Classical binary nucleation theory applied to the real mixture n-nonane/methane at high pressures

Transcription

1 Classcal bnary nucleaton theory appled to the real mxture n-nonane/methane at hgh pressures K. N. H. Loojmans, a) C. C. M. Lujten, G. C. J. Hofmans, and M. E. H. van Dongen Department of Appled Physcs, Endhoven Unversty of Technology, 5600 MB Endhoven, The Netherlands Receved 6 September 1994; accepted 5 December 1994 A thermodynamc model of the formaton free energy of a droplet, based on a real equaton of state, has been mplemented n the bnary classcal nucleaton theory to analyze homogeneous nucleaton of mxtures of n-nonane and methane n the coexstence regon at hgh pressures. The composton of the crtcal nucleus s computed by solvng the Kelvn equatons, the chemcal potentals, and molar volumes beng evaluated from the Redlch Kwong Soave equaton of state. Real gas behavor appears to have a strong effect on nucleaton due to mutual nteractons between methane and nonane molecules. Numercal calculatons show an ncreasng concentraton of methane n crtcal clusters wth ncreasng pressure at fxed temperature and supersaturaton. As a consequence, the surface tenson of the crtcal droplets, whch s evaluated by means of the Parachor method, s lowered, havng a strong ncreasng effect on the nucleaton rate; a 10 bar ncrease of total pressure leads to an ncrease of the nucleaton rate of several orders of magntude Amercan Insttute of Physcs. I. INTRODUCTION Durng handlng and transport of natural gas, t s observed that large and sudden pressure drops may nduce mst formaton. Mutjens et al. 1 showed that the condensaton n the gas mxture s a nonequlbrum phase change of the hgher alkanes, such as nonane and decane, whch are present n the mxture besdes the man components methane and ntrogen. From the strong devaton from vapor lqud phase equlbrum of the onset ponts of condensaton measured n a Wlson cloud chamber, t was concluded that droplets were formed by a homogeneous nucleaton process. The measured onset ponts were located nsde the coexstence regon of the gas mxture n the p T phase dagram, above the crtcal temperature of the man component methane. Ths coexstence regon s characterzed by the appearance of real gas effects n condensaton, when pressure s above a few bars. The real gas effects become vsble n densty dependent solubltes affectng partal equlbrum vapor pressures, and n so-called retrograde condensaton,.e., condensaton nduced by a decrease of pressure. Up to now, no examples of applcaton of nucleaton theory to nucleaton n the coexstence regon at hgh pressures are known from lterature. In ths paper we present numercal calculatons based on classcal bnary nucleaton theory of the mxture n-nonane/methane. Snce natural gas conssts of many components, most of them beng heaver hydrocarbons, we chose the n-nonane/methane mxture as a model mxture to study homogeneous nucleaton. At pressures rangng from 10 to 200 bar correspondng to exploraton pressures of natural gas, the coexstence regons of natural gas and of the mxture mentoned above wth a molar n-nonane concentraton of about 10 4, largely concde n a Address all correspondence to: Endhoven Unversty of Technology, Faculty of Appled Physcs, P.O. Box 513, 5600 MB Endhoven, The Netherlands. the p T phase dagram and therefore are expected to exhbt smlar real gas effects n condensaton. From now on, we shall consder the n-nonane/methane mxture. The real gas effects due to the mutual nteractons of the nonane and methane molecules lead to a solublty of methane n lqud nonane that ncreases wth pressure. Snce the droplets wll contan a sgnfcant amount of methane besdes the nonane, the modelng of homogeneous nucleaton requres a bnary nucleaton model. The theory of homogeneous condensaton goes back to the work of Volmer and Weber 2 and Becker and Dörng. 3 They developed the classcal nucleaton theory CNT of pure vapors whch predcts the nucleaton rate,.e., the rate at whch crtcal nucle are formed. A major role n ths theory s played by the capllary approxmaton; propertes of mcroscopc droplets are taken as f the droplets were of macroscopc szes. Ress 4 extended CNT to bnary vapors. The calculaton of the knetc prefactor was mproved by Stauffer. 5 Later, the correctness of the mplementaton of the Gbbs Duhem dentty n bnary nucleaton theory was dscussed by Rennnger et al., 6 Doyle, 7 and Wlemsk. 