wax, definition and thermodynamic modelling · 2010-09-26 · wax, definition and thermodynamic...

WAX, DEFINITION AND THERMODYNAMIC MODELLING

INTRODUCTION

The paraffin wax molecules are straight-chain alkanes that contain more than 15 carbon atoms and have very little branching. Under the most favourable conditions, n-paraffins form clearly defined orthorhombic crystals, but unfavourable conditions and the presence of impurities lead to hexagonal and/or amorphous crystallisation. The gelation characteristics are also affected the same way. Alex et al[DoPE1].1 have said that the crude oils with a high n-paraffin content are considered waxy. Rønningsen et al.2 [DoPE2] pointed out that wax may range from predominantly low molecular weight (M) n-alkanes (C20-C40) to high proportion of high M iso-alkanes.

Wax study has been going on for many years (since early 1900’s), mainly in the experimental aspect. Many wax deposition studies have been conducted and reported in the literature2-14. There are different methods for the determination of the onset temperature of wax crystallisation in petroleum fluids. These methods, depending on their capability, could measure wax appearance temperature (WAT), Cloud Point Temperature (CPT) some times called wax precipitation temperature (WPT) and wax disappearance temperature (WDT). The most common methods are as follows; Standard methods, Polarisation Microscopy, Differential Scanning Calorimetry (DSC), Viscometry and Quartz Crystal Microbalance (QCM). The standard methods are not considered good enough for the onset temperature measurement of black oils due to the limitation of the applied liquids.

Until Won15 presented his first thermodynamic model, almost all studies were aimed at experimentally determining the wax formation conditions and the influence of various parameters. Minchin3 had conducted a study on the properties of petroleum waxes in term of their

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compositions. In 1949, Mazee4 studied and discussed the properties of paraffin waxes in solid state. Bern et al.7 studied wax deposition in North Sea subsea crude-oil pipelines. Burger et al.8 investigated the mechanism of wax deposition in the Trans-Alskan pipeline (TAPS). They concluded that deposition occurs on the pipe wall as a consequence of the lateral transport of both dissolved and precipitated wax molecules during cooling of the flowing oil. Since Won15 many authors have developed wax models13-14, 16-20. Among these, Hansen and Co-workers16 have used polymer solution, and Lira et al.13 have considered the solidification of each component or pseudocomponent as an individual solid phase. Pedersen20 considered the ideal solid solution method.

In this study a new model was developed to predict wax appearance temperature (WAT) and the amount of wax precipitation. The fugacity coefficient of liquid was found by the cubic EOS for pure component, i at pressure and temperature of the system. Regular solution theory was used to perform the calculation of the solid phase, and a cubic (Valderama, PR or SRK) EOS was used for the calculation of liquid andvapour phases (if exists). The reported experimental values were used in this study to correlate the imperative parameters and validate the model. The binary mixtures are taken from phase diagram data reported by Peters et al.21-24, Puri and Kohan6, Mazee4 and Domanska et al.25-28. Regarding the multi-component systems, many mixtures representing solid (wax)-liquid have been reported14, 16, 29. As far as the three-phase (S-L-V) data are concerned, only a few experimental data are available.

WAX MODEL

The essential step in determining the distribution of component i in solid phase is the calculation of fugacities of each component i in that

phase and employing the concept of the equality of fugacities (Equation-2.27), the fugacity of solid ( ) of each component is defined by,

(1)

where is the activity coefficient of component i, is the mole fraction of component i and is standard state fugacity of component i and superscript S is solid phase.



More detailed description of the wax model is discussed elsewhere30-31. In this study a regular solution theory, is considered for the solid phase. In order to calculate the fugacity of each component in the solid phase, the standard state fugacity of pure components as liquid phase was required, which was calculated by EOS. Besides, in order to perform wax calculation a number of properties are requisite to be

evaluated. These properties are fusion temperatures ( ), latent heat of fusion ( ), solubility parameters (δi) and heat capacity of fusion (∆Cpi). In the following sections evaluations of aforementioned parameters are discussed and equations which are used in this study are presented.

