A steady-state heat-transfer model for solids deposition from waxy mixtures in a pipeline

Fuel 137 (2014) 346–359

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A steady-state heat-transfer model for solids deposition from waxy mixtures in a pipeline Samira Haj-Shafiei, Dalia Serafini, Anil K. Mehrotra ⇑ Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada

h i g h l i g h t s  We present a mathematical model for solids deposition from paraffinic mixtures in a pipeline.  The model is based on steady-state heat-transfer considerations.  The predicted deposit thickness increases in the hot flow regime and decreases in the cold flow regime.  The cold flow regime involves two-phase flow with wax crystals suspended in the liquid phase.  The modeling approach and predictions will be useful to flow assurance engineers.

a r t i c l e

i n f o

Article history: Received 26 April 2014 Received in revised form 31 July 2014 Accepted 31 July 2014 Available online 19 August 2014 Keywords: Solids deposition Waxy oil Cold flow Hot flow Heat transfer

a b s t r a c t A steady-state heat-transfer model is presented for the formation of a deposit-layer from wax–solvent ‘waxy’ mixtures in a pipeline under turbulent flow. The waxy mixture is taken to enter the pipeline under the single-phase hot flow regime (where the average mixture temperature is higher than its wax appearance temperature, WAT) and, upon gradual cooling, the mixture transitions into the cold flow regime (where its average temperature is lower than its WAT). The cold flow regime is characterized by two-phase flow, in which solid particles are suspended in the liquid phase. The effect of deposit aging is incorporated via a shear-induced deformation approach proposed in the literature. The model predictions are reported for the deposit thickness, waxy mixture temperature, pressure drop and the rate of heat loss in the hot flow and cold flow regimes for a range of inlet mixture temperature, surrounding temperature, and the Reynolds number. The predicted deposit thickness is shown to increase axially in the hot flow regime, to reach a maximum as the liquid temperature approaches the WAT of the wax– solvent mixture, and to decrease gradually to zero in the cold flow regime. The trends in the model predictions compare satisfactorily with those reported from bench-scale experimental studies as well as the predictions from an unsteady state moving boundary problem formulation. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Crude oils are complex mixtures of hydrocarbons, which are often subjected to large temperature gradients when flowing in pipelines, particularly in subsea environments. High molecular weight paraffin waxes (with carbon number ranging from 18 to 65) are soluble in crude oil at high temperature and pressure environments such as in petroleum reservoirs [1,2]. When highly paraffinic crude oil, called ‘‘waxy’’ crude oils, are subjected to a colder environment, the solubility of paraffin waxes decreases, resulting in their separation from the crude oil. The separated waxy solids can precipitate, deposit and build up as wax-like ⇑ Corresponding author. Tel.: +1 403 220 7406. E-mail address: [email protected] (A.K. Mehrotra). http://dx.doi.org/10.1016/j.fuel.2014.07.098 0016-2361/Ó 2014 Elsevier Ltd. All rights reserved.

species onto the pipeline wall, which cause major operating issues for the petroleum industry. The consequences of such phenomena are flow assurance problems, including reduced process efficiencies as a result of decrease in effective pipe diameter, an increase in pressure drop, and potential temporary or permanent shutdown of the operation [3]. A further concern is the difficulty in re-starting the pipeline filled with congealed crude oil, which often requires extremely high pressures or might not even be possible [4]. The transported crude oil cools down gradually as it flows through the pipeline until it reaches the surrounding temperature. The deposition starts when the pipe-wall temperature becomes less than the crude oil’s wax appearance temperature (WAT), which is the temperature at which the first wax particles separate from the crude oil. The deposit forming at the pipe-wall is a gel consisting of a waxy solid network entrapping a liquid phase [2,5]. The

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347

Nomenclature Ac

surface area of outer pipe wall in contact with cold surroundings (m2) Ah area of liquid–deposit interface (m2) Ai inside pipe surface area (m2) b tilting angle of cubical cage (degree) C mix p;l or C wax–solvent mixture specific heat capacity (J/kg K) D inside pipe diameter (m) d deposit thickness (m) Df change in solid wax phase mass fraction DHm latent heat of fusion (J/kmol) q rate of heat transfer (W) qd rate of heat transfer across deposit layer (W) qh rate of heat transfer from wax–solvent mixture (W) F volumetric flow rate (m3/s) g mole fraction hc convective heat transfer coefficient of coolant (W/m2 K) hh convective heat transfer coefficient for wax–solvent mixture (W/m2 K) (DHm)i enthalpy of melting or fusion for component i (J/kmol) mix k or kl thermal conductivity of wax–solvent mixture (W/m K) kd thermal conductivity of the deposit layer (W/m K) (kl)i thermal conductivity of wax (W/m K) (kl)j the thermal conductivity of solvent (W/m K) km thermal conductivity of pipe material (W/m K) L pipe length (m) DL length of axial pipe element for steady state calculations (m) m molar volume (m3/kmol) m mass flow rates of wax–solvent mixture (kg/s) l viscosity of wax–solvent mixture (Pa s) Dp pressure drop (Pa) qmix wax–solvent mixture density (kg/m3) l qi paraffin wax and solvent densities (kg/m3) Pr Prandtl number R inside pipe radius (m)

compositions of solid and liquid phases in the deposit vary across the deposit layer thickness as a result of changes in shear stress (caused by the flowing crude oil), temperature, and the composition of the flowing liquid. Over time, the solid wax deposit forms a stronger continuous wax crystal network resulting in a stiffer material [6]. The gel networks have a complex morphology, and their characteristics are greatly affected by wax composition [7], process conditions (i.e., temperature, shear) [8–10], and time [6,7]. The wax deposit shows non-Newtonian characteristics compared to the crude oil, which behaves as a Newtonian fluid at temperatures higher than WAT [6,11]. A number of experimental and modeling studies have been reported on the flow and rheological behavior of the oil and wax components in crude oil [2,6,11]. In general, when a warm ‘‘waxy’’ crude oil (at an average temperature higher than its WAT) flows through a pipeline in a cold environment (at a temperature lower than the WAT), it may experience two flow regimes, known as hot flow and cold flow [12–14]. In the hot flow regime, the average waxy mixture or crude oil temperature is higher than the WAT, where solids deposition takes place on the pipe wall. The deposit thickness increases axially as the temperature of the waxy mixture decreases until a maximum deposit thickness occurs when the crude oil temperature approaches the WAT. The cold flow regime is considered to start when the temperature of the crude oil becomes less than the WAT, resulting in the formation of wax crystals. In this regime, the precipitated solid crystals stay in the flowing waxy mixture

Ro Re Rei Dt Tc Td Th Thin Thout TL Tm Twi Two Td Ui

outside pipe radius (m) Reynolds number inlet Reynolds number time interval or residence time (s) surrounding temperature (K) liquid–deposit interface temperature (K) average wax–solvent mixture temperature (K) inlet wax–solvent mixture temperature (K) outlet wax–solvent mixture temperature (K) liquidus temperature (K) melting-point temperature (K) inside pipe wall temperature (K) outside pipe wall temperature (K) average deposit temperature (K) overall heat transfer coefficient based on inside area (W/m2 K) Vs solid phase volume fraction in deposit Vl liquid phase volume fraction in deposit VS solid phase volume fraction in untilted cubical cage VL liquid phase volume fraction in untilted cubical cage VSb solid phase volume fraction in tilted cubical cage VLb liquid phase volume fraction in tilted cubical cage w29 mass fraction of wax in wax–solvent mixture (mass%) w⁄29 liquid phase mass fraction of C29 wi mass fraction of component i (mass%) x rate of cooling (°C/min) y mass fraction of wax in wax–solvent mixture (mass%) z axial location (m) Zi (=n/a) ratio of the edge thickness to the side of the cubical cage /i and /j superficial volume fraction of paraffin wax and solvent Acronyms WAT wax appearance temperature (K) WPT wax precipitation temperature (K)

(or crude oil), giving rise to a suspension and the deposit thickness decreases [12,15]. This phenomenon is an important attribute of the cold flow technology for decreasing or preventing wax deposition [13]. There are several other commercial methods considered by flow assurance groups to prevent or remediate wax deposition, including chemical additives, thermal insulation, and mechanical treatments. All of these methods have underlying shortcomings, including high costs, environmental issues and viability [7]. The development of a more precise control strategy and technology would avoid risks for offshore pipelines, including the risk of their inspection and cleaning operations [7]. The ‘‘cold flow’’ technology has been proposed as an alternative approach for the transportation of waxy crude oils [12,13,15]. Merino-Garcia and Correra [13] reviewed existing patents on the methods for creating wax–oil suspension mixture for the cold flow technology. Several possible explanations have been suggested for the decreased deposition under cold flow conditions, including a lower thermal driving force between crude oil and surrounding temperature, crystallization of wax on the suspended wax particles (which act as nucleation sites), and a lowering of the wax appearance or cloud point temperature as the heavier paraffins gradually precipitate out of the solution [13]. Among these, the latter explanation suggests that, as a result of the preferential precipitation of wax constituents with higher carbon numbers, the liquid phase becomes leaner consequently with a lower WAT. It is pointed out that the suspended wax crystals in the liquid phase do not contribute to solids deposition [12,15].

