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A Semiempirical Approach for Estimating the Water Content of Natural Gases

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A Semiempirical Approach for Estimating the Water Content of Natural Gases
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   A Semiempirical Approach for Estimating the Water Content of Natural Gases  Amir H. Mohammadi, †  Antonin Chapoy, ‡ Bahman Tohidi,* ,† and Dominique Richon ‡ Centre for Gas Hydrate Research, Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K., and Centre d’Energe ´ tique, Ecole Nationale Supe ´ rieure des Mines de Paris, CENERG/TEP, 35 Rue Saint Honore ´  , 77305 Fontainebleau, France  A semiempirical approach for estimating the equilibrium water content of natural gases in theliquid water - vapor and ice - vapor regions has been developed. This method estimates the watercontent of natural gases using water/ice vapor pressure and water/ice molar volume as well aspressure and temperature of the system. The approach has been developed using theexperimental water content data for methane at a temperature range of 273.15 and 377.59 K and pressures up to 13.81 MPa. Experimental data for the water content of a hydrocarbon gasmixture (94% methane  +  4% ethane  +  2%  n- butane) have been generated at a temperaturerange of 288.15 to 313.14 K and pressures up to 17.56 MPa using a static-analytic apparatus,taking advantage of a pneumatic capillary sampler for fluid sampling. These independent datahave been used in examining the reliability of the semiempirical approach. The predictions of this approach are in good agreement with the experimental data generated in this work andthose reported in the literature. 1. Introduction Natural gases are normally saturated with water atreservoir conditions. During production, transportation,and processing, dissolved water in the gas phase mayform liquid water phase, ice, and/or gas hydrates.Forming a liquid water phase may lead to corrosion and/ or two-phase flow problems. Gas hydrates and/or iceformation may cause blockage during production andtransportation. To predict and avoid such problems,predictive methods are necessary. On the other hand,estimating the water content by predictive techniquesis crucial to design and selection of operating conditionsin natural gas facilities. General methods for calculationinclude the following: (1) empirical or semiempiricalcorrelations and charts of the water content and cor-rections for the presence of acid gases (such as hydrogensulfide and carbon dioxide), heavy hydrocarbons, andsalts in the system and (2) thermodynamic modelswhich are based on equality of chemical potential of various components in different phases.The main advantage of empirical or semiempiricalcorrelations and charts is the availability of input dataand the simplicity of the calculations, which can beperformed by using charts or hand-held calculators. Thecorrelations/charts have still kept their popularityamong engineers in the petroleum industry. Althoughmost available thermodynamic models could be installedon typical laptop computers, there seem to be a needfor simple, yet robust, predictive methods for quickestimation of the water content of natural gases.The available correlations and charts are generallybased on limited data and with limited application. Ingeneral, the available correlations/charts can predict thewater content of gases with good accuracy at high-temperature conditions. While in predicting the watercontent at low-temperature conditions, the availablemethods have lower accuracy and need further verifica-tion at low-temperature conditions. In fact, during thedevelopment of the srcinal correlations/charts, experi-mental data describing the phase equilibrium in water - hydrocarbons systems for temperatures typically lowerthan 298.15 K were not available. Due to this shortcom-ing, the water content for low temperatures calculatedby the correlations/charts might not be accurate.To develop new correlations/charts and to