Gelation and Breakage Behavior of Model WaxâOil Systems

Article pubs.acs.org/IECR

Gelation and Breakage Behavior of Model Wax−Oil Systems: Rheological Properties and Model Development Yansong Zhao,* Lalit Kumar, Kristofer Paso, Hassan Ali, Jamilia Safieva, and Johan Sjöblom Ugelstad Laboratory, Department of Chemical Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway ABSTRACT: Gelation and breakage of model wax−oil systems is investigated using microscopy, densitometry, rheometry, and XRD. Various model waxy oils are prepared, including 1 to 20 wt % (w/w) macrocrystalline wax in dodecane, 1 to 20 wt % microcrystalline wax in dodecane, and 5 wt % macrocrystalline wax in Primol 352. The influence of shear history, thermal history, and fluid composition is ascertained. A novel gel breakage model is introduced which spans the entire mechanical response of wax gels from initial Hookean behavior at low strains to equilibrium slurry flow at the infinite strain limit. The local rheology model will benefit shut-in and restart processes of waxy crude pipelines via application of standard fluid mechanics relationships.

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INTRODUCTION Crude oil is a complex mixture of hydrocarbons, containing saturates, aromatics, naphthenes, asphaltenes, and resins.1 Shutin and restart of waxy crude transport pipelines is an important consideration for crude oil flow assurance strategies. Restart models have been developed by several scientists.2−5 These flow models are vital to designing and managing shut-in and restart operations of waxy crude oil pipelines. During flow shutin processes in low temperature environments, paraffins may form strong physical gels, which require applied pressure to rupture and allow flow commencement. Phillips et al. state novel analysis approaches to waxy crude oil shut-in and restart and presented excellent experimental results as well as promising numerical simulations.6,7 Crude oil gelation and breakage are central themes in waxy crude oil transportation strategies. Operational and emergency outages in transport pipelines often result in cooling of the produced fluid to temperatures below the pour point temperature (PPT). Precipitation of paraffin wax at the low temperature conditions leads to the formation of strong solid−liquid gels in which a volume-spanning network of crystal−crystal interactions entraps the remaining liquid phase among the crystals. Gelation mechanisms have been investigated by several researchers using experimental and modeling methods.1,8−10 Differential scanning calorimetry, rheometry, and microscopy are common methods used for performing gelation research. It is reported that shear history, thermal history, wax content, and wax type influence the gelation process. In addition to gel formation processes, gel breakage mechanisms are also important elements of flow modeling strategies for waxy petroleum fluids. Davenport and Somper assume first order Bingham breakdown kinetics and apply the model to the restart of a laboratory-scale pipeline.11,12 Rheological behavior of wax−oil gels is often ascribed to the following dependencies: τ = τ(λ , γ )̇

fraction of intact crystal−crystal linkages remaining during the gel breakage process. De Kee et al. report a time-dependent mathematical rate of gel breakage:13 dλ = −aγ ḃ (λ − λe)n dt

where λ ranges from 0 to 1, λe is the equilibrium value of intact crystal−crystal bonds, corresponding to the slurry flow state, γ̇ denotes the shear rate, and a, b, and n are degradation rate constants. Paso et al. established a third order gel breakage model and applied the model to macrocrystalline paraffin wax−oil gels, crude oil gels, and crude oil emulsion gels. Experimental results showed excellent agreement with the third order model at moderate strain values.11,14

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EXPERIMENTAL SECTION Materials. The chemicals used in this work were obtained from Sigma-Aldrich Co. LLC., Sasol Wax Co., and Exxon Mobil Co. Primol 352 is a high viscosity purified mixture of liquid saturated hydrocarbons. The properties of Primol 352 are shown in Table 1.15 The purity of dodecane (CAS Registry Number 112-40-3) is ≥0.990. Macrocrystalline wax (Sasolwax 5405), microcrystalline wax (Sasolwax 3971), and crude oil (internal code UL-YS1) are used in this work. Macrocrystalline wax consists primarily of linear paraffin components.11 Microcrystalline wax consists of saturated hydrocarbons and has a significantly elevated content of iso- and cycloalkanes.16 Sample Preparations. Samples of 1 to 20 wt % (w/w) macrocrystalline wax in dodecane and 1 to 20 wt % microcrystalline wax in dodecane are prepared. In order to investigate the influence of solvent type, 5 wt % macrocrystalReceived: Revised: Accepted: Published:

(1)

where τ, γ̇, and λ represent shear stress, shear rate, and a structural parameter, respectively.11 λ is often equated to the © 2012 American Chemical Society

(2)

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February 9, 2012 May 17, 2012 May 22, 2012 May 22, 2012 dx.doi.org/10.1021/ie300351j | Ind. Eng. Chem. Res. 2012, 51, 8123−8133

