âWaxyâ Mixtures in - American Chemical Society

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Modeling of Solids Deposition from “Waxy” Mixtures in “Hot Flow” and “Cold Flow” Regimes in a Pipeline Operating under Turbulent Flow Sridhar Arumugam, Adebola S. Kasumu, and Anil K. Mehrotra* Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4 ABSTRACT: Solids deposition from “waxy” mixtures under turbulent flow in a pipeline was modeled as a moving boundary problem involving liquid−solid phase transformation. The developed model is applicable for the “hot flow” regime (i.e., with the mixture temperature above its wax appearance temperature, WAT) and the “cold flow” regime (i.e., with the mixture temperature below its WAT, resulting in solid particles suspended in the liquid phase). A recently proposed correlation for the wax precipitation temperature (WPT) as a function of the wax concentration and the cooling rate was used to predict the transition from the “hot flow” regime to the “cold flow” regime. Predictions obtained for both radial and axial deposit growth in the pipeline with time in the “hot flow” and “cold flow” regimes were found to be in agreement with the trends observed in the laboratory deposition results reported in the literature. The predicted deposit thickness in the axial direction increased under the “hot flow” regime, reached a maximum as the liquid temperature approached the WAT of the wax−solvent mixture, and decreased subsequently under the “cold flow” regime. The axial location for the transition from the “hot flow” regime to the “cold flow” regime was predicted to shift with changes in the inlet mixture temperature, pipe wall temperature, and Reynolds number. The predicted maximum deposit thickness was also impacted by these variables. The predictions in this study indicate that solids deposition in pipelines carrying “waxy” mixtures could be decreased by maintaining the flow under the “cold flow” regime. This study shows that solids deposition from “waxy” mixtures can be modeled satisfactorily as a thermally driven process involving partial solidification.

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INTRODUCTION Crude oils are complex mixtures containing aromatics, paraffins, naphthenes, asphaltenes, and resins. Waxes are high-molecularweight paraffins dissolved in crude oil at high temperatures and pressures, typically encountered in a reservoir environment. When waxes are subjected to much lower temperatures and pressures during transportation of crude oil, their solubility decreases, and the crude oil tends to form macro- and microcrystalline structures of wax crystals that deposit on the cold walls of the pipeline.1 The temperature at which the first wax particles precipitate out of the crude oil upon cooling is called the wax appearance temperature (WAT). It has been shown that a “waxy” mixture containing as little as 2 mass % of wax is sufficient to undergo deposition upon cooling below the WAT.2 Wax deposition is a major flow assurance problem faced by all facets of the petroleum industry from crude oil production to transportation through pipelines to downstream process operations. The precipitated wax imparts complex nonNewtonian and nonlinear characteristics to the flow properties of the oil.3 The deposition of solids is undesirable as it causes plugging and increases the energy consumption during transportation and processing of the crude oil. Several methods to mitigate wax deposition have been tried over the years, such as chemical treatment,4,5 mechanical methods,6 and thermal methods.7 Attempts to control wax deposits have also been made, with limited success, by using bacterial injection, electromagnetic fields,8 piezoelectric energy,9 vacuum-insulated tubing,10 and fused chemical reactions.11 All such methods have considerable associated costs and limitations. Methods such as © 2013 American Chemical Society

chemical treatment are also highly selective for the particular “waxy” mixture considered.12 Another approach proposed for mitigating or avoiding solids deposition is “cold flow”, which involves pipeline transportation of “waxy” mixtures and crude oils in the form of a slurry or suspension at a temperature below the WAT. Several possible reasons have been suggested for the decreased deposition under “cold flow” conditions, including a lower thermal driving force; preferential crystallization of wax onto the suspended wax particles, which act as nucleation sites; and the lowering of the cloud-point temperature as a result of the precipitation of some of the waxes from the solution.13−16 The lowering of the WAT as a result of precipitation of wax from “waxy” mixtures is a predominant reason for the decreased deposition under the “cold flow” regime. As a result of the preferential precipitation of heavier paraffins, the residual or remaining liquid phase becomes more dilute or “leaner”, which causes a lowering of its WAT. Even though wax crystals exist in the liquid phase, yielding a suspension or slurry in the “cold flow” regime, these have not been found to contribute to solids deposition.14 Several mechanisms for predicting wax deposition in pipelines have been proposed, such as molecular diffusion, heat transfer, shear dispersion, Brownian diffusion, and gravity settling. Molecular diffusion17−24 and heat transfer14,25−39 are considered to be the most prevalent mechanisms. Among other differences in modeling of the deposition process, these two Received: July 10, 2013 Revised: October 24, 2013 Published: October 28, 2013 6477

