Measurement and Prediction of the Phase Behavior of Wax− Solvent

C40, with the mean carbon number of 28) dissolved in n-hexadecane (C16) and Norpar13 (a paraffinic solvent, C11-C15). For these mixtures, the WAT valu...
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Ind. Eng. Chem. Res. 2004, 43, 3451-3461

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Measurement and Prediction of the Phase Behavior of Wax-Solvent Mixtures: Significance of the Wax Disappearance Temperature Nitin V. Bhat and Anil K. Mehrotra* Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4

This study investigates the phase behavior of wax-solvent mixtures below their liquidus temperature, particularly between the wax disappearance temperature (WDT, recorded while heating the mixture) and the wax appearance temperature (WAT, recorded while cooling the mixture). The prepared wax-solvent mixtures comprised 6-22 mass % of a paraffinic wax (C20C40, with the mean carbon number of 28) dissolved in n-hexadecane (C16) and Norpar13 (a paraffinic solvent, C11-C15). For these mixtures, the WAT values were lower than the WDT values by 3.2 ( 0.6 °C. Gas chromatograph analyses of the liquid phase, at temperatures between the WDT and the pour-point temperature, were used to calculate the wax solubility and the composition of the dissolved wax. At temperatures below the WDT, the wax solubility decreased with decreasing temperature, and the liquid-phase concentrations of C20-C27 were higher than those in the whole wax and those of C29+ were lower. The experimental data compared satisfactorily with predictions from an available UNIQUAC-based model for the liquidus temperature as well as for the composition of C20+ constituents in the liquid phase. This study establishes that the WDT rather than the WAT is closer to the liquidus or saturation temperature of “waxy” mixtures. Introduction Many crude oils and crude oil products contain substantial fractions of petroleum wax, which is a major constituent of most “waxy” solid deposits from crude oils. Typically, paraffin waxes are mixtures of n-alkanes and constitute about 40-60% of the average crude oil deposits. The precipitation and deposition of paraffin waxes from crude oils are commonly observed in production, transportation, and processing operations.1 Wax precipitation is undesirable because it causes plugging of pipelines, reservoirs, and process equipment. Wax deposition results in a reduction of the flow rate or an increase in the pressure drop, involving substantial expenditures for its control and remediation. Despite compositional differences, most “waxy” crude oils at sufficiently high temperatures behave as Newtonian liquids. Upon cooling below a certain temperature, known as the wax appearance temperature (WAT), wax crystals start to appear and become suspended in the liquid phase. As the temperature is decreased further, wax crystals grow in size and precipitate from the solution. When cooled further toward the pour-point temperature (PPT), the individual wax crystals give rise to an interlocking structural network, which ultimately turns the crude oil into a gelled, solidlike state. It is pointed out that the wax deposit is not entirely solid; i.e., it consists of liquid and solid phases. Several studies have reported measurements for the WAT, also referred to commonly as the cloud-point temperature (CPT) or the wax appearance point, of mixtures of n-paraffins2-5 and waxy crude oils.6-9 Mirante and Coutinho10 used the Wilson, nonrandom two liquid, and UNIQUAC methods to estimate the WAT of fuels and fuel blends and found that the * To whom correspondence should be addressed. Tel.: (403) 220-7406. Fax: (403) 284-4852. E-mail: [email protected].

predictive UNIQUAC method of Coutinho11 gave WAT values within experimental uncertainties. In studying the “aging” of wax-oil deposits, Singh et al.12 used the UNIQUAC method for predicting a critical carbon number for wax-oil mixtures. The critical carbon number was defined such that paraffin molecules below it diffuse out of the deposit, while those above it diffuse into the deposit through a counterdiffusion process.12 In a study on the aging of wax deposits, Coutinho et al.13 reported that wax deposits from crude oils undergo recrystallization and proposed Ostwald ripening as an additional aging mechanism. Previous studies have also examined wax deposition mechanisms.14-21 Burger et al.17 provided a detailed study of wax deposition mechanisms, identifying the main mechanisms to be molecular diffusion, shear dispersion, and Brownian motion. Most deposition models use molecular diffusion as the sole mechanism for wax deposition; however, in a recent review,22 it was pointed out that not enough experimental evidence exists to support this because the role of Brownian diffusion of wax crystals is not well established. Other studies have examined the modeling of wax deposit removal by shear,14,23,24 deposit aging,15,24,25 and deposition under multiphase-flow conditions.26 A significant difference may exist between the temperature at which the first wax crystals are formed (during cooling) and the temperature at which the last crystals are melted (during heating) even at very slow rates of heating or cooling. Gimzewski and Audley27 reported that, at the same rate of temperature change, the amount of superheating during melting/dissolution is considerably less severe than the amount of supercooling during crystallization. Consequently, the dissolution temperature of waxes [i.e., wax disappearance temperature (WDT), which is recorded during heating] may be a better representation of the true phase change

10.1021/ie0400144 CCC: $27.50 © 2004 American Chemical Society Published on Web 05/20/2004

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temperature than the wax crystallization temperature (i.e., WAT, which is recorded during cooling). However, quantitative differences between the WAT and the WDT of paraffin-solvent “waxy” mixtures have not been established. In this study, measurements of WDT, WAT, and PPT were made on prepared wax-solvent mixtures. Experiments were also performed to study the liquid-phase composition changes at temperatures lower than WDT. These results, together with those reported on waxC12 mixtures by Tiwary,28 have been compared with thermodynamic predictions for the liquidus (i.e., the saturation or the liquid-solid-phase change) temperature as well as the liquid-phase composition. Thermodynamic Relations for Multiphase SolidLiquid Mixtures. A number of thermodynamic techniques for predicting the onset of wax crystallization have been proposed. The following generalized relationship for solid-liquid equilibrium relates, for each component of the mixture, the composition in both phases with nonideality of the phases and the purecomponent thermophysical properties