Industrial & Engineering Chemistry Research

Article

Subsequently, the sample is cooled quiescently to 4 °C at a cooling rate of 1 °C/min. The temperature is maintained at 4 °C for 10 min under quiescent conditions. Subsequently, the formed gel is broken at a constant shear rate of 0.1 s−1. Similar experiments are performed at shear rates ranging from 0.001 to 10 s−1. b. Shut-in Flow Investigation Protocol. The protocol is similar to protocol a except that a constant shear stress is imposed on the sample during the gel formation process, in order to emulate pipeline fluid flows during the shut-in process. Experiments are performed at various imposed shear stresses during the gelation process from 0 to 40 Pa. c. Thermal History Investigation Protocol. The protocol is similar to protocol a. Experiments are performed at various cooling rates ranging from 0.2 to 20 °C/min. d. Composition Investigation Protocol. The protocol for macrocrystalline wax in dodecane and microcrystalline wax in dodecane is identical to protocol a. The experimental protocol for 5 wt % macrocrystalline wax in Primol 352 is as follows: The sample is cooled quickly to 45 °C at a cooling rate of 20 °C/min. Subsequently, the sample is cooled to 4 °C at a cooling rate of 1 °C/min. Finally, the temperature is maintained at 4 °C for 10 min to ensure solid−liquid equilibrium before breakage is applied. e. Wax Content Investigation Protocol. The protocol is similar to protocol a. Rheological experiments are performed with solutions of 4 to 10 wt % macrocrystalline wax in dodecane. f. High Strain Investigation Protocol. Rheological experiments are performed using the concentric cylinder geometry. The thermal protocol is similar to protocol a. Shear rates range from 10 to 1000 s−1. g. Crude Oil Protocol. Rheological experiments are performed using the cone-and-plate geometry. The thermal protocol is similar to protocol a. The crude oil gel is broken at a constant shear rate of 0.1 s−1. Microscopy. Crystal structures are investigated at cooling rates ranging from 1 to 20 °C/min. Micrographs are obtained using a camera system attached to the microscope.1,16 The microscope protocol is as follows: A sample of model waxy oil is loaded onto a glass slide. Prior to loading, the sample is heated to a temperature above the WAT to avoid gel formation. The sample is maintained at 50 °C for 5 min. Subsequently, the

Table 1. Properties of Primol 35215 properties

test value

appearance kinematic viscosity (40 °C, mm2/s) kinematic viscosity (100 °C, mm2/s) dynamic viscosity (20 °C, mPa·s) density (20 °C, kg/m3) pour point (°C) flash point (°C) hydrocarbon (with 25 carbons and less) (wt %) carbon type (paraffinic/naphthenic/aromatic) (wt %)

clear and bright 65.0−75.0 8.5 165−220 860−870 −12 240 ≤5 66/34/0

line wax in Primol 352 is prepared as a complement to the 5% macrocrystalline/dodecane solution. Density. Densities of macrocrystalline wax in dodecane and microcrystalline wax in dodecane are measured using an Anton Paar DMA 5000 (Austria). Measurement temperatures range from 293.15 to 343.15 K, at 10 K intervals. Density measurements are calibrated with ultrapure water and dry air. Rheological Experiments. Rheometric experiments are performed using an Anton Paar Physica 301 instrument equipped with a 2° cone-and-plate geometry of 4 cm diameter. The contact surface of the cone was previously roughened by sandblasting to reduce slippage effects.11 Anton Paar Physica 301 is nominally a controlled stress instrument. However, the rheometer has a computerized feedback mechanism to provide for accurate controlled shear rate measurements. Hence, in practice, the rheometer is a dual-mode instrument, providing both controlled stress measurements and controlled shear rate measurements. Effects of shear history, thermal history, wax content, and wax type on the gelation and breakage of model wax−oil systems are investigated. In order to investigate rheological properties of model waxy oils at high shear strain conditions, a concentric cylinder geometry with an inner radius of 13.33 mm and an inner cylinder length of 40 mm is used. The following rheological protocols are used. a. Shear Rate Investigation Protocol. A waxy oil sample is loaded between the cone and the plate. Prior to loading, the sample is heated to a temperature of at least 20 °C above the wax appearance temperature (WAT) to ensure complete melting of wax nucleates and fragments. The sample is cooled quiescently to 40 °C at a cooling rate of 20 °C/min.