improve theaccuracy of the estimated water content of gases,experimental data are required (which could also beused for validation of correlations/charts). Mohammadiet al. 1 gathered most of the water content data for themain components of natural gases from literature andconcluded that most of the water content data forhydrocarbons and for non-hydrocarbon gases (e.g. N 2 ,CO 2 , and H 2 S) at low-temperature conditions are ofteninconsistent. These types of uncertainties can lead tolarge deviations for correlations, when using thesescattered data for regressing.The aim of this work is to develop a semiempiricalapproach based on equality of water fugacity in equi-librium phases for estimating the water content of natural gases in equilibrium with liquid water or ice.For this purpose, a quick review is made on the mostfamous correlations/charts existing in the open litera-ture. Then a set of experimental data for the watercontent of a hydrocarbon gas mixture (94% methane + 4% ethane  +  2%  n- butane) is generated at a tempera-ture range of 288.15 to 313.14 K and pressures up to17.56 MPa using a static-analytic apparatus, taking advantage of a pneumatic capillary sampler for fluidsampling. 1 - 3 To develop the approach, experimental data for thewater content of methane are used; however, as men-tioned earlier, most of the water content data at low-temperature conditions are inconsistent, and further * To whom correspondence should be addressed. Tel.: + 44 (0)131 451 3672. Fax:  + 44 (0)131 451 3127. E-mail:Bahman.Tohidi@pet.hw.ac.uk. † Institute of Petroleum Engineering. ‡ Ecole Nationale Supe´rieure des Mines de Paris. 7137  Ind. Eng. Chem. Res.  2004,  43,  7137 - 7147 10.1021/ie049867m CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 09/29/2004  experimental work is necessary for developing newpredictive methods at low-temperature conditions. Toevaluate the capability of this approach, the results arecompared with the experimental data generated in thiswork and other literature data as well as other predic-tive tools. The results are in good agreement demon-strating the capability of the approach and data gen-erated in this work. 2. Predictive Methods Many thermodynamic models and correlations/chartsare available, which can calculate the phase equilibriumin water - hydrocarbon systems. Thermodynamic modelsuse different approaches in order to model the fluid, ice,and gas hydrates phases. For example, some thermo-dynamic models use activity coefficient and Henry’sconstant approaches for modeling the aqueous phase;however, other models use the equation of state (  EoS )approach. Although thermodynamic models are usefultools in phase behavior calculations, they may not beavailable easily. Correlations and charts are moresimple tools, and because of their ease of use, they areof interest to engineers in petroleum industry. Variouscorrelations and charts with different capabilities havebeen reported for estimating the water content/waterdew point of gases. The srcinal correlations/charts arenormally applicable to dry and sweet gases. Generally,these correlations/charts have been developed for theliquid water - vapor (  L w - V  ) region and interpolating theresults to the hydrate - vapor (  H  - V  ) and ice - vapor (  I  - V  ) regions may be questionable. In this section, a reviewof the most famous correlations and charts in thenatural gas industry are presented:(1) The  Ideal  model (Raoult’s law) is expressed by thefollowing expressionwhere  y ,  x , and  P  are the mol fraction in the vaporphase, mol fraction in the liquid phase, and pressure,respectively, subscripts  w  and  g  relate to water and gas,and the superscript  sat  relates to the saturation state.In this equation, the gas solubility in the water  x  g  canbe ignored for hydrocarbons, as hydrocarbons are weaklysoluble in water and solubility will decrease by increas-ing the molecular weight of hydrocarbons; however, foracid gases (CO 2  and H 2 S) the solubility can be signifi-cant, even at relatively low pressure. 