[

ln

(

)

(

)]

∆Htr Ttr sγs ∆Hm Tm ) + 1 -1 RTtr T xγl RTm T

(1)

i

where s and x are mole fractions of component i in the solid and liquid phases, respectively; γ is the activity coefficient; Tm and Ttr are the melting and solid-solid transition temperatures, respectively; and ∆Hm and ∆Htr are molar enthalpies of the respective phase transformations. Equation 1 is the basis for most models reported in the literature; however, the models differ in the approach for describing the nonideality of solid and liquid phases. A liquid phase can be considered ideal for lowpressure systems.29,30 Some studies have considered a free-energy model to describe the moderate nonideality of the liquid phase.6,31-33 For high-pressure systems (e.g., live oils), the liquid phase is described using an equation of state.32,34-36 The solid-phase modeling has also received much attention. In earlier studies, all compounds in the mixture were assumed to become part of the wax phase;6,33 however, the predictions overestimated the amount of precipitated wax as well as the WAT. Spectroscopic and calorimetric studies reported by Snyder et al.37-39 suggested the existence of more than one solid phase. Models were also developed by assuming that the solid phase is comprised of independent components or pseudocomponents.32,34,40 The thermophysical properties when the solid phase is poorly defined are usually fitted to a set of experimental data. Liquid-Phase Nonideality. The activity coefficient model used for the liquid phase is the Flory free-volume model41

) ln ln γcomb-fv i

φi φi +1xi xi

(2)

with

φi )

xi(Vi1/3 - Vwi1/3)3.3

∑j xj(Vj

1/3

1/3 3.3

- Vwj )

(3)

where Vi is the molar volume and Vwi is the van der Waals volume of component i. Solid-Phase Nonideality Predictions from the UNIQUAC Method. The solid-phase nonideality is described by the following equation in the predictive UNIQUAC model,11 which is a modified version of the original UNIQUAC model:

gE

n

)

RT

∑ i)1

()

xi ln

Φi

+

xi

Z

θi

n

∑qixi ln Φ 2 i)1

[

n

-

(

i

n

qixi ln ∑θj exp ∑ i)1 j)1

)]

λij - λii qiRT

(4)

with

Φi )

xiri n

; θi )

xjrj ∑ j)1

xiqi n

(5)

xjqj ∑ j)1

where r and q are structural parameters in the UNIQUAC model. Coutinho11 modified r and q for the solid phase to account for a decreased flexibility of larger molecules in crystals by assuming the interaction unit to be proportional to half the size of the paraffin molecule. For the carbon number range of n-alkanes between 15 and 30, an interaction unit with 10 of the “original” methylene units was chosen, and the modified r and q values were obtained by dividing the original UNIQUAC r and q expressions by the r and q values corresponding to 10 methylene units. For n-alkanes in the solid phase, Coutinho11 provided the following semiempirical equations in terms of carbon number, n

ri ) 0.1ni + 0.0672

(6)

qi ) 0.1ni + 0.1141

(7)

The predictive local composition concept allows an estimation of the interaction energies, λij, used in these models. The pair interaction energies between two identical molecules are estimated from the heat of sublimation of an orthorhombic crystal of the pure component, i.e.

2 λii ) - (∆Hsblmi - RT) Z

(8)

where Z ) 6 is the coordination number for orthorhombic crystals. The heat of sublimation, ∆Hsblm ) ∆Hvap + ∆Hm + ∆Htr, is calculated at the melting temperature of the pure component. Coutinho11 recommended the use of correlations by Lindeloff for ∆Hm and ∆Htr and the Morgan-Kobayashi42 correlation for ∆Hvap. The pair interaction energy between two nonidentical molecules is given by

λij ) λjj

(9)

where j denotes the shorter-chain-length n-alkane in the pair ij. Thus, the model is predictive for the calculation of the phase behavior because it is based only on purecomponent properties. Calculation Algorithm. The predictive UNIQUAC model has been used successfully to predict multiple solid and liquid phases.11 The following multiphase

Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3453

algorithm, proposed by Heidemann and Abdel-Ghani,43 was used for equilibrium calculations. The input quantities in these calculations were the mixture composition, pressure, and temperature. Initially, the maximum number of phases was set to be one more than the number of components, NC. The initial estimates of the phase amount βk for phase k were provided. The iterative procedure involved alternating between the determination of phase distribution β and the reevaluation of fugacity coefficients. The equilibrium constant, Kij, for each component i in each phase j was compared. If the values were within the tolerance limit, the phase of interest was merged with the preceding phase, and the total number of phases, NPH, was reduced by 1. Using the stability criterion44-46 and the tangent plane distance concept,43 the number of phases and the composition of each phase were determined. The following convergence criterion was used in the iterative procedure:43 NC NPH

∑ ∑ i)1 j)1 N

Gij2

PH

NC

< 10-12

(10)

where Gij is a function relating the fugacity of component i in phase j, the reference fugacity of component i, and the minimum tangent plane distance.43 The Kij values were updated using the following relationship:

) ln Kold ln Knew ij ij - Gij

(11)