Table 2. Density (ρ) of Macrocrystalline Wax in Dodecane and Microcrystalline Wax in Dodecane for T = 293.15−343.15 K ρ (g·cm−3) T (K)

0 wt %

1 wt %

5 wt %

10 wt %

20 wt %

0.751 808 0.744 493 0.737 252 0.729 993 0.722 697 0.715 823

0.756 226 0.748 483 0.742 083 0.735 425 0.727 243 0.719 959

0.757 584 0.750 820 0.745 670 0.739 259 0.732 120 0.724 950

0.753 964 0.744 281 0.736 523 0.728 099 0.722 377 0.714 022

0.754 898 0.747 974 0.737 753 0.732 395 0.724 802 0.715 513

0.756 167 0.748 631 0.742 185 0.734 513 0.727 564 0.716 921

Sasolwax 5405 293.15 303.15 313.15 323.15 333.15 343.15

0.748 760 0.741 510 0.734 224 0.726 910 0.719 560 0.712 169

0.749 333 0.742 092 0.734 821 0.727 520 0.720 185 0.712 801

293.15 303.15 313.15 323.15 333.15 343.15

0.748 760 0.741 510 0.734 224 0.726 910 0.719 560 0.712 169

0.749 732 0.742 055 0.735 130 0.727 798 0.720 461 0.713 079

Sasolwax 3971

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sample is cooled to 20 °C at a specific cooling rate ranging from 1 to 20 °C/min. Finally, the temperature is maintained at 20 °C for 5 min and the micrographs are obtained. X-ray Diffraction. Powder diffraction diagrams are recorded using a diffractometer with Bragg−Brentano geometry, D8 FOCUS (Bruker, Germany), utilizing Cu Kα radiation. The wax−oil gel quantity is approximate 5 mg. Diffraction diagrams are recorded with a step of 0.008° at room temperature, in 2θ ranging from 5 to 50°.

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RESULTS AND DISCUSSION Density of Model Wax−Oil. Measured densities of various model waxy oils are listed in Table 2 and are regressed by the equations as follows:17 ρ = A 0 + A1T + A 2 T 2

(3)

n

ADD =

1 ∑ |ρ − ρi ,calcd | n i = 1 i ,exptl

Figure 1. Repeated experiments of 5 wt % macrocrystalline wax in dodecane at a cooling rate of 1 °C/min and a shear rate of 0.1 s−1.

(4)

where ρ denotes density, A0, A1, and A2 are quadratic fitting parameters with the least-squares method, and ADD is the average absolute deviation between calculated and experimental results. Densities of 1 to 20 wt % macrocrystalline wax in dodecane and microcrystalline wax in dodecane are fitted by eq 3, and ADD can be calculated by eq 4. Values of A0, A1, A2, and ADD are listed in Table 3. Thermal density dependencies are

1 °C/min. The gels are broken at a shear rate of 0.1 s−1. The samples used in the study are as follows: 1, 5, 10, and 20 wt % macrocrystalline wax in dodecane; 1, 5, 10, and 20 wt % microcrystalline wax in dodecane; and 5 wt % macrocrystalline wax in Primol 352. Experimental results of 5 wt % macrocrystalline wax in Primol 352, 5 wt % macrocrystalline wax in dodecane, and 5 wt % microcrystalline wax in dodecane are shown in Figure 2. The yield stress is defined as the

Table 3. Parameters of eq 4 and AADs for Density Correlation of Macrocrystalline Wax in Dodecane and Microcrystalline Wax in Dodecane wt %

A0

0 1 5 10 20

0.9460 0.9460 0.9990 0.8799 0.7669

0 1 5 10 20

0.9460 0.9604 1.2486 0.9590 0.6642

A1 × 104 Sasolwax 5405 −6.2258 −6.2019 −9.4718 −1.7003 4.89856 Sasolwax 3971 −6.2258 −7.1000 −24.6000 −6.2738 12.3000

A2 × 107

AAD × 100

−1.7161 −1.7339 3.5446 −8.6303 −17.8321

0.0016 0.0040 0.0072 0.0028 0.0278

−1.7161 −0.3089 26.4000 −2.3346 −31.3339

0.0016 0.0104 0.0834 0.0763 0.0505

similar to those from the work of Lee et al.17 ADDs listed in Table 3 are between 0.0016 and 0.0834%, demonstrating that the densities at any other temperature between 293.15 and 343.15 K can be predicted accurately by the values of A0, A1, and A2. Repeatability and Accuracy of Rheometer. In order to confirm the repeatability and accuracy of the rheometric measurements, experiments are performed at a shear rate of 0.1 s−1, using the protocol for shear rate investigation. The sample used is 5 wt % macrocrystalline wax in dodecane. Results are shown in Figure 1, from which it is ascertained that the curves of the two repeated experiments are very close, illustrating that the repeatability and accuracy are very good. Effect of Wax and Solvent Type. Two kinds of wax (Sasolwax 5405 and Sasolwax 3971) and two kinds of solvents (dodecane and Primol 352) are used in this work. Rheological experiments are performed with gels formed at a cooling rate of

Figure 2. Comparison of rheological properties of various model wax− oil systems at a cooling rate of 1 °C/min and a shear rate of 0.1 s−1.