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In recent studies, heat transfer and molecular diffusion have been considered to be the two most significant mechanisms responsible for the wax deposition [12,16–18]. These two mechanisms for modeling the wax deposition involve a few contradictory assumptions. The molecular diffusion approach assumes a radial concentration gradient of the wax molecules, which is considered to be caused by a temperature gradient, between the (warmer) flowing crude oil and the (cooler) pipe wall. A pseudo-steady-state form of the Fick’s diffusion equation, along with an energy balance, is used for modeling the time-dependent, unsteady-state growth of wax deposition [10,19]. The liquid–deposit interface temperature is estimated from the energy balance and it is predicted to increase gradually from the pipe-wall temperature initially to the WAT at steady state. Thus, an inherent assumption in the molecular diffusion mechanism is that the deposition occurs while the liquid–deposit interface temperature increases, and the deposit formation stops once the interface temperature becomes equal to WAT. However, this important assumption in the molecular diffusion approach, that the liquid–deposit interface temperature is variable, has not been verified experimentally. On the other hand, the heat-transfer modeling approach considers the deposition process to be analogous to a liquid-to-solid phase transformation or (partial) freezing process. It is based on the assumption that the liquid–deposit interface temperature remains constant and equal to the WAT of the liquid phase throughout the deposit growth process. This assumption of constant liquid–deposit interface temperature has been validated experimentally with prepared mixtures in batch cooling experiments under both static and sheared cooling conditions [20,21]. Several bench-scale experimental studies have reported the liquid–deposit interface temperature to be equal to the WAT over varying deposition times [20–24]. Furthermore, several experimental investigations have reported a relatively fast approach to the thermal steady-state during the deposition process, under both laminar and turbulent flow, which is a characteristic of thermallydriven processes [3,22,23,25]. However, despite several differences between the molecular diffusion and heat transfer mechanisms, the steady-state liquid–deposit interface temperature is taken to be equal to the WAT in both approaches. In the heat transfer mechanism, two calculation approaches have been used for the wax deposition process at unsteady state and steady state. The unsteady state heat-transfer approach is based on the moving boundary problem formulation, which has been used extensively to model freezing and melting processes, for which the liquid–solid interface temperature is defined [26]. This approach has been used to predict the deposit growth process in a pipeline under the hot flow regime and static conditions [14,27], laminar flow conditions [17,27] and turbulent flow conditions [14,28]. Recently, a moving boundary problem formulation was described for both the hot flow and cold flow regimes [14]. The steady state heat-transfer model has been used to analyze the bench-scale experimental wax deposition results for short tube lengths under laminar and turbulent flow conditions [12,22,23,27,29–32]. However, the steady-state heat-transfer modeling approach has not been applied to wax deposition in a long pipeline. This study extends the steady state heat-transfer modeling approach to predict the deposit thickness from a waxy mixture flowing in a pipeline under turbulent flow in both the hot flow and cold flow regimes. The proposed model is based on energy balance and heat transfer relationships. It is used to predict the deposit thickness profile and pressure drop in a pipeline. Although the calculations are made with a pseudo-binary wax–solvent mixture for simplifying the numerical calculations, the proposed model could be easily extended to deal with multicomponent mixtures or waxy crude oils.

1.1. Previous studies Some of the fundamental observations from the experimental and modeling studies reported by different research groups are summarized below. These findings have been used to corroborate the trends in the model predictions presented in this study. There are a number of important variables that affect the deposit-layer thickness (d) at steady state when a warm ‘‘waxy’’ oil or mixture exchanges thermal energy with the cold surroundings. These include the Reynolds number (Re), the temperature of waxy mixture (Th), the temperature of the surroundings (Tc), the temperature of the pipeline surface in contact with the oil (Twi), the liquid– deposit interface temperature (Td), the heat transfer coefficients (hh and hc), the average deposit thermal conductivity (kd), and the average pipe wall thermal conductivity (km). The effects of these variables on solids deposition have been investigated through experimental and modeling approaches [6,12,14,17,27]. The wax precipitation and deposition on the pipe wall are different, but related, processes. The wax precipitation process involves liquid-to-solid phase transformation and it is driven primarily by thermodynamic or equilibrium considerations, whereas the wax deposition on the pipe wall is a predominantly transport process which requires a thermal gradient [12,14]. Experimental studies have shown that the precipitated wax crystals may not contribute to deposit formation or growth [12,15]. Furthermore, it has been established that, at steady-state conditions, the deposit thickness is smaller under turbulent flow than under laminar flow [22,28]. The deposit thickness decreases with an increase in the Reynolds number (Re) under both laminar and turbulent flow conditions [12,14,22–25]. Also, the solid content of the deposit increases with time and Re as the deposit becomes enriched with heavier paraffin waxes and depleted in lighter paraffin waxes [22,23,28,29,33]. Even though the solid fraction in the deposit increases with Re and deposition time, the steady-state deposit thickness of the aged deposit was reported to be dependent on the heat transfer and phase equilibrium considerations [28]. The overall thermal driving force for heat transfer across the deposit layer is the difference between the bulk crude oil temperature and the surrounding temperature. The rate of heat transfer is also influenced by the sum of convective thermal resistances (due to the flowing crude oil and the surroundings) and conductive thermal resistances (due to the deposit layer and the pipe wall) [12]. Even though the overall temperature difference is important for the rate of heat transfer, the deposit-layer thickness is dictated by the ratio of the temperature difference across the deposit to the overall temperature difference or thermal driving force [22,23,25]. The important role of heat transfer in wax deposition from waxy crude oil has been identified in a number of studies [3,16,17,22,23,25,27,30,31,34–38]. For example, Merino-Garcia et al. [16] investigated the wax deposition phenomenon using a mass transfer approach and the gelation process with a heat transfer approach. They concluded that the gelation process is much faster when compared to the predictions from the mass transfer diffusion mechanism. Mehrotra and Bidmus [36] demonstrated that wax deposition could be prevented altogether if the flowing crude oil was maintained at a very high temperature. That is, even with a large thermal driving force due to a high crude oil temperature, no deposition could occur. It was also shown that, in the cold flow regime, when the liquid temperature approached the surrounding temperature, the deposit mass decreased substantially [12,14]. Whereas the liquid–deposit interface temperature is equal to the WAT of the flowing liquid in the hot flow regime, it has been found to be equal to the lowered WAT of the (diluted) liquid phase in the cold flow regime. Furthermore, in the cold flow regime, with a gradual decrease in the deposit thickness, the difference between the average liquid mixture temperature, Th, and the liquid–deposit