4,5 In this case, thewater content can be expressed by the following expres-sion:The above relation assumes the water content of a gasequals to the ratio of the water vapor pressure and totalpressure of the system. A more accurate form of the  Ideal  model can beexpressed by taking into account the  Poynting  correctionwhere  v, R,  and  T   are molar volume, universal gasconstant, and temperature of the system, respectively,and the superscript  L  stands for the liquid state. The  Ideal model  and its  Poynting correction  are simple toolsfor predicting the water content of natural gases.However, these methods can be used at low-pressureconditions (typically up to 1.4 MPa 4 ).(2) Bukacek 6 developed a method similar to the  Ideal model, which only requires information on the watervapor pressure and the temperature and pressure of thesystem. This correlation is one of the most used methodsin the natural gas industry for calculating the watercontent of dry and sweet natural gases. This correlationin  American Engineering units  is given as follows 4,5 where the water content (  y w ) and  t  are in  lbm/MMscf  and the temperature is in °  F.  As can be seen, thiscorrelation uses an ideal contribution and a deviationfactor. The above relation is reported to be accurate fortemperatures between 288.15 and 511.15 K and forpressures from 0.1 to 69 MPa. 4,5 This correlation isaccurate to about  ( 5% within the stated range, 4,5,7 asthis is about as accurate as the water content can bemeasured.(3) Sharma and Campbell 8 - 10 provided a relativelycomplicatedmethodinordertocalculatetheequilibriumwater content of sweet and sour gases in the  L W  - V  region. In this method, the water content is calculatedas belowwhere  k, Z,  and  f   are a correction factor, the compress-ibility factor, and the fugacity, respectively, and sub-script  g  refers to the gas phase. The compressibilityfactor  Z  should be calculated using a suitable method.The correction factor  k  can be calculated from a figure(provided by the authors) or by the following equationwhere  f  w sat and  f  w  are fugacity of water at saturationconditions ( T   and  P w sat ) and the fugacity of water atpressure and temperature of the system ( T   and  P ). Theyprovided a chart for calculating the fugacity of water. As mentioned before, this method is relatively compli-cated; however, Campbell 10 mentioned that the consis-tency of the results of this method is high.(4) Behr 11 proposed the following equation for pres-sure ranging from 1.379 to 20.679 MPawhere  y w  is in  lbm/MMscf   and  A 0  to  A 7  are constantsbased on fitting the natural gas dew point versus thewater content data of ref 6.(5) Later Kazim 12 proposed an analytical expressionfor calculating the water content of sweet natural gases.The expression in the  American Engineering units  is  y w ) (1 -  x  g )  P w sat  P  (1)  y w )  P w sat  P  (2)  y w )  P w sat  P  exp ( v w L (  P -  P w sat )  RT   )  (3)  y w ) 47484  P w sat  P  + B (4) log(B) )- 3083.87459.6 + t  + 6.69449 (5)  y w )  k (  f  w sat  f   g ) Z (6)  k ) (  P w sat  P  )(  f  w sat  /   P w sat  f  w  /   P  )(  P P w sat ) 0.0049 (7)  y w ) exp(  A 0 +  A 1 (1/  T  ) 2 +  A 2 (1/  T  ) 3 +  A 3 ( lnP ) +  A 4 ( lnP ) 2 +  A 5 ( lnP ) 3 +  A 6 ( lnP  /  T  ) 2 +  A 7 ( lnP  /  T  ) 3 ) (8) 7138  Ind. Eng. Chem. Res., Vol. 43, No. 22, 2004  where  y w  is in  lbm/MMscf   and  p  is the pressure in  psia ,and  a i s and  b i s are constants reported in the srcinalpublication. These two correlations are similar as inwhich, they srcinate from regression methods to ex-press the water content of natural gases as a functionof temperature and pressure and require many con-stants, which may reduce their applications for calculat-ing the water content of natural gases in comparisonwith the Bukacek 6 correlation.