maximum in the stress versus strain curve. The yield stress of 5 wt % macrocrystalline wax in dodecane is larger than that of 5 wt % microcrystalline wax in dodecane, demonstrating that branched and cyclic alkanes reduce gel strength. It is shown that gels of 5 wt % macrocrystalline wax in dodecane and 5 wt % macrocrystalline wax in Primol 352 have dissimilar yield stresses, manifesting that rheological properties also depend on solvent type. Effect of Wax Content. Samples used consist of 4 to 10 wt % macrocrystalline wax in dodecane. Experimental results are shown in Figure 3, plotting yield stress as a function of total wax content. Figure 3 illustrates that yield stress increases with 8125



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Figure 3. Effect of macrocrystalline wax content in dodecane on yield stress at a cooling rate of 1 °C/min and a shear rate of 0.1 s−1.

increasing wax content. Venkatesan et al. correlate yield stress (τy) as a function of solid fraction (ω) as follows:18 τy ∝ ωn

(5)

In general, yield stress increases with increasing solid content. The present experiments confirm that yield stress increases with increasing total wax content. However, the relationship between yield stress and wax content is also highly dependent on the type of wax as well as solvent quality. Effect of Shear Rate. The effect of shear rate on model wax−oil gel breakage is investigated at shear rates ranging from 0.001 to 10 s−1. Results are shown in Figure 4a. Yield stress values obtained at shear rates of 0.1, 0.01, and 0.001 s−1 are nearly identical. Higher maximum stress values are observed at shear rates of 1 and 10 s−1. The measured rheological behavior is similar to that in the work of Paso et al.11 In order to investigate high shear rate rheology, experiments are performed using a concentric cylinder geometry at shear rates ranging from 10 to 1000 s−1 with 5 wt % macrocrystalline wax in dodecane and 5 wt % macrocrystalline wax in Primol 352. As shown in Figure 4b, shear stress increases with increasing shear rate ranging from 10 to 1000 s−1 in samples of 5 wt % macrocrystalline wax dodecane and 5 wt % macrocrystalline wax in Primol 352. Upper horizontal portions of data curves in Figure 4b reflect the upper torque limit of the rheometer. Therefore, temporary shear rate deviations are evident for the most viscous model fluids during the initial breakage process. Effect of Shearing during Gelation. The influence of imposed shear stresses during the gel formation process is investigated with the solution containing 5 wt % macrocrystalline wax in dodecane, and the formed gels are broken at a shear rate of 0.1 s−1. Imposed shear stresses during gelation range from 0 to 5 Pa. Results are shown in Figure 5a,b. The gel strength is reduced when shearing is applied during the entire gel formation process. Imposed shear stresses (during the gelation process) up to 1 Pa result in a small reduction in the resultant gel strength. Gel strength is rapidly reduced when the imposed shear stress (during gelation) increases from 1 to 2 Pa. Yield stress trends measured in this work do not indicate obvious mass transfer limitations in the crystallization rate. In order to confirm the results for more viscous gels, 5 wt % macrocrystalline wax in Primol 352 is prepared. Gel strength

Figure 4. Effect of shear rate on model wax−oil gel breakage. (a) Experiments of 5 wt % macrocrystalline wax in dodecane at shear rates from 0.001 to 10 s−1. (b) Experiments of 5 wt % macrocrystalline wax in dodecane and 5 wt % macrocrystalline wax in Primol 352 at shear rates from 10 to 1000 s−1.

values are plotted against strain, and the results are shown in Figure 5c,d. A continuous reduction in gel strength is observed with increasing values of imposed stress during gelation, demonstrating the same overall trend as for experiments performed with 5 wt % macrocrystalline wax in dodecane. When shear stress values greater than 10 Pa are imposed on the model fluid during crystallization, there is no yield stress observed experimentally during the constant shear rate portion of the measurement protocol. The rheological response is almost entirely independent of absolute strain. Strainindependent behavior indicates a cold slurry state of the waxy fluid. Measured stress values confirming the slurry state are plotted against strain in Figure 5e. The experiments are repeated (0−8 Pa imposed stress) to confirm the mechanism in the presence of shearing flows during the gelation process of 5 wt % macrocrystalline wax in Primol 352. The repeated experiments show excellent agreement with the plots in Figure 5c, providing additional confirmation of the mechanism. Effect of Cooling Rate. Thermal history conditions have a large influence on crystal morphology and the resultant gel strength. Gel breakage experiments are performed with samples 8126



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Figure 5. Effect of imposed shear stress on gel breakage. (a) Experiments of 5 wt % macrocrystalline wax in dodecane at imposed shear stresses from 0 to 1 Pa. (b) Yield stress of 5 wt % macrocrystalline wax in dodecane at imposed shear stresses from 0 to 5 Pa (repeated one time). (c) Experiments of 5 wt % macrocrystalline wax in Primol 352 at imposed shear stresses from 0 to 8 Pa. (d) Yield stress of 5 wt % macrocrystalline wax in dodecane at imposed shear stresses from 0 to 40 Pa. (e) Experiments of 5 wt % macrocrystalline wax in Primol 352 at imposed shear stresses from 10 to 14 Pa.