dx.doi.org/10.1021/ef401315m | Energy Fuels 2013, 27, 6477−6490

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sufficient to explain the deposition behavior.14,15,29−31 The deposit thickness or mass has been shown to increase with a decrease in the difference between the bulk liquid temperature and the interface temperature, which has been shown to be close to the WAT. Lower crude oil temperatures (above the WAT) and lower pipe wall temperatures (below the WAT) favor an increase in solids deposition. An increase in the wax concentration of a crude oil results in a higher WAT, which increases the extent of solids deposition. Heat-Transfer-Based Modeling Approach. In the heattransfer approach for wax deposition, two calculation methods have been presented for the unsteady-state and steady-state cases. For the steady-state case, the total thermal resistance corresponding to the overall thermal driving force is divided into individual thermal resistances as follows: two convective resistances (for the oil and the coolant) and two conductive resistances (for the deposit and the pipe wall).14,26,29−31,34,37 The amount of and thermal resistance offered by the deposited wax are then related to the fractional temperature difference across the deposit layer. The steady-state modeling approach was validated with the laboratory results obtained from benchscale flow loop apparatuses under laminar flow conditions14,29,31 and turbulent flow conditions.34,37,42 The unsteady-state modeling approach for wax deposition32,33,35,36,38,39 is based on the moving boundary problem formulation, which is characterized by the presence of an interface where the liquid−solid phase transformation of the “waxy” mixture occurs. The moving boundary formulation was used to predict the extent of wax deposition under static conditions,32,39 laminar flow conditions,32,33 and turbulent flow conditions.38 It should be noted that the previous heat-transfer-based modeling studies have considered solids deposition only in the “hot flow” regime. This study provides an important extension of the unsteady-state modeling approach based on the moving boundary formulation to the prediction of solids deposition in both the “hot flow” and “cold flow” regimes. The transition from the “hot flow” regime to the “cold flow” regime was accomplished by incorporating the effect of cooling rate on the precipitation of wax crystals in the bulk liquid phase. The model includes mass and energy balances in the radial and axial directions. The growth of the deposit layer in both radial and axial directions is predicted at different times, leading to a steady-state deposit layer profile. The calculations were performed with a pseudobinary mixture comprising n-C13H28 (denoted as C13) and n-C29H60 (denoted as C29) to represent the lighter and heavier fractions of the crude oil, respectively. The developed model is used to explore the solids deposition behavior of a “waxy” mixture in both the “hot flow” and “cold flow” regimes. Predictions of the model are presented for the effects of important parameters, such as the pipe wall temperature, inlet mixture temperature, and inlet Reynolds number, on the predicted deposit thickness profile axially in a 4 km long pipeline at different residence times as well as at steady state. The trends in the model predictions are validated with the observations from experimental results obtained in the “hot flow” regime14,29,31,34,37 and the “cold flow” regime.14 The model predictions are also in general agreement with the predictions for solids deposition from “waxy” mixtures in the “hot flow” regime under turbulent flow conditions.38