8 Wlemsk 8 10 ntroduced the model of a droplet consstng of a bulk and a surface layer wth excess molecules, 11,12 and at the same tme ponted out the shortcomngs of ths model. Because of the rather unsatsfactory assumpton of the capllarty approxmaton for droplets that consst of typcally molecules, several attempts have been made to mprove CNT Refs wth varyng success. Numercal studes of bnary homogeneous condensaton wth CNT performed so far manly refer to aqueous solutons and mxtures of alcohols. Nucleaton rates were determned as a functon of actvtes of the speces present n the mxture, and attenton was pad to the composton of the crtcal cluster. 17 In the followng sectons we wll present an extenson of CNT allowng calculaton of nucleaton rates n mxtures showng real gas effects. Results obtaned wth the model are presented. J. Chem. Phys. 102 (11), 15 March /95/102(11)/4531/7/$ Amercan Insttute of Physcs 4531

2 4532 Loojmans et al.: Classcal bnary nucleaton theory II. BINARY HOMOGENEOUS NUCLEATION THEORY Snce the ntroducton of bnary nucleaton theory, dfferent models and nterpretatons have been publshed n the lterature. Therefore we shall gve a bref outlne of the theory as t s mplemented n our calculatons. The nucleaton rate s the number of droplets formed per unt tme and space. It s expressed n the followng form: J K e G/k B T. 1 K s a knetc prefactor, G n the exponental s the free reversble energy of formaton of a crtcal droplet, k B s Boltzmann s constant, and T the temperature. The energy of formaton G(n 1,n 2 ) s a functon of the numbers of partcles of speces 1 and 2 n the droplet. When the gas vapor mxture s n a supersaturated state, the G(n 1,n 2 ) plane exhbts a saddle pont. The poston of ths saddle pont s consdered to represent the crtcal nucleus. The crtcal nucleus s n unstable equlbrum wth the surroundng vapor, and the saddle pont forms the lowest passage over an energy barrer. Clusters passng over ths barrer wll grow to macroscopc droplets. Dfferent models have been employed to calculate G. In CNT, Gbbs capllarty approxmaton s used to descrbe the clusters on the bass of classcal thermodynamcs. A. Classcal droplet model We adopt the bnary classcal theory as descrbed by Wlemsk. In ths macroscopc model of a droplet and surroundng vapor, the droplet conssts of bulk lqud wth numbers of bulk molecules n 1 and n 2 and a surface layer wth l l numbers of surface molecules n s 1 and n s 2. Bulk and surface are consdered to be n thermodynamc equlbrum and the droplet s assumed to be ncompressble. The number of surface molecules s an excess number, defned as n s n n l n v, where n s the total number of molecules n the system, and n v s the number of molecules n the vapor. The excess molecules n s do not contrbute to the droplet volume. The presence of the surface layer n the droplet model mplctly takes nto account the effect of surface enrchment; the concentraton of speces 1 and 2 near the surface can dffer from the nteror composton. Accordng to ths droplet model, the free formaton energy s 8 G p p l V l A l v l n s v n s. 2 Chemcal potentals l and v are evaluated at pressures p l n the droplet and p v of the vapor, respectvely. V l s the droplet volume, A s the surface area, and the surface tenson. Because bulk lqud and surface are assumed to be n equlbrum, s l, bulk and surface molecules can be taken together n Eq. 2 ; n t n l n s. If we defne the dfference n chemcal potental between droplet and vapor as l ( l,t) v (p v,t), and furthermore assume that the formaton of clusters does not lead to a change of vapor pressure p.e., p v p, then G can be wrtten as G p v p l V l A n t. The saddle pont of ths G plane s determned by G n t sp 3 0,. 4 Combnng Eqs. 3 and 4 together wth the Gbbs Duhem relaton for the bulk, S l dt V l dp l n l d l p l,t 0, and the Gbbs adsorpton equaton for the surface, S s dt Ad n s d s 0, we arrve at 0. In dervng the last result, we used the Laplace relaton p l p v 2 /r, and the expressons for the volume and surface area of a sphercal droplet, V l n l v 4 3 r 3 and A 4 r 2. Usng the ncompressble flud approxmaton l (p l,t) l (p v,t) v (p l p v ) n whch v s the partal molecular volume of speces, one arrves at the so-called Gbbs Thomson or Kelvn equatons, * 2 v 0. r In ths equaton * l (p v,t) v (p v,t). As stated by Wlemsk, 9 all varables n these equatons are functons of bulk composton only. Therefore, at the saddlepont only the bulk composton can be obtaned from these equatons. The method descrbed by Laaksonen et al. 17 to calculate n s l and n cannot be appled here because the surface molecules do not contrbute to the droplet volume n ths model. The free energy of formaton of a crtcal cluster at the saddle pont follows from Eq. 3, wth 0, because the crtcal cluster s n unstable equlbrum wth the surroundng vapor, G sp 1 3 A. B. Knetcs The knetc prefactor K of Eq. 1 was frst derved by Ress 4 and later modfed by Stauffer. 5 The result obtaned by Stauffer s gven by the followng equatons: K NR av Z, where K s the product of the number densty N of monomers n the gas, an average growth rate R av and the Zeldovch factor Z. R av s gven by