Upon convergence, the model yielded the phase distribution and the composition of each component in all phases under equilibrium. The composition results were used to compare with the experimental results for solubility and the compositional differences for the wax constituents. The difference between the experimental results and predictions was expressed as the average absolute deviation (AAD) or as the average absolute relative deviation (AARD). For estimation of the liquidus (or saturated) temperature, equilibrium calculations were repeated in steps of 0.1 °C decrease in temperature until a solid phase was predicted. Experimental Section Materials. A sample of paraffin wax (mp 59-62 °C) was obtained from Sigma-Aldrich (Oakville, Ontario, Canada). This same wax sample was also used by Tiwary28 for measurements with wax-C12 and waxC16 mixtures. n-Hexadecane (C16), used as one of the solvents, was of 99+% purity, and it was obtained from Sigma-Aldrich. The other solvent, Norpar13 with 97%+ n-paraffins, was obtained from Imperial Oil Ltd. (Toronto, Ontario, Canada). Samples of pseudobinary mixtures were prepared by dissolving the wax in the two solvents. Gas Chromatograph (GC) Analyses. All GC analyses were performed using a HP6890 GC equipped with a nonpolar fused-silica column (10 m × 0.53 mm × 0.88 µm) and a hydrogen flame ionization detector. The GC was calibrated using the ASTM D2887 extended method, using a C5-C66 hydrocarbon standard SD-SS3E-05 (Separation Systems Inc., Gulf Breeze, FL). The GC analysis of the wax sample is shown in Figure 1. The average molar mass of the wax sample was calculated to be 390.8 kg/kmol, which corresponds to a mean carbon number of 28. The results of GC analysis

Figure 1. Carbon number distribution of Norpar13, the whole wax, and wax-solvent mixtures (on a solvent-free basis). Table 1. Selected Properties and Composition of Norpar13 specific gravity at 15.6 °C vapor pressure at 20 °C (kPa) normal boiling point (°C) freezing/melting point (°C) kinematic viscosity at 25 °C (mm2/s) average molar mass (kg/kmol) composition (mass %) n-C9H20 n-C10H22 n-C11H24 n-C12H26 n-C13H28 n-C14H30 n-C15H32 n-C16H34

0.76 0.01 221-248 0 2.37 187 0.03 0.2 1.8 13.3 51.3 32.8 0.6 0.01

on Norpar13, along with its selected properties, are provided in Table 1. The carbon number distributions for Norpar13 and the wax sample are shown in Figure 1. These results show that Norpar13 is a mixture of n-paraffins ranging from C9 to C16, with C13 and C14 being the two predominant constituents with concentrations of 51.3 and 32.8 mass %, respectively. The average molar mass of Norpar13 is 185.7 kg/kmol, which corresponds to an equivalent carbon number of 13.1. Note that Figure 1 shows no overlap in carbon number distributions for the wax and Norpar13 samples, which was necessary for calculating the composition of C20+ n-alkanes in the liquid phase, on a solvent-free basis, as described below. The GC analyses of the prepared wax-solvent mixtures were also performed. The three prepared samples for this purpose were 18 mass % wax in Norpar13, 10 mass % wax in C16, and 22 mass % wax in C16. From these GC analyses, the solvent-free compositions of C20+ n-alkanes were calculated, which were then normalized for wax constituents, i.e., for the C20-C40 range. The solvent-free wax analyses are compared with that for the whole (or undiluted) wax in Figure 1. Table 2 provides the carbon number distributions for the whole wax and those obtained for wax-solvent mixtures on a solvent-free basis. These results show that, whereas the three analyses for the solvent-free wax are in good agreement, small differences exist between the solventfree and whole wax analyses. The solvent-free analyses do not show any concentrations of n-alkanes beyond C34, whose combined concentration in the wax sample is less

3454 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 Table 2. Results of GC Analysis: Whole and Solvent-Free Wax in Solution (Normalized)

Table 3. Measured WDT, WAT, and PPT for Wax-Norpar13 and Wax-C16 Mixtures

solvent-free wax, normalized (mass %) carbon whole wax 18 mass % 10 mass % 22 mass % number (mass %) wax in Norpar13 wax in C16 wax in C16 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.05 0.29 1.07 2.87 5.09 8.16 13.42 14.47 12.45 12.22 10.70 8.05 6.01 2.17 1.38 0.53 0.41 0.21 0.19 0.11 0.15

0.05 0.32 1.16 3.42 6.48 9.43 15.12 15.07 11.33 10.48 10.38 7.53 5.58 1.90 1.26 0.47