dodecane decreases at higher cooling rates. The gels formed at lower cooling rates are stronger than the gels formed at higher cooling rates. Lower cooling rates yield larger crystals with a lower number density, increasing the overall strength of the gel. The effect of thermal history is also investigated for the gel formed from 5 wt % macrocrystalline wax in Primol 352 at nonquiescent conditions. The formed gels are broken at a shear

formed at various thermal history conditions, and the formed gels are broken at a shear rate of 0.1 s−1. The solution first used is 5% macrocrystalline wax in dodecane. A shearing flow is imposed on the sample during the crystallization process by applying a constant shear stress of 1 Pa during cooling. The cooling rate range is from 1 to 20 °C/min. Results are shown in Figure 6. The yield stress of 5 wt % macrocrystalline wax in 8127



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ages. Hence, a yield stress corresponds to a complete volumespanning crystal−crystal network. Subsequently, thermal history investigations are performed under quiescent conditions. No shearing is present to rupture crystal−crystal linkages during wax precipitation. As shown in Figure 8, effects of the cooling rate on model wax−oil gelation

Figure 6. Effect of cooling rate on gel breakage of 5 wt % macrocrystalline wax in dodecane under constant imposed shear stress (1 Pa).

rate of 0.1 s−1. As shown in Figure 7, a constant shear stress of 14 Pa is imposed during the gelation process. At high cooling Figure 8. Effect of cooling rate on gel breakage of 5 wt % macrocrystalline wax in dodecane under quiescent gelation process.

and breakage are investigated at cooling rates ranging from 0.2 to 5 °C/min. The yield stress of 5 wt % macrocrystalline wax in dodecane is reduced at higher cooling rates. At cooling rates of 0.2 and 0.5 °C/min, nearly identical yield stress values are attained. The yield stress decreases rapidly when the cooling rate increases from 1 to 5 °C/min. The results confirm that stronger gels are formed under slower cooling. Microscopy. Micrographs of 20 wt % macrocrystalline wax in dodecane and 5 wt % macrocrystalline wax in Primol 352 are obtained at cooling rates ranging from 1 to 20 °C/min. As shown in Figures 9 and 10, crystal sizes decrease with increasing cooling rates. The effect of wax content on crystal morphology is investigated with 1 to 20 wt % macrocrystalline wax in dodecane and 1 to 20 wt % microcrystalline wax in dodecane. As shown in Figure 11, crystal sizes are reduced at higher wax contents. X-ray Diffraction. XRD diffractograms are obtained to establish the crystal structure at various compositional conditions. In real petroleum fluids, it is assumed that fluid viscosity does not influence the paraffin crystal structure. However, crystal structure may be modified by polar colloidal components or polymeric additives which have the potential to cocrystallize with paraffin chains. Therefore, XRD diffractograms are obtained using a single solvent, dodecane, with and without the presence of a polymeric additive (3000 ppm) based on an alkyl ester functionality. The additive was obtained from Champion Technologies. The additive concentration of 3000 ppm was selected to ensure interfacial saturation in the event of a strong adsorption mechanism. The added paraffin is 20 wt % macrocrystalline wax. In both cases, X-ray diffraction patterns are shown to belong to an orthorhombic lattice structure, which is common for aliphatic compounds, as shown in Figure 12. The results show no influence of inhibitor on the parameters of the orthorhombic lattice of the model macrocrystalline wax−oil

Figure 7. Effect of cooling rate on gel breakage of 5 wt % macrocrystalline wax in Primol 352 at constant imposed shear stress of 14 Pa.

rates of 10 and 20 °C/min, a yield stress is present in the formed gel. However, at a low cooling rate of 1 °C/min, no yield stress is evident in the formed slurry. At a slow cooling rate of 1 °C/min, the crystallization process and the crystal− crystal network formation process is sufficiently slow such that the applied shearing is able to completely prevent the formation of a volume-spanning network of crystal−crystal linkages. Mechanistically, the applied shearing serves to rupture crystal− crystal linkages at low linkage number densities, preventing the formation of a percolating volume-spanning network of crystal−crystal anchoring bonds. The resultant slurry therefore consists of dispersed crystal aggregates in which no volumespanning linkage network exists. At higher cooling rates (10 and 20 °C/min), the crystallization process and crystal−crystal network formation process are sufficiently fast such that the applied shearing is unable to prevent the formation of a volume-spanning network of percolating crystal−crystal link8128



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Figure 11. Micrographs of microcrystalline wax in dodecane at cooling rate of 10 °C/min: (a) 5 wt %; (b) 20 wt %.