Fusion Temperature, Tf

The fusion temperature plays an important part in calculating the solid (wax)-liquid equilibria (i.e. solid fugacity). Therefore, its accurate estimation is vital to the calculations of wax solid formation.

Won15 has suggested the following forms,

(2)

(3) (4)

where M is the molecular weight of each component and T is in Kelvin. Equation-2 is used for n-paraffin when , Equation-3 is used for and Equation-4 is used for non-paraffinic hydrocarbons. Others have presented different correlations

Study conducted by different authors32-35 indicated that the characteristics of n-alkanes according to their carbon numbers (odd/even)significantly differ from one another. Hence, in this study correlations were developed for normal alkanes of different (odd/even) carbonnumbers, using the data reported by Broadhurst32,



(5)

where Cn is the carbon number. Coefficients a, b, c, d, e and f are listed in Table-1. The above equation-(5) was correlated for even carbon numbers from nC4 and for odd carbon numbers from C9. Also a similar equation was correlated for iso-paraffins and naphtenes with one less coefficient (Table-.1).

The comparison between measured and predicted Tf for these correlations is shown in Figures-1 and 2. These correlations are mostly valid for carbon numbers above C10 to around C45. Our correlations have been extrapolated up to C100 for normal paraffins.

The following equation was also developed for aromatics,

(6)

Latent Heat of Fusion,

The latent heat of fusion of low molecular weight hydrocarbons depends very much on their number of carbon atoms as well as on the parityof the carbon numbers. For instance, the latent heat of fusion for undecane is 5330 cal/g-mole, as for the teradecane is 10772 cal/g-mole. Won15 presented the following equation for the heat of fusion, by lumping all phase transition heats into one latent heat of fusion,

(7)

Other authors presented similar equation for the heat of fusion11,14,19. Since the characteristics of the odd and even carbon numbers differfrom each other and their accurate values could lead to better results, therefor we developed the following equations for the heat of fusionbased on the odd/even carbon numbers of n-paraffins. Experimental data used for the correlation of these equations were taken from



Broadhurst32,

(8)

where the values for coefficients A, B, C, D and E are given in Table-2. The fit of the above mentioned equations were about 99% for bothcases. The comparisons of the aforementioned correlations have been shown in Figure-3. As it can be seen using the odd/even method, the latent heat of fusion was the best fitted as compared to the other correlations.

Solubility Parameter of Solid, δS

One of the parameters involved in calculating the fugacity of a component in the solid phase is the solubility parameter. A good approximation of the solubility parameters could very well lead to a better prediction of the WAT. Different authors have presented different correlations for δS. Using different binary mixtures thefollowing equation was developed for δS.

(9)

where δs is the solubility parameter of solid (wax) (cal/cc)1/2.

The correlation proposed by Pedersen et. al11 was used in this study for this model and for the Pan model, which was recommended by the author,

(10)



where ∆Cp is in cal/(gmole.K) and T is in K, and M is molecular weight.

We have also generated the following equation for the molar volume of the solid (wax) components, based on the density correlation given byWarth36,

(11)

VALIDATION OF THE MODEL

In order to validate the model, different fluids were taken according to the availability of the data. These include binary mixtures as well as multicomponent mixtures for Solid (Wax)-Liquid hydrocarbon (S-L2) and Solid (Wax)-Liquid hydrocarbon-Vapour (S-L2-V).

Binary Mixtures

Sets of different binary mixtures were chosen and the model was tested against the experimental data. One chosen sample was C21H42-

C23H48 n-alkanes for which the liquid-solid phase envelope has been reported37. Other binary mixtures to predict WAT were C6-C16, C6-C18, C6-C20, C2-C20 and Cyclopentane-C16 . As an example Figure-4. the model has been able to predict the WAT for C6-C16, C6-C18, C6-C20 and C2-C20 with a very good accuracy, using odd/even correlation. When binary mixture of Cyclopentane-C16 was used, the developed model shows significant deviations from the experimental values for mixture (Figure-4) . There seems to be more discrepancies as the mole fraction of Cyclopentane increases and Hexadecane decreases Tabel-4 represents the percent average absolute deviation between experimental and predicted WAT using this model.



