Fgure 9. Densty Errors Assocated wth the Orgnal PR Equaton for Hydrogen Sulfde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.31 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 35

36 Fgure 10. Densty Errors Assocated wth the PR-Peneloux Equaton for Hydrogen Sulfde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.31 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 36

37 Fgure 11. Densty Errors Assocated wth the PR-Mathas Equaton for Hydrogen Sulfde % Error 0.0 Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.31 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 37

38 Fgure 1. Densty Errors Assocated wth the Orgnal PT Equaton for Hydrogen Sulfde % Error Pr = (lqud) Pr = (vapor) Pr = (lqud) Pr = (vapour) Pr = Pr =.31 Pr = Pr sat (lqud) Pr sat (vapor) crtcal pont Tr Boyle and Carroll 38

39 Fgure 13. Densty Errors Assocated wth Each Equaton for the mol% Carbon Doxde mol% Hydrogen Sulfde Mxture at Tr = (Supercrtcal Regon) % Error SRK SRK-Peneloux PR PR-Peneloux PR-Mathas PT Pr Boyle and Carroll 39

40 Fgure 14. Densty Errors Assocated wth Each Equaton for the mol% Carbon Doxde mol% Hydrogen Sulfde Mxture at Tr = 1.66 (Supercrtcal Regon) % Error SRK SRK-Peneloux PR PR-Peneloux PR-Mathas PT Pr Boyle and Carroll 40

41 Fgure 15. Densty Errors Assocated wth Each Equaton for the mol% Carbon Doxde mol% Hydrogen Sulfde Mxture at Tr = 0.91 (Lqud Regon) % Error SRK SRK-Peneloux PR PR-Peneloux PR-Mathas PT Pr Boyle and Carroll 41