A Mini Pilot-Scale Flow Loop Experimental Study of Turbulent Flow

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A Mini Pilot-Scale Flow Loop Experimental Study of Turbulent Flow Wax Deposition by Using a Natural Gas Condensate Amrinder Singh, Ekarit Panacharoensawad,*,† and Cem Sarica Tulsa University Paraffin Deposition Projects, McDougal School of Petroleum Engineering, 2450 East Marshall, Tulsa, Oklahoma 74110, United States ABSTRACT: Wax deposition is one of the important flow assurance problems in oil and gas production for both offshore and onshore fields. In this study, we experimentally investigated the turbulent flow wax deposition phenomena under single-phase turbulent flow conditions. The experiments were conducted in a mini pilot-scale flow loop by using Garden Banks condensate from the Gulf of Mexico. Condensate velocities of 1.89, 2.83, 3.78, and 4.72 m/s and testing durations of 2.67 h (2 h, 40 m), 8 h, 16 h, and 24 h were used for the deposition test. The initial value of the Reynolds number, wall shear stress, and heat-transfer coefficient corresponding to these four flow velocities were in the ranges of 7300−19 000, 12−60 Pa, and 1200−2900 W/(m2 K), respectively. Deposit thickness data based on the mass and pressure drop measurements were obtained. The deposit composition was analyzed by using high-temperature gas chromatography (HTGC). The results revealed that more than 45% of the total wax mass flux (from the bulk oil to the deposit interface) contributed to the aging of the deposit. Solubility and heat- and masstransfer analogy methods were used to calculate the theoretical lower and upper limits of the total wax mass flux, respectively. A new method to calculate the ratio of aging wax mass flux to the total wax mass flux directly from experimental data was developed. A new concept on the superficial wax crystal aspect ratio was introduced. It was found that the superficial wax crystal aspect ratio has an abnormally decreasing trend, with respect to the deposit wax fraction. The experimental wax mass flux was found to be significantly less than the minimum theoretical limit. These findings of the abnormal trend in the superficial wax crystal aspect ratio and the experimental incoming wax mass flux are the indication of another factor, in addition to the conventional diffusion−deposition, which impacts the deposition behavior. The deposit composition and thickness data from this study also increased the available paraffin deposition data, which was very limited but required for a reliable deposition model development.

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laminar flow yields a thicker final deposit, compared to the turbulent flow cases. Matzain33 did not measure the deposit wax fraction (F̅w). Thus, the thicker deposit from laminar flow tests cannot necessarily indicate more wax mass (total deposit mass × deposit wax fraction), compared to the turbulent flow cases. In 1999, Matzain34 reported flow loop test results for single-phase and multiphase flow wax deposition. The deposit was analyzed by using high-temperature gas chromatography (HTGC), but the detailed deposit composition was not provided;34 instead, the deposit oil percentage (100 − wax%) was reported. The paraffin deposition model for single-phase and multiphase turbulent flow based on Fick’s law of diffusion was proposed with the introduction of the shear effect term into the model. Yet, the wax fraction was predicted by an empirical correlation, which is only valid for the case of South Pelto oil and the operating conditions (pipe diameter, temperature, etc.) that Matzain34 used. Nevertheless, Matzain34 provided the experimental techniques, the flow loop data, and the visual observation for single-phase and multiphase flow wax deposition, which are important for further model development. In 2000, Singh et al.6 reported the study on laminar flow wax deposition. They introduced the aging mechanism in a systematic way by using Fick’s law with energy and mass conservation

INTRODUCTION Paraffin or wax deposition is a complicated transport phenomenon that has been studied for years, because of its complexity and severity in both onshore1 and offshore2 pipeline transportation. The deposit reduces the fluid flow area in subsea pipeline and can cause a pig to be stuck inside the pipeline during a pigging operation. The estimation of the subsea pipeline capital and operational costs related to wax deposition problems requires a reliable wax deposition prediction. Despite the significant efforts in experimental and theoretical studies during the past few decades,2−30 currently available models cannot be directly used as predictive tools for the field conditions. The prediction of these models for laboratory cases, where operating conditions (oil type, flow rate, pipe diameter, etc.) are different from the model base cases, is also questionable. The main reason for this shortcoming is the unavailability of the closure relationship to calculate unknowns in a model for the field conditions, as pointed out in Sarica and Panacharoensawad.31 Furthermore, the mechanism causing the wax mass flux to be lower than the theoretical limit (proposed by Lee3), as found in the works of Venkatesan,7 Mirazizi,32 and Panacharoensawad and Sarica,11 are not completely understood and more experimental investigations are required to fully predict turbulent flow wax deposition phenomena in all oil types and operating conditions. Additional experimental data are also required to develop and validate a wax deposition model. The flow loop experiment for turbulent flow wax deposition was performed as early as 1996.33 Matzain33 reported that © XXXX American Chemical Society

Received: October 12, 2016 Revised: January 30, 2017 Published: February 1, 2017 A

DOI: 10.1021/acs.energyfuels.6b02125 Energy Fuels XXXX, XXX, XXX−XXX

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increased rapidly within the first hour and roughly reached a constant thickness within 4 h. Fong and Mehrotra27 (hot flow cases) reported the rapid formation of a deposit that had reached the thermal steady state (Ti = WAT) within 20− 30 min. Tiwary and Mehrotra29 (hot flow cases) also reported the same 20−30 min time period was needed for the thermal steady state to be reached in wax deposition experiments. Bidmus and Mehrotra24 (hot and cold flow cases) reported that their deposition tests reach a thermal steady state within ∼15 min. Kasumu and Mehrotra conducted hot flow wax deposition with various water percentages and also reported the time required to reach the thermal steady state to be 10− 20 min in all of their experiments. The above flow loop deposition data from Mehrotra and co-workers22,24,27,29,38 used a model oil system (Parowax with a solvent that was typically Norpar 1324,27,29,38 or Linpar 1416 V22). Parowax is an n-alkane wax compound with the distribution of the n-alkane in the range of C19−C60, and Norpar/Linpar solvent is an n-alkanes solvent with the carbon number in the range of C9−C16. The direct measurement of the deposit interface temperatures used to validate that T ≅ WAT are from the cold finger and Couette−Taylor apparatus.25,30 The model oil Paraffin wax− Norpar 13 were also used in these direct measurements.25,30 With the direct measurement and deposition data from Mehrotra and co-workers,22,24,25,27,29,30,38 it is clear that the deposit interface temperature reaches WAT rapidly for the hot flow, and the deposit thickness increases very rapid, reaching the steady state within ∼30 min to 1 h with no significant changes thereafter. However, wax deposition from crude oil occurs at a much slower rate. Panacharoensawad and Sarica (cold flow case) showed that the deposit formed from South Pelto crude oil becomes thicker without reach a constant thickness, even after 48 h. In the field condition, Singh et al.39 showed that the field deposit became thicker without reaching a constant thickness, even after 7 days. For a hot flow case, Singh et al.6 used a model oil system with only 0.67% (w/w) wax to achieve a slow deposition rate, mimicking the field conditions. The quick deposition growth in Singh et al.6 occurred within 1 day, not within 20−30 min as in many model oil flow loop data.22,27,29,38 The model oil content was 7% (w/w) or more in the tests performed by Parthasarathi and Mehrotra38 and Fong and Mehrotra.27 Tiwary and Mehrotra29 used 10% (w/w) or more wax to create their model oil mixture while 6% (w/w) wax was used in the Kasumu and Mehrotra22 model oil. This high percentage of wax in oil led to rapid deposition in the laboratory and did not allow the slow deposition process as in the field case39 to be simulated. Moreover, even though Singh et al.6 (hot flow case) showed no direct measurement of Ti, it is a self-evident that Ti was not at WAT, at least for the first 24 h. This is because, if Ti was at WAT, there will be no further thermal driving force and no concentration gradient driving force as a consequence. Then, the deposition should stop within 20−30 min, as in the case of Mehrotra and co-workers.22,27,29,38 Yet, this was not the case in Singh et al.6 and the field condition.39 In fact, Singh et al.39 showed that the pigging operation in the field was conducted before the steady state (Ti = WAT) was reached. We note that the flow loop results where the thermal steady state is reached within 30 min cannot be directly compared to the transient deposition cases,3,5,7,9−11,13,40−42 including the field data39 (where the thermal steady state was not reached and the deposit thickness continued to increase with time). Nevertheless, the heat-transfer approach used by Mehrotra and co-workers19,21,37 serves as the prediction of the