Figure 9. Micrographs of 20 wt % macrocrystalline wax in dodecane at various cooling rates: (a) 1 °C/min; (b) 20 °C/min.

inhibitor. The XRD data strongly demonstrate that the crystal form is independent of fluid composition and serve to extend the validity of the observed crystal form to real produced petroleum fluids containing additives, other surface active components, and/or polar colloidal fractions. An identical crystal form even for real produced petroleum fluids implies that similar rheological behavior may be expected, even for complex waxy crude oil systems. Because an identical crystal form is observed both with and without the alkyl ester inhibitor, the active mechanisms of the inhibitor may be based on interfacial adsorption, morphological alternation, and/or steric repulsion. Model Wax−Oil Gels Breakage Modeling. According to the work of De Kee et al.13 and Paso et al.,11 crystal breakage is assumed to follow eq 2. The definition of the shear rate (γ̇) is as follows: γ̇ =

dγ dt

(6)

where γ is the absolute strain and t is time. According to the chain rule dλ = −aγ ḃ − 1(λ − λe)n dγ

Figure 10. Micrographs of 5 wt % macrocrystalline wax in Primol 352 at various cooling rates: (a) 1 °C/min; (b) 20 °C/min.

system. Parameters for normal long-chain paraffins (orthorhombic lattice parameters Pnam) are a = 7.455 00, b = 4.966 00, c = 2.589 00, α = 90.000, β = 90.000, and γ = 90.000. Obtained XRD patterns show the correspondence of the structure of analyzed waxy systems to these parameters. Lattice parameters are not affected by the presence of inhibitor. Several conclusions may be drawn from the XRD data: An orthorhombic structure is observed for the macrocrystalline wax, with unit cell parameters independent of the presence of

(7)

When the gel state is a point function of absolute strain (b = 1) or, alternatively, for constant applied shear rate conditions, the following holds: dλ = −k(λ − λe)n dγ

(8)

where k is a constant parameter. Integration and rearrangement of eq 8 yields 8129



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Figure 12. XRD diagrams of 20 wt % macrocrystalline wax in dodecane (upper plot) and 3000 ppm alkyl ester based inhibitor in 20 wt % macrocrystalline wax in dodecane (lower plot). Blue lines show peaks corresponding to n-paraffin (CH2)x.

⎡ 1 ⎢ λ − λe = ⎢ ⎣ (n − 1)kγ +

1 (λ 0 − λe)n − 1

⎤1/(n − 1) ⎥ ⎥ ⎦

Hence, a residual term captures the slurry flow condition at the large strain limit. τ = h(λ − λe)n γ + τs

(9)

Experimentally, λ may be defined in a simple analysis as the ratio of a measured stress value to the yield stress value. The nth order model (eq 9) may be fit to experimental λ values using the simple assumption of λ = τ/τs. Figure 13a shows modeling results of λ at a shear rate of 10−1 s−1, according to eq 9, where corrected time values are shown. Corrected time values are calculated with respect to the measured time at the shear stress maximum. As a more advanced theory, crystal−crystal bonds in wax−oil gels may be represented as Hookean springs at low deformation. Crystal−crystal anchoring bonds resist deformation similar to a hard spring. Crystal−crystal bond strengths vary due to size and contact area distributions, as well as variations in junction bending strengths. Hence, crystal−crystal bonds do not break simultaneously, but rather continuously with imposed deformation. We assume the gel deformation stress response is Hookean: τ = h(λ − λe)n γ

(11)

Equation 11 captures high strain deformation as well as low deformation physics. However, it predicts finite shear stress values in the absence of deformation. It is necessary to implement a correction term in the residual stress term. However, the correction is significant only at low deformation, where crystal−crystal network bonds remain intact. At high deformation, the correction term vanishes. ⎡ (λ − λe)n ⎤ τ = h(λ − λe)n γ + τs⎢1 − ⎥ (λ 0 − λe)n ⎦ ⎣

(12)

In this relation, τs denotes the equilibrium shear stress which establishes the gel slurry state. Inserting (λ − λe) from eq 9 gives ⎡ 1 ⎢ τ=⎢ ⎣ (n − 1)kγ +

(10)

In this relation, a more advanced functionality is assumed between shear stress and the structural parameter λ. Specifically, the parameter h is a Hookean preconstant. The term (λ − λe)n captures the effect of crystals present which resist deformation, where the superscript n is due to the complex structure of the crystal network. The Hookean relation captures the mechanical response in the case in which crystal bonds are unbroken. At large deformations, crystal−crystal anchoring bonds rupture, and equilibrium slurry flow results.