3 Loojmans et al.: Classcal bnary nucleaton theory 4533 det R R av R 11 sn 2 R 22 cos 2 2R 12 sn cos ; 11 R av depends on the growth rate tensor R and on the angle of the drecton of the current of clusters n the saddle pont. We shall assume that no cluster cluster nteractons take place, but only monomers collde wth the crtcal cluster, then the elements of R are gven by 5 R R 11 R 12 R 21 R 22, R 11 A n 1,n 2 1, 12 R 12 R 21 0, R 22 A n 1,n 2 2, where A(n 1,n 2 ) s the cluster surface area, and 1 and 2 are the mpngement rates of molecules of substance 1 and 2. The expresson for the Zeldovch factor reads Z 1 2 G x,y 2 x 2 det D 1/2. 13 The tensor D conssts of the dervatves of G wth respect to the total number of molecules n the cluster. In combnaton wth CNT, a problem arses n the calculaton of R av and Z, because G s not known as a functon of the overall cluster composton, but only as a functon of bulk composton. Therefore D can not be determned exactly from ths model. However, as already remarked by Wlemsk, 9 ths wll not play a crucal role snce the nfluence of D s lmted to the relatvely unmportant knetc prefactor. III. A REAL GAS MODEL FOR G Calculaton of the nucleaton rate accordng to CNT requres a thermodynamc descrpton for the gas and lqud mxture takng nto account all real gas effects nvolved n the nucleaton process. Frst, the bulk composton of the crtcal droplet can be found by solvng an equaton that follows from a lnear combnaton of the Kelvn equatons: v 2 * 1 v 1 * For gven p and T, all quanttes n ths equaton are known functons of droplet bulk composton only, whch follow from the EOS. Second, t s not possble to evaluate mxture propertes from the propertes of the pure substances; methane s n a state n whch no pure lqud methane can exst. Surface tenson, whch s a very mportant parameter determnng the heght of the energy barrer G n Eq. 1, wll thus be estmated by usng an emprcal correlaton Parachor method. From several avalable equatons of state we selected the Redlch Kwong Soave RKS and the Lee Kesler Plöcker LKP equatons, because they proved to be well applcable to alkanes. 18 The RKS equaton s of the cubc type and therefore relatvely smple from a calculatonal pont of vew. The LKP equaton of state s a transcendental one, thus computatonally more complex. It s applcable, however, over a wder range of reduced temperatures. To compare both EOS to test ther sutablty to our applcaton, we calculated the equlbrum molar volume of the lqud as a functon of temperature and pressure. The molar volume s a very mportant parameter n our model because of ts appearance n the Kelvn equatons and n the surface tenson correlaton. The results obtaned were compared wth expermental data of Shpman et al. 19 Although RKS s known for ts poor predcton of lqud molar volume, results obtaned by RKS when Peneloux correcton 18 was ncluded turned out to be far better than LKP results n predctng the molar volumes RKS, less than 0.8% devaton from expermental data; LKP, up to 6% devaton. For ths reason, the RKS equaton of state was used for all thermodynamc calculatons throughout ths work. The RKS equaton has the followng form: p RT V RKS a b V RKS V RKS a, 15 where and a RT c p c b R2 T c p c wth 1 f 1 T 2 T c, f In these equatons, R s the unversal gas constant, V RKS s the molar volume subscrpt m s left out to avod confuson wth mxture propertes n the remander of ths text, s Ptzer s acentrc factor, and the subscrpt c refers to the crtcal pont. For a mxture, a and b are evaluated from the pure component values usng the mxng rules a m b m y a, j y y j b b j 1/2 1 k j. The quanttes y n the above expressons denote the molar fractons of the components, both n the vapor and lqud phases. From now on, however, we wll use x for the lqud molar fractons. In the last mxng rule, the nteracton parameter k j s ntroduced. Its value s determned by a ft to experments. 20 The RKS equaton s known to yeld too large values for the lqud molar volume V. The correcton term proposed by Peneloux 18 reads V V RKS c, where V s the corrected molar volume and c Z RA RT c p c 16