equations in their wax deposition model. This approach is also applicable for turbulent flow, as many researchers used it in turbulent and multiphase flow wax deposition models.2−5,7,8 Singh et al.6 introduced the wax mass flux calculation by using the Chilton−Colburn analogy. In 2002, Hernandez14 reported turbulent flow wax deposition results and the shear stripping effect causing the patchwork pattern on the deposit surface. She introduced the shear stripping term and the model was tuned to match the predictions with her experimental results.14 Thus, Hernandez14 model cannot be used as a predictive tool with confidence unless her model is verified by other experimental data. The model14 used the Chilton−Colburn analogy,35 as in Singh et al.6 case, to estimate the wax mass flux to the interface for the turbulent flow cases. Later in 2004, Venkatesan and Fogler36 commented on the wax mass flux calculation based on the Chilton−Colburn analogy,35 that it is equivalent to the case where temperature and the dissolved wax concentration profiles are developed independently. Venkatesan7 estimated wax mass flux by assuming that the dissolved wax concentration profile is completely governed by the temperature profile. This results in the incoming wax mass flux from the bulk to the deposit interface (qin) to be lower than the incoming wax mass flux (qin), calculated by using the Chilton−Colburn analogy35 approach. Lee3 proposed that the dissolved wax concentration profile is governed by the temperature profile and wax precipitation rate for turbulent flows. As the precipitation rate (kr) approaches infinity, the concentration profile is completely governed by the temperature profile, and qin in the Lee model3 will be equivalent to that of the Venkatesan model.7 This is the theoretical lower limit for the incoming wax mass flux, given that the impact of shear is excluded. In 2013, Arumugam et al.19 proposed the unsteady-state wax deposition model based on the heat-transfer approach used previously by Bidmus and Mehrotra.37 The heat-transfer approach proposed by Mehrotra and co-workers19,21,37 is different from the heat- and mass-transfer analogy approach used in Singh et al.6 Mehrotra and co-workers19,21,37 categorized the flow of waxy oil mixture into a hot flow (oil temperature > wax appearance temperature (WAT)) and a cold flow (oil temperature < WAT). For a hot flow case, the heat-transfer approach of Mehrotra and co-workers19,21,37 was performed by setting the deposit interface temperature (Ti) to be at WAT. The deposit thickness (δ) that allows Ti to be WAT then was calculated. For a cold flow case, the heat-transfer models19,21 calculated the deposit thickness by setting Ti to be the WAT of the cold flow liquid phase. The cold flow liquid phase has less wax dissolved and has a lower WAT than the case of hot flow, because some wax crystals precipitated from the solution already. Singh et al.6 experiments are all in the hot flow regime. They6 did not set the deposit interface temperature to be at WAT. The heat- and mass-transfer analogy was performed by calculating the Sherwood number from the same formula of the Nusselt number with the replacement of Prandtl number by Schmidt number. In fact, Singh et al.6 showed that even with a deposition time of 5 days, the deposit did not reach a constant thickness (see Figure 25 in Singh et al.6). Haj-Shafiei et al.21 later modified the model reported by Arumugam et al.19 to be a steady-state model that considers the aging of the deposit. Mehrotra and co-workers19,21,37 supported their argument of Ti = WAT via direct measurement of the deposit interface temperature.25,30 Their deposition data also showed that the deposit thickness increases rapidly with time. Parthasarathi and Mehrotra38 (hot flow cases) showed that the deposit thickness B

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Energy & Fuels maximum possible deposit thickness, by assuming that the thermal steady state is reached. Their experimental data contribute to the limited deposition data available in the literature and can be used to validate the fast deposition cases (Ti ≈ WAT within 30 min). The flow loop data from Mehrotra and co-workers22,24,27,29,38 have a long enough flow developing length (23 cm in the Parthasarathi et al.38 study and 57 cm in the Fong and Mehrotra27 study) with a 10.16-cm-long deposition section (10.8 cm in the Parthasarathi et al.38 study), and an inner diameter (ID) of 2.54 cm. These flow loop data22,24,27,29,38 are meaningful and valuable for a further up-scaling study. Mehrotra and co-workers modeled21,23,28 the shear effect via aging of the wax deposit. The simplified cubical cage structure was used to model the aging phenomena of wax deposit without the need to exactly matching the complicated depositgel structure. This cubical cage deforms, once it is exposed to a shear field.21,23,28 In their model,21,23,28 the deformation causes the liquid phase to come out from the deposit structure and the deposit solid fraction increases. The purpose of their model was to quantify the impact of shear on the deposit solid fraction. Unlike the heat-transfer deposition model,19,21,37 some diffusion−deposition models4,5,7 consider the shear effect to have a direct impact on decreasing the wax mass flux toward the deposit. Yet, many diffusion-deposition models, such as those of Lee,3 Huang et al.,2 and Zheng et al.,8 have no explicit shear term in their deposition models. The unknown role of shear on wax deposition motivated Dwivedi,43 Mirazizi,32 and Panacharoensawad4 to further investigate the impact of shear (from the fluid to the deposit surface) on wax deposition. Dwivedi43 and Mirazizi32 provided turbulent flow experimental data that are crucial for any model development. Mirazizi32 showed the wax mass flux analysis by comparing the experimental wax mass flux to the output from his in-house software. He showed that the experimental wax mass fluxes are lower than the fluxes calculated from Fick’s law (which is equivalent to the solubility model used in he work of Venkatesan7). Panacharoensawad4 confirmed that the experimental wax mass flux can be lower than the flux calculated from the solubility method. Panacharoensawad and Sarica11 showed that the wall shear stress has an impact on the deposit composition independently from the heat flux across the deposit. They found that the early time (4 h) deposit wax fraction increases as the wall shear stress increases. The deposit thickness versus time trends from different velocity cases were found to crossover each other at a certain time.11 The crossover among the thickness versus time trend lines (from various flow rate cases) did not occur at the same time. On the early deposition period (less than 4 h before any crossover occurred), they11 found that the deposit thickness increased as the flow rate increased. For the late deposition time (e.g., 48 h, after all crossover occurred), they11 found that the deposit thickness decreased as the flow rate increased. Thus, as flow rate increased, the value of δ can be seen to have increased, reached a maximum, and decreased, if the comparison was done after some (but not all) trend lines crossover occurred, as in the case of Dwivedi.43 We refer to the work of Panacharoensawad and Sarica11 for more details on the impact of flow rate on δ, before, during, and after the crossover period. Panacharoensawad and Sarica44 also showed that the patchwork pattern found in the work of Hernandez could be a localized random phenomenon. They showed that the turbulent flow wax deposit surface is visually smooth away from the inlet but there was a no-deposition zone and a patchwork zone near the test section inlet. Furthermore, Rittirong et al.12,40

recently confirmed that the single-phase turbulent and twophase gas-oil slug flow had the visually smooth deposit surface for the region away from the test section inlet. In this case, they used a much longer test section and Garden Banks condensate (same fluid as in this study), instead of South Pelto crude oil.12 Even though the impact of shear on turbulent flow wax deposition is not completely understood, recently, there were the attempts to (1) predict the deposit composition8 and (2) use the data from a high-pressure Taylor−Couette device to up-scale the model prediction to the field conditions.18,45 Zheng et al. model8 successfully predicted the impact of the operating conditions on the trend of the deposit carbon number distribution. The Zheng et al. model8 extrapolated the Hayduk−Minhas equation46 from the binary mixture system to the multicomponent system. They also extrapolated the use of Fick’s law of diffusion from isothermal to nonisothermal cases. These extrapolations should lead to some error in the prediction because the Hayduk−Minhas equation46 and Fick’s law are not valid for nonisothermal multicomponents systems to begin with, as noted earlier in the work of Hoteit et al.16 Nevertheless, Zheng et al.8 showed that their model can predict the trend of the change in the deposit composition, even though “absolutely no tuning was performed” in their study. It should be emphasized that the Zheng et al.8 model does not include the impact of shear on the wax deposit composition, as found earlier in Panacharoensawad and Sarica.11 It is also known that the precipitation rate kr (originally used in the work of Lee3) cied in the work of Zheng et al.8 will change with the Reynolds number (Re), but the behavior of kr, which varies with Re, pipe diameter, or other operating conditions, is not completely known. A closure relationship for kr was not provided either. Furthermore, the Michigan Wax Predictor8 requires the relationship between the wax fraction aspect ratio, α (required in the effective diffusivity calculation, the Cussler et al.47 equation), and the deposit wax fraction, F̅w, similar to the case observed with the Singh et al.6 model. Singh et al.6 proposed the linear relationship between α and F̅w. They reported that the final wax crystal aspect ratio (αfinal) decreased as the flow rate increased.6 Lee3 proposed the square root form for the relationship between α and F̅w. Zhen et al.8 did not give the relationship between α and F̅w, which is used in their thickness and wax fraction calculation. Therefore, even though Zheng et al.8 can successfully predict the deposit composition trend, their model is not yet ready to be used for other cases, because the unknown closure relationships in their model (for kr and α calculations) were not given. In addition, it is also not known on how these unknown closure relationships will change with the pipe diameter or oil type. Yet, the Zheng et al.8 model can serve as a theoretical basis, which requires additional efforts to develop the closure relationships of unknown constants used in their model, before it can be fully used as a predictive tool under both laboratory and field conditions. The attempt in using the data from a high-pressure Taylor− Couette device to scale-up the model prediction to the field conditions, as in thw work of Eskin et al.,18,45 poses some theoretical concerns as noted in depth earlier in the work of Panacharoensawad.1 Eskin et al.18,45 assumed that the unknown parameters in their model could be determined by matching the wall shear stress, pressure, and temperature to the field conditions by conducting experiments in a Taylor−Couette test cell. The unknown parameters in the work of Eskin et al.18,45 are (1) precipitation rate, (2) initial deposit porosity at the wall, (3) initial deposit porosity at the deposit interface, (4) the C