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interface temperature, Td, was found to be relatively constant with an average difference of about 2 °C [12]. 2. Model development and calculations In the following sections, the steady-state modeling approach is described for the deposit formation from a waxy mixture over the hot flow and cold flow regimes in a pipeline under turbulent flow. This approach is different from the unsteady-state moving boundary problem formulation [16]. The model is used to predict the effects of inlet mixture temperature (Thin), surrounding temperature (Tc), and inlet Reynolds number (Rei), on the deposit thickness, wax–solvent mixture temperature, pressure drop and rate of heat loss along axial direction (i.e., the pipe length). The transition from the hot flow regime to the cold flow regime is achieved by incorporating the effect of cooling rate on the precipitation of wax crystals in the bulk liquid phase. The effect of aging on the deposit solid is incorporated by using the shear-induced deformation approach [29]. Deposit aging has been reported to yield a stronger and stiffer deposit layer, with a higher solid phase content, which also results in an increase in the deposit-layer thermal conductivity [18,19,23,39–41]. The trends in the model predictions have been validated with the previous experimental results obtained in the hot and cold flow regimes as well as the predictions from the unsteady-state moving boundary problem formulation for the hot flow and cold flow regimes. A schematic of the wax deposit thickness profile in the axial direction and temperature profile through thermal resistances are given in Fig. 1. The steady-state heat transfer model is developed for a pipeline of length L and inside diameter D, which is exposed to cooler surroundings at a constant temperature, Tc (
_ q ¼ mCðT hin  T hout Þ ¼ U i Ai ðT h  T c Þ

ð1Þ

_ and C are the mass flow rate and average specific heat where m capacity of the wax–solvent mixture, respectively; Thin and Thout are the inlet and outlet temperatures across DL; Ui and Ai are the overall heat transfer coefficient and surface area for radial heat

transfer at the inner pipe wall, respectively; Th is the average temperature of the wax–solvent mixture (i.e., Th = (Thin + Thout)/2); and Tc is the temperature of the surroundings. When a hot wax–solvent mixture (above its WAT) flows through a pipe, with the pipe-wall temperature below the WAT, an outward radial transfer of its thermal energy would occur as a result of the radial temperature difference. This would lead to the formation of a wax deposit layer that would grow in thickness until a thermal steady-state is reached. The wax deposit layer would offer an additional thermal resistance to the transfer of thermal energy from wax–solvent mixture to the surroundings. At the thermal steady-state condition, the rates of heat transfer from the flowing wax–solvent mixture and across the deposit layer, the pipe wall and the surroundings would become equal. That is, once the deposit thickness stops increasing, all thermal resistances would become constant in magnitude; these thermal resistances are offered by the flowing wax–solvent mixture (convection), the deposit layer (conduction), the pipe wall (conduction), and the surroundings (convection) [3,22,23,25]. The overall thermal resistance can be expressed as the sum of the four individual thermal resistances in series:

1=½U i Ai  ¼ 1=½hh Ah  þ ½lnðR=ðR  dÞÞ=½2pkd DL þ ½lnðRo =RÞ=½2pkm DL þ 1=½hc Ac 

where R is the inside pipe radius; Ro is the outside pipe radius; d is the deposit thickness; hh and hc are the convective heat transfer coefficients for wax–solvent mixture and surrounding, respectively; kd and km are the thermal conductivities of deposit layer and pipe wall, respectively; and Ai, Ah and Ac are the areas for radial heat transfer at the inner pipe-wall, at the liquid–deposit interface, and at the outer pipe-wall in contact with the surroundings, respectively. Note that, for a clean pipe with no deposit layer, the second thermal resistance would not exist and Ah  Ai. At the thermal steady state, with an average deposit-layer thickness, d, for the pipe length, DL, the equality of heat fluxes through each thermal resistance is expressed as follows:

q=Ai ¼ ½hh ðT h  T d Þ=½R=ðR  dÞ ¼ ½kd ðT d  T wi Þ=½R lnðR=ðR  dÞÞ ¼ ½km ðT wi  T wo Þ=½R lnðRo =RÞ ¼ ½hc ðT wo  T c Þ=½R=Ro 

Hot Flow

Td

T wi

Cold Flow

R Ro

δ: Deposit Thickness

Tc R

δ Pipe Thickness

ð3Þ

where Two is the outside pipe-wall temperature. The thermal resistances due to the pipe-wall and the surroundings typically have relatively small contributions to the rate of heat transfer when compared to those due to the wax–solvent mixture and the deposit layer [3,25]. This is because of the relatively large magnitudes of km and hc. Therefore, to simplify the iterative calculations, only the two thermal resistances corresponding to the wax–solvent mixture (convection) and the deposit layer (conduction) were

Th

T wo

ð2Þ

L

Ro Fig. 1. Schematic representation of steady-state deposit thickness profile in hot flow and cold flow regimes and temperature profile across thermal resistances.

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considered in this study. Note that this simplification implies Twi  Two  Tc. Accordingly, the following expressions for the rate of heat transfer, based on the individual thermal resistances, were obtained by rearranging Eq. (3):

qh ¼ hh ðR  dÞð2pDLÞðT h  T d Þ

ð4Þ

qd ¼ ½kd ð2pDLÞðT d  T c Þ=½lnðR=ðR  dÞÞ

ð5Þ

The three expressions for the rate of heat transfer over a pipe length, DL, given by Eqs. (1), (4) and (5), could be solved simultaneously to predict Thout and d from the estimated values of other _ and C. Note that, for the hot flow regime, variables, such as hh, kd, m Td is equal to WAT whereas Td would decrease gradually in the cold flow regime as the wax particles precipitate out to form a suspension, as explained later in more detail. The estimated value of Thout was also used to determine the rate of cooling, x, as follow:

x ¼ ðT hin  T hout Þ=Dt

ð6Þ

where Dt is the average residence time for the axial element, DL, which was obtained by dividing the volume of the liquid phase by the volumetric flow rate, F.

Dt ¼ ½pðR  dÞ2 DL=F

ð7Þ

For the turbulent flow of wax–solvent mixture, the heat transfer coefficient, hh, was estimated using the following Dittus–Boelter correlation [42]:

hh ¼ ð0:023Re0:8 Pr0:3 kÞ=ðD  2dÞ

ð8Þ

where k is the thermal conductivity of wax–solvent mixture, Re is the Reynolds number, and Pr is the Prandtl number. Re and Pr were estimated as follow:

Re ¼ ½4qmix l F=½plðD  2dÞ

ð9Þ

Pr ¼ C l=k

ð10Þ

mix l ,

where q l and C are the density, viscosity and average specific heat capacity of the wax–solvent mixture, respectively (for which the estimation methods are included in Appendix A). The frictional pressure drop for the axial element DL was calculated by accounting for a decrease in pipe diameter and assuming a uniform deposit layer with smooth surface as follows [43,44]: n

5n 2n Dp ¼ ½ð2cqmix Þ½l=qmix l DLÞ=ððD  2dÞ l  ½4F=p

ð11Þ

where Dp is the pressure drop, and n = 0.2 and c = 0.046 for turbulent flow. 2.1. Calculation procedure – hot flow regime The input quantities for the numerical calculations were the inlet temperature of wax–solvent mixture, Thin, composition as the mass fraction of wax, w29, the constant inside pipe-wall temper_ Note ature, Twi (=Tc), the pipe diameter, D, and the mass flow rate m. that Thin for the very first DL is the same as the temperature of wax– solvent mixture entering the pipe, Thin. The estimation methods for various properties (e.g., C, qmix l , l, kd) are reported in Appendix A. It is pointed out that the rate of thermal energy lost (q) by the wax–solvent mixture, given by Eq. (1), is a function of Thout. The rate of convective heat transfer from the wax–solvent mixture, given by Eq. (4), is a function of Thout and d. Finally, the rate of conductive heat transfer across the deposit layer, given by Eq. (5), is a function of d. Therefore, an iterative procedure was used to solve Eqs. (1), (4) and (5) for obtaining q, Thout and d for each axial element, DL. All the deposit properties were estimated at its average temperature. For the next axial element, the inlet temperature, Thin, was taken as Thout of the preceding element. The same calculations