(6) Several charts have been reported in order tocalculate the equilibrium water content of gases. Themost commonly used is the McKetta-Wehe 13 chart,which is used for sweet natural gases containing over70% methane and small amounts of heavy hydrocar-bons. 13,14 This chart was first published in 1958 15 andwas based on experimental data available at that time. 13 Gas Processors Associations ( GPA ) and Gas ProcessorsSupplier Associations ( GPSA ) have reproduced thischart for many years. In this chart the water contentof a sweet gas is presented in a semilogarithmic plotversus temperature at different pressures. Two correc-tion factors have been provided in order to take intoaccount the presence of heavy hydrocarbons in the gasphase and salts in the liquid water. In this chart, themeta-stable  L W  - V   equilibrium is assumed rather thanthe  H  - V   equilibrium in the hydrate formation regionwith limited justification. 16 However the actual watercontent in the  H  - V   region is lower than the calculatedwater content by assuming the  L W  - V   equilibrium.Furthermore, reading the water content from thissemilogarithmic chart may be slightly difficult. If usedwith care, this chart can calculate the water content of sweet gases with less than 5% error. 4 (7) Ning et al . 17 proposed the following correlationbased on the McKetta-Wehe 13 chart: 4,5 For the above equation, a complicated figure and tablehave been provided in order to calculate the pressuredependent coefficients  a 0  , a 1  ,  and  a  2  for pressures up to100 MPa. It seems that this correlation is not a simpletool due to the complicated dependency of coefficientsto the pressure. Their correlation takes into account theeffect of gas gravity by the following correction factor 4 where  F   and  SG  are correction factors due to thepresence of heavy hydrocarbons and gas gravity, andsubscripts  HC ,  heavy , and  light  relate to hydrocarbon,heavy, and light components, respectively.The above correlations/charts (except for the Sharmaand Campbell method 8 - 10 ) assume that the watercontent of dry and sweet natural gases is independentof the gas composition. However, when acid gases andheavy hydrocarbons and/or salts are present in thesystem, their accuracy is reduced, and some correctionsshould be used in addition to the above correlations/ charts.Both hydrogen sulfide and carbon dioxide containmore water at saturation than methane or sweetnatural gas mixtures, and the relative amounts varyconsiderably with temperature and pressure. 13 Thewater content of carbon dioxide and hydrogen sulfideplays an important role in phase equilibria calculations,e.g., enhanced oil recovery (  EOR ) processes, corrosionprevention, flow assurance, gas hydrates inhibition, etc.There are some methods for estimation of the watercontent due to the presence of acid gases in the gasphase. The correlations should be applied when the gasmixture contains more than 5% hydrogen sulfide and/ or carbon dioxide, especially at high pressures. 13 Several graphs are reported for estimating the watercontent of pure carbon dioxide 5,13,18 - 23 and pure hydro-gen sulfide. 5,13,20,24 There are also other graphs forestimating the water content of different mixturescontaining acid gases. 13,22,25 - 29 The Robinson et al . , 25 - 27 Maddox et al ., 28 and Wichert - Wichert 29 methods correct the water content of sweetgases due to the presence of acid gases.Robinson et al . 25 - 27 reported a series of charts toestimate the water content of sour natural gases. Thesecharts were calculated based on an equation of a statebased model. They used an equivalent mol fraction forH 2 S for their charts, which is calculated by the following expression 26,27 where  z  is the mol fraction in the natural gas, thesubscripts  CO  2  and  H   2 S  refer to carbon dioxide andhydrogen sulfide, respectively, and the superscript  equi refers to equivalent H 2 S. This method is applicable for  z  H2S equi < 0.4 (mol fraction), 283.15 < T  < 450.15 K and2.07  <  P  <  69 MPa. In addition, using these charts isslightly difficult due to the need for interpolations.In the Maddox et al . 