C3H8-C34H70 was another binary mixture of which the solid (wax) phase boundary was reported by Peters et al.18. Using this mixture, Solid (wax)-Liquid hydrocarbon (S-L2) and Solid (Wax)-Liquid hydrocarbon-Vapour (S-L2-V) phase boundaries were predicted by using two andthree-phase boundary calculations. Figures-5 represents the phase boundary comparison for this mixture. The S-L2 phase boundary was at 0.3969 mole fraction of C34H70 at different pressures. As it can be seen, the change of WAT point with pressure is small, but the model isquite capable of predicting the effect. Also in the same figure, the S-L2-V phase boundary represents points of intersection between two and

three-phase equilibrium at different compositions, ranging between = 0.3696 to = 0.0112. The over all percent average error in predicting the S-LHc phase boundary was less than 1 K and in predicting S-L2-V phase boundary was about 2 K. As it can be seen, predicted results showed a good unanimity with the reported data.

Multi-component Mixtures

In order to test the model against the actual fluids, several multi-component systems were chosen. The characterisation of the heavy part ofthese systems was described in using a continuous function. Mixtures with different characteristics were used in this study are as follows,

• Stock thank oils with one WAT reported for each,

• a synthetic oil,

• live oils (volatile, lighter and heavy) with reported WAT at different pressures,

• a North Sea gas condensate mixture,

Stock Tank Oil

Three of these samples had reported WAT at 1 atm and the amount of wax precipitated as the temperature was reduced below the WATpoint16. These fluids are reported to be biodegraded, aromatic for the Mixture-1, paraffinic Mixture-2 and waxy oil for the Mixture-3. The developed model was used to predict the WAT. Then the flash calculation below the WAT temperatures, all at 1 atm was carried out. Figures- is an example of the result of the flash calculation for two-phase (S-L2). As it can be seen even though the model was able to predict the WAT with a very good accuracy, in general it over predicted the amount of wax build up. This is especially true after the early stages of



the temperature drop. Also Table- presents the reported and predicted values for the WAT using the developed model, the Won and the Panmodels. As it can be seen, the suggested model has been able to predict the WAT with a very good accuracy. It has to be noted that for both the Won and the Pan models only parameters that have been presented by the respected authors were used, but the method of calculationswere similar to the one used for this study.

Synthetic Oil

A synthetic oil mixture of C1-C10-Heavy fraction37, with the heavy included components from n-C18 to n-C30 was the other chosen mixture. For this mixture, the phase boundary of S-L2 and S-L2-V were reported. Figure-6 presents the comparison of the measured and predicted two-phase (L2-V and S-L2) and three-phase (S-L2-V) phase boundary. As shown the model developed in this study was able to predict the phase envelope with a good accuracy.

Reservoir Oil

These mixtures were light, volatile and heavy oil samples. The WAT was measured and reported at different saturation pressures for eachmixture 14. Figures-7 to 10 compare the predicted values for these oils using the developed model, the Won and the Pan models. The developed model has predicted the WAT with a very good accuracy for the second oil, which was a volatile oil, while slightly over predictedthe WAT for the first and third mixtures. The Pan model has under predicted the WAT for the first oil, and shows a very good prediction of WAT for the second oil, while over predicted the WAT for the third oil. The Won model has over predicted the WAT in all three cases. It has to be pointed out that the same characterisations were implemented using all three models. Table-5 represents the average absolute deviation between experimental and predicted using all three models. It has to be noted that these data has been reported by Pan et al.14, which were used in defining their correlations.

North Sea Gas Condensate (GC)

Another mixture used in this study was a North Sea gas condensate with reported value of the WAT and the amount of wax precipitated whentemperature was lowered38. Figures-11 presents the amount of wax precipitated for a three-phase (S-L2-V) system. This North Sea gas condensate mixture was reported to have the WAT of around 309 K at 13.61 atm and the developed model predicted a value of 314 K. The Pan model under predicted the WAT, by about 5.4 K while the Won model has over predicted by more than 15 K. The amount of wax build



up was over predicted by both this model and the Won model. The Pan model has under predicted the wax precipitation at the early stages but with only three Kelvin inside the wax zone it over estimated the wax build up. As can be seen the developed model was able to predict the WAT and the amount of wax predicted for this oil with a fairly good accuracy.