were performed iteratively on successive axial elements. With gradual heat loss, Th decreased and approached the WAT (=Td) of the wax–solvent mixture, the thermal driving force for the convective term decreased, which in turn decreased the rate of change of temperature of the wax–solvent mixture. An important effect of the rate of change of temperature, or the cooling rate, of the wax–solvent mixture on phase transformation is described in the next section. 2.2. Transition from hot flow regime to cold flow regime The transition from hot flow to cold flow was investigated experimentally by Bidmus and Mehrotra [12,14,20,21]. There are two factors that need to be considered as the flow transitions from the hot flow to the cold flow. Whereas the wax–solvent mixture exists as a single-phase liquid in the hot flow regime, it becomes a two-phase suspension (i.e., precipitated solid crystals suspended in liquid) in the cold flow regime. As a result of the precipitation of the solid phase (comprising the relatively heavier paraffins), the remaining liquid phase becomes leaner and consequently its WAT is decreased (depending upon the extent of wax precipitation). Thus, whereas the WAT remains constant in the hot flow regime, the WAT of the liquid phase would decrease with gradual cooling and increased precipitation of wax crystals. Furthermore, one significant change in the thermal behavior takes place as the flow transitions from hot flow to cold flow. In the hot flow regime, the flowing liquid exists as a single-phase solution and the latent heat is released at the liquid–deposit interface. In the cold flow regime, the flowing stream is actually a two-phase suspension (with wax crystals suspended in a dilute solution), for which the latent heat is released during crystal formation as well as at the interface; i.e., there are two latent heat terms. This would be expected to result in a discontinuity in the deposit thickness profile. The second factor that influences the phase transition is the cooling rate, which affects the kinetics of crystallization. It has been shown that the onset temperature of crystallization decreases as the cooling rate is increased, which is referred to as the supercooling effect that is attributed to the nucleation and crystallization kinetics [14,45,46]. By conducting experiments on two waxy mixtures, Paso et al. [45] showed that the wax precipitation temperature, WPT, increases as the cooling rate is decreased. In a recent study by Kasumu et al. [47], experimental results were provided for several prepared solutions of paraffin wax in a multi-component solvent. They correlated their results with the following relationship for the WPT in terms of the cooling rate (x, varying from 0.05 to 0.4 °C/min) and the wax concentration (y, varying from 2 to 20 mass%).

WPT ¼ 24:17  0:21  4:16  0:49ðxÞ þ 6:68  0:09 lnðyÞ

ð12Þ

Eq. (12) indicates that, for a given wax solvent mixture considering constant WPT, the transition from hot flow (single liquid phase) to cold flow (two phases of liquid and solid) occurs with a decrease in the cooling rate. The significance of the cooling rate in transitioning from the hot flow regime to the cold flow regime has been explained fully elsewhere [14]. For Eq. (12), the term wax precipitation temperature (WPT), which is measured at a constant and controlled cooling rate, was used to distinguish it from the wax appearance temperature (WAT), which is measured while cooling the sample in an uncontrolled stepwise manner [14]. The above experimental finding (Eq. (12)) was used to model the transitioning from the hot flow regime to the cold flow regime. For each DL, the cooling rate was calculated and compared to that estimated for the phase transition from Eq. (12). When the cooling rate became less than the transition point from a single liquid phase to the two phases of solid and liquid, it was taken to be the first appearance of a solid phase and a transition from the hot flow regime to the

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cold flow regime. The appearance of a solid phase necessitated changes in the calculation method, as described in the next section. 2.3. Calculation procedure – cold flow regime As mentioned previously, the cold flow regime is characterized by the presence of precipitated wax crystals in a suspension form. Throughout the cold flow regime, the average temperature of the flowing suspension, Th, would be less than the WAT of the original wax–solvent mixture. Also, in the cold flow regime, the wax concentration in the liquid phase of the wax–solvent mixture would decrease gradually over the pipe line length. The precipitation of waxes would also decrease the WAT of the liquid phase along the pipe length (due to a decreased dissolved wax concentration). Thus, in the cold flow regime, the WAT would decrease for each successive element, DL. The energy balance relationship, Eq. (1), was modified to include a latent heat term for the phase transformation (corresponding to the precipitated wax crystals), as follow:

_ _ Df DH m q ¼ mCðT hin  T hout Þ þ m

ð13Þ

where Df is the incremental change in solid phase fraction (calculated from lever rule, Appendix A) and DHm is the latent heat of fusion for C29. Furthermore, for each DL in the cold flow regime, even though Th was taken to be the same as the WAT, the liquid– deposit interface temperature, Td, was taken to be below the WAT. The assumption of Td < WAT is supported by the experimental results of a bench-scale flow-loop study by Bidmus and Mehrotra [12], who reported that the difference between Th and Td in the cold flow regime remains constant with an average difference of about 2 °C; i.e., (Th  Td)  2.0. Considering that the wax crystals precipitate out into the wax–solvent mixture while the cooling occurs in each DL, Th should be at WAT of the liquid phase for such precipitation to occur, whereas a thermal driving force, (Th  Td), is necessary for the transfer of sensible and latent heat energy across the convective thermal resistance (from the liquid region to the interface). Note that, in a recent modeling study [14], a much lower value of (Th  Td) = 0.1 was used. Referring to Eq. (4), the rate of heat flow, qh, across the convective thermal resistance is proportional to (Th  Td). To highlight the importance of this term, the predictions are reported from additional calculations with a few selected values of (Th  Td). Eqs. (8) and (11) were also used for estimating the heat transfer coefficient (hh) and pressure drop (Dp) in the cold flow regime. The presence of solid crystals is expected to impart non-Newtonian characteristics depending on the size, shape and concentration of solid particles; however, this information is presently not available for the wax–solvent mixture under consideration. Hence, the predictions for pressure drop in the cold flow region may have significant uncertainties and these should be used with caution. The cold flow regime calculations were continued until the deposit thickness became zero and Th approached Twi (=Tc). 2.4. Inclusion of deposit aging effects Additional calculations were undertaken to study the effect of deposit aging on the deposit thickness in both the hot flow and cold flow regimes. As mentioned previously, aging refers to the growth of a stronger and stiffer solid wax network in the deposit. The main effect of aging is that it causes the deposit to become enriched in heavier paraffins and depleted in lighter paraffins, which also results in an increase in the deposit thermal conductivity [18,19,23,39–41]. The deposit aging was incorporated by using the shear-induced deformation approach proposed by Bhat and Mehrotra [29] and used subsequently in the moving boundary problem approach for solids deposition in a pipeline

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[33]. Their model considers the tilting of a one-dimensional cubical cage, which causes the release of a fraction of liquid from the deposit matrix, leading to wax enrichment and a shift in the carbon number distribution in the deposit. Even though this simplified representation was not intended to reproduce the crystalline structure of the deposit or the configuration of solid and liquid phases in the gel-like deposit, it offered a simplified framework to account for the effect of shear stress on the deposit [29,33]. In the shear-induced deformation approach [33], it is assumed that, in the absence of any shear stress, the deposit has a lattice-like structure with a unit cell that resembles a cubical cage of side a and volume a3. Each edge of the cubical cage is assumed to be rectangular in cross-section, n  n. The solid fraction is assumed to form the edges of the cubical cage while the liquid fraction fills all of the void space. The ratio of the edge thickness to the side of the cubical cage is: Zi = n/a. The solid and liquid phase volume fractions in the cubical cage are [35]:

V S ¼ ð12Z 2i  16Z 3i Þ

ð14Þ

V L ¼ 1  ð12Z 2i  16Z 3i Þ

ð15Þ

The value of Zi was calculated from VS and VL, which were obtained from the lever rule (Appendix A). The effect of shear stress (in the flow direction) on the deposit is manifested by a tilting of the cubical cage by an angle b, which causes the release of a portion of the liquid phase trapped in the cubical cage. The solid and liquid volume fractions, following the tilting of the cubical cage, are given by [35]:

V Sb ¼ ðV S VÞ=V b ¼ ð12Z 2i  16Z 3i Þ=ðcos bÞ

ð16Þ

V Lb ¼ 1  ð12Z 2i  16Z 3i Þ=ðcos bÞ

ð17Þ

Using experimental compositional data from analysis of deposit samples collected from a flow-loop apparatus (using 10 mass% and 15 mass% wax–solvent mixtures), a correlation was developed for the effect of Re and aging time on the deformation angle, b (in degrees) [22]. For the steady-state case (with aging time ? 1), the simplified correlation for 10 mass% wax–solvent mixture is:

b ¼ 79:394 þ 7:256=½1 þ expððlog Re  4:411Þ=0:220Þ

ð18Þ

In the calculations, Zi was estimated from the phase equilibrium considerations. Knowing Re for the axial element, b was estimated from Eq. (18), which was substituted into Eqs. (16) and (17) to obtain VSb and VLb for the aged deposit layer. Next, VSb and VLb were used for estimating the deposit thermal conductivity and the corresponding deposit thickness. 3. Results and discussion As described previously, the three equalities for the rate of heat transfer in the hot flow regime, given by Eqs. (1), (4) and (5), were solved iteratively for each incremental pipe length, DL. Once the rate of cooling for the wax–solvent mixture decreased below the transition point corresponding to the phase transition from a single liquid phase to the two phases of solid and liquid, the flow was taken to shift from hot flow regime to cold flow regime. Thereafter, the three equalities for the rate of heat transfer in the cold flow regime, given by Eqs. (13), (4) and (5), were solved for each DL. These calculations were continued until the deposit thickness in the axial element was predicted to be zero, at which point the temperature of the wax–solvent mixture approached Twi (=Tc). A set of base case conditions, summarized in Table 1, was chosen for the calculations. Table 1 also includes the values of other parameters considered for comparison with the base case conditions and previous studies.

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Table 1 List of conditions used for the base-case scenario and the other cases for the steady state wax deposition model in the hot and cold flow regimes. WAT refers to the original wax–solvent mixture WAT. Variables

Base-case

Other cases

Inside pipe diameter (D) Wax content of mixture (mass% w29) Inlet wax–solvent mixture temperature (Thin, K)

0.1 m 10% WAT + 15

Inside pipe wall temperature (Twi, K)

WAT  15

Inlet Reynolds number, Rei

15,000

Th  Td

0.5

– – WAT + 5 WAT + 10 WAT  5 WAT  10 25,000 30,000 0.1 1 2

3.1. Predictions for the base case conditions Presented in Fig. 2(A) are the predictions for the variation of wax deposit thickness and wax–solvent mixture temperature as a function of axial distance (z/D) for the base case scenario. Fig. 2(B) presents the predictions for the pressure drop and the rate of heat loss as a function of axial distance (z/D). In Fig. 2(A), the temperature of the wax–solvent mixture (Th) decreases in the axial direction as the thermal energy is transferred to the colder surroundings. At approximately, z/D  1140, Th approaches the WAT. With a decreased cooling rate, the transition from the hot flow regime to the cold flow regime is predicted to take place at this axial location. In the cold flow regime, at z/D > 1140, the rate of decrease of Th is predicted to be lower because of the transfer of both the sensible heat and latent heat (due to the crystallization

(A)

15 10

0.3

5 0

0.2 -5 0.1

Th -WAT (K)

Deposit Thickness (δ/R)

0.4

-10 -15

0.0 0

2000

4000

6000

8000

10000

(B)

Pressure Drop (∆P/∆L)

140

500

120 400

100

300

80 60

200

40 100

20 0 0

2000

4000

6000

8000

Rate of Heat Loss (q/∆L)

600

160

0 10000

Axial Distance (z/D) Fig. 2. Predicted results for the base case set of conditions as a function of axial distance: (A) wax deposit profile (left axis) and wax–solvent mixture temperature (right axis) and (B) pressure drop (in Pa/m) (left axis) and rate of heat loss (in W/m) (right axis).

of a fraction of waxes) as well as a decreasing rate of heat transfer due to a gradually lower thermal driving force, (Th  Tc). The cooling process continued until Th approaches Tc. The deposit thickness in Fig. 2(A) shows a graduate increase in the hot flow regime, reaching a maximum as Th approaches WAT at the transition (i.e., z/D  1140), and then a gradual decline in the cold flow regime. Note that, in the hot flow regime, the deposit thickness is predicted to increase axially with a decrease in the overall temperature difference, (Th  Tc), which would correspond to a lower rate of heat transfer. This shows that the deposit thickness is not dependent on the rate of heat transfer. The maximum deposit thickness is predicted to be d/R  0.4 at the transition point. For the base case set of conditions, the deposit in the hot flow and cold flow regimes is predicted to occur over z/D  8420, or about 842 m of the 0.1-m diameter pipeline. Beyond 842 m of the pipe length, there is no deposit formation because Th approaches Tc, and the wax–solvent mixture would continue to flow as a solid + liquid suspension. It is emphasized that the trend of the deposit-thickness profile in Fig. 2(A) agrees well with the predictions from the moving boundary problem formulation for wax deposition [14]. Fig. 2(B) presents the predictions for the pressure drop per unit length (Dp/DL) along the pipe length (z/D) for the hot flow and cold flow regimes. The predictions for the pressure drop follow the same pattern as the deposit thickness profile. According to Eq. (11), pressure drop is inversely proportional to effective pipe diameter (D–d). Thus, in the hot flow regime, when the effective pipe diameter gradually decreases as a result of increasing deposit thickness, the pressure drop increases. However, the pressure drop decreases in the cold flow regime as the effective pipe diameter gradually increases with the decrease in the wax deposit thickness. The pressure drop also depends directly on the liquid velocity, which increases as the deposit thickness increases. In addition, pressure drop is also dependent on the wax–solvent mixture viscosity, which increases with a decrease in Th. Fig. 2(B) also presents the predictions for the rate of heat loss (q/DL) as a function of axial distance (z/D). The hot flow and cold flow regimes are distinguished in this plot by the change in the slope of the curve. The rate of heat loss decreases sharply in the hot flow regime from q/DL  536 W/m initially to q/DL  23 W/m at z/D  1140. In the cold flow regime, q/DL decreases from about 23 W/m to about 13 W/m at z/D  7278. These predictions indicate that the rate of heat loss is much higher in the hot flow regime. In the cold flow regime, a lower rate of heat loss corresponds to a slower rate of decrease in Th, as shown in Fig. 2(A). Fig. 3(A) presents the experimental deposition results from a bench scale flow-loop apparatus operated in the hot flow and cold flow regimes for 3 mass% and 6 mass% wax solutions at two flow rates [12]. The abscissa of Fig. 3(A) is the difference between the wax–solvent mixture temperature and the corresponding WAT, such that the results to the left of the WAT are for the hot flow regime (Th > WAT) and to the right are for the cold flow regime (Th < WAT). In Fig. 3(B), the results of Fig. 2(A) have been re-plotted in a similar manner as Fig. 3(A). The predictions in Fig. 3(B) match with the trends in the experimental deposition data in Fig. 3(A), especially considering that the data in Fig. 3(A) were obtained using a bench-scale apparatus with a short deposition section, a short entrance region, and at lower Reynolds numbers. In the hot flow regime, the wax concentration of the wax– solvent mixture has a constant value of 10 mass% C29, which is the same as inlet mixture. Note that, at steady state without any accumulation, the composition of the wax–solvent mixture will be constant throughout. In the cold flow regime, however, the (dissolved) wax concentration of the wax–solvent mixture was found to decline steadily to about 1 mass% C29 for the base case scenario, which is due to the precipitation of wax crystals giving

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Deposit Thickness (δ/R)

0.4

(A)

and Mehrotra [12]. They reported an average difference value of about 2 °C between Th and Td ((Th  Td)  2.0). The rate of heat flow, qh, across the convective thermal resistance (Eq. (4)) is proportional to (Th  Td). To highlight the importance of this term, Fig. 5 presents the predictions for the variation of wax deposit thickness and wax– solvent mixture temperature as a function of axial distance (z/D) for four values of (Th  Td), i.e., (Th  Td) = 2, (Th  Td) = 1, (Th  Td) = 0.5, and (Th  Td) = 0.1. In Fig. 5(A), the axial distance for the hot flow regime is approximately the same for all values of (Th  Td), while, in the cold flow regime, the axial distance varies significantly. For example, the deposit thickness is predicted to approach zero at z/ D  2500, 5000, 8300 and 27,000 for (Th  Td) = 2, (Th  Td) = 1, (Th  Td) = 0.5, and (Th  Td) = 0.1, respectively. In addition, the predicted maximum deposit thickness at the transition from the hot flow to the cold flow regimes are also significantly different for different values of (Th  Td). The effect of (Th  Td) on the predicted temperature profile of the wax–solvent mixture is shown in Fig. 5(B). Again, despite a similarity in the hot flow regime, the profiles in the cold flow regime are different for the four cases, and Th approaches Tc at different axial locations. These predictions indicate that the temperature difference across the convective thermal resistance, (Th  Td), significantly affects the deposit thickness and temperature profiles in the cold flow regime, which is a topic for future experimental studies.