28 method, the water content of sour gases is calculated using the following expressionwhere subscript  HC  refers to hydrocarbon. In the aboveequation, the contribution to the sweet gas can becalculatedusinganappropriatecorrelationorchart.Theacid gas contributions can be calculated by either thecorrespondingchartsorequations.Theabovecorrelationis applicable to acid gas concentrations of less than 40mol %, a pressure range of 0.7  <  P  <  20.7 MPa, and atemperature range of 300.15  <  T   <  344.15 K for CO 2 and 300.15  <  T   <  411.15 K for H 2 S.Wichert and Wichert 29 proposed a new chart basedon temperature, pressure, and equivalent H 2 S contentin order to calculate a correction factor (  F   sour ). They usedthe definition of Robinson et al . 26,27 for the equivalentH 2 S content. Using this correction factor, the watercontent of sour natural gases can be calculated by using the following expression:  y w )  A  ×  B t (9)  A ) ∑ i ) 14 a i (  p - 350600  ) i - 1 (10)  B ) ∑ i ) 14 b i (  p - 350600  ) i - 1 (11)  y w ) exp( a 0 + a 1 T  + a 2 T  2 ) (12)  F   HC ) 1.01532 + 0.011 ( T  -273.15) - 0.0182  SG  g  - 0.0142  SG  g ( T  -273.15) (13)  y w , heavy )  F   HC  y w , light  (14)  z  H2S equi )  z  H2S + 0.75  z CO2  (15)  y w )  y w,HC  ×  z  HC +  y w , CO2  ×  z CO2 +  y w ,  H2S  ×  z  H2S  (16)  y w ,  sour )  F   sour  y w ,  sweet  (17) Ind. Eng. Chem. Res., Vol. 43, No. 22, 2004  7139  In the above equation, the subscripts  sour  and  sweet relate to the sour and sweet natural gases. The Mc-Ketta-Wehe 13 chart is recommended for calculating   y  sweet in the above equation. This method is applicable for  z  H2S equi <  0.55 (mol fraction), 283.15  <  T   <  450.15 K,and 1.4  <  P  <  69 MPa. In addition, using this methodis easier than the method suggested by Robinson etal . , 25 - 27 as there is no need for interpolation.In addition, determining the water content in acid/ sour gases is a very complex topic. According to  GPSA , 13 an accurate determination of the water content requiresa careful study of the existing literature and availableexperimental data. In most cases, additional experi-mental data is the best way to verify the predictedvalues. Even the most sophisticated  EoS  techniquesmay give results of questionable reliability.Figure 1 shows a typical pressure-temperature dia-gram for a water - hydrocarbon system. 1,30  As can beseen, the  I  - V   equilibrium for sweet natural gases withvery low nitrogen content can be reached at relativelylow-pressure conditions. The maximum pressure atwhich the  I  - V   equilibrium can be reached is around2.563 MPa, which corresponds to hydrate formationconditions for methane at around quadruple point. The  Poynting  correlation can be used for estimating thewater content of sweet natural gases with very lownitrogen content in equilibrium with ice. Katz 31,32 alsoreported a chart in the temperature and pressure rangeof 222.04 < T  < 273.15 K and 0.1 <  P < 2.757 MPa forcalculating the water content of natural gases in equi-librium with ice.The water content of natural gases in equilibriumwith gas hydrates is lower (typically less than 0.001 molfraction 16 ) than the water content of natural gases inequilibrium with meta-stable liquid water and thereforedifficult to measure, as hydrate formation is a time-consuming process and the water content of gases inthe hydrate region is a strong function of composi-tion. 13,16 In other words, a gas-phase saturated withwater can form gas hydrates in the  H  - V   region from astrictthermodynamicstandpoint;however,thequestionof the accumulation of a hydrate phase is a question of kinetics, dependent upon the time necessary for hydratenuclei to attain a critical size. 16 This time may be inexcess of that available for laboratory study but mayoccur in processes, which operate over extended periodsof days, months, or years. 16 On the other hand, limitedexperimental data have been reported in this region.Therefore, a comprehensive correlation/chart for calcu-lating the water content of gases in equilibrium withgas hydrates would be problematic. 16 Few mathematicalrelations/charts for this region have been developed inthe literature. 