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Facility Description and Testing Procedures. The mini pilotscale flow loop designed and used by Panacharoensawad and Sarica11 was slightly modified by adding a pipe-in-pipe heat exchanger section (93 cm long) prior to the deposition section (see Figure 1 for the brief

diffusivity inside the deposit, and (5) the shear effect. The Eskin et al.18,45 approach to scale-up the unknown parameters to the field conditions is not valid, because their parameters can be a function of both wall shear stress and the heat flux across the pipe wall at the same time. However, the Taylor−Couette highpressure cell can only match either wall shear stress or the heat flux across the pipe wall of the field condition one at a time, but not both shear and heat flux at the same time.1 Nonetheless, the Eskin et al.18,45 work shows the need to have closure relationships for a few unavoidable unknowns in the wax deposition model. In this study, we conducted flow loop experiments by controlling the bulk fluid temperature and the initial interface temperature to be constant while changing the flow velocity. The interface temperature changed as the deposit grows but the initial value of the interface temperature was controlled to be the same for every tests (A01−A16) by decreasing the coolant temperature as the oil flow rate increases. We tackled the complexity of the wax deposition phenomenon by analyzing the experimental aging wax mass flux and the experimental total incoming wax mass flux from the bulk fluid to the deposit interface. This analysis was based on the diffusion−deposition model and has no assumption of Ti = WAT. The heat-transfer approach was not selected because the condensate deposit thickness was determined to conitnue to increase with time. This, by itself, is the self-evidence that the WAT was not reached within the testing duration. No direct measurement of the deposit interface temperature was performed. By analyzing these parameters, we gained an in-depth understanding of the role of wax mass flux in the aging process. This helps to improve the modeling of the wax deposition process. The provided flow loop data also significantly contribute to the very limited available deposition databank, which is vital for a model development and verification.

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Figure 1. Mini pilot-scale facility11 with the modification by adding the pipe-in-pipe heat exchanger prior to the test section. The letters “F”, “P”, and “T” in the figure represent the flow meter, pressure, and temperature transmitters. Test section 1 (not used in this study) and 2 (used in this study) are the rectangular duct and a pipe-in-pipe copper test section, respectively, as fully described previously.11 description of the facility). The deposition section used in this study has the same dimensions as in Panacharoensawad and Sarica.11 The deposition section length, inner pipe ID, inner pipe outside diameter (OD), and the outer pipe OD are 84 cm, 0.651 in., 0.75 in., and 0.995 in, respectively. Both inner and outer pipes of the deposition section are made of copper. Prior to the deposition section (jacketed part) is the velocity profile developing section of 52.3 cm (31.6 diameter). The inner pipe (copper) of the heat exchanger section has ID and OD of 0.995 in. and 1.125 in., respectively. The outer pipe (copper) of the heat exchanger section has length, ID, and OD of 93 cm, 1.481 in., and 1.625 in., respectively. The heat exchanger section was used to cool the oil prior to the deposition testing and to achieve the inlet temperature stability during the test. This allows the inlet temperature of each test to be within ±0.6 °F over the testing duration. The oil inlet temperatures of all tests combined for the entire experimental program were determined to be 77.4 ± 1.2 °F. Based on the deposition results in the test section, the depletion of wax mass inside the heat exchanger section during wax deposition test was estimated to be 0 (as in the assumption of this study), δ0 and F̅w versus time functions as t is approaching zero are approximately linear or dδ = constant. This means that dFw̅

⎛ jage ⎞ is approaching zero, or lim⎜ j ⎟ = 0. The discrete t → 0⎝ in ⎠ data point of jage/jin at 1.33 h is calculated by using δ and F̅w data at t = 0 and 2.67 h, where dδ/dF̅w is approximately the same as δ/F̅w at this point (t = 1.33 h). Therefore, the jage/jin ratios of all discrete data cases are ∼0.5 at t = 1.33 h, as shown in Figure 14. The continuous wax mass flux ratio lines shown in Figure 14 are calculated based on the curve-fitting functions (from Figures 3 and 9) for δDP and F̅w data. The criterion for selecting these curve-fitting functions is that the function must be a monotonically increasing function. Then, the fitting functions to be used for δDP and F̅w data are not unique. This causes the term dδ/dF̅w to be sensitive to the functions being used, because different functions give the different dδ/dt and dF̅w/dt values, which result into some degree of uncertainty in the dδ/ dF̅w calculation, depending on the selected fitting functions. Intuitively, the aging wax mass flux should decrease as time progresses, because the wax crystal aspect ratio in the deposit increases with time and hinders the diffusion inside the deposit.2,3,6−8 However, the calculation based on the experimental data of this study (Figure 14) counterintuitively shows that the ratio of the aging to incoming wax mass flux can increase as time progresses. We note that the experimental program of this study was conducted under the constant volumetric flow rate condition. The bulk oil temperature is lower than the WAT during the test at any location in the system. Thus, there is no thermal restriction to prevent the deposit to grow thicker than the point where Ti > WAT. Yet, the deposit cannot grow and reach the center of the pipe, because the pump was set to flow at a constant volumetric flow rate and will break any soft gel deposit layer trying to block the pipe. As the deposit grows, the flowing area decreases, the average velocity increases, and the shear stress at the deposit interface increases, because the pump was set at a constant volumetric flow rate. Eventually, the deposit thickness will reach the point where it cannot tolerate the shear force provided by the pump. Hence, the wax mass flux still exists, because of the concentration gradient imposed by the temperature gradient at the deposit interface where Ti > WAT. Then, it is possible for the jage/jin ratio to increase with time, which contradicts the MWP2 theory. Since the growth of the deposit can be hindered by the shear from the pump, and the wax molecules, which are already migrated to the deposit inter-

(

F ̅ dδ lim w t → 0 δ dFw̅

Figure 14. Aging wax mass flux to the incoming wax mass flux ratio versus time calculated based on the experimental data. The continuous lines are the wax mass flux ratio calculated based on eq 6 and the curve-fitting functions in Figures 3 and 9.

The continuous lines in Figure 14 are the wax mass flux ratio calculated based on the curve-fitting functions used in Figures 3 and 9. The term dδ/dF̅w in eq 6 used for the continuous wax mass flux ratio line (in Figure 14) is calculated by performing the central finite difference approximation on the continuous line plotted between δ and F̅w (shown in Figure 15). It should be

Figure 15. Growth and aging characteristic curve for all flow rate cases. The continuous lines are based on curve-fitting functions used in Figures 3 and 9. The color definition are identical to those described in Figure 3.

emphasized here that eq 6 contains no assumption on the precipitation rate of wax in oil. Yet, it assumes that there is no loss of either aging or incoming wax mass flux due to shear. The discrete points in Figure 14 are based on the actual deposit thickness and wax fraction data without using any J

−1

)

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because this interpretation is done without the assumption on the precipitation rate of wax. This interpretation is used to confirm the jage/jin continuous line result shown in Figure 14. To understand the wax mass flux ratio more, the expression for the theoretical wax mass flux ratio is written by neglecting the shear effect as follows:

face (due to the concentration gradient) must exist somewhere, it is possible that those molecules diffuse into the deposit, making the deposit stronger. The above discussion is plausible for the observed result to be different from the conventional transport equation-based theory2,57 (MWP theory). In addition to the calculation of dδ/dF̅w, the δ vs F̅w graphs in Figure 15 (used previously4 to analyze the wax deposition characteristics) can also be used to qualitatively understand the growth and aging characteristic of wax deposition. Figure 15 shows that δ vs F̅w are the monotonically increasing function for all flow rate cases. However, the curve for the 20 kg/min case is approximately a monotonically increasing convex function, but other flow rate cases are monotonically increasing concave functions. The continuous line in Figure 15 deviates from the data points because the curve-fitting function in Figure 9 does not perfectly fit with the F̅w data and δDP does not completely match with δmass. It may seem that the 20 kg/min case is an outlier at first glance. The deposition behavior in the case of South Pelto oil4 was checked. The similar convex function was reported in a plot of δ vs F̅w − F̅w0 for a single-phase turbulent flow low flow rate case (Figure 5.186 in the previous study4). Panacharoensawad4 observed that the δ vs F̅w − F̅w0 curve can change from a convex function to a concave function as the flow rate increases.4 Figure 15 shows that the deposit has a tendency to grow (increase in δ) instead of age for the 20 kg/ min case but the opposite behavior occurs for all other higherflow-rate cases. This behavior is consistent with the previous study in the South Pelto oil case, in that the decrease in flow rate has a tendency to cause the deposit to grow instead of age.4 Previously, Panacharoensawad4 showed the S-shaped curve for δ vs F̅w data in many South Pelto crude oil cases. The 48 h deposition duration was selected in the previous study4 but 24 h of the deposition duration is used in this study. Thus, the 20 kg/min trend in Figure 15 should not be extrapolated. This is because direct extrapolation of the 20 kg/min curve (in Figure 15) cannot predict the S-shaped curve in the δ vs F̅w graph. In a previous study with South Pelto oil,4 the S-shape in the δ vs F̅w curve was not always observed within 48 h of the wax deposition test. The low flow rate of the single-phase South Pelto case (13 kg/min)4 was found to be an increasing convex function without showing the S-shape at all, although the testing duration was 48 h. Based on the previous result,4 if the deposition time is extended, it is possible to find the change in the sign of d2δ/ dF̅w2 as time increases. It was hypothesized in the previous work4 that the deposit has a tendency to age more instead of grow as time increases so that the deposit can be strong enough to withstand the increment in τi as the deposit grows. This hypothesis4 needs more proofs. Yet, it can be used to explain that the deposit in a lower flow rate case has less increment in τi as the deposit grows, so that it does not need to age much before it can grow, because there is not much increment in τi to withstand, compared to the higher-flow-rate cases. The curvature in the wax deposition characteristic curve (Figure 15) is also an indication of the wax mass flux ratio. The monotonically increasing convex functions of δ vs F̅w have Fw̅ dδ δ dFw̅

jage jin

vz

ri+

∂C

−+wo ∂r

ri−

(10)