mechanisms differ in the treatment of the liquid−deposit interface temperature. In the molecular diffusion approach, a pseudo-steady-state form of Fick’s diffusion equation is used to model the time-dependent growth of wax deposition. In this approach, the liquid−deposit interface temperature is backcalculated from an energy balance, which predicts a gradual increase in its value from close to the pipe wall temperature initially to the WAT at steady state. Thus, an inherent assumption in the molecular diffusion approach is that the interface temperature is variable and less than the WAT in order for the deposition to take place. It is pointed out that this important assumption has not been verified experimentally. It should be noted that according to the molecular diffusion approach, the deposit layer would cease to grow once the interface temperature becomes equal to the WAT. Furthermore, the diffusion-based modeling approach does not consider the precipitation of wax crystals in the liquid phase while the liquid-phase temperature near the liquid−deposit interface is below the WAT. In contrast, the heat-transfer approach assumes that the interface temperature remains constant and equal to the WAT of the liquid phase throughout the deposition process. Experimental results have confirmed that the liquid−deposit interface temperature for wax−solvent prepared mixtures remains close to the WAT of the liquid phase during the deposit-growth process.40,41 In these batch cooling experiments, the deposit layer continued to grow even after the interface temperature approached the WAT of the liquid phase. Observations from Previous Experimental Studies. Some of the main observations from the experimental studies undertaken by Mehrotra and co-workers14,29,31,34,37,42 are summarized below, and these have been used to validate the trends in the model predictions presented in this study. In all of the experiments with prepared “waxy” mixtures performed with different mixtures and at different deposition or residence times, the liquid−deposit interface temperature was always found to be close to the WAT.14,29,31,34,37,42 This important observation was also confirmed with batch cooling experiments under static and sheared conditions.40,41 It has been shown that wax deposition and wax precipitation are related but different processes involving the liquid-to-solid phase transformation with crystallization.14 Furthermore, experimental results have confirmed that wax deposition may not occur even when precipitated wax crystals are present (i.e., suspended in the liquid phase) in the “cold flow” regime.14 The mass of deposited solids, or the deposit thickness, has been found to decrease with an increase in the flow rate or Reynolds number under both laminar and turbulent flow conditions.14,31,34,37,42 This decrease in the mass of deposited solids is necessary to maintain the same rate of heat flow, at steady state, through all of the convective and conductive thermal resistances that exist in series.29,31,34,37 Whereas the thermal resistances corresponding to the pipe wall (conduction) and the outer fluid (convection) remain constant, those due to the waxy liquid (convection) and the deposit (conduction) vary with the extent of solids deposition.14,42 The heat-transfer approach can also be used to show that a higher thermal conductivity of the deposit yields an increased deposit thickness. The wax deposition process cannot occur without a thermal driving force between the liquid phase and the cooler surface, which must be held at a temperature below the WAT of the liquid phase. However, the overall thermal driving force is not

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MODEL DEVELOPMENT In this section, a mathematical model for the wax deposition process in a pipeline based on heat-transfer considerations is 6478


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Figure 1. Schematic of a pipeline showing the deposit thickness profile in the “hot flow” and “cold flow” regimes.

by an imaginary thin circular disk that keeps the two liquids separated from each other both physically and thermally, that is, it is assumed that the “waxy” mixture entering the pipeline is separated from the nondepositing liquid by an imaginary disk of negligible thickness and very large thermal resistance. The imaginary disk is pushed axially with an average velocity u, thereby displacing the nondepositing liquid by the depositing “waxy” mixture. This scenario allows the pipeline to be maintained under flowing conditions at all times, thereby avoiding any momentum or thermal startup of a pipeline.33 The steady-state predictions for both startup scenarios were found to be the same for a set of base-case values of the mixture and pipe wall temperatures, mixture composition, pipe diameter, and Reynolds number.33 In this study, the initial condition for the deposition process is derived from the second startup scenario. In the development of the mathematical model, the liquid− deposit interface was assumed to be hydrodynamically smooth. The liquid-to-solid phase transformation was assumed to be instantaneous and governed only by thermodynamic considerations in the “hot flow” regime. In the “cold flow” regime, the effect of the cooling rate on the wax precipitation temperature (WPT) was also included. Any mass-transfer resistance due to molecular diffusion within the deposit or convective diffusion across the liquid−deposit interface was not taken into consideration. The model does not account for the effect of shear stress on the formation and growth of the wax deposit. It was also assumed that there is no shear stripping of the deposit layer at high flow rates. Finally, no deposit aging effects have been considered in this study; such effects could be included in a future study following the shear-induced time-dependent viscoplastic deformation approach developed by Mehrotra and co-workers.35,37,38 Mass and Energy Balance Equations. The unsteady-state energy balance equation for the liquid region across an axial element, Δz, under “hot flow” of “waxy” mixtures is32,33