4,16,32 - 36 Where experimental data arelimited, utilization of a thermodynamic model canprovide an estimate of the water content in equilibriumwith hydrates.In addition, in many standards, the Bukacek 6 cor-relation and the McKetta-Wehe 13 chart are recom-mended to estimate the water content of sweet naturalgases in equilibrium with liquid water. However, theBukacek 6 correlation and the McKetta-Wehe 13 chartmay not describe real phase behavior in water - hydrocarbon systems at low-temperature conditions. Inother words, the Bukacek 6 correlation should be usedat temperatures higher than 288.15 K, 4,5 and the watercontents obtained from the McKetta-Wehe 13 chart attemperatures below hydrate formation conditions cor-respond to the meta-stable  L W  - V   equilibrium ratherthan the  H  - V   equilibrium. 3. Experimental Section  A description of the commonly used methods formeasuring the water content/water dew point of gasesis reported elsewhere. 1 Based on the laboratory experi-ence and on the existing laboratory equipment a methodbased on gas chromatography ( GC)  is selected to per-form the analyses in this study. A detailed descriptionof the experimental setup and experimental procedurehas been reported previously. 1 - 3 3.1. Materials.  The gas mixture, 94% methane, 4%ethane ( ( 2%, i.e., 3.92 - 4.08), and 2%  n- butane ( ( 2%,i.e., 1.96 - 2.04) was purchased by  Messer Griesheim .Helium from  Air Liquide  is pure grade with traces of water (3  ppm ) and of hydrocarbons (0.5  ppm ). Further-more, helium was dried by means of molecular sievesplaced at the outlet of the cylinder. 3.2. Apparatus.  The apparatus used in this work isbased on a static-analytic method with vapor phasesampling. The analytical work was carried out using a GC  ( VARIAN model CP-3800 ) equipped with two detec-tors connected in series, a thermal conductivity detector( TCD ) and a flame ionization detector (  FID ), connectedto a data acquisition system (  BORWIN   ver. 1.5, from  JMBS, Le Fontanil , France). The  FID  was utilized todetect the hydrocarbons. It was repeatedly calibratedby introducing known amounts of the gas mixturethrough a gas syringe in the injector of the  GC . Thecalculation of the amount of water is carried out using equilibrium (equality of fugacity) and mass balancerelations as summarized by the following expressionswith An exact relationship is obtained: Figure 1.  Typical pressure - temperature diagram for a water(limiting reactant) - single (pure) hydrocarbon system. 1,30  y w ) γ w L  x w  P w sat  P dilutor φ w sat φ wV   exp (( υ  L  RT  ) (  P -  P  sat ) )  (18)  y w ) n w n T  (19) n w ) γ w L  x w  P w sat  P dilutor φ w sat φ wV   (  PVol ZRT  ) loop exp (( υ  L  RT  ) (  P -  P  sat ) )  (20) 7140  Ind. Eng. Chem. Res., Vol. 43, No. 22, 2004  In the above relations,  γ  ,    , n, Z,  and  Vol  are activitycoefficient, fugacity coefficient, number of moles, com-pressibility factor, and volume of the loop, respectively.The superscripts and subscripts  V, loop, T  , and  dilutor correspond to the vapor phase, loop, total, and dilutor,respectively. 3.3. Experimental Uncertainty.  The temperatureuncertainty is estimated to be not higher than  ( 0.02K. The uncertainty in the pressure measurements isestimated to be ( 5 kPa in the operating pressure range.The experimental accuracy of the TCD  water calibration(from 9  ×  10 - 10 mol to 1.2  ×  10 - 8 ) is estimated in theworst case at ( 5%. The methane, ethane, and  n -butanecalibration uncertainty are estimated to be within ( 1%, ( 3%, and  ( 3%, respectively. 