1 ∂⎡ ∂C ∂C ⎤ = ⎢r(εM + +wo) ⎦⎥ − kr(C − C(T )) ⎣ r ∂r ∂z ∂r

(11)

The precipitation rate (kr) has a tendency to increase, approaching infinity as the flow becomes more turbulent.2,3 As kr increases, approaching infinity, the precipitation term (also known as the sink term), kr(C − C(T)), will cause any deviation of C from C(T) to vanish. Thus, by assuming the limit of kr approaching infinity, the instantaneous precipitation will occur or the dissolved wax concentration will be controlled by the temperature profile. The term ∂C/∂r becomes ∂C/∂T × ∂T/∂r in this case.7 By assuming the instantaneous precipitation (same as using the solubility method approximation), we have jage jin

∂C ∂T ∂r r + i

−+e ∂T =

∂C ∂T ∂r r − i

−+wo ∂T

(12)

The heat of crystallization of wax can be neglected in the temperature profile calculation3 (also see Appendix D). The growth and aging of wax deposition is relatively slow, compared to the time required for the radial temperature profile in oil and deposit to be developed. Thus, the immediate development of the radial temperature profile (in oil and deposit) is assumed. The justification of this assumption is given in Appendix E. With these assumptions, the heat flux to and from the deposit interface are equal, or −ko

∂T ∂r

ri−

= − kd

∂T ∂r

ri+

(13)

Then, eq 12 becomes jage jin

=

+eko +wokd

(14)

Conventionally, the deposit thermal conductivity can be approximated using the Maxwell correlation,54 by assuming that oil is a continuous phase wherein wax particle are embedded.2−4,6−8 In this case, the deposit thermal conductivity can be written as kd =1+ ko

3Fw̅ k w + 2k o k w − ko

− Fw̅

(15)

It is shown here that the Maxwell correlation approximation, widely used in several models,2−7 are relatively the same as using a linear interpolation between the oil and wax thermal conductivity for the case of F̅w < 0.35, as in this study (see Figure 16). The Maxwell correlation is originally for the case of

increasing with time or the jage/jin ratio decreases with time, per eq 6. For all other monotonically increasing concave funcFw̅ dδ δ dFw̅

=

The precipitation rate defined through the transport equation of the dissolved wax concentration can be written as2,3

(roughly, the slope at that point divided the overall slope)

tions in Figure 15, the term

∂C

−+e ∂r

decreases or the jage/jin ratio

increases with time. The above interpretation is important K

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Figure 16. Dimensionless deposit thermal conductivity (kd̃ ), as a function of the wax fraction calculated based on the Maxwell correlation and the linear interpolation between wax and oil thermal conductivity values. At most, the Maxwell correlation gives kd to be 3.2% lower than that from the linear interpolation approach. kw and ko values of 0.25 and 0.1413 W/(m K), respectively, are used in these calculations of kd and kd̃ calculation.

Figure 18. Diffusivity ratio as a function of F̅w according to eqs 16 and 18 at various kα.

used in eq 14 is not expected to cause the opposite trend in +e/+wo in Figure 17. Then, the assumption that shear plays no role on wax deposition used earlier to obtain Figure 17 is expected to be invalid. Therefore, the contradiction between the calculated value of +e/+wo, based on the experimental data and the theoretical trend of +e/+wo is the indication of the shear effect on wax deposition. This indication also agrees with the shear-effect existence-indication, reported previously.11 Since we cannot quantify the impact of shear on wax deposition, we defined the obtained diffusivity ratio, shown in Figure 17, as the superficial diffusivity ratio, instead of the true experimental diffusivity ratio. The indication of the impact of shear and the contradiction between the diffusivity ratio and the expected one demonstrate the need for a closure relationship to capture the shear effect in a transport-based wax deposition model. Next, we show the potential use of the modified wax crystal aspect ratio to capture the impact of shear. The superficial wax crystal aspect ratio (αs) is calculated from the superficial diffusivity ratio. αs is defined as the wax crystal aspect ratio calculated based on the assumption of the instantaneous precipitation and no shear effect. By using the expression given in the work of Singh et al.6 (the Cussler et al.47 array barrier membrane), we have

the negligible amount of spherical particles embedded in a continuous solid phase. The particle volume fraction must be small enough to not interact with the continuous phase thermally for the Maxwell correlation to be valid.54 Wax deposit is a three-dimensional (3D) network of wax crystal in oil, not the negligible amount of spherical particles in a continuous solid phase. Yet, the Maxwell correlation is widely used to approximate the deposit thermal conductivity.6 With the value of the thermal conductivity ratio from the Maxwell correlation and the wax mass flux ratio calculated earlier, eq 14 is used to calculate the diffusivity ratio and the results are shown in Figure 17. These diffusivity ratio results are

+e 1 = 2 2 +wo 1 + αs Fw̅ /(1 − Fw̅ )

(16)

or αs = Figure 17. Superficial diffusivity ratio calculated from eq 14.

⎛ +wo ⎞ 1 − Fw̅ − 1⎟ ⎜ ⎝ +e ⎠ Fw̅ 2

(17)

Figure 19 shows the superficial wax crystal aspect ratio as a function of time. The superficial wax crystal aspect ratio was found to decrease monotonically with time in the 30, 40, and 50 kg/min cases. This does not mean that the actual wax crystal aspect ratio decreases with time, because two assumptions used in the αs calculation could be inaccurate. Originally, Singh et al.6 proposed that the wax crystal aspect ratio can be assumed to be a monotonically increasing linear function of F̅w or

not as expected, because the diffusivity restriction in the deposit (as F̅w increases) should cause +e to decrease. The actual +e/+wo ratio is expected to be a decreasing function of t and F̅w, based on the theory given in Singh et al.,6 as shown in Figure 18. The difference in the theoretical diffusivity ratio6 and the calculated ratio based on the experimental data is expected to be caused by the assumptions made in calculation. These assumptions in the diffusivity ratio calculations are that (1) the shear effect is neglected and (2) the precipitation rate is infinitely large. Venkatesan,7 Lee,3 and Huang et al.2 suggested that the precipitation rate increases and can cause complete precipitation of wax in the bulk oil, as the flow becomes more turbulent. Thus, the assumption of the complete precipitation

α = 1 + kαFw̅

(18)

By using the idea that α is a function of F̅w, Figures 20 and 21 surprisingly show that the αs vs F̅w curves are merged into one curve as F̅w increases for the 30, 40, and 50 kg/min cases in L

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Energy & Fuels

correlation to predict the wax mass flux ratio or to compensate for the shear term in other flow rate cases. The αs correlation has the potential to be used in developing a closure relationship for the wax crystal aspect ratio, which is missing from all wax deposition models, or to compensate for the impact of shear on the wax mass flux ratio. The formula for the converged curve and the 20 kg/min case (in Figure 20) is given in Appendix C. The analysis of αs suggests that the merging of the αs curves possibly occur as the shear stress is high enough and it is expected to not be valid for the low-flow-rate or low-shear cases. The deviation of αs vs F̅w of the 20 kg/min case is

Figure 19. Superficial wax crystal aspect ratio, as a function of time.