4 4534 Loojmans et al.: Classcal bnary nucleaton theory Z RA s the Rackett compressblty factor of the substance under consderaton. For a mxture, the correcton term s obtaned by usng the conventonal mxng rule c m x c. From the above equatons, an expresson for the chemcal potental can be derved n analytcal form n a standard way by ntegraton of pressure wth respect to volume obtanng the free energy F as a result and then takng the dervatve wth respect to n. In ths way an expresson of the form f (T,V,y ) s obtaned. For the lqud, of course, y s replaced by x. The partal molecular volumes are found by substtutng the mxng rules nto the RKS equaton and applyng standard expressons from thermodynamcs for the partal volumes N A denotes Avogadro s number, N A v 1 V RKS x 2 V RKS x 2 c 1, 17 V RKS N A v 2 V RKS x 1 c x In these equatons the Peneloux correcton has been used. Because of the use of the ncompressble flud approxmaton n dervng the Kelvn equatons 8, the molecular lqud volume has to be calculated at pressure p v outsde the droplet. Usng chemcal potentals and molecular volumes obtaned from Eqs n Eq. 14, a soluton for the droplet bulk composton s found, and * and v are known. Wth ths result, t s possble to calculate the droplet radus from ether one of the Kelvn equatons wrtten n the form r 2 v *. 19 The crtcal free energy barrer, G sp, s now obtaned from Eq. 9, provded that the surface tenson s known. Several correlatons exst for the surface tenson, most of whch need the pure component values as ther nput. Snce relevant temperatures are above the crtcal temperature of methane, these correlatons can not be used for the mxture under consderaton. A relaton that can be used for our purposes s the Macleod Sugden correlaton, 18 an expresson obtaned by the best ft to experments. It reads P x V l y v 20 V 1/4 n whch superscrpts l and v refer to lqud and vapor, respectvely. The parameters P are the so-called parachors of the components. Orgnally, Macleod suggested to calculate these from molecular structures; however, better agreement wth experments can be obtaned by emprcally fttng the parachor values to measurements. For our calculatons, the values obtaned by Deam and Maddox 21 are used. We note that the parachor method s a ft on bulk composton. Ths s consstent wth the fact that, as a soluton of the Kelvn equatons, bulk compostons are obtaned whch are substtuted nto Eq. 20 n order to fnd the surface tenson of the droplet. A lmtaton of ths method s the lack of a proper correcton for the curvature of the droplet surface whch s stll subject of dscusson n lterature What remans to be calculated, s the knetc prefactor K of expresson 1 n whch the second dervatves of G at the saddle pont poston wth respect to the total numbers of t t molecules, n 1 and n 2 are present. As was already ponted out n Sec. II B, we have no nformaton concernng the excess s numbers n 1 and n s 2. Accordng to Wlemsk, 9 the best one can do s usng Ress orgnal expresson for G,.e., G n l 1,n l 2 n l * A 21 keepng constant at the value of the crtcal droplet. The shape of the G surface obtaned n ths way s approxmately correct n the neghborhood of the saddle pont. For the mpngement rates present n the growth rate tensor R Eq. 12, the deal expresson / k B T/2 m s used, where s the number densty of monomers of speces and m s the molecular mass. It s expected that the values of only have a slght effect on the nucleaton rate. No specal attenton has been pad to the lmtng case to unary nucleaton. It may be expected that a small overestmate of the nucleaton rate appears, when the poston of the saddlepont s too close to the n 1 -axs. Ths mght occur n the low pressure lmt, n whch only a lttle methane wll be dssolved n the droplets. A more extensve treatment of ths matter s gven by Wlemsk. 25 Fnally, we ntroduce a supersaturaton S, whch wll be used n presentng results of our calculatons S y 1, 22 y 1,eq where y 1 and y 1,eq are the molar vapor fractons of nonane n the supersaturated state and at equlbrum, respectvely. The equlbrum molar fracton s calculated by equatng the FIG. 1. Contourplot of the free energy surface n the vcnty of the saddle pont. T 240 K, p 40 bar, and y (S 8). The value of G at the saddle pont s 65.5 k B T; contours correspond to ncrement steps of 1 k B T.