0.10 0.31 1.22 3.67 6.83 9.79 15.60 15.29 11.42 10.40 10.40 7.54 5.50 1.94

0.06 0.33 1.23 3.63 6.70 9.54 15.35 15.07 11.27 10.44 10.27 7.48 5.52 1.90 1.23

than 1.5 mass %. This observation was noted for all GC analyses performed on the wax-solvent mixtures, indicating the inability of the GC technique to detect small concentrations of C34+ n-alkanes in the wax-solvent mixtures. Therefore, for an effective comparison of the solvent-free and whole wax analyses, the whole wax analysis was reestimated by ignoring the small concentrations of C34-C40 in Table 2, followed by normalizing the concentrations of C20-C33. WAT, PPT, and WDT Temperatures. Mixtures of 6, 10, 14, 18, and 22 mass % wax were prepared with n-hexadecane and Norpar13 as solvents. Each waxsolvent mixture, held in a pour-point tube, was maintained at 70 °C for 1 h to allow the dissolution of wax and the removal of any thermal history. All experiments were performed at atmospheric pressure in a temperature-regulated bath equipped with a programmable temperature controller. The procedures used for measuring the WAT, PPT, and WDT of wax-solvent mixtures are described in the following sections. (a) Measurement of WAT. When a wax-solvent mixture is cooled, the heavier paraffins begin to separate as solid crystals once the solubility limit is exceeded. Experimentally, the WAT is the highest temperature at which the first sign of turbidity or haziness occurs when a wax-solvent mixture is cooled gradually. The ASTM D2500-99 method for measuring the CPT or WAT involves its visual determination while cooling the sample under a prescribed schedule. Tiwary28 modified the method slightly by cooling the sample in steps of 1 °C and holding the sample at each constant temperature for 15 min before checking visually for the appearance of wax crystals. For the wax-C12 and waxC16 mixtures, the WAT values obtained from the modified visual method were in agreement with those from other methods (such as differential scanning calorimetry and cross polar microscopy); hence, it was considered to be a reliable method for WAT measurements.28 In the modified visual method used in this study, all mixtures were cooled at a rate of 10 °C/h from 70 to 50 °C. After reaching 50 °C, the temperature was decreased in steps of 1 °C and the sample held at that temperature for 15 min. At each constant temperature, the samples

liquid sampling temperatures wax concn (mass %)

WDT (°C)

WAT (°C)

PPT (°C)

6 10 14 18 22

Wax-Norpar13 Mixtures 26 23 17 30 28 25 34 31 29 37 33 31 39 35 33

6 10 14 18 22

29 33 35 38 41

Wax-C16 Mixtures 25 20 30 27 33 30 35 33 37 35

T1 (°C)

T2 (°C)

21 27 32 34 36

25 30 33 36 38

21 29 32 35 37

26 31 34 37 40

were checked visually for any sign of turbidity. The highest temperature at which the sample showed turbidity was recorded as the WAT. The results are summarized in Table 3, which show WATs ranging from a high of 37 °C (for 22 mass % wax in C16) to a low of 23 °C (for 6 mass % wax in Norpar13). (b) Measurement of PPT. The PPT is defined as the lowest temperature at which any movement of the liquid sample can be observed when the pour-point tube is held horizontally for 5 s. The PPT of a “waxy” crude oil depends on the wax concentration, the crystallization habit of wax, and the shear stability of different wax structures. All PPT measurements were performed by following a slightly modified ASTM D-97 method, which specifies the cooling of samples in steps of 3 °C. Similar to the procedure for WAT measurements, the wax-solvent mixtures were cooled from 70 to 50 °C at a rate of 10 °C/h. Subsequently, the samples were cooled in steps of 1 °C at the same cooling rate of 10 °C/h, and after 30 min is allowed for the sample temperature to reach the bath temperature, the sample “pourability” was checked. As shown in Table 3, the highest measured PPT was 35 °C (for 22 mass % wax in C16), while the lowest PPT was 17 °C (for 6 mass % wax in Norpar13). (c) Measurement of WDT. A standard procedure for measuring the WDT of “waxy” crude oils does not exist; hence, the following procedure was developed. After reaching the lowest PPT of all mixtures (i.e., 17 °C for 6 mass % wax in Norpar13, as mentioned above), the samples were heated in steps of 1 °C. With a gradual increase in the temperature higher than the corresponding PPT, each wax-solvent sample divided into two distinct layers: a clear liquid layer at the top and a solid layer consisting of settled wax particles at the bottom. The samples were stirred to facilitate the contact between the liquid and solid phases under isothermal conditions. The temperature at which the last traces of the solid phase disappeared was recorded as the WDT. In Table 3, the highest WDT is 41 °C (for 22 mass % wax in C16), while the lowest WDT is 26 °C (for 6 mass % wax in Norpar13). (d) Liquid-Phase Samples at Temperatures between PPT and WDT. A procedure was developed for measuring the solubility of wax in solvents at temperatures below the WDT. A total of 20 wax-solvent mixture samples were used for this purpose (two for each of the 10 mixtures identified in Table 3). As mentioned above for measuring the WDT, the heating of wax-solvent mixtures at temperatures higher than

Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3455

Figure 2. Variation of WDT, WAT, and PPT values with wax concentration: (a) wax-C12 mixtures;28 (b) wax-Norpar13 mixtures; (c) wax-C16 mixtures.

its PPT destroyed the gel structure. That is, by increasing the temperature of each wax-solvent mixture above its PPT, the mixture divided into a clear liquid layer at the top and a solid layer consisting of settled wax particles at the bottom. The liquid-solid-phase separation was monitored by observing the height of the liquid layer above the settled solid phase. When the height of the liquid layer did not change with time (monitored typically for about 30 min), equilibrium between the two phases was assumed at that temperature. At this point, a sample of the liquid phase was carefully decanted and saved for GC analysis. The experiment was resumed by continuing to heat all other wax-solvent mixtures in 1 °C steps. Using this method, liquid samples of each wax-solvent mixture were collected at two sampling temperatures between the PPT and the WDT. The two sampling temperatures for each wax-solvent mixture, labeled as T1 and T2, are listed in Table 3. The solubility of wax in the solvent at the sampling temperature was calculated by mass balance from GC analysis of the liquid-phase sample. The mass fractions for components heavier than C19 were normalized to obtain the composition of wax constituents in the liquid phase. Results and Discussion WAT, PPT, and WDT Results. Figure 2 shows the variation of the measured values of WAT, PPT, and WDT with the wax concentration in Norpar13 and C16. Also shown in Figure 2, for comparison, are the previously reported data for wax-C12 and wax-C16 mixtures.28 The two sets of data for WAT, PPT, and WDT of the wax-C16 mixture are in good agreement. The average difference between WDT and WAT for waxNorpar13 and wax-C16 mixtures is 3.2 ( 0.6 °C. All of the results show that WDT > WAT > PPT. Hammami et al.47 had made a similar observation in a review of the wax deposition models.