Fw̅ dδ term for δ dFw̅ Fw̅ dδ the δ dF term of w̅

expected because the

the 20 kg/min case (in

Figure 15) is >1, but

other curves in the same

figure are 0.1. The decrease in αs is expected to compensate for the missing shear term. This interesting behavior allows one to use the superficial wax crystal aspect ratio curve as an empirical M

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Energy & Fuels Incoming Wax Mass Flux Analysis. Overview. The theoretical lower limit of the incoming wax mass flux and the experimental incoming wax mass flux are calculated and analyzed in this section. The theoretical lower limit of jin is calculated based on the instantaneous precipitation case (limit as kr approaches ∞) and neglecting any shear effect. It was found that the experimental jin is lower than its theoretical limit. Thus, this finding suggests the existence of a shear effect, as suggested earlier in previous studies.4,7,10,11,13,32,43 Different researchers propose the impact of shear on wax deposition differently. One common description on the impact of shear is that shear stress on the deposit decreases wax mass flux toward the deposit interface. The upper and lower theoretical limits of incoming wax mass flux (jin,h/m and jin,eq) are also analyzed and compared against the experimental wax mass flux at the end of this section. Derivation and Results. The total wax mass flux from the bulk to the deposit interface (jin) at time t over the length ΔL is defined as ⎡ 1 dmw ⎤ jin |t ≡ ⎢ ⎥ ⎣ 2πriΔL dt ⎦

Figure 22. Experimental incoming wax mass flux estimated from eq 23. The color definition for flow rate values are identical to those described in Figure 3. Discrete data points are the values calculated based on the mass thickness and the HTGC wax fraction data. The continuous lines are based on the curve-fitting functions between F̅w(R2 − ri2) vs t (given in Appendix C), where ri is obtained from the DP thickness data.

in eq 23. The experimental incoming wax mass fluxes calculated by using eq 23 are shown in Figure 22. The discrepancy between the discrete data points and the continuous lines occurs due to the difference between the DP thickness and mass thickness data. Figure 22 is replotted in Figure 23 as the

(19)

t

where mw is the wax mass of the deposit (not including the oil part). The wax fraction data are discrete data. They are available only at t = 2.67, 8, 16, and 24 h. For this case, the dmw/dt term at time tn+1/2 can be estimated from dm w dt

≅ tn + 1/2

(Fw̅ md )|tn+1 − (Fw̅ md )|tn tn + 1 − tn

(20)

where md is the total deposit mass (including the oil part). The incoming wax mass flux at tn+1/2 then can be calculated from jin |tn+1/2 ≡

(Fw̅ md )|tn+1 − (Fw̅ md )|tn tn + 1 − tn

1 2πriΔL

tn + 1/2

(21)

The term ri|tn+1/2 is the average value between the ri value at tn and the ri value at tn+1. The deposit mass (md) can be expressed as md |tn = π ΔLρd (R2 − ri 2)|tn

(22)

Figure 23. Experimental incoming wax mass flux versus Reynolds number (Re). The continuous lines are the linear interpolation of the jin values calculated from δDP. The discrete points are the jin values calculated from δmass. Symbol legend: blue diamond (◆), jin = 1.34 h; burnt orange square (■), jin = 5.36 h; green triangle (▲), jin = 12 h; and purple cross symbol (×), jin = 20 h. The exception is that, for ṁ = 50 kg/min (Re ≈ 18 600), the reported jin values are 11 and 19 h, instead of 12 and 20 h, respectively.

By substituting eq 22 into eq 21, we have jin |tn+1/2

(Fw̅ (R2 − ri 2))|tn+1 − (Fw̅ (R2 − ri 2))|tn 1 ≡ ρd tn + 1 − tn 2ri

tn + 1/2

(23)

The values of δDP and δmass were used in the jin calculation via eq 23. The Garden Banks deposit density (ρd) data from Rittirong5 (only for single-phase cases) is used as an approximate deposit density in the calculation of eq 23. This deposit density value for eq 23 is approximated to be 856 kg/m3 in this study. The δmass value used in eq 23 was already calculated by excluding the excess residual oil film mass on the deposit surface. The δDP value was used in eq 23 to calculate jin by, first, plotting F̅w (R2 − ri2) vs t and obtain the curve-fitting functions (HTGC data are used for F̅w values). The slope of the curvefitting function is given as slope =

experimental incoming wax mass flux versus Reynolds number (Re). This was completed by setting the time (abscissa) of Figure 22 to be a constant, then reading and replotting the incoming wax mass flux data at that particular time in Figure 23. Both Figures 22 and 23 show that jin data based on δDP are approximately the same for the 30, 40, and 50 kg/min cases (Re > 11 000). The jin data based on δmass did not agree with cases of jin data based on δDP. This shows that the uncertainty from the experimental result is not low enough to precisely quantify the impact of flow rate on jin, because jin data from different sources do not have the same trend. Here, note that jin, when backcalculated from the experimental value, can, in fact, increase, reach a maximum, and decrease, as shown in

(Fw̅ (R2 − ri 2))|tn+1 − (Fw̅ (R2 − ri 2))|tn tn + 1 − tn N

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Energy & Fuels Figure 23 for the 5.36 h cases where jin based on δDP and δmass have the same trends (reaching maximum and decreasing). This is possible because the thickness versus time trend for different flow rates can cross over each other. Thus, a snapshot at a particular time can be seen as the increasing of thickness, reaching a maximum and decreasing as the flow rate increases (also previously found South Pelto oil tests11). jin is calculated based on the thickness value and inherits this trend from the thickness versus flow rate behavior. The theoretical incoming wax mass flux at time t from the equilibrium method was calculated from jin,eq

⎛ ∂C ∂T ⎞⎟ = −⎜+wo ⎝ ∂T ∂r ⎠ t , r −

t

calculated by using the trapezoidal numerical integration method. Cb − Ci were calculated from C b − C i = (fs ρo )|Ti̅ − (fs ρo )|Tb̅

The local value of hH along the z-direction was calculated from the Gnielinski correlation60 for the Nusselt number, which is given as NuH, z =

(30)

where the space-averaged bulk and interface temperature are calculated from the expression T̅ =

1 L

∫0

L

T dz

by using the trapezoidal numerical integration method. The temperature profile (local Tb, TC, and Ti) along the pipe length was calculated by using the same spreadsheet that was used in the work of Panacharoensawad and Sarica11 (see Appendix B in the previous work11) with the adjustment in the fluid properties and the deposit thickness to match the case of Garden Banks condensate at any deposition time t. The term ∂C/∂T was estimated from the slope of the precipitation curve as ⎛ ∂ρ ∂f ⎞ ∂(ρ f ) ∂C = − o s = −⎜fs o + ρo s ⎟ ∂T ⎠ ∂T ∂T ⎝ ∂T

(25)

where fs is the precipitated solid fraction. The values of ρo and fs, as a function of T for estimating the slope of each parameter, are given in Appendix C. The term ∂T/∂r was estimated from the overall heat flux from the hot side to the cold side as ⎛ ∂T ⎞ −⎜ko ⎟ ≅ hH̅ (Tb̅ − Ti̅ ) ⎝ ∂r ⎠r −

⎡ ⎛ D ⎞2/3⎤ ⎢1 + ⎜ ⎟ ⎥ ⎝ z ⎠ ⎥⎦ 1 + 12.7(f /2)1/2 (Pr 2/3 − 1) ⎢⎣ (f /2)(Re − 1000)Pr

where f is the Fanning friction factor. The +wo term in eq 24 was estimated from the Hayduk− Minhas equation46 by using viscosity and temperature at the deposit interface (T̅ i). The molar volume (VA) used in the diffusivity correlation46 was estimated to be 523.3 cm3/g mol, based on the average VA value of the CCN+ fraction of n-alkane in oil.53 Based on the CCN+av data of this study and the previous theoretical study of Zheng et al.,8 it is known that the average chain lengths of n-alkane that diffuses into the deposit change with time. To understand the magnitude of the error introducing by the approximation of VA, the VA value of n-C45H92 (∼669.5 cm3/g mol, PVTSim61 approximation data) was used to estimate +wo. It was found that, at 23 °C (the typical value of the interface temperature at 24 h, where VA is expected to be greatest) +wo is overestimated by 20% if 523.3 cm3/g mol is used instead of 669.5 cm3/g mol. Here, note that the VA value of n-C45H92 is the extreme case, because the experimental result suggests that CCN+av will be 4000 for all cases. The heat- and mass-transfer analogy to approximate mass transfer then can be written as

⎛ Sc ⎞1/3 Sh =⎜ ⎟ Nu ⎝ Pr ⎠

(27)

Figure 24. Equilibrium method incoming wax mass flux versus time. The interface temperature used in the theoretical jin calculation was based on the experimental δmass and F̅w.

where Nu, Pr, Sc, and Sh are the Nusselt, Prandtl, Schmidt, and Sherwood numbers as defined in the nomenclature section. After substituting the formulas of Nu and Sh to eq 27, we have jin,h/m

t

⎡ ⎤ ⎛ h ̅ ⎞⎛ Sc ⎞1/3 = ⎢+wo|Tb̅ ⎜⎜ H ⎟⎟⎜ ⎟ (C b − C i)⎥ ⎢⎣ ⎥⎦ ̅ ⎠⎝ Pr ⎠ ⎝ ko,b

t

The experimental incoming wax mass flux is shown against jin,eq with a −20% error bar in Figure 25 (the +wo value based on a VA value of 523.3 cm3/g mol is the maximum value). It can be seen that the experimental jin data were found to be well below the theoretical lower limit of the incoming wax mass flux (jin,eq at the limit of kr → ∞). This jin,eq theoretical limit does not consider the impact of the shear stress from the flow to the deposition phenomena. The experimental jin value with the theoretical value of jin,eq suggests that there is at least one

(28)

where k̅o,b is the bulk oil thermal conductivity calculated based on T̅ b and hH̅ =

1 L

∫0

L

hH dz O

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Energy & Fuels or jin,h/m jin,eq

= t

(fs ρo )|Tb̅ − (fs ρo )|Ti̅ ⎛ Sc ⎞1/3 1 ⎜ ⎟ ⎝ Pr ⎠ d(fs ρo )/dT Tb − Ti

t

(32)