5 Loojmans et al.: Classcal bnary nucleaton theory 4535 FIG. 2. Nucleaton rate n the mxture n-nonane/methane as a functon of supersaturaton wth the total pressure as a parameter, T 240 K. FIG. 4. Methane lqud fracton as a functon of total pressure. The sold curve represents the composton of the crtcal droplet at supersaturaton S 20. The dashed curve refers to equlbrum lqud; T 240 K. chemcal potentals n the vapor and the lqud for both components at gven p and T. We emphasze that ths defnton s not used n the calculaton of nucleaton rates. The reason for usng the supersaturaton rato n representng the results s that S s consdered to be the drvng force behnd the condensaton process. The calculatons descrbed are mplemented n a FOR- TRAN numercal code. Calculatons can be performed for n-alkanes up to n-dodecane wth several carrer gases. The numercal values of the mxture n-nonane/methane, used n the present calculatons, are lsted n the Appendx. The program code s avalable upon request. IV. RESULTS AND DISCUSSION A plot of a typcal free energy surface accordng to Eq. 21 for the n-nonane/methane mxture s shown n Fg. 1. The condtons are T 240 K, p 40 bar, and the molar fracton of n-nonane n the gas y correspondng to a saturaton rato S 8. The crtcal cluster conssts of 66 nonane molecules and 11 methane molecules. Durng nucleaton, the saddle pont s passed n the drecton ndcated by the arrow. Fgures 2 and 3 show the nucleaton rate as a functon of supersaturaton wth the pressure as a parameter for two dfferent temperatures. The model predcts a large effect of total pressure on the nucleaton rate. At fxed supersaturaton, ncreasng pressure nduces an ncrease of nucleaton rate by several orders of magntude demonstratng real gas effects n homogeneous nucleaton. Apart from ths phenomenon, the generally observed exponental dependence of nucleaton rate on supersaturaton at gven total pressure and temperature can also be seen n the fgures. The nteracton of nonane and methane n the nucleaton process also comes forward when the composton of the crtcal cluster s consdered. The saddle pont of Fg. 1 already ndcated the presence of methane n the crtcal cluster. In Fg. 4 the molar fractons of methane n the crtcal cluster and n the correspondng equlbrum lqud state are plotted as a functon of pressure at fxed supersaturaton and temperature. The methane molar fracton ncreases from approxmately 0 at low pressure to 0.11 at 40 bar. Ths s less than the equlbrum molar fracton, but, as can be seen from Fg. 5, t has sgnfcant consequence for the surface tenson. The ncreasng amount of methane n the crtcal cluster lowers the surface tenson from N/m to N/m at 40 bar. The decrease of surface tenson has a drect FIG. 3. Nucleaton rate n the mxture n-nonane/methane as a functon of supersaturaton wth the total pressure as a parameter, T 260 K. FIG. 5. Surface tenson as a functon of pressure for fxed values of supersaturaton and temperature. The sold curve corresponds to the crtcal droplet composton, the dashed curve refers to the equlbrum composton.