4.5. CONCLUSIONS

Expressions to determine the parameters of the wax model, that is Tf, ∆hf, δs and V were developed in this study. The model and the developed expressions were validated using several binary and muti-component mixtures. These mixtures were chosen within different categories of reservoir fluids that would potentially form waxes, i.e. gas condensate, volatile oil, synthetic oil, black oil and dead oil. The above fluids enabled us to evaluate the model for two-phase (S-L2) and three-phase (S-L2-V) systems. The overall results indicate that the developed model is capable of predicting the WAT and wax deposition with a good accuracy. The developed correlations and wax parameters were tested against those of other models and showed their superiority in predicting the WAT and the amount of wax precipitated.

REFERENCES

1. Alex R.F., Fuhr B.J., and Klein L.L. (1991), “Determination of Cloud Point for Waxy Crudes Using a Near-Infrared/Fiber Optic Technique,” Energy & Fuel, American Chemical Society, 5.

2. Rønningsen H.P., Bjørndal B., Hansen A.B. and Pedersen W.B. (1991), “Wax Precipitaion from North Sea Crude Oils. 1.Crystallization and Dissolution Temperatures, and Newtonian and Non-Newtonian Flow Properties,” Energy & Fuels, , 5, 895-908.

3. Minchin S.T. (Aug. 1948), “An Account of Some Solid Properties of Petroleum Waxes in Terms of Their Composition.” Journal of Institute of Petroleum, 34, No. 296, pp. 543-601.

4. Mazee W.M. (1949) “On the Properties of Paraffin Wax in the Solid State,” J. Inst. Petrol., 35, 97-102.

5. Barmby D.S., Bostwick L.G. and Huston, Jr. J.A. (1963), “Some Studies on the Physics of Paraffin Wax and Wax/Polymer Systems,” Proc. 6th World Petroleum Congress, Section VI, 21, pp.16.

6. Puri S. and Kohn J.P. (1970), “Solid-Liquid-Vapor Equilibrium in Methane-n-Eicosane and Ethane-n-Eicosane Binary Systems,”Journal of Chemical and Engineering data, 15, No. 3.



7. Bern P.A., Withers V.R. and Cairns R.J.R. (1980), “Wax Deposition in Crude Oil Pipelines,” EUR 206, European Offshore Petroleum Conference & Exhibition.

8. Burger E.D., Perkins T.K. and Striegler J.H. (June 1981), “Studies of Wax Deposition in the Trans Alaska Pipeline,” Journal of Petroleum Technology.

9. Pedersen W.B., Hansen A.B., Larsen E, Nielsen A.B. and Rønningsen H.P. (1991), “Wax Precipitaion from North Sea Crude Oils. 2. Solid-Phase Content as function of Temperature Determined by Pulsed NMR,” American Chemical Society.

10. Hansen A.B., Larsen E., Pedersen W.B., Nielsen A.B. and Rønningsen H.P. (1991), “Wax Precipitaion from North Sea Crude Oils. 3. Precipitation and Dissolution of Wax studied by Differential Scanning Calorimetry,” American Chemical Society.

11. Pedersen K.S., Skovborg P. and Rønningsen H.P. (1991), “Wax Precipitation from North Sea Crude Oils. 4. ThermodynamicModelling,” American Chemical Society.

12. Hsu J.J.C., Santamaria M.M. and Brubaker J.P. (Sept. 1994), “Wax Deposition of Waxy Live Crudes Under Turbulent Flow Conditions,” SPE 28480, 69th SPE Annual Technical Conference.

13. Lira-Galeana C., Firoozabadi A. and Prausnitz J.M. (1995) “Thermodynamics of Wax Precipitation in Petroleum Mixtures”, AIChE, 42, 239.