Figure 3. Comparison of measured WDTs with predictions for TL: (a) wax-C12 mixtures; (b) wax-Norpar13 mixtures; (c) waxC16 mixtures. Table 4. Comparison of the Predicted Liquidus Temperature (TL) with WDT and WAT

wax-C12 mixtures28 wax-Norpar13 mixtures

wax-C16 mixtures

wax concn (mass %)

predicted TL (°C)

TL - WDT (°C)

TL - WAT (°C)

6 10 20 30 6 10 14 18 22 6 10 14 18 22

25.9 29.8 37.1 41.9 28.4 31.6 34.6 37.2 39.4 31.7 35.7 38.6 40.9 42.9

1.9 1.3 1.1 1.4 2.4 1.6 0.6 0.2 0.4 2.7 2.7 3.6 2.9 1.9

3.8 4.1 3.9 5.4 3.6 3.6 4.2 4.4 6.7 5.7 5.6 5.9 5.9

A further discussion of the experimental results for WDT, WAT, and PPT as well as the GC analyses, and their comparison with predictions, is given in the following section. Comparison of the Results with Predictions from the UNIQUAC Model. For all wax-solvent mixtures tested in this study, multiple solid phases were predicted at temperatures at least 10 °C below the corresponding liquidus temperature. However, only one solid phase together with one liquid phase was predicted for all wax-solvent mixtures over the WDT-PPT temperature range considered in this study. (a) Prediction of the Liquidus Temperature. The experimental WDTs and the predicted liquidus temperatures (TL) for the wax-solvent mixtures are compared in Figure 3. The data for wax-C12 mixtures shown in Figure 3a were reported by Tiwary.28 The predictions for TL match satisfactorily the trends shown by the WDT data in all three cases, particularly for wax-C12 and wax-Norpar13 mixtures. The experimental WDT and the predicted TL are listed in Table 4 for all mixtures. The predicted liquidus temperatures for 6-22 mass % wax-Norpar13 mixtures varied from 28.4 to

3456 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004

Figure 4. Comparison of measured and predicted solubilities of the wax in Norpar13.

Figure 5. Comparison of measured and predicted solubilities of the wax in C16.

39.4 °C, and for 6-22 mass % wax-C16 mixtures, they varied from 31.7 to 42.9 °C. For the same wax-solvent ratio, the solvent with a higher molar mass gave a higher WDT as well as a higher TL value. For all waxsolvent mixtures, both WDT and TL values increased with an increase in the wax concentration. From the results given in Table 4, it is apparent that the predicted TL values are higher than both the WDT and WAT values. The differences between TL and WDT were 1.4 ( 0.2 °C for wax-C12 mixtures and 1.0 ( 0.8 °C for wax-Norpar13 mixtures but somewhat larger, 2.8 ( 0.4 °C, for wax-C16 mixtures. In contrast, the TL - WAT differences are much larger in all cases, i.e., 3.9 ( 0.1 °C for wax-C12 mixtures, 4.2 ( 0.5 °C for waxNorpar13 mixtures, and 6.0 ( 0.3 °C for wax-C16 mixtures. That is, the WDTs are closer to the liquidus temperature of all wax-solvent mixtures than the WATs. As noted above, the predictions for TL are closer to the experimental WDT values for wax-solvent mixtures prepared with lighter solvents, C12 and Norpar13, than those with C16. This trend is unexpected because the predictions would be expected to improve as the solvent molar mass approaches that of the wax. Considering that the thermodynamic model used here has been validated in the past with data on lighter liquids, comprising predominantly C8-C13,11,12 its predictive capability for heavier solvents needs further investigation. (b) Wax Solubility at Temperatures below the WDT. As mentioned before, experimental measurements showed a decrease in the amount of the solid phase when the wax-solvent mixtures were heated gradually from PPT to WDT. This indicated that the wax solubility in each solvent increased with an increase in the temperature until reaching the WDT, at which point the mixture became a homogeneous liquid phase. The experimental and predicted values of wax solubility (as mass percent of wax) in the five waxNorpar13 mixtures are plotted in Figure 4. Figure 5 shows similar results for the wax-C16 mixtures. For all mixtures, the wax solubility increased with the temperature until reaching the mixture composition at the liquidus temperature. As shown in Figures 4 and 5, the effect of a change in the temperature on the wax solubility below the liquidus temperature is more pro-