6 4536 Loojmans et al.: Classcal bnary nucleaton theory FIG. 6. Pressure dependence of the equlbrum gas composton of the mxture n-nonane/methane. Dashed curve, nonane molar fracton n the gas phase at vapor lqud equlbrum at 240 K as a functon of the mxture total pressure. Sold curve correspondng equlbrum n-nonane vapor densty. consequence for the energy barrer G Eq. 9, whch s also reduced. Ths largely explans the enhanced nucleaton rate wth pressure. In Fgs. 2 and 3 the nucleaton rate was plotted as a functon of supersaturaton S. It has to be realzed here that the saturated nonane vapor densty tself depends on the mxture total pressure, as s shown n Fg. 6. Due to real gas effects, the solublty of nonane n the gas phase ncreases, resultng n an ncrease of equlbrum nonane concentraton y 1,eq and therefore an even stronger ncrease of equlbrum nonane vapor densty 1,eq, at pressures above 20 bar. So, S s not a measure for the actual nonane vapor densty when nucleaton rate curves at dfferent pressures are compared. In Fg. 7 the nucleaton rate s plotted as a functon of nonane vapor densty 1 whch s obtaned from the relaton 1 y 1 /V v, where V v s the molar volume of the vapor mxture. An ncrease of total pressure above 10 bar at fxed nonane vapor densty and temperature appears to reduce the nucleaton rate. Ths can also be observed n Fg. 8. In ths plot the nucleaton rate s gven as a functon of pressure sold curve wth a fxed nonane vapor densty of mol m 3. The dashed curve shows the nucleaton rate f not FIG. 8. Nucleaton rate as a functon of pressure for a fxed nonane vapor densty mol m 3. Sold curve, n-nonane/methane mxture calculated wth bnary nucleaton theory. Dashed curve, CNT for n-nonane wth an nert carrer gas; T 240 K. methane, but an nert carrer gas s added. The pronounced dfference between bnary nucleaton under real gas condtons and unary nucleaton wth an nert carrer gas s clear. Addng methane to nonane frst shows a strong ncrease of nucleaton rate, due to the lowerng of surface tenson caused by the presence of methane n the crtcal cluster. So, ntally ths effect domnates the effect of decreasng supersaturaton wth pressure. At hgher pressures, the stuaton s reversed and fnally, when pressure s hgh enough, thermodynamc equlbrum s attaned agan. Addng an nert component to a nonane vapor of fxed densty shows a monotoncally decreasng nucleaton rate wth nert gas pressure, ths beng a result of a hgher energy barrer due to the ncrease of lqud chemcal potental, whle vapor chemcal potental s not changed much. Recently, a reducton of the nucleaton rate wth ncreasng total pressure of 1-propanol n helum and n hydrogen, was expermentally found by Hest et al. 26 n a specal hgh pressure dffuson cloud chamber. Fnally, n Fg. 9 a p T phase dagram s shown of a FIG. 7. Nucleaton rate n the mxture n-nonane/methane as a functon of nonane vapor densty for dfferent total pressures at T 240 K. FIG. 9. p T dagram wth lnes of constant nucleaton rate for a gas mxture wth molar fracton n-nonane y The outer curve corresponds to vapor lqud equlbrum for gven nonane fracton. The labels denote the nucleaton rate n cm 3 s 1.