For the case where fs is linear within the range between Ti and Tb, and ρo is a weak function with T, compared to the functionality between fs and T, then, we have

jin,h/m jin,eq

≅ t

⎛ Sc ⎞1/3 ⎜ ⎟ ⎝ Pr ⎠

t

(33)

where Sc/Pr is evaluated at the bulk fluid temperature. T̅ b in this study was within 77.4 ± 1.2 °F. Within this T̅ b range, (Sc/Pr)1/3 is in the range of 6.82 ± 0.04. The ratio of jin,h/m to jin,eq was found to be 6.3, which is close to the theoretical approximation value of 6.8 (shown above). They are not identical, because of the error in approximating the function fsρo to be a perfectly linear function. This consistency with the theoretical approximation value provides additional confidence in the calculation result in this section. Assumptions. Equation 19 contains no assumption on the shear effect, precipitation rate, temperature profile, or concentration profile. The assumptions in eq 24 are from those of the equilibrium method,2 which are (1) kr approaches infinity (completely precipitated, the C profile is governed by the T profile) and (2) there is no impact of shear on the incoming wax mass flux. The ∂T/∂r term in eq 24 was estimated with the assumption that the temperature profile development is more than an order of magnitude faster than the growth in the deposit thickness or change in the deposit wax fraction. The impact of the enthalpy of crystallization on the temperature profile was neglected (justified in Appendix D). +wo in eq 24 is the extrapolation of the Hayduk−Minhas equation46 from a binary mixture to a multicomponent system. The assumption in eq 28 is that the temperature and concentration profiles are developed independently36 and shear has no effect on the incoming wax mass flux.

Figure 25. Experimental incoming wax mass flux and theoretical equilibrium method incoming wax mass flux (symbols with −20% error bar and connected using a dashed line). The experimental wax mass flux are symbols without a dashed line (calculated based on δmass) and solid lines (calculated based on δDP). Color definitions are identical to those described in Figures 3 and 22.

additional factor that causes the actual incoming wax mass flux to be lower than the theoretical limit value. Previous studies of Venkatesan7 and Panacharoensawad4 suggest that this factor is the shear stress from the flow to the deposit interface. Flow shear stress may cause the incomplete in the formation of the newly form deposit layer, or the partial failure of wax particles to participate in the deposition process. Moreover, the precipitation rate (kr) alone cannot cause jin to be lower than jin,eq. The finding that jin can be lower than the theoretical lower limit is also consistent with the earlier observation4 for the case of South Pelto crude oil. Last, but not least, jin,h/m was calculated by using eq 28. jin,h/m and jin,eq are plotted against each other and shown in Figure 26.

■

CONCLUSIONS Paraffin deposition experiments in a mini pilot-scale flow loop were conducted to further elucidate the deposition behavior. A Garden Banks condensate was the test fluid. A total of 16 independent deposition tests with the testing duration of 2.67 h (2 h, 40 min), 8 h, 16 h, and 24 h were successfully completed. The tests were conducted by controlling the initial bulk oil temperature and the initial interphase temperature to be constant throughout the testing program by slightly adjusting the coolant temperature. The experimental data on the deposit thickness, deposit wax fraction, critical carbon number, average carbon number of the CCN+ fraction, and wax mass are reported. These data significantly contribute to the limited flow loop data, which are required for future model development and validation. Furthermore, the aging wax mass flux and incoming wax mass flux are analyzed. The approach used to analyze the experimental aging and incoming wax mass flux is new. It was derived from the transport-equations-based model originally developed in Fogler’s research group,2,3,6−8,57 which has been widely accepted and implemented in many later studies.1,4,5,10−13,18,31,32,43−45,62−66 The mass flow rate, Reynolds number (Re), velocity, wall shear stress heat-transfer coefficient, and Nusselt number (Nu)

Figure 26. Theoretical incoming wax mass flux graph calculated by using heat- and mass-transfer analogy and equilibrium method. The interface temperature used in the theoretical jin calculation was based on the experimental δmass and F̅w values.

As expected, the heat- and mass-transfer analogy approach yields higher incoming wax mass flux values, compared to those obtained from the solubility method and the experimental jin value. A linear relationship between jin,h/m and jin,eq was observed, and jin,h/m was determined to be ∼6.3 times higher than jin,eq. Theoretically, the ratio between jin,h/m and jin,eq can be written based on eqs 24 and 28, as jin,h/m jin,eq

= t

C b − C i ⎛ Sc ⎞1/3 1 ⎜ ⎟ dC /dT Tb − Ti ⎝ Pr ⎠

t

(31) P

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Energy & Fuels are coupled in this study. The DP thickness result confirms the thickness crossover behavior of the thickness versus time trends reported in the previous study.11 The CCN of all tests where the initial inner wall temperature was controlled to be 61.5 ± 0.7 °F was ∼30. The total wax mass (deposit mass × wax fraction) was found to increase with time and flow rate without the crossover between the wax mass versus time trends. The aging to incoming wax mass flux ratio based on the experimental result was analyzed. The analysis shows that the aging wax mass flux was ∼45% or more of the incoming wax mass flux for most cases, except the low-flow-rate 20 kg/min cases. This finding can be used to compare against model prediction for model validation purposes. The superficial wax crystal aspect ratio was calculated based on the aging wax mass flux ratio. It was found that the superficial wax crystal aspect ratio decreased with F̅w, instead of increase with F̅w as expected, based on the previous models3,6 that contain no shear term. This indicate that there is at least one additional factor, such as shear stress, that was not considered in the Lee model3 that causes the abnormally decreasing trend in the superficial wax crystal aspect ratio value. The experimental and theoretical incoming wax mass fluxes were analyzed based on the obtained experimental results. The analysis shows that the incoming wax mass flux are generally lower than the theoretical limit provided by the solubility method in the Venkatesan study7 (or kr → ∞ in the Lee model3). This finding confirms the abnormality in jin that is lower than the theoretical limit observed previously4 for the South Pelto crude oil case. The abnormality in jin shown in this study is also important for developing the wax deposition model. It suggests that at least one additional term, such as the shear effect term,4,7 is needed to account for the reduction of jin that is lower than the theoretical limit. The provided procedure for estimating the aging to incoming wax mass flux ratio from experimental results is new. This procedure serves as a tool for future studies to analyze the deposition wax mass flux ratio and compares the finding to the result of this study, or to validate a new wax deposition model. Last but not least, it should be emphasized here that not all deposition data in the literature can be used for model validation purposes. This is because some necessary information for wax deposition simulation (such as fluid properties) are missing in many literatures. However, this article provides all required information for developing wax deposition modeling in the appendices. This information includes fluid properties, fluid composition, and curve-fitting functions. The information is sufficient to the point where it can be used as reliable experimental results for future model development purposes.

μo [cP] =

1000

a

(10(10 ) − 0.7)

(A2)

where a = −3.9932 log10(T[°R]) + 10.7408

(A3)

The viscosity correlation given in eq A2 is based on the value reported in the previous works.51 It was confirmed by comparing the measured pressure drop and the calculated pressure drop based on the eq A2 viscosity function, and the match between the measured data and the predicted value was found. The wall shear stress in the pipe flow for rechecking eq A2 was in the range of 0.4−2.8 Pa (ref 5) and 12−60 Pa (ref 53). ⎡ J ⎤ Cp̂ ⎢ ⎥ = 3.816T[°C] + 1951.34 ⎣ kg K ⎦

(A4)

. ⎡ W ⎤ k⎢ = −2.18674 × 10−4T[°C] + (1.44927 × 10−1) ⎣ m K ⎥⎦ (A5)

fs [kg/kg] = 1.02290 × 10−5T[°C]2 − 7.10294 × 10−4T[°C] + (1.33214 × 10−2)

(A6)

Please note that eq A6 fits with the fs value but it is not a good estimate for dfs/dT. dfs ⎡ kg/kg ⎤ −7 2 ⎢ ⎥ = −2.34244 × 10 T[°C] dT ⎣ °C ⎦ + 3.00846 × 10−5T − (8.05855 × 10−4) (A7)

+wo[m 2/s] =

1.33 × 10−11 0.71

Va [cm 3 /g mol] × (μ[cP])γ

γ=

× (T[°C] + 273.15)1.47 (A846)

10.2 − 0.791 Va [cm 3 /g mol]

and Va [cm 3/g mol] = 5.23279 × 102

(A9)

The Garden Banks composition is given in Tables A1 and A2.