14. Pan H. and A. Firoozabadi (1996), “Pressure and Composition Effect on Wax Precipitation: Experimental Data and Model Results,”SPE 36740.

15. Won K.W. (1986), “Thermodynamics for Solid solution-Liquid-Vapour Equilibria: Wax phase Formation from Heavy Hydrocarbon Mixtures”, Fluid Phase Equilibria, 30, 255-279.

16. Hansen J.H., Fredenslund Aa., Pedersen K.S. and Rønningsen H.P. (Dec. 1988), “A Thermodynamic Model for Predicting Wax Formation in Crude Oils”, AIChE Journal, 34, No. 12.

17. Chung F.T.H. (1992), “Modelling Heavy Organic Deposition”, NIPER-555.

18. Narayana L., Leontarities and Darby R. (Jan. 1993), “A Thermodynamic Model for Predicting Wax Deposition From Crude Oils”, Core Laboratory Division, Western Atlas Division, Inc.

19. Erickson D.D., V.G. Niesen, and T.S. Brown (Oct. 1993), “Thermodynamic Measurement and Prediction of Paraffin Precipitationin Crude Oil”, SPE 26604.



20. Pedersen K.S. (Feb. 1995), “Prediction of Cloud Point Temperature and Amount of Wax Precipitation”, SPE Production & Facilities.

21. Peters C.J., de Roo J.L. and Lichtenthaler R.N. (1991), “Measurements and Calculations of Phase Equilibria in Binary Mixtures ofethane+eicosane: Part 2: Solid+Liquid Equilibria,” Fluid Phase Equilibria, 69,135-143.

22. Peters C.J., de Roo J.L. and Lichtenthaler R.N. (1991), “Measurements and Calculations of Phase Equilibria in Binary Mixtures ofethane+eicosane: Part 3: Three-Phase Equilibria,” Fluid Phase Equilibria, 69,51-66.

23. Peters C.J., de Roo J.L. and de Swaan Aron J. (1987), “Three-Phase Equilibria in (ethane-pentacosane),” J. Chem. themodynamics, 19, 265-272

24. Peters C.J., de Roo J.L. and de Swaan Aron J. (1987), “Measurements and Calculations of Phase Equilibria in Binary Mixtures of propane+teratriacontane,” Fluid Phase Equilibria, 72, 251-266

25. Domanska U. and Jolanta Rolinska (1984), “Solid-Liquid Equilibria in Some Binary Mixtures,” International Data Series, 4, 269-276.

26. Domanska U. and K. Domanski (1991), “Correlation of the Solubility of Hexacosane in aliphatic alcohols,” Fluid Phase Equilibria, 68, 103-11.

27. Domanska U. and Kniaz K. (1990), “Solid-Liquid Equilibria in Some Binary Mixtures,” International Data Series, 2, 83-93.

28. Domanska U. and Kniaz K. (1990), “Solid-Liquid Equilibria in Some Binary Mixtures,” International Data Series, 3, 194-206.

29. Elsharkawy A.M., Al-sahhaf T.A., Fahim M.A. and Alzabbai W. (April 1999), “Determination and Prediction of Wax Deposition from Kuwaiti Crude Oils”, SPE 54006.

30. Ali R. Tabatabaei, Ali. Danesh, Bahman Tohidi, and Adrian C. Todd (1999), “A Consistent Thermodynmic Model for PredictingCombined Wax-Hydrate in Petroleum Reservoir Flyuids”, Presented at the Third International Conference on Gas Hydrate, held in Salt Lake City, Utah, 18-22.

31. S.A.R. Tabatabaei-nejad (1999), Phase Behaviour Modelling of Petroleum Wax and Hydrate PhD Dissertation.

32. Broadhurst M.G. (1962), “An Analysis of the Solid Phase Behaviour of the Normal Paraffins,” Journal of Research of the National Bureau of Standard, 66, No. 3.