+ τs

1 (λ 0 − λe)n − 1

⎤n /(n − 1) ⎡ ⎤ τs ⎥ ⎢hγ − n⎥ ⎥ (λ 0 − λe) ⎦ ⎣ ⎦ (13)

Equation 13 is highly complex and contains several parameters. In order to reduce the complexity, an approximation is implemented without losing generality. The structure of the denominator may be expressed as 8130



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rheology at low strains. Equation 16 captures the physics of the very high strain regime where all crystal−crystal bonds are broken and flow is defined by a residual stress term (τs) which may be shear rate dependent. Modeling results of τ at γ̇ = 10−1 s−1 are obtained according to eq 15, and are shown in Figure 13b for the case of 5 wt % macrocrystalline wax in dodecane. Equation 15 is able to adequately correlate stress data acquired after a measuring time of 0.1 s. Equation 15 is unable to model the sharp yield stress peak, due to the unbounded dependence of stress on strain in eq 10. Future models may incorporate a bounded dependence on absolute strain for improved modeling predictions. Modeling results of τ at γ̇ = 10−3 s−1 are shown in Figure 14, demonstrating adequate correlation.

Figure 13. Comparison of experimental and fitted λ and shear stress values at a shear rate of 10−1 s−1 for quiescently formed 5 wt % macrocrystalline wax in dodecane. (a) R2 = 0.9975, n = 3.23; corrected time values are shown. (b) R2 = 0.9856, n = 2.93; nominal time values are shown.

Figure 14. Comparison of experimental and fitted shear stress values at a shear rate of 10−3 s−1 sample of 5 wt % macrocrystalline wax in dodecane under quiescent gelation process. R2 = 0.9872; n = 3.35.

Equation 15 requires data from the entire strain regime. Therefore, the nth order model (eq 9) is better suited to model the high-strain breakage data. Figure 15 shows direct

1 = (n − 1)kγ (λ 0 − λe)n − 1 1 1 ≈ (n − 1)kγ + n − 1 n−1 λe λ0 1− λ

(n − 1)kγ + +

(

λ0n− 1 as

0

)

λe ≪1 λ0

(14)

Equation 13 becomes ⎡ 1 ⎢ τ=⎢ − γ+ ( n 1) k ⎣

⎤n /(n − 1) ⎡ τ ⎤ ⎥ ⎢hγ − sn ⎥ + τs 1 ⎥ λ0 ⎦ ⎣ λ0n − 1 ⎦

(15)

The generalized shear stress relationship reduces to the following for third order (n = 3) degradation kinetics: ⎡ 1 ⎢ τ=⎢ 2 k γ + ⎣

⎤3/2 ⎡ τ ⎤ ⎥ ⎢ hγ − s3 ⎥ + τs 1 ⎥ λ0 ⎦ 2 ⎦ ⎣

λ0

(16)

Figure 15. Comparison of experimental and fitted τ values at γ̇ = 1000 s−1 for 5 wt % macrocrystalline wax in Primol 352 under quiescent gelation process. R2 = 0.9916; n = 3.50. Nominal time values are shown.

The degradation portion of eq 16 retains an −0.5 power dependency on γ at the high strain regime, while it also contains Hookean physics and accurately captures the correct 8131



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application of the nth order model to the high-strain breakage kinetics of 5 wt % macrocrystalline wax in Primol 352 at a shear rate of γ̇ = 1000 s−1. The nth order model adequately correlates shear stress values acquired after a measuring time of t = 0.5 s (γ = 344.6). According to the above results and discussions, it is shown that both eqs 9 and 15 can predict λ and τ values when the order n is approximately 3−3.5. The benefit of the new theory is that it spans the entire strain regime. Finally, experimental and modeling results of UL-YS1 crude oil are shown in Figure 16, based on eq 15. It is observed that

cooling rates, larger crystals are formed which yield stronger gels. XRD data show an identical crystal lattice form even under the presence of a powerful wax inhibitor, lending validity to the observed rheological responses, even for complex waxy crudes containing highly surface active components. A novel model of gel breakage is established according to rheological experiments and previous gel breakage models. The novel model is excellent for predicting gel breakage rheology over the entire strain regime from linear Hookean behavior to equilibrium slurry flow.

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AUTHOR INFORMATION

Corresponding Author

*Address: Ugelstad Laboratory, Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), Sem Sælands vei 4, Trondheim 7491, Norway. Email: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Research Council of Norway for financial support.