3.4.ExperimentalProcedure. The equilibrium celland its loading lines are evacuated down to 0.1 Pa, andthe necessary quantity of the preliminary degassedwater (approximately 10 cm 3 ) is introduced using anauxiliary cell. Afterward, the desired amount of gas isintroduced into the cell directly from the cylinder or viaa gas compressor. The sampling is carried out using acapillary sampler injector 37 for each phase. The with-drawn samples are swept to a  Varian 3800 GC  foranalysis. For each equilibrium condition, at least 10points are withdrawn using the pneumatic samplers  ROLSI   and analyzed in order to check for measurementrepeatability. As the volume of the withdrawn samplesis very small compared to the volume of the vapor phasepresent in the equilibrium cell, it is possible to withdrawmany samples without disturbing the phase equilibri-um. 4. Semiempirical Approach The vapor - liquid equilibrium ( VLE ) of a system iscalculated, using the following equationwhere  N   is the number of components. The equality of fugacities can be calculated using the following relation-ship:In the intermediate pressure range, liquid water isan incompressible fluid, and gas solubility is very smallcompared to unity for hydrocarbons and some gasessuch as nitrogen (solubility of hydrocarbons in waterare, in general, considerably less than water in hydro-carbons 13 ) and to an approximation activity coefficientof water can be taken unity .  However, the nonidealityof the liquid phase and gas solubility become importantat high-pressure conditions. Therefore, the mol fractionof water in the gas phase can be estimated, using thefollowing equation: As can be seen, the water content is determinedprimarily by the fugacity coefficient of water ( φ w ) in thegas phase, temperature, and pressure. In other words,the nonideality of the gas phase is the critical factordetermining the water content in the intermediatepressure range. The fugacity coefficient of water ( φ w )in the gas phase up to intermediate pressures may becalculated as belowwhere  B  and C are a function of temperature. Thefollowing relations for  B  and  C  seem to be satisfactorywhere  a, b, c , and  d  are constants and can be calculatedfor every water - gas system by regressing the watercontent data for that system. To estimate the vaporpressure and the molar volume of water in eq 23, therelations reported by Daubert and Danner 38 and McCain 7 are used, respectively where,  T, t, p, P w sat  , and v w L arein  K  , °  F, psia ,  MPa,  and  ft  3 /lbmol,  respectively, and F w  , D w , and  ∆ V  wt  are water density in  lb m /ft  3 , formationvolume factor, and volume change due to temperature,respectively. Eqs 30 and 31 are valid at  t < 260 °  F  , and  p < 5000 psia even over a wide range of salt concentra-tion. 39 To find constants  a, b, c,  and  d  in eqs 25 and 26,the water content data are used as input for a multi-dimension regression procedure, to reduce the averageabsolute deviation (  AAD ) between experimental andcalculated data.The above approach can be used for estimating thewater content of gases in equilibrium with ice. For thispurpose, the following relations for the molar volumeof ice and ice vapor pressure can be used 40 where superscript  I   and  sub  refer to ice and sublimation,respectively. In the above equations, T,  v w I  , and  P w sub -are in  K, m  3 /kgmol,  and  mmHg , respectively. 5. Results and Discussions To find the constants  a, b, c , and  d  for the methane - water system, the data reported in Table 1 are used. As can be seen, the temperature range is from 273.15to 377.59 K, and the pressures are up to 13.81 MPa,respectively.The  AAD%  amongalltheexperimentaland  f  iV  )  f  i L i ) 1,  N   (21)  y i φ i  P )  x i γ i  P i sat  exp ∫  P i sat  P  v i L dP RT   (22)  y w ) (1 -  x  g )  P w sat φ w  P  exp ( v w L (  P -  P w sat )  RT   )  (23) φ w )  exp(BP + CP  2  )  (24)  B ) a +  bT   (25)C ) c +  dT   (26)  P w sat ) 10 - 6  exp(  73.649 - 7258.2 /T  -7.3037 ln(T) + 4.1653 × 10 - 6 T   2 ) (27) v w L ) 18.015/  F w  (28) F w ) 62.368/   D w  (29)  D w ) (1 + ∆ V  wt ) (30) ∆ V  wt )- 1.0001  ×  10 - 2 + 1.33391  ×  10 - 4 t + 5.50654  ×  10 - 7 t  2 (31) v w I  ) (19.655 + 0.0022364  ×  ( T  -273.15))/10 3 (32)log (P w sub ) )- 1032.5576407/  T  + 51.0557191  × log( T  ) - 0.0977079751  ×  T  + 7.035711316  × 10 - 5 ×  T  2 -98.5115496 (33) Ind. Eng. Chem. Res., Vol. 43, No. 22, 2004  7141
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