33. Turner W.R. (1971) “Normal Alkanes,” Ind. Eng. Chem. Prod. Res. Dev., 10, No. 3.

34. Srivastava S.P., Handoo J., Agrawal K.M. and Joshi G.C. (1993) “Phase-Transition Studies in n-Alkanes and Petroleum-Related Waxes- A Review,” J.Phys. Chem. Solids, 54, No. 6, pp 639-670.

35. Coutinho J.A.P., Stenby E.H. (1996), “Predictive local composition models for solid-liquid and solid-solid equilibrium in n-alkanes: Wilson equation for multicomponent systems,” Ind. Eng. Chem. Res., 35, 918-925.

36. Warth A.H, (1956), The chemistry and Technology of Waxes, Sec. Ed., Reinhold publishing corporation, New York.

37. Mazee W.M. (1960), Erdöl Kohle, 13, 88-93.

38. Daridon J.L., Xans P., Montel F. (1996), “Phase Boundary Measurement on a Methane+Decane+Multi-Paraffins System”, Fluid Phase Equilibria, 117, 241-248

39. Affens, W.A., Hall J.M. Holt, S.and Hazlett, R. N. (1984), Fuel, 63, 543-547.

Table-1. Coefficients of Equation (5).

* Iso-paraffins and Naphtanes

Table-2. Coefficients of Equation (8).

Carbon numbers

a B c d e f

Even -426.659 -16.560 0.1238 690.670 - 326.469Odd 156.871 -2.086 0.01223 -775.598 - 76.219

Iso-P& N* -62.401 -2.543 - - 1287.38 140.370

Carbon numbers

A B C D E

Even -1.843 1.886E-01 -2.016E-04 - 7.284E-01



Table-3. Comparison of average absolute deviation (AAD) between experimental and predicted liquid and solid phase boundary usingdifferent models for C21H42-C23H48 Mixture(in K) .

* LPB is average deviation of liquid phase boundary.

+ SPB is average deviation of solid phase boundary.

Table-4. Average absolute deviation(AAD) between

measured and predicted WAT for binary mixtures

at different composition (in K).

Odd 4.520 3.741E-01 -2.317E-03 6.903E-06 -2.901

Won Pan odd even newLPB* 0.79 0.93 0.28 0.66 0.74 SPB+ 0.88 1.10 0.13 0.53 0.90

Mixture

AAD/K

C6H14-C16H34 0.18

C6H14-C18H38

0.61

C6H14-C20H42

0.52

C2H6-C20H42

1.23



Table-5. Experimental and predicted WAT points (in K) for three stock

tank oil using different model.

Table-6. Average absolute deviation (AAD) between

experimental and predicted WAT points for three oil

mixtures using different models (in K) .

* these data are reported by Pan et al.14

Cyclopentane-C16H34

1.88

Exp. Won Pan This Work Mix-1 308 318 300 309 Mix-2 314 328 306 315 Mix-3 289 304 294 292

Won Pan This Work

Oil-1* 15.19 3.95 10.22

Oil-2* 24.59 4.18 2.54

Oil-3* 12.14 3.94 4.94



[DoPE1] Alex R.F., Fuhr B.J., and Klein L.L., “Determination of Cloud Point for Waxy Crudes Using a Near-Infrared/Fiber Optic Technique”, Energy & Fuel, American Chemical Society, 1991, 5

[DoPE2] Rønningsen H.P., Bjorndal B., Hansen A.B., Pedersen W.B., “Wax Precipitaion from North Sea Crude Oils. 1. Crystallization and Dissolution Temperatures, and Newtonian and Non-Newtonian Flow Properties.”, Energy & Fuels, 1991, 5, 895-908

Cn Mw Mw Tf Won Pan N-1 N-2 Mw Exp. Kcal/mole Kcal/mole

1 Cn DHF (WON) Pan

2 58.12 113.6 5 0.995219 1.1034210 MinMax 1 MinMaxChart 1 -1 "Figure-3.3 Comparison of the prediction of Tf for Non-Paraffinic using different models." Arial 12 None None 169.218402099609363835.1437866210936630.5612036132812462.62481689453125 FullPage Arial 10 0 07124.9853515625 5009.9853515625 1.10264587402343751.3568267822265625520.12393066406253363.223937988281255762.33190307617223142.5133117675782