nounced for wax-solvent mixtures with higher wax concentrations. Table 5 summarizes the solubility results for all three wax-solvent mixtures. The predicted solubilities for a 20 mass % wax-C12 mixture are higher than the data28 with an AARD of 5.2 ( 2.1%. For wax-Norpar13 mixtures, the AARD is 2.6 ( 2.4%, which indicates a better agreement between the experimental and predicted wax solubilities. However, the AARD of 9.7 ( 6.4% for the wax-C16 mixtures is larger than the AARDs for the other two lighter solvents. It is pointed out that these AARDs between experimental and predicted wax solubilities follow the trend for the TL WDT differences in Table 4. That is, the deviations are the least for wax-Norpar13 mixtures and the largest for wax-C16 mixtures. (c) Liquid-Phase Composition at Temperatures below the WDT. As mentioned previously, GC analyses were performed on liquid-phase samples collected for each wax-solvent mixture listed in Table 4. The mass fraction data for C20+ constituents were compared with the average solvent-free wax analysis from Table 2. For the 20 mass % wax-C12 mixtures,28 the liquid-phase analyses for C20+ (on a solvent-free basis) at 32 and 35 °C are compared with the predicted compositions in Figure 6. Note that both of these temperatures are below the WDT of 36 °C (and the predicted TL of 37.1 °C) for a 20 mass % wax-C12 mixture. With a decrease in the temperature from 35 to 32 °C, both the data and predictions show a shift in the carbon number distribution to the left, which indicates that heavier n-alkanes have relatively lower solubilities at lower temperatures. For a more effective comparison of carbon number distributions, the results for wax-C16 mixtures were expressed on a differential basis as follows. The change in concentrations for each n-alkane was calculated by subtracting the (normalized) liquid-phase concentration from that in the whole wax. These changes in mass percent values for all n-alkanes from C20 to C33 are plotted in Figure 7 as a function of the carbon number. The experimental data (shown by symbols) for all waxC16 mixtures at both temperatures show negative differences for C20-C27 and positive differences for C28C33. The minimum and maximum differences occur at carbon numbers of about 26 and 29, respectively. Also, as expected, the differences decrease in magnitude at

Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3457 Table 5. Comparison of Experimental Results with Predictions for Wax Solubility and C20+ Composition in the Liquid Phase at Temperatures below WDT mixture composition (mass % wax) 20

6 10 14 18 22

6 10 14 18 22

temp (°C)

wax solubility (mass %) preexptl dicted

solubility deviationa (%)

AADb for composition (mass %)

Wax-C12 Mixtures (Data of Tiwary28) 32 16.6 18.0 -8.5 33 17.5 18.7 -6.9 34 18.4 19.3 -4.7 35 18.6 19.6 -5.6 36c 20.0 19.9 0.5

1.0 ( 0.2 1.0 ( 0.1 0.6 ( 0.1 0.3 ( 0.1 0.7 ( 0.2

Wax-Norpar13 Mixtures 21 5.8 5.3 8.6 25 6.4 6.0 6.3 26c 6.0 6.0 0.2 27 8.9 9.4 -5.6 30 10.0 9.9 1.0 32 13.1 13.7 -4.6 33 13.8 13.9 -0.7 34c 14.0 14.0 0.2 34 16.8 17.3 -3.0 36 17.7 17.9 -1.1 37c 18.0 18.0 0.1 36 20.0 20.9 -4.5 38 21.9 21.8 0.5 39c 22.0 22.0 0.2

0.6 ( 0.1 0.7 ( 0.1 0.8 ( 0.2 0.9 ( 0.2 0.6 ( 0.2 0.9 ( 0.2 0.6 ( 0.2 0.8 ( 0.2 0.8 ( 0.2 0.7 ( 0.2 0.8 ( 0.2 0.9 ( 0.2 0.7 ( 0.2 0.8 ( 0.2

Wax-C16 Mixtures 4.1 2.1 5.4 4.6 6.0 5.8 8.0 7.3 9.3 8.6 10.0 9.6 12.0 10.5 12.9 12.2 14.0 13.0 15.7 14.4 17.0 16.4 18.0 17.1 19.9 17.8 21.6 20.9 22.0 21.6

0.7 ( 0.1 0.6 ( 0.2 0.6 ( 0.2 0.6 ( 0.2 0.5 ( 0.2 0.6 ( 0.2 0.7 ( 0.2 0.7 ( 0.2 0.5 ( 0.2 0.8 ( 0.2 0.9 ( 0.2 0.6 ( 0.2 0.8 ( 0.2 0.9 ( 0.2 0.7 ( 0.2

21 26 29c 29 31 33c 32 34 35c 35 37 38c 37 40 41c

48.8 14.8 4.0 8.8 7.5 3.7 12.5 5.4 7.5 8.3 3.5 4.8 10.6 3.2 2.0

a Solubility deviation ) 100(expt - pred)/expt. b AAD ≡ (100/ 34 15)Σi)20 |expti - predi|. c Sample at the WDT.

Figure 6. Data28 and predictions for the shifting of the carbon number distribution in the liquid phase for 20 mass % wax-C12 mixtures at 32 and 35 °C.

the higher temperature as the wax-C16 mixture approaches the corresponding WDT. In Figure 7a, for a 6 mass % wax-C16 mixture, the differences are relatively larger at 21 °C because this temperature is 5 °C below

Figure 7. Data and predictions for the change in liquid-phase concentrations of C20-C33 in wax-C16 mixtures at temperatures below the WDT: (a) 6 mass % wax; (b) 10 mass % wax; (c) 14 mass % wax; (d) 18 mass % wax; (e) 22 mass % wax.

the WDT, compared to temperatures of only about 2-3 °C below the WDT for all other cases. The predicted changes in compositions for C20-C33 in Figure 7 are in satisfactory agreement with the trends shown by the data; they show negative differences for carbon numbers from 20 to 27 and positive differences for carbon numbers from 29 to 33. The experimental and predicted difference is approximately zero for the carbon number of 28, which as indicated by the whole wax analysis in Table 2 is also its mean carbon number. In Figure 7, the predicted minimum difference is for C26, but the maximum difference is predicted for C32, as opposed to C29 shown by the data. Moreover, the magnitudes of the predicted minimum and maximum differences in Figure 7 are smaller than those indicated by the experimental data. Although not presented here, a similar analysis of the results for all wax-Norpar13 mixtures showed similar trends, including the predicted minimum and maximum differences being less than those for the experimental data. These small discrepancies between the data and predictions, particularly for carbon numbers larger than 28 in wax-C16 mixtures, could be attributed to either experimental errors in GC analyses or the model being inadequate for heavier solvents. The last column in Table 5 lists the AAD in the composition for all carbon numbers from C20 to C33. The AADs range from 0.3 to 1.0 mass %, with a standard deviation of about 0.1-0.2 mass %. Overall, the UNIQUAC-based model provides satisfactory predictions for carbon number distributions, and their shift