Re=1030 - 2650 (3 mass%) Re=1680 - 4200 (3 mass%) Re=1030 - 2650 (6 mass%) Re=1680 - 4200 (6 mass%)

0.3

0.2

0.1

0.0

Deposit Thickness (δ/R)

0.4

(B)

Re= 15000 (10 mass%)

0.3

0.2

0.1

0.0 2

0

-2

-4

-6

-8

-10

3.2. Effect of inlet temperature, Thin

Th-WAT (K) Fig. 3. Experimental results and model predictions for deposit thickness as a function of wax–solvent mixture temperature in hot flow and cold flow regimes: (A) experimental results for 3 mass% and 6 mass% wax (data re-plotted from [12]) and (B) model predictions for 10 mass%.

rise to a solid + liquid suspension. Fig. 4 shows the predicted increase in the (solid) wax content of the suspension for different surrounding temperatures (Tc). For example, for the base case scenario, with Tc = WAT  15, the suspension has 9 mass% C29 ultimately in the cold flow regime, (i.e., only 1 mass% C29 is remaining in the wax–solvent mixture). In Fig. 4, the solid wax content in the suspension decreases with an increase in Tc, which is because the solubility of the wax increases with temperature. In the hot flow regime, the liquid–deposit interface temperature, Td, was taken to be the same as WAT of the wax–solvent mixture. On the other hand, in the cold flow regime, even though Th was taken to be the same as the WAT, the Td was taken to be less than the WAT. The assumption of Td < WAT is supported by the experimental results of a bench-scale flow-loop study by Bidmus

Fig. 6 presents the predicted results for three different wax–solvent mixture inlet temperatures (i.e., Thin = WAT + 5, WAT + 10, and WAT + 15), while all of the other parameters were held at their base case values. Fig. 6(A) shows the predictions for the deposit thickness as a function of pipe length (z/D) for different Thin. The axial distance (z/D) for the hot flow regime is shorter for a lower Thin. For example,

0.8

Deposit Thickness (δ/R)

4

(A)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

10000 15000 20000 25000 30000

15

10

(B)

10

8

Th-WAT (K)

Solid Wax Content (mass%)

5000

6 4

5 0 -5 -10

2

-15 0

0 0

2000

4000

6000

8000

10000

5000

10000 15000 20000 25000 30000

Axial Distance (z/D)

Axial Distance (z/D) Fig. 4. Predicted results for solid content in the suspension for different values of Tc. Tc = WAT – 15 —, Tc = WAT – 10 – – , and Tc = WAT – 5 – – –.

Fig. 5. Effect of difference between Th and Td (Th  Td) in the cold flow regime along the axial direction of a pipeline: (A) deposit thickness profile and (B) wax–solvent mixture temperature. Th  Td = 2 —, Th  Td = 1 – , Th  Td = 0.5 – – –, Th  Td = 0.1 – – .

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0.45

(B)

10

0.35 0.30

Th-WAT (K)

Deposit Thickness (δ/R)

15

(A)

0.40

0.25 0.20 0.15 0.10

5 0 -5 -10

0.05

-15

0.00 0

2000

4000

6000

Pressure Drop (ΔP/ΔL)

140

8000

0

10000

(C)

120 100 80 60 40 20

2000

4000

6000

8000

1000 0

600

Rate of Heat Loss (q/ΔL)

160

(D)

500 400 300 200 100 0

0 0

2000

4000

6000

8000

1000 0

Axial Distance (z/D)

0

2000

4000

6000

8000

10000

Axial Distance (z/D)

Fig. 6. Effect of Thin in the axial direction of a pipeline: (A) deposit thickness profile; (B) wax–solvent mixture temperature; (C) pressure drop (in Pa/m); and D) rate of heat loss (in W/m). Thin = WAT + 15 —, Thin = WAT + 10 – – , and Thin = WAT + 5 – – –.

the hot flow regime is predicted to end at z/D  936, 839 and 648 for WAT + 15, WAT + 10 and WAT + 5, respectively. A higher Thin requires more pipe distance to lose the additional sensible heat in order for the liquid to cool to the WAT. According to Fig. 6(A), a lower Thin is also predicted to yield a somewhat larger deposit thickness in the hot flow regime. However, the effects are opposite in the cold flow regime, where a higher Thin is predicted to yield a slightly larger deposit thickness. The maximum deposit thickness at the transition from the hot to cold flow regimes is slightly larger for a higher Thin. These results are in agreement with the predictions using the moving boundary problem formulation [14]. The predicted temperature of the wax–solvent mixture, in terms of (Th  WAT), as a function of the axial distance (z/D) is shown in Fig. 6(B). Here, the transition from the hot flow regime to the cold flow regime occurs as (Th  WAT) ? 0. Despite a variation in the hot flow regime, the changes in (Th  WAT) in the cold flow regime are similar for all the three cases, and the Th value approaches Tc at approximately the same axial location. Fig. 6(C) presents the predictions of the pressure drop in the pipeline for different inlet temperatures Thin. In the hot flow regime, the pressure drop is higher with lower Thin and vice versa. This is a result of larger deposit thickness for a lower Thin in the hot flow regime. The pressure drop predictions follow the same trend as that for the deposit thickness in Fig. 6(A). Fig. 6(D) presents the rate of heat loss for the different Thin values. As expected, the rate of heat loss at the beginning of the hot flow regime is greater for a higher Thin due to a larger thermal driving force. For example, at the beginning of the hot flow regime, q/DL corresponding to Thin = WAT + 15, WAT + 10 and WAT + 5 is predicted to be 537 W/m, 371 W/m and 196 W/m, respectively. As mentioned previously for the base case scenario, the changes in the rate of heat transfer are much larger in the hot flow regime when compared to the cold flow regime. In the cold flow regime, a change in Thin does not seem to significantly affect the rate of heat loss.

3.3. Effect of surrounding temperature, Tc For this set of calculations, only the surrounding temperature, Tc, was varied while all of the other parameters were held at their base case values. Presented in Fig. 7 are the predictions for the variation of wax deposit thickness, wax–solvent mixture temperature, pressure drop, and the rate of heat loss as a function of axial distance (z/D) for three Tc values, i.e., Tc = WAT  5, Tc = WAT  10, and Tc = WAT  15. In Fig. 7(A), the predicted deposit thickness is seen to increase significantly with a lowering of Tc in both the hot flow and cold flow regimes. For example, the maximum deposit thickness at the transition from the hot flow regime to the cold flow regime for the lowest Tc (=WAT  15) is approximately two-times that for the highest Tc (=WAT  5). These trends for the effect of Tc on the deposit thickness are in good agreement with the previously reported experimental results as well as the predictions from the unsteady state moving boundary problem formulation [3,6,14]. In the cold flow regime, a lowering of Tc is predicted to increase the pipe length needed for the deposit thickness to approach zero. For instance, the deposit thickness is predicted to be negligible at z/D  8424 when Tc = (WAT  15), whereas z/D  3897 when Tc = (WAT  5). Despite the differences in the deposit thickness, the predictions in Fig. 7(B) do not show any significant effect of Tc on the temperature profile for the wax–solvent mixture. However, the effect of Tc on the pressure drop is quite significant, as seen in Fig. 7(C), and the variation in pressure drop is similar to that for the deposit thickness in Fig. 7(A) over both the hot flow and cold flow regimes. The predictions for the rate of heat loss in Fig. 7(D) do not show any noticeable effect of Tc. Even though a lowering of Tc implies an increase in the overall thermal driving force for heat transfer, which is expected to increase the rate of heat transfer. A lower Tc would also yield an increase in the deposit thickness (and consequently its thermal resistance), which would decrease the rate of