■

APPENDIX B The insignificant depletion of wax during the test is discussed in this section. The n-C30+ of the Garden Banks fluid is 0.4 wt %. The total oil mass in the facility was 243.5 lb (or 110.4 kg). This means that there was 0.4 × 110.4/100 = 0.442 kg or 442 g of n-C30+ in the system. The test where the maximum wax mass occurred is the 50 kg/min, 24 h test (condition A16, see Table 1), as shown in Figure 12. This maximum wax mass is 3.85 g (deposit mass × wax fraction = wax mass). The inner diameter (ID) and length of the heat exchanger section are 0.995 in. and 93 cm, respectively. Thus, the deposition surface in the heat exchanger is π × 0.995 × 2.54 × 93 = 738.4 cm2. The deposition surface for the test section case is π × 0.651 × 2.54 × 84 = 436.4 cm2. The coolant temperature and flow rate used in the case of the heat exchanger section are 50 °F and 7.35 kg/min. Whereas the coolant temperature and flow rate

■

APPENDIX A The information on fluid properties and all related parameters required for using this experimental result for model development purposes is given in this section. The Garden Banks condensate fluid properties, as a function of temperature, are given in eqs A1−A9. The given fluid properties are valid only in the range of temperature and pressure used in the experiment. The extrapolation of the fluid property functions to the condition outside the tion of this study is discouraged. The formulas are taken from both PVTsim61 calculation and measurement. The detailed discussion of each parameter can be found from the works of Singh53 and Rittirong.5 ρo [kg/m 3] = − 4.0977 × 10−1T[°F] + 859.79

ρo [kg/m 3]

(A1) Q

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Energy & Fuels Table A1. Garden Banks Condensate Composition32 component

amount (wt %)

component

amount (wt %)

C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26

0.43 1.43 3.40 5.88 6.11 5.80 5.44 4.47 4.80 5.17 3.40 4.14 2.96 3.37 3.74 2.58 2.46 2.31 2.17 1.92 1.76 1.71

C27 C28 C29 C30 C31 C32 C33 C34 C35 C36 C37 C38 C39 C40 C41 C42 C43 C44 C45 C46 C47 C47+

1.49 1.45 1.48 1.34 1.23 1.45 1.30 0.70 0.95 0.91 0.74 0.90 0.63 0.84 0.56 0.52 0.47 0.51 0.94 0.26 1.05 4.81

738.4

a

amount (wt %)

component

amount (wt %)

C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26

0.2036 0.7285 1.2544 1.4204 1.3588 1.2131 1.0843 0.9457 0.8632 0.7275 0.7886 0.6019 0.4992 0.4290 0.4694 0.3521 0.2905 0.2464 0.1839 0.1808 0.1825 0.1143

C27 C28 C29 C30 C31 C32 C33 C34 C35 C36 C37 C38 C39 C40 C41 C42 C43 C44 C45 C46 C47

0.0976 0.0678 0.0630 0.0642 0.0562 0.0488 0.0398 0.0376 0.0306 0.0225 0.0188 0.0124 0.0120 0.0099 0.0121 0.0073 0.0088 0.0098 0.0090 0.0037 0.0039

29

77.6 − 54.6

impact of temperature on wax deposition. The reason is that this multiplier is based on the assumption that the flow of coolant in both the heat exchanger and the test section are both turbulent and have a comparable Nu values. However, the flow in the heat exchanger section was laminar (lower Nu), so the actual impact of temperature on wax mass should be less than the ratio of 1.2. Moreover, v0, Re0, and τ0 in the test section for A16 were 4.68 m/s, 18 485, and 59.9 Pa, respectively, these values in the heat exchanger section are much less (v0 = 2.1 m/s, Re0 = 12 376, and τ0 = 13.3 Pa). With lower momentum transfer (lower v0, Re0, and τ0), it is expected that the wax mass per unit area of the heat exchanger case should be either equal or less than the case of A16, as can be roughly estimated from Figure 12. Yet, the mass of 9.45 g estimated earlier does not account the impact of this lower momentum transfer, so this value is the conservative approximation of wax depletion in the heat exchanger section. Therefore, at most, 9.45 + 3.85 = 13.30, or 3.0% of wax depleted during the deposition test. This value serves as the maximum depletion and the depletion should be less for other cases because the deposit mass can be either equal to or less than the A16 case. For the case of the maximum depletion, the n-C30+ in the system becomes 0.39% after the depletion, instead of 0.4%. This amount of change causes no visible change in the precipitation curve when TUWAX is used to estimate the solids fraction isobar (∼3% less concentration driving force or 3% thinner deposit). Moreover, if this 3% error is added into either Figure 20 or Figure 25, there is no impact on the conclusion drawn from those findings. Therefore, this uncertainty of 3% due to wax depletion during the test is negligible.

Table A2. Extended Nenninger n-Alkane Analysis for the Garden Banks Condensatea component

77.6 − 50

3.85 × 436.4 × 77.6 − 54.6 × 24 = 9.45 g . Approximately 3−5 h was required for the oil to cool from the melting temperature to the test temperature. The ratio of 29/24 is to account for the time that the heat exchanger was in contact with oil (29 h) and the time that the test section was in contact with oil (24 h). Yet, only some part of this 3−5 h cooling period can cause wax deposition. This is because the inner pipe wall temperature was not below WAT for the entire cooling period. The ratio of (77.6 − 50) to (77.6 − 54.6) is present to conservatively account for the impact of the coolant temperature in the heat exchanger part. It is well-known that laminar flow has a much smaller Nusselt number (Nu), compared to the turbulent flow case. Thus, the ratio 77.6 − 50 = 1.2 already overestimates the

■

APPENDIX C

Curve-fitting software67 was used to search the best function to fit δ and F̅w data. The curve-fitting functions for the deposit thickness are in the form of δ = A ln(t + B) + C, where δ and t are given in units of millimeters and hours, respectively. The constants A, B, and C in a scientific format for the deposit thickness cases are shown in Table C.1. The curve-fitting function and coefficient values for the deposit wax fraction cases are shown in Tables C.2 and C.3, where F̅w is dimensionless (w/w) and t is given in units of hours. The curve-fitting functions between F̅w and αs (in Figures 20 and 21) are given in eqs C1 and C2. The curve fitting functions for the combined cases of 30, 40, and 50 kg/min (the dashdotted line) is given as

Data taken from ref 32.

used in the test section for the maximum wax mass case (50 kg/min, 24 h, condition A16) are 54.7 °F and 19.1 kg/min, respectively. The laminar flow of water in the jacket of the heat exchanger was found (Re = 1800) for the case of condition A16. The coolant in the test section part was under turbulent flow conditions for all cases. To estimate the maximum possible deposition in the heat exchanger section, the deposit mass in the heat exchanger section can be estimated from the condition A16 deposit and the deposition area as R

DOI: 10.1021/acs.energyfuels.6b02125 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels Table C.1. Constant Values Used in the Curve Fitting of the δ vs t Function Value flow rate = 20 kg/min

constant

flow rate = 30 kg/min

−1

1.5043 × 10 1.4506 × 100 −6.3397 × 10−2

A B C

1.0906 × 10 7.5448 × 10−1 3.4487 × 10−2

Table C.2. Formula Used in the Curve Fitting of the F̅w vs t Functiona flow rate [kg/min] 20

Fw̅ = Fw,0 ̅ + A[1 − exp(Bt )]

30

Fw̅ = Fw,0 ̅ +

40

Fw̅ = Fw,0 ̅ + Fw̅ = Fw,0 ̅ +

A+t B + Ct

+ Dt

−B + B2 − 4A(C − t ) 2A −B + B2 − 4A(C − t ) 2A

F̅w,0 used in all formulas is 4.4318 × 10−3, which is the estimated wax fraction of oil. a

log10 αs =

A + (E − Fw̅ ) B + C(E − Fw̅ )2

+ D(E − Fw̅ )

(C1)

where A, B, C, D, and E are −7.403858 × 10−3, 1.357448, −9.748105, 2.566838, and 3.422391 × 10−1, respectively. The r-squared value (RSQ function in MS-Excel) for the curve fitting described by eq C1 is 0.9845. The curve-fitting function for the case of 20 kg/min is given as log10 αs =

1 + D ln(Fw̅ ) A(Fw̅ + B)2 + C

(C2)

where A = −4.130140 × 103, B = −7.439084 × 10−2, C = 1.978731 × 101, and D = −4.035097 × 10−1. The r-squared value of the curvefitting described by eq C2 is 0.9801. The curve-fitting functions for the continuous line of jin versus time in Figure 22 are given in eqs C3−C6. The functions for the case of 20, 30, 40, and 50 are given in eqs C3, C4, C5, and C6, respectively. jin,20 kg/min = −0.0132618t + 0.836985

(C3)

jin,30 kg/min = −0.0346002t + 1.37712

(C4)

jin,40 kg/min = −0.0301207t + 1.33134

(C5)

jin,50 kg/min = −0.0342754t + 1.34961

(C6)