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REFERENCES

(1) Venkatesan, R.; Ö stlund, J. A.; Chawla, H.; Wattana, P.; Nydén, M.; Fogler, H. S. The effect of asphaltenes on the gelation of waxy oils. Energy Fuels 2003, 17 (6), 1630−1640. (2) Davidson, M. R.; Dzuy Nguyen, Q.; Chang, C.; Rønningsen, H. P. A model for restart of a pipeline with compressible gelled waxy crude oil. J. Non-Newtonian Fluid Mech. 2004, 123 (2−3), 269−280. (3) Vinay, G.; Wachs, A.; Agassant, J. F. Numerical simulation of weakly compressible Bingham flows: The restart of pipeline flows of waxy crude oils. J. Non-Newtonian Fluid Mech. 2006, 136 (2−3), 93− 105. (4) Davidson, M. R.; Nguyen, Q. D.; Rønningsen, H. P. Restart model for a multi-plug gelled waxy oil pipeline. J. Pet. Sci. Eng. 2007, 59 (1−2), 1−16. (5) Magda, J. J.; El-Gendy, H.; Oh, K.; Deo, M. D.; Montesi, A.; Venkatesan, R. Time-dependent rheology of a model waxy crude oil with relevance to gelled pipeline restart. Energy Fuels 2008, 23 (3), 1311−1315. (6) Phillips, D. A.; Forsdyke, I. N.; McCracken, I. R.; Ravenscroft, P. D. Novel approaches to waxy crude restart: Part 2: An investigation of flow events following shut down. J. Pet. Sci. Eng. 2011, 77 (3−4), 286− 304. (7) Phillips, D. A.; Forsdyke, I. N.; McCracken, I. R.; Ravenscroft, P. D. Novel approaches to waxy crude restart: Part 1: Thermal shrinkage of waxy crude oil and the impact for pipeline restart. J. Pet. Sci. Eng. 2011, 77 (3−4), 237−253. (8) Paso, K.; Senra, M.; Yi, Y.; Sastry, A. M.; Fogler, H. S. Paraffin polydispersity facilitates mechanical gelation. Ind. Eng. Chem. Res. 2005, 44 (18), 7242−7254. (9) Lionetto, F.; Coluccia, G.; D’Antona, P.; Maffezzoli, A. Gelation of waxy crude oils by ultrasonic and dynamic mechanical analysis. Rheol. Acta 2007, 46 (5), 601−609. (10) Tinsley, J. F.; Jahnke, J. P.; Dettman, H. D.; Prud’home, R. K. Waxy gels with asphaltenes 1: Characterization of precipitation, gelation, yield stress, and morphology. Energy Fuels 2009, 23 (4), 2056−2064. (11) Paso, K.; Kompalla, T.; Oschmann, H. J.; Sjö blom, J. Rheological degradation of model wax-oil gels. J. Dispersion Sci. Technol. 2009, 30 (4), 472−480. (12) Davenport, T. C.; Somper, R. S. The yield value and breakdown of crude oil gels. J. Inst. Pet. 1971, 57, 86−105.

Figure 16. Experimental and fitted shear stress values at a shear rate of 10−1 s−1 in crude oil UL-YS1 under quiescent gelation process. (a) Repeated experiments of crude oil. (b) R2 = 0.9980, n = 3.35, τs = 6.57 Pa, k = 72, and h = 1384 Pa.

the modeling results of τ for crude oil UL-YS1 are excellent, lending validity to the new theory for real produced petroleum fluids. The elegant agreement between theory and experiment, spanning the entire strain regime, demonstrates that the novel gel breakage model presented in this work holds substantial promise for providing accurate predictions of pipeline restart.

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CONCLUSION In this work, gelation and breakage behavior of model wax−oil systems are investigated. Effects of shear history, thermal history, wax type, and wax content on gelation and breakage are elucidated. The gel strength is reduced when shearing is applied to the sample during the gel formation process. At lower 8132



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(13) De Kee, D.; Code, R. K.; Turcotte, G. Flow properties of time? Dependent foodstuffs. J. Rheol. 1983, 27 (6), 581−604. (14) Paso, K.; Silset, A.; Sørland, A.; Gonçalves, M. A. L.; Sjöblom, J. Characterization of the formation, flowability, and resolution of Brazilian crude oil emulsions. Energy Fuels 2009, 55, 471−480. (15) Exxon Mobil Corporation Home Page. http://www.exxonmobil. com/Denmark-English/Specialties/PDS/GLXXENSPCEMPrimol_ 352.aspx (accessed April 19, 2012). (16) Sasol Wax Corporation Home Page. http://www.sasolwax.com/ sasolwaxmedia/Personal+Care+ab+2011/Brosch%C3%BCre/SAS_ PC_PI_Micro_en.pdf (accessed April 19, 2012). (17) Lee, S.; Choi, S. I.; Maken, S.; Song, H. J.; Shin, H. C.; Park, J. W.; Jang, K. R.; Kim, J. H. Physical properties of aqueous sodium glycinate solution as an absorbent for carbon dioxide removal. J. Chem. Eng. Data 2005, 50 (5), 1773−1776. (18) Venkatesan, R.; Nagarajan, N. R.; Paso, K.; Yi, Y. B.; Sastry, A. M.; Fogler, H. S. The strength of paraffin gels formed under static and flow conditions. Chem. Eng. Sci. 2005, 60 (13), 3587−3598.

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Gelation and Breakage Behavior of Model WaxâOil Systems

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