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0.45

0.35

10

0.30

5

Th-WAT (K)

0.25 0.20 0.15

-5 -10

0.05

-15 0

2000

4000

6000

8000

0

10000

160

2000

4000

6000

8000

10000

600

(C) Rate of Heat Loss (q/ΔL)

140

Pressure Drop (ΔP/ΔL)

0

0.10

0.00

(B)

120 100 80 60 40 20 0

500

(D)

40

Rate of Heat Loss (q/ΔL)

Deposit Thickness (δ/R)

15

(A)

0.40

400 300 200

30

20

10

0 0

100

2000

4000

6000

8000

10000

Axial Distance (z/D)

0

0

2000

4000

6000

8000

10000

Axial Distance (z/D)

0

2000

4000

6000

8000

10000

Axial Distance (z/D)

Fig. 7. Effect of Tc in the axial direction of a pipeline: (A) deposit thickness profile; (B) wax–solvent mixture temperature; (C) pressure drop (in Pa/m); and (D) rate of heat loss (in W/m). Tc = WAT – 15 —, Tc = WAT  10 – – , and Tc = WAT – 5 – – –.

heat transfer. Thus, a variation in Tc is predicted to cause a significant effect on the deposit thickness and the pressure drop, but only a small effect on the temperature profile of the wax–solvent mixture and the rate of heat loss.

3.4. Effect of inlet Reynolds number, Rei Fig. 8 presents the predictions for the variation of deposit thickness, wax–solvent mixture temperature, pressure drop and the rate of heat loss along the pipe length for three values of the inlet Reynolds number, Rei. The Reynolds number, Re, varies along the axial distance with changes in the effective inside pipe diameter and liquid properties due to variations in deposit thickness and temperature. The Rei values used for these calculations were 15,000, 25,000 and 30,000, while all the other parameters were held at their base case values. As seen in Fig. 8(A), the predicted deposit thickness decreases with an increase in Rei in the hot flow regime. This effect has also been shown in previous experimental and modeling studies [3,12,17,22–25,27,37,48]. A comparison of Eqs. (4) and (5) reveals that an increase in Re would yield a higher heat transfer coefficient, hh, which would correspond to a lower convective resistance, that must be matched by a lower conductive resistance offered by the deposit and consequently a smaller deposit thickness, d. The maximum deposit thickness at the transition from the hot flow to cold flow also decreases with an increase in Rei. The predicted maximum deposit thickness values are d/ R  0.28, d/R  0.32 and d/R  0.40 for Rei = 30,000, 25,000 and 15,000, respectively. However, the axial location where d ? 0, in the cold flow regime, is predicted to be shorter corresponding to a lower Rei. On the other hand, as shown in Fig. 8(B), Th is predicted to approach the WAT somewhat faster at a lower Rei. For example,

the transition from the hot flow regime to cold flow regime occurs at z/D  1125, z/D  1077 and z/D  936 for Rei = 30,000, 25,000 and 15,000, respectively. The pressure drop predictions for different Rei values in Fig. 8(C) indicate that a lower Rei would cause a smaller pressure drop, in both the hot flow and cold flow regimes, which is attributed to a lower liquid velocity. Fig. 8(D) presents the predictions for the effect of Rei on the rate of heat loss. A higher Rei is predicted to yield a higher rate of heat loss in both the hot flow and cold flow regimes. The effect of Reynolds number is more significant at the pipeline inlet, where for Rei = 30,000, 25,000 and 15,000, the rate of heat loss, q/DL, is predicted to be 928, 803 and 536 W/m, respectively.

3.5. Effect of overall temperature difference, (Th  Tc) As mentioned previously, the overall temperature difference, (Th  Tc), is the thermal driving force for heat transfer from the wax–solvent mixture at Th to the surroundings at Tc. Even though (Th  Tc) influences the rate of heat transfer, it is the temperature difference or thermal driving force across each individual thermal resistance, such as (Th  Td) and (Td  Twi)  (Td  Tc), which is a significant controlling parameter for the deposition process [3,22,23,25]. In this regard, it is incorrect to presume that the deposit thickness would depend directly on the overall thermal driving force. It has been shown that the deposit layer could be eliminated by maintaining Th Tc, despite a very high (Th  Tc) leading to a very high rate of heat transfer [48]. The deposit layer could also be eliminated by maintaining Th = Tc, which would yield (Th  Tc) = 0, leading to no heat transfer, which actually is the basis for the cold flow technology. It is also important to point out that an increase in overall temperature difference, (Th  Tc), could be accomplished by either

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0.45 15

(B)

(A)

0.35

10

0.30

5

Th-WAT (K)

Deposit Thickness (δ/R)

0.40

0.25 0.20 0.15 0.10

0 -5 -10

0.05

-15

0.00 2000

4000

6000

8000

0 1000

10000 12000

2000

4000

6000

8000

900

(C) Rate of Heat Loss (q/ΔL)

Pressure Drop (ΔP/ΔL)

0

240 220 200 180 160 140 120 100 80 60 40 20 0

10000 12000

(D)

800 700 600 500 400 300 200 100 0

0

2000

4000

6000

8000

10000 12000

0

2000

Axial Distance (z/D)

4000

6000

8000

10000 12000

Axial Distance (z/D)

Fig. 8. Effect of Rei in the axial direction of a pipeline: (A) deposit thickness profile; (B) wax–solvent mixture temperature; (C) pressure drop (in Pa/m); and (D) rate of heat loss (in W/m). Rei = 30,000 – – –, Rei = 25,000– –, Rei = 15,000 —.

0.45 Thi=WAT+5, Tc=WAT-15

Deposit Thickness (δ/R)

0.40

Thi=WAT+15, Tc=WAT-5

0.35

(A)

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

2000

4000

6000

8000

10000

0.45 Thi=WAT+10, Tc=WAT-15

0.40

Deposit Thickness (δ/R)

increasing Th, or by decreasing Tc, or by increasing Th while decreasing Tc. If either Th or Tc were to be changed (i.e., but not both), then the effect on the deposition process would only be that of the changed temperature, Th or Tc, and not necessarily of the overall temperature difference, (Th  Tc). As shown in Figs. 6 and 7, the deposit thickness in the hot flow regime is predicted to decrease with either an increase in Th or an increase Tc, and vice versa. Fig. 9 presents two sets of predictions for the variation of deposit thickness with the same value of the overall thermal driving force, (Th  Tc). Fig. 9(A) shows two deposit thickness profiles with the same overall thermal driving force of (Thin  Tc) = 20 K. The solid line represents (Thin = WAT + 5, Tc = WAT  15) and the dashed line represents (Thin = WAT + 15, Tc = WAT  5). Even though (Thin  Tc) is the same, the predictions for the lower Thin together with the lower Tc indicate a much larger deposit thickness in both the hot flow and cold flow regimes. The maximum deposit thicknesses at the transition from the hot flow regime to cold flow regime are predicted to be d/R  0.4 (for the solid line) and d/ R  0.2 (for the dashed line). This is because the deposit thickness increases when Th and/or Tc is decreased. Fig. 9(B) presents similar predictions for a different overall thermal driving force of (Thin  Tc) = 25 K. In Fig. 9(B), the solid line represents (Thin = WAT + 10, Tc = WAT  15) and the dashed line represents (Thin = WAT + 15, Tc = WAT  10). Note that the two deposit thickness profiles in Fig. 9(B) are closer to each other because the selected values for both Thin and Tc differ only by 5 K, as opposed to a difference of 10 K for the two profiles in Fig. 9(A). The results in Fig. 9 confirm that the overall thermal driving force does not influence the deposit thickness; instead, the deposition tendency is related to the thermal driving force across each individual thermal resistance.