3 50 -215.32 60.07125 -89.2138 227.2244

15 Solid Narrow None Arial 11 ScatterMarker None 0 1 3 -1" Exp. " Exp. None Circle 6 31 31 0 '[FIG-1.XLS]Sheet1'!$H$5:$H$26 0 '[FIG-1.XLS]Sheet1'!$I$5:$I$26 0 -1 "Won " Won 24 DashDotDot MediumNone 0 '[FIG-1.XLS]Sheet1'!$C$7:$C$31 0 '[FIG-1.XLS]Sheet1'!$D$7:$D$31 1 -1 "Pan" Pan 25 Solid Narrow None 0'[FIG-1.XLS]Sheet1'!$C$7:$C$31 0 '[FIG-1.XLS]Sheet1'!$E$7:$E$31 2 -1 "This work" This work 0 Dot Wide None 0 '[FIG-1.XLS]Sheet1'!$C$7:$C$31 0 '[FIG-1.XLS]Sheet1'!$F$7:$F$31 Bottom 0 0 -1 "Mw/ g/mol" Arial 11 NoneNone 1 Arial 11 0 Value Left 1 1 -1 "Tf/K" Arial 11 NoneNone 0 Arial 11 0 Value Horizontal Custom 4043.42918701171861881.2494995117188

6 1.752649 1.631942



1920 820 Arial 10

4 58.12 75 -9.21333 112.6296 52.30883 202.4824 70.135 107.9 7 2.51152 2.2214385 72.146 100 93.84 155.0838 127.5825 202.4824 70.135 104.7 8 3.271839 2.8598976 86.172 150 196.8933 217.076 208.3459 202.296 72.151 113.3 9 4.033615 3.5372487 100.198 200 248.42 257.5237 251.9903 209.6108 86.178 119.5 10 4.796852 4.2450668 114.224 250 279.336 283.9143 279.6052 219.2185 100.205 149.4 11 5.561548 4.9763379 128.25 300 299.9467 301.1333 298.6252 229.2615 100.205 138.7 12 6.327709 5.72524

10 142.276 350 314.6686 312.368 312.4052 239.03 114.232 164 13 7.095335 6.48697111 156.302 400 325.71 319.6982 322.7061 248.2567 114.232 152.2 14 7.864426 7.25758212 170.328 450 334.2978 324.4809 330.5515 256.8581 114.232 147 15 8.634985 8.03386213 184.354 500 341.168 327.6015 336.5834 264.8295 128.242 192.8 16 9.407007 8.81320914 198.38 550 346.7891 329.6375 341.2275 272.1996 128.242 171 17 10.1805 9.59352915 212.406 600 351.4733 330.966 344.7789 279.0097 142.286 206 18 10.95546 10.3731716 226.432 154.297 224 19 11.73188 11.1508417 240.458 154.297 215 20 12.50977 11.9255518 254.484 168.324 220 21 13.28913 12.6965519 268.51 182.351 229 22 14.06996 13.4633120 282.536 210.421 251.1 23 14.85226 14.2254521 296.562 210.421 269.4 24 15.63602 14.9827522 310.588 252.286 278 25 16.42125 15.7350823 266.513 282 26 17.20795 16.4824124 280.54 290 27 17.99612 17.2248625 25.07974 28 18.78575 17.9623126 27.83462 29 19.57685 18.695127 31.62505 30 20.36943 19.4233228 34.3274 31 21.16347 20.1471929 408.77 32 21.95897 20.8668930 422.796 33 22.75595 21.5826431 436.822 34 23.55439 22.2946632 450.848 35 24.35431 23.00317



33 464.874 36 25.15569 23.7083934 478.9 37 25.95854 24.4105335 492.926 38 26.76285 25.1097936 506.952 39 27.56864 25.8063937 520.978 40 28.37589 26.500538 535.004 41 29.13133 27.2386239 549.0340 563.05641 577.08242 591.108



wax, definition and thermodynamic modelling · 2010-09-26 · wax, definition and thermodynamic...

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