described. The mathematical model is based on the moving boundary formulation, which is characterized by the presence of an interface where the liquid−solid phase transformation of “waxy” mixture occurs. The location of the interface changes with heat transfer during the phase transformation; however, the interface location is not known a priori, which makes the numerical solution procedure challenging. Numerous applications of the moving boundary formulation have been reported for freezing and/or melting studies, especially related to metallurgical systems that involve melting and solidification of metals and alloys. As mentioned previously, the moving boundary approach has also been used for predicting solids deposition from wax−solvent mixtures.32,33,35,36,38,39,43,44 When a “waxy” liquid mixture or crude oil in a pipe is exposed to lower pipe wall temperatures, a solid layer should start to deposit on the inner pipe surface via a partial freezing process. The deposit layer thickness should increase with time as the thermal energy (including both the sensible heat and the latent heat of phase change) is transferred radially outward. The phase transformation associated with the solidification of the liquid layer adjacent to the deposit layer has been assumed to be controlled primarily by the rate of heat transfer. As illustrated schematically in Figure 1, the “waxy” mixture enters the pipeline, of length L and inside diameter D, at an inlet temperature Thi (>WAT of the wax−solvent mixture), while the cooler pipe wall is held at a constant temperature Tw ( WAT and in the “cold flow” regime when Tl < WAT is predicted with time until a steadystate deposit thickness is attained at the chosen pipeline length of 4 km, corresponding to L/D = 40 000. Bhat and Mehrotra33 proposed two scenarios for initiation of the deposition process in a pipeline, which define the initial conditions for solving the partial differential equations. In the first scenario, the deposition process is assumed to commence when the pipe wall temperature at all axial locations is suddenly lowered to a temperature less than the WAT. The second scenario mimics the pipeline “batching” of immiscible liquids by assuming that an imaginary “nondepositing” liquid is initially flowing through the pipeline while the pipe wall is held at Tw at all axial locations (z > 0). For t > 0, the nondepositing liquid is displaced by the “waxy” (depositing) mixture. The nondepositing and depositing liquids are assumed to be separated

ṁ 1Cp ,l(Tl‐in − Tl‐out) = hAd (Tl − Td) + (πs 2Δz)ρl Cp ,l Tl ≥ WAT

∂Tl , ∂t (1)

in which Ad = 2πsΔz is the area for heat transfer at the liquid− deposit interface, Tl = (Tl‑in + Tl‑out)/2 is the average temperature of the liquid mixture, and s = R − δ is the effective radius of the pipeline available for flow of the liquid mixture, where R = D/2 is the radius of the pipe and δ is the deposit layer thickness. 6479


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mentioned previously, it is assumed that the pipeline is initially carrying a nondepositing liquid in a thermal steady state:

The unsteady-state energy balance equation for the liquid region across an axial element Δz under “cold flow” of “waxy” mixtures, resulting in the precipitation of solid particles suspended in the liquid phase (and accounting for the associated latent heat), is43 ⎡ k ∂ ⎛ ∂T ⎞ df ⎤ ṁ 1Cp ,l(Tl‐in − Tl‐out) + ⎢ 1 ⎜r l ⎟ − ρλ l ⎥Vl dt ⎦ ⎣ r ∂r ⎝ ∂r ⎠ ∂Tl = hAd (Tl − Td) + ρl VC , Tl < WAT l p ,l ∂t

Tl = g (z),

s = R, (2)