7 Loojmans et al.: Classcal bnary nucleaton theory 4537 n-nonane/methane mxture wth a nonane molar fracton of The outer curve, on whch the gas mxture wth gven nonane concentraton s n equlbrum wth the lqud phase, forms the boundary of the two-phase coexstence regon. Insde ths envelope, lnes of constant nucleaton rate are drawn, lnes representng hgher nucleaton rates are found further nwards the coexstence regon. Furthermore, lnes of gven nucleaton rate form retrograde curves, and they all appear to converge to the crtcal pont of the mxture. V. CONCLUSIONS The classcal bnary nucleaton theory has been extended wth a thermodynamc model takng nto account real gas effects. For calculatng chemcal potentals and molar volumes the RKS equaton of state was appled, surface tenson was modeled wth the Parachor method. Ths model enables the theoretcal study of homogeneous nucleaton of real gas mxtures n the coexstence regon, such as the n-nonane/ methane mxture, subject of the calculatons performed n ths paper. The model predcts for ths mxture an ncrease of nucleaton rate wth pressure when supersaturaton and temperature are fxed. Due to real gas effects, an ncreasng amount of methane s present n the crtcal droplet wth ncreasng pressure, as also occurs n equlbrum condensaton. The presence of methane n the crtcal cluster lowers the surface tenson, and therefore the free energy of formaton s lowered, leadng to an enhanced nucleaton rate. We have only presented calculatons for the n-nonane/ methane mxture. However, t s to be expected that smlar results are found n other mxtures at hgh pressures, when nteractons between the mxture components are present. An example s the n-nonane/ntrogen mxture. Up to now no expermental data of homogeneous nucleaton n the coexstence regon at hgh pressures have been publshed n lterature. Expermental work wth a nucleaton pulse expanson wave tube s n progress. 27,28 Results wll be publshed separately. APPENDIX Numercal values of the n-nonane/methane mxture used n the calculatons are lsted below. n-nonane: crtcal pressure p c 22.9 bar, crtcal temperature T c K, Ptzer s acentrc factor 0.445, Rackett compressblty factor Z RA , molar mass M g/mol, and parachor P p g 1/4 cm 3 s 1/2 mol 1. Methane: crtcal pressure p c 46.0 bar, crtcal temperature T c K, Ptzer s acentrc factor 0.011, Rackett compressblty factor Z RA , Molar mass M g/mol, and parachor P 81.0 g 1/4 cm 3 s 1/2 mol 1. The nteracton parameter for the mxture k j All lsted data can be found n the book of Red, Prausntz, and Polng, 18 except for the bnary nteracton parameter whch comes from Knapp et al M. J. E. H. Mutjens, V. I. Kalkmanov, M. E. H. v. Dongen, A. Hrschberg, and P. A. H. Derks, Rev. Inst. Fr. Pét. 47, M. Volmer and A. Weber, Z. Phys. Chem. 119, R. Becker and W. Dörng, Ann. Phys. 5, H. Ress, J. Chem. Phys. 18, D. Stauffer, J. Aerosol Sc. 7, R. G. Rennnger, F. C. Hller, and R. C. Bone, J. Chem. Phys. 75, G. J. Doyle, J. Chem. Phys. 75, G. Wlemsk, J. Chem. Phys. 80, G. Wlemsk, J. Phys. Chem. 91, G. Wlemsk, J. Chem. Phys. 88, J. W. Gbbs, The Scentfc Papers of J. W. Gbbs Volume 1 Dover, New York, J. S. Rowlnson and B. Wdom, Molecular Theory of Capllarty Clarendon, Oxford, J. Lothe and G. M. Pound, J. Chem. Phys. 36, A. Dllmann and G. E. A. Meer, J. Chem. Phys. 94, X. C. Zeng and D. W. Oxtoby, J. Chem. Phys. 95, H. R. Kobrae and B. R. Anderson, J. Chem. Phys. 94, A. Laaksonen, M. Kulmala, and P. E. Wagner, J. Chem. Phys. 99, R. C. Red, J. M. Prausntz, and B. E. Polng, The Propertes of Gases and Lquds McGraw Hll, New York, L. M. Shpman and J. P. Kohn, J. Chem. Eng. Data 11, H. Knapp, R. Dörng, L. Oellrch, U. Plöcker, and J. M. Prausntz, Vapor Lqud Equlbra for Mxtures of Low Bolng Substances Deutsche Gesellschaft für Chemsches Apparatewesen, Frankfurt am Man, J. R. Deam and R. N. Maddox, J. Chem. Eng. Data 15, R. C. Tolman, J. Chem. Phys. 17, S. M. Thompson, K. E. Gubbns, J. P. R. B. Walton, R. A. R. Chantry, and J. S. Rowlnson, J. Chem. Phys. 81, M. Njmejer, C. Brun, A. B. van Woerkom, A. F. Bakker, and J. M. J. van Leeuwen, J. Chem. Phys. 96, G. Wlemsk, J. Chem. Phys. 62, R. H. Hest, M. Janjua, and J. Ahmed, J. Phys. Chem. 98, K. N. H. Loojmans, P. C. Kresels, and M. E. H. Van Dongen, Exp. Fluds 15, K. N. H. Loojmans, J. F. H. Wllems, and M. E. H. v. Dongen, n Proceedngs of the 19th Internatonal Symposum on Shock Waves Sprnger, Berln, n press.