Thi=WAT+15, Tc=WAT-10

0.35

(B)

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

2000

4000

6000

8000

10000

Axial Distance (z/D) Fig. 9. Effect of overall thermal driving force (Thin  Tc) on solids deposition profile: (A) (Thin  Tc) = 20 K and (B) (Thin  Tc) = 25 K.

S. Haj-Shafiei et al. / Fuel 137 (2014) 346–359

Re=15000 Re=30000 Re=15000(Aged) Re=30000 (Aged)

Deposit Thickness (δ/R)

0.4

0.3

(A) 0.2

0.1

0.0 0

2000

4000

6000

8000

10000 12000

15

(B)

Th-WAT (K)

10 5 0 -5 -10 -15 0

2000

4000

6000

8000

10000 12000

Axial Distance (z/D) Fig. 10. Predicted results for the effect of deposit aging at different Rei: (A) deposit thickness profile and (B) wax–solvent mixture temperature.

3.6. Effect of deposit aging As mentioned previously, the effect of deposit aging on the deposit thickness was modeled with the shear-induced deformation approach [29]. As a result of changes in the deposit composition, deposit aging causes an increase in the average solid phase fraction in the deposit [23,29,33]. Since the thermal conductivity of solid paraffins is higher than liquid paraffins [23,33], the aging process would cause an increase in the average deposit thermal conductivity, kd, which according to Eq. (5) would change the rate of heat transfer across the deposit layer. An increase in kd, therefore, would require an increase in the deposit thickness in order to maintain the same rate of heat transfer in Eqs. (1), (4) and (5) (in the hot flow regime) or in Eqs. (13), (4) and (5) (in the cold flow regime). That is, one consequence of the deposit aging process is to cause an increase in the deposit thickness. Fig. 10 presents the predictions for the aged (i.e., with shear effect) and un-aged (i.e., without shear effect) cases at two values of Rei, 15,000 and 30,000. As explained above, the predictions in Fig. 10(A) show the deposit thickness to slightly increase as a result of aging in both the hot flow and cold flow regimes for both Rei. The predictions in Fig. 10(B), however, do not show a significant effect of aging on the temperature profiles because the deposition process changes the deposit thickness while maintaining the same rate of heat transfer. 4. Conclusions A mathematical model, based predominantly on steady-state heat-transfer considerations, was presented for solids deposition from a ‘waxy’ crude oil or mixture under both hot flow and cold flow regimes. The ‘waxy’ crude oil was approximated as a pseudo

357

binary mixture, comprising C13H28 (C13, representing the lighter solvent fraction) and C29H60 (C29, representing the heavier wax fraction). The developed model was used to predict the axial variations in deposit thickness, liquid temperature, pressure drop and the rate of heat transfer in a pipeline maintained under turbulent flow conditions at steady state. The predictions, in both hot flow and cold flow regimes, were shown to be in agreement with the trends and observations in experimental results as well as predictions from a moving boundary problem formulation reported in the literature. The calculations for the steady state modeling approach were much faster (<30 min) when compared to the moving boundary problem formulation (>3 h) as only the final deposit thickness was considered to satisfy the equality of the rate of heat transfer across different thermal resistances. On the other hand, the transient moving boundary problem formulation considered a layer-by-layer wax deposit formation to arrive at the final deposit thickness. The deposit thickness along the pipeline length was predicted to increase in the hot flow regime, to reach a maximum at the transition from hot flow to cold flow, and then to decrease in the cold flow regime. Once the liquid temperature approached the surrounding temperature, there was no deposition and the waxy mixture continued to flow as a solid + liquid suspension. The solid content of the solid + liquid suspension was predicted to increase in the axial direction as a result of precipitation of the solid wax from the liquid phase. The predictions indicated that the axial location for the transition from hot flow to cold flow shifted down the pipeline with an increase in the inlet liquid temperature (Thin), an increase in the surroundings temperature (Tc), and an increase in the (inlet) Reynolds number (Rei). The main effect of deposit aging was a small increase in the deposit thickness, in both hot flow and cold flow regimes, which was attributed to an increase in the deposit thermal conductivity. The predictions indicated that, in the hot flow regime, the deposit thickness decreased with an increase in Tc and an increase in Rei. The predictions in the cold flow regime indicated that the deposit thickness was decreased with an increase in Tc and a decrease in Rei. A change in the inlet liquid temperature (Thin) was predicted to yield similar deposit thickness profiles in both the hot flow and cold flow regimes, although a lower Thin resulted in a slightly larger deposit thickness in the hot flow regime and a slightly smaller deposit thickness in the cold flow regime. The predictions confirmed that the overall thermal driving force, (Th  Tc), does not influence the deposit thickness directly; instead, the deposit thickness was explained in terms of two temperature differences, namely (Th  Td) and (Td  Tc), with Td = WAT of the liquid mixture in the hot flow regime and Td = WAT of the (diluted) liquid phase in the cold flow regime. A reduction in the deposit thickness could be accomplished by increasing either Thin or Tw. The deposit thickness predictions in the cold flow regime were noted to be sensitive to the temperature difference across the convective thermal resistance of the flowing liquid (Th  Td). Future experimental studies could provide an improved estimate of this important temperature difference. In addition, the predictions indicated that the rate of heat loss was much higher in the hot flow regime. A lower rate of heat loss in the cold flow regime corresponded to a slower rate of decrease in Th as a result of existence of both the sensible heat and latent heat transfer (due to the crystallization of a fraction of waxes) as well as a decreasing rate of heat transfer due to a gradually lower thermal driving force, (Th  Tc). This study provided further confirmation that solids deposition from ‘waxy’ mixtures or crude oils, in both hot flow and cold flow regimes, can be modeled satisfactorily as a thermally-driven heattransfer process involving partial freezing and crystallization of wax particles.

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Acknowledgments

References

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Centre for Environmental Engineering Research and Education (CEERE), and the Department of Chemical and Petroleum Engineering, the University of Calgary, is gratefully acknowledged.

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Appendix A The methods for estimating the properties of wax–solvent mixture and deposit were described elsewhere [27] and are summarized in the following table. Estimation of properties and thermodynamic consideration P 1 Density of the i=1 qmix ¼ ; i wi qi l wax–solvent (solvent), 2 (paraffin) mixture, qmix i Specific heat capacity of the wax–solvent mixture, C

C mix p;l

or

C ¼ C mix p;l ¼ C p;l ¼

P

i wi ðC p;l Þi

3 2C CH p;l

þ ðN 

2 2ÞC CH p;l

3 C CH ¼ 17:33 þ 0:0455T h p;l 2 C CH ¼ 30:41 þ 0:01479T h p;l

Viscosity of wax– solvent mixture, l

l = 10–3 exp

Thermal conductivity of the wax– solvent mixture, k or

k ¼ kl

Cp,l: pure component specific heat capacity N: number of carbon atoms in the alkane

(4.75 + 1720/Th) mix

¼

PP i

j /i /j kij

1

1 1

kij ¼ 2ððkl Þi Þ þ ððkl Þj Þ P /n = gnmn/( ngnmn), n = i, j

i: paraffin wax j: solvent

mix

kl

Thermal conductivity of the deposit, kd,

kd ¼ kl

Freezing point depression for an ideal binary mixture The volume fractions were determined using the lever rule. The wax (C29) and solvent (C13) components were assumed to form an ideal mixture, without any heat of mixing and volume change

ln wi = [(DHm)i/R] [(1/(TL)i)  (1/(Tm)i)], i = 1, 2

(DHm)C13: 28.5 MJ/ kmol

f = (w29  w⁄29)/(1  w⁄29) at Th and Td = (Twi + Td)/2

(DHm)C29: 106.6 MJ/ kmol (Tm)C13: 267.8 K (Tm)C29: 335.4 K f : solidphase mass fraction

mix

V l þ kwax;s V s

kwax,s: thermal conductivity of pure solid wax

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