3458 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004

Figure 8. Predictions for the change in liquid-phase concentrations of C20+ at the measured WAT of wax-solvent mixtures: (a) wax-Norpar13 mixtures; (b) wax-C16 mixtures.

with temperature, for C20-C33 in all three wax-solvent mixtures considered in this study. (d) Predictions for Compositional Changes of C20-C33 at the WAT. Additional calculations were undertaken to study the extent of predicted compositional changes for C20+ constituents in the liquid phase as the mixture temperature is lowered from TL to WAT. The predicted results for wax-Norpar13 and wax-C16 mixtures at 6-22 mass % wax concentrations are presented in Figure 8. The results in Table 4 indicated the WAT to be lower than the corresponding TL by 4.2 ( 0.5 °C for wax-Norpar13 mixtures and by 6.0 ( 0.3 °C for wax-C16 mixtures. In Figure 8a, the changes in composition for wax-Norpar13 mixtures are dependent on the wax concentration in the mixture; the lower the wax concentration, the larger the change. The minimum and maximum changes are about -1 and +1%, respectively, for the 6 mass % mixture. However, as shown in Figure 8b, the changes are much larger for wax-C16 mixtures; the minimum and maximum changes for the 6 mass % mixture are about -2.5 and +2.5%, respectively. In Figure 8b, the predicted curves for 10, 14, 18, and 22 mass % are not much different. Sensitivity of TL Predictions to Structural Parameters q and r. The results in Table 4 indicate all values of the predicted TL to be higher than the corresponding WDT values. Between the three waxsolvent mixtures, the predictions for TL were higher than those for the WDT by 1-2 °C for the wax-C12 and wax-Norpar13 mixtures but by 2-4 °C for the waxC16 mixtures. The following sensitivity analysis was undertaken to possibly obtain “optimum” values of q and r for improving the agreement between the predicted TL and the experimental WDT. As mentioned previously, eqs 6 and 7 were proposed by Coutinho11 by empirically modifying the original UNIQUAC equations for q and r that are estimated by the group contribution method. Effectively, this modification yielded multiplication factors of 0.185 and 0.148 for the original (group contribution method) values of q and r, respectively. Following the same approach, Singh et al.12 used 16 methylene units instead for their “heavier” wax sample, which yielded multiplication factors of 0.116 and 0.0927 for the original q and r, respectively, or one multiplication factor of 0.625 for both q and r in eqs 6 and 7.

Figure 9. Sensitivity of the predicted liquidus temperature (TL) to variations in multiplier R for the modified11 UNIQUAC structural parameters q and r: (a) wax-C12 mixtures; (b) waxNorpar13 mixtures; (c) wax-C16 mixtures.

All of the predicted results in Figures 3-8 and Tables 4 and 5 were obtained by the use of q and r that were obtained from eqs 6 and 7. With the mean carbon number of 28 for the wax sample used in this study, an interaction unit comprising 14 methylene units would have yielded multiplication factors of 0.132 and 0.106 for the original q and r, respectively, or one multiplication factor of 0.714 for q and r in eqs 6 and 7. This sensitivity analysis was focused on the effect of a change in q and r on the predicted TL. Preliminary calculations indicated that the predicted TL was more sensitive to changes in q than to changes in r. For simplicity, however, only one multiplier, R, was selected for both q and r in eqs 6 and 7. Note that the use of R ) 5.40 would revert q to its original UNIQUAC value; likewise, R ) 6.744 would make r equal to its original value. For each wax-solvent mixture, the TL value corresponding to R ) 1 was subtracted from the predicted TL for different values of multiplier R. Figure 9 presents the predictions of (TL)R - (TL)R)1 for 6, 10, and 18 mass % wax in all three wax-solvent mixtures. The results show an interesting and complex relationship between three quantities. In addition to the multiplier R, the solvent type and the wax concentration also affect the predicted TL. In all cases, the predicted TL actually increases when R < 1, which is not desirable when attempting to decrease the predicted TL to match the experimental WDT. On the other hand, R > 1 could possibly yield the desired result; however, selecting a value of R > 1 implies that the number of methylene units in the modification by Coutinho11 would be less than 10. For wax-C12 mixtures in Figure 9a, R > 1 resulted inasmuch as a 1 °C decrease in the predicted TL but only for lower wax concentrations of 6 and 10 mass %. That is, R > 1 did not affect the predicted TL for the 18

Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3459

mass % wax-C12 mixture. Similarly, in Figure 9b, for wax-Norpar13 mixtures, R > 1 resulted inasmuch as a 1 °C decrease in the predicted TL but only for the lowest wax concentration of 6 mass %. That is, R > 1 did not affect the TL value for the 10 and 18 mass % wax-Norpar13 mixtures. The results for wax-C16 mixtures in Figure 9c indicate that R > 1 did not affect the predicted TL for the wax concentration range of 6-18 mass % wax-C16 mixtures. That is, a decrease in the predicted value of TL for the wax-C16 mixtures could not be achieved by increasing or decreasing the values of q and r. On the basis of the results presented in Figure 9, it was concluded that no significant benefit would be realized by altering eqs 6 and 7 for the structural parameters q and r. Comparison of the WDT and WAT with the Liquidus Temperature. All of the various temperatures used for characterizing “waxy” crude oils, such as CPT, WAT, and PPT, have their significance in crude oil production and transportation operations. Numerous studies have reported these values for “waxy” crude oils from many petroleum reservoirs worldwide. However, measurements for these temperatures for “waxy” crude oils are made while cooling the sample. Their values depend significantly on the cooling rate as well as the shearing conditions used during the measurement, particularly for the case of PPT;28 hence, these temperatures may not have unique values for the same “waxy” crude oil. Previous studies on the prediction of the liquid-solidphase transformation temperature of wax-solvent mixtures or “waxy” crude oils have used the WAT value for assessing the accuracy of thermodynamic predictions. However, the results of this investigation under relatively slow cooling rates have shown that, because of the less severe effects of superheating (when heating) than supercooling (when cooling), the WDT measurement would offer a superior approximation of the liquid-solid-phase change temperature, TL, than the WAT measurement. That is, the WDT rather than the WAT is closer to the true liquidus or saturation temperature, which is the lowest temperature for a waxsolvent mixture to exist as a single liquid phase. In this context, the observed overpredictions of the liquidsolid-phase change temperature in previous studies, such as those by Hansen et al.6 and Won,33 may be attributed partially to the use of experimental WATs for model validation. Furthermore, recent studies have highlighted the difference between wax precipitation and the process of wax deposition. It is pointed out that wax deposition would occur only when paraffinic mixtures, such as “waxy” crude oils, are exposed to a cooling environment, for which WAT would be the temperature of importance. Recently, Bidmus and Mehrotra48 showed the wax deposition process to be controlled primarily by heat transfer, which requires the existence of a thermal driving force and a surface temperature below the WAT. On the other hand, the WDT represents a more appropriate measure of the liquidus temperature (TL) of wax-solvent mixtures, i.e., the highest equilibrium temperature for the liquid and solid phases to coexist. Conclusions This study investigated the liquid-solid-phase transformation of wax-solvent mixtures containing 6-22 mass % wax. The multicomponent wax sample used in

this study comprised n-alkanes from C20 to C40, with a mean carbon number of 28. The two solvents used were C16 and Norpar13 (a mixture of C9-C16). In addition to measuring the WAT and the PPT of wax-solvent mixtures, a procedure was developed for the WDT that was recorded while gradually heating the mixture. The measured WAT values were lower than the WDT values by >3 °C. Below the WDT, the wax solubility in all three solvents was found to decrease with decreasing temperature. In addition, at temperatures below the WDT, the liquid-phase composition, when compared to that of the whole wax, showed an increase in the concentrations of lighter n-alkanes and a decrease in the concentrations of heavier n-alkanes. Experimental results were validated with equilibrium calculations from a UNIQUAC-based model, which gave satisfactory predictions for the liquid-solid-phase transformation temperature as well as the compositional changes in the liquid phase at temperatures below the WDT. The predictions for wax-C12 and wax-Norpar13 mixtures were closer to the data than those for the case of wax-C16 mixtures. The predicted liquidus temperatures for wax-C12 and wax-Norpar13 mixtures were higher than the WDT by about 1-2 °C but by about 3-4 °C for wax-C16 mixtures. A sensitivity analysis for the UNIQUAC structural parameters q and r showed that the predicted liquidus temperature depended not only on the number of methylene units used for defining the solid-phase interaction unit but also on the mixture composition and the molar mass of the diluent used for preparing wax-solvent mixtures. The measurements and predictions presented in this study suggested that the WDT, as opposed to the WAT, is a more appropriate measure of the liquidus or saturation temperature of wax-solvent mixtures or “waxy” crude oils. Acknowledgment Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Department of Chemical and Petroleum Engineering, University of Calgary, is gratefully acknowledged. We thank Ms. Elizabeth Zalewski of the In Situ Combustion Group at the University of Calgary for GC analyses. Nomenclature Cn ) n-alkane (≡normal CnH2n+2) g ) Gibbs free energy, J/kmol Gij ) minimization function used in eq 10 H ) molar enthalpy, J/kmol K ) equilibrium constant n ) carbon number NC ) number of components NPH ) number of phases q ) UNIQUAC structural parameter r ) UNIQUAC structural parameter R ) universal gas constant, 8314 J/kmol‚K s ) mole fraction in the solid phase T ) temperature, K TL ) liquidus (or saturation) temperature, °C V ) molar volume, m3/kmol Vw ) van der Waals volume, m3/kmol x ) mole fraction in the liquid phase Z ) coordination number

3460 Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 Greek Letters R ) empirical multiplier for parameters q and r (eqs 6 and 7) β ) phase mole fraction Φ ) parameter in eq 4 φ ) parameter in eq 2 γ ) activity coefficient λ ) pair interaction energy, J/kmol θ ) parameter in eq 4 Subscripts i ) ith component j ) jth phase m ) melting sblm ) sublimation tr ) solid-phase transition vap ) vaporization Superscripts E ) excess property l ) liquid s ) solid Acronyms AAD ) average absolute deviation (for composition) [≡(100/ 34 15)Σi)20 |expti - predi|], mass % AARD ) average absolute relative deviation [≡(100/m) m Σi)1 |expti - predi|/expti], % CPT ) cloud-point temperature, °C GC ) gas chromatograph PPT ) pour-point temperature, °C WAT ) wax appearance temperature, °C WDT ) wax disappearance temperature, °C

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Received for review January 6, 2004 Revised manuscript received April 7, 2004 Accepted April 16, 2004 IE0400144