−2

■

8.9525 × 10 6.5671 × 10−1 4.5357 × 10−2

flow rate = 50 kg/min 6.1888 × 10−2 2.6699 × 10−1 7.5897 × 10−2

APPENDIX D

The justification to neglect the heat of crystallization in the heat-transfer calculation of the deposition process is given in this section. The enthalpy of crystallization of multicomponent n-alkane can be lower than that of the shortest n-alkane component participating in the crystallization process when cocrystallization of n-alkane components occurs.64 The exothermic heat of crystallization of wax in the deposit is typically 200 J/g31 (roughly in the range of 100−300 J/g for wax in North Sea crude oil).68 For a single n-alkane component, the specific exothermic heat of crystallization (J/g) increases with the n-alkane chain length.64 The number of 600 J/g (3 times of the typical enthalpy of crystallization) is used to show that the heat released from the crystallization process is significantly less than the heat loss to the environment, even though the enthalpy of crystallization is overpredicted by a factor of 3. In this experimental program, the wax mass (given as deposit mass × F̅w) is 3.85 g at most. Therefore, the heat of 600 × 3.85 = 2310 J is released over 24 h or 2310/(24 × 3600) = 0.0267 W (averaged over 24 h). For the A16 case, where the highest wax mass was obtained, the initial heat flux at time zero was given as h(Tb − Ti0), or 2840 × (77.6 − 61.7)/1.8 = 25 086 W/m2 (values are taken from Table 1). The deposition area is roughly defined as π × 0.651 × 2.54 × 84 = 436.4 cm2 = 0.43636 m2. Therefore, the initial heat transfer is given as 25 086 × 0.043636 = 1094.7 W. If we assume that there is no heat loss at time t = 24 h (perfect insulation, no further deposition after 24 h), which is not true, this gives the most conservative approximation of the (lowest) average heat loss (in watts). Based on this conservative assumption, we can compute the average heat loss over 24 h to be (1094.7 + 0)/2 = 547.34 W. The heat released (in watts) due to the enthalpy of crystallization is 0.0267/547.34 × 100 = 0.004885% (≈ 0.0049%) of the heat loss to the coolant side, on average. This value is negligible. Therefore, the above estimation shows that the enthalpy of crystallization can be neglected in the calculation of heat flux at the interface, and can also be neglected in the temperature profile calculation.

formula

50

flow rate = 40 kg/min

−1

■

APPENDIX E The justification to approximate the temperature profile responses to the change in δ and F̅w instantly, with respect to the slow change in δ and F̅w, is given in this section. As time progress by 1 s, the deposit become thicker by a small increment, ε1s, and heat can penetrate into a stationary medium

The correlation coefficient (R2) values of eqs C3, C4, C5, and C6 are 0.9982, 0.9993, 0.9994, and 0.9997, respectively. The units used for jin and t in eqs C3−C6 are [mg/(ms s)] and hours, respectively.

Table C.3. Constant Values Used in the Curve Fitting of the F̅w vs t Function Value constant

flow rate = 20 kg/min

A B C D

1.3667 × 10−1 −2.3708 × 10−1 n/a n/a

flow rate = 30 kg/min 6.6864 1.8827 6.0264 4.1769

× × × ×

10−4 101 100 10−3 S

flow rate = 40 kg/min

flow rate = 50 kg/min

2.4557 × 102 7.7522 × 100 −6.3870 × 10−2 n/a

2.0248 × 102 1.4179 × 100 3.4697 × 10−3 n/a DOI: 10.1021/acs.energyfuels.6b02125 Energy Fuels XXXX, XXX, XXX−XXX

Energy & Fuels with the distance of ζ1s. For a stationary semi-infinite medium, the thermal penetration depth (ζ) can be calculated from ζ = 4 αT t

or (E2)

(see Bird et al. page 375, eqs 12.1−12.9), where αT is the thermal diffusivity of the medium and t is the time for heat to transfer. From Figure 3, the maximum growth rate of the deposit thickness can be estimated to be 0.1 mm per 1 h or 2.778 × 10−8 m/s. Therefore, for 1 s, at most, the deposit will grow ∼2.778 × 10−8 m or ε1 s = 2.778 × 10−8 m. The 1 s thermal penetration depth can be estimated by using typical values of oil phase, where k = 0.12 W/(m K), ρo = 835 kg/m3, and Ĉ p = 2000 J/(kg K). With these values, αT = k ̂ = 7.186 × 54

10 m /s and ζ1 s = 4 7.186 × 10 × 1 = 1.072 × 10−3 m. By comparing ζ1 s and ε1 s, it can be seen that heat transfer −3 occurs at a speed of 1.072 × 10−8 = 3.86 × 104 times faster than

NOMENCLATURE

Greeks Symbols

αT = thermal diffusivity; αT = k/(ρĈ p) [m2/s] δ = deposit thickness [m] ε1s = deposit thickness increment over 1 s [m] μ = viscosity [Pa s] ρ = density [kg/m3] τ = shear stress [Pa] ζ = thermal penetration depth [m] ζ1s = thermal penetration depth over 1 s [m]

2.778 × 10

the growth of the deposit. The thermal conductivity and thermal diffusivity of the deposit are higher than those of oil. Thus, ζ1 s in the deposit is higher than the ζ1 s value of the oil phase, as per eq E2. These findings indicate that the temperature profile developed is more than 10 000 times faster than the growth rate of the deposit. Therefore, it can be assumed that the temperature profile responds to the change in the deposit thickness immediately. At time zero, the deposit wax fraction was analyzed to be F̅w,0 s = 0.004074, as shown previously. From the aging data (Figure 9), F̅w was ∼0.07−0.08 at t = 1 h for the fastest aging case. The conservative F̅w value of 0.1 at t = 1 h (higher than the actual value) is used to analyze the impact of aging on the heat-transfer rate. With these F̅w values at 0 and 1 h, the value of F̅w at t = 1 s can be estimated as F̅w,1 s = 0.1 − 0.004074 + 3600 0.004074 = 0.004101. By assuming kw = 0.25 and ko = 0.12, kd was calculated from the Maxwell equation (eq 15) and found to be 0.120392 and 0.120390 for F̅w,1 s and F̅w,0 s, respectively. As per eq E2, the 1 s heat penetration depths of the F̅w,1 s and F̅w,0 s cases are 1.073991 and 1.073980 mm, respectively. It can be seen that aging the deposit for >1 s causes the thermal penetration depth to be 1.073991 − 1.073980 × 100 = 0.0011% (≈ 0.001%)

Operators

⟨X⟩ = time-averaged value of parameter X, averaged over the entire deposition time X̅ = space-averaged value of parameter X, averaged over the entire deposition section X̃ = normalized value of parameter X, based on its maximum and minimum values Variables

C = concentration of dissolved wax in oil [kg/m3] CCN = critical carbon number CCN+ = carbon number equal to or above the CCN CCN+av = average carbon number of the CCN+ fraction Ĉ p = specific heat capacity [J/(kg K)] D = diameter [m] +e = effective diffusivity of wax inside the deposit [m2/s] +wo = wax in oil diffusivity [m2/s] fs = precipitated solid fraction [kg precipitated wax/kg solution] F̅w = wax fraction of the deposit, averaged in the radial direction [kg/kg] h = convective heat-transfer coefficient [W/(m2 K)] j = wax mass flux [kg/(m2 s)] k = thermal conductivity [W/(m K)] km = convective mass-transfer coefficient [m/s] m = mass [kg] ṁ = fluid mass flow rate [kg/min] q = heat flux [W/m2] R = radial coordinate measured from the center of the pipe outward radially [m] R = inner radius of the test section [m] t = time [h] T = temperature [°F] ΔT̅ eff = Tb − Ti V = velocity [m/s]

1.073980

faster (or 11.4 nm longer) than the case where there is no aging; this value is negligible. Therefore, the temperature profile development for >1 s is negligibly impacted by the change of the deposit wax fraction over 1 s. In other words, the temperature profile can be assumed to be an immediate response to the change in the deposit wax fraction.

■

■

Nu = Nusselt number; Nu = hD/k Pr = Prandtl number; Pr = μĈ p/k Sc = Schmidt number; Sc = μ/(ρ +wo) Sh = Sherwood number; Sh = (k mD)/+wo Re = Reynolds number; Re = ρVD/μ

−8

2

ACKNOWLEDGMENTS

Dimensionless Quantities

ρCp

−8

■

We would like to thank Tulsa University Paraffin Deposition Projects members (BP, Chevron, CNOOC, ConocoPhillips, DSME, GE, Halliburton, Nalco-Champion, Petrobras, Total, and YPF) for their financial support and suggestions during the course of this study. The authors gratefully acknowledge Dr. Ramachandran Venkatesan from Chevron for providing the DSC precipitation curve information on Garden Banks condensate.

(E1)

ζ1 s = 4 αT

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Ekarit Panacharoensawad: 0000-0003-2556-3427 Present Address †

Bob L. Herd Department of Petroleum Engineering, Texas Tech University, 807 Boston Avenue, Box 43111, Lubbock, TX 79409, USA. Notes

The authors declare no competing financial interest. T

DOI: 10.1021/acs.energyfuels.6b02125 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels Subscripts

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0 = value at the initial time (at t = 0 s) age = aging av = average b = bulk fluid of the hot side C = coolant side of the test section d = deposit eq = equilibrium method H = hot side of the test section (test fluid side) h/m = heat- and mass-transfer analogy i = interphase between the deposit and oil (for t > 0 s) or between the inner pipe wall and oil (for t = 0 s) in = incoming from the bulk to the deposit interface inlet = at the inlet (the coolant inlet is at the same axial location as the oil outlet) max = maximum value min = minimum value o = oil w = wax part (not include oil in the deposit) z = value at location z Superscripts

x− = approaching x value from the left-hand side (the side that is slightly less than x) x+ = approaching x value from the right-hand side (the side that is slightly more than x)

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DOI: 10.1021/acs.energyfuels.6b02125 Energy Fuels XXXX, XXX, XXX−XXX