∂z 2

s < r < R, z > 0

where the modified thermal diffusivity in the deposit phase, αδ′, is given by

(4)

in which fδ is the mass fraction of solid phase in the deposit, λ is the latent heat of fusion, and kδ is the thermal conductivity of the deposit at the deposit temperature Tδ. In the “hot flow” regime, Tδ is equal to the liquid−deposit interface temperature Td (=WAT) at r = s and to Tw at r = R. The axial heat conduction in the deposit layer was observed to be negligible; however, in the “cold flow” regime, the interface temperature Td at r = s varies with the axial position z, and the term ∂2Tδ /∂z2 was found to be non-negligible. Heat-Transfer Considerations. In the moving boundary formulation, the growth of the deposit layer is governed by the heat transfer at the liquid−deposit interface. The energy balance at the liquid−deposit interface is given by kδ

∂Tδ ds − h(Tl − Td) = ρλfs , ∂r dt

r = s, z > 0

(5)

where fs is the equilibrium solid-phase fraction at the liquid− deposit interface (i.e., at r = s), corresponding to the interface temperature Td. The heat-transfer coefficient in eqs 1 and 2 was estimated using the Dittus−Boelter equation for the turbulent flow forced convection heat transfer in smooth tubes:45 Nu = 0.023Re 0.8Pr 0.3

(7b)

Tδ = Td ,

r = s , t > 0, ut > z > 0

(8a)

Tδ = Tw ,

r = R , t > 0, ut > z > 0

(8b)

Transition from “Hot Flow” to “Cold Flow”. Previous experimental studies by Bidmus and Mehrotra14,40,41 in both the “hot flow” and “cold flow” regimes indicated that two important changes take place when the flow transitions from “hot flow” to “cold flow” in a pipeline. Under “cold flow”, the flowing liquid actually becomes a two-phase suspension or slurry with the precipitated wax crystals suspended in the liquid phase. Because a fraction of the wax constituents has precipitated out of the solution, the residual liquid phase becomes “leaner”, and its WAT is less than that of the original mixture. This means that the WAT of the waxy mixture remains constant during “hot flow”, whereas the WAT of the liquid phase starts to decrease after the flow transitions from “hot flow” to “cold flow”. The actual decrease in the WAT in the “cold flow” regime depends on the extent of wax constituent precipitation (as predicted from thermodynamic equilibrium considerations). Another factor that influences the phase transition temperature is the cooling rate, which is known to affect the kinetics of crystallization. The effect of the cooling rate on the phase change temperature for waxy mixtures and crude oils has been described in several studies. Increased cooling rates have been reported to give lower temperatures for the onset of crystallization because of supercooling effects and the roles of nucleation and crystallization kinetics.46−50 Hammami and Mehrotra46 reported the effect of the cooling rate on the phase change temperature for paraffins. Paso et al.50 reported experimental WPT results for two waxy mixtures that showed an increase in WPT with a decrease in the cooling rate. In a recent study, Kasumu et al.51 reported similar experimental results for the effect of the cooling rate on the WPT for several prepared solutions of a wax in a multicomponent solvent, Norpar13. They provided the following correlation for the effect of the cooling rate, x = |dT/dt| (varying from 0.05 to 0.40 °C min−1), and the wax concentration, y (varying from 2 to 20 mass %), on the measured WPT:

(3)

ρλ ∂fδ 1 1 = − αδ′ αδ kδ ∂Tδ

t = 0, z > 0

For t > 0, the pipe wall temperature is maintained at a constant temperature Tw < WAT at z > 0. The liquid−deposit interface temperature T d was set equal to the WAT corresponding to the liquid-phase composition at all times during the deposition process. At each time step Δt, the depositing mixture would displace the nondepositing liquid (which moves with an average velocity u) while the pipe wall temperature is held at Tw. The boundary conditions are as follows:

where Vl = πs Δz is the volume of the liquid across the differential element Δz and h is the heat-transfer coefficient. The second term on the left-hand side of eq 2 accounts for heat transfer by conduction due to the precipitated wax particles and the thermal energy released by precipitation in the bulk liquid phase. Thus, the liquid mixture is in a single phase in the “hot flow” regime, whereas it is a liquid−solid two-phase mixture (i.e., a suspension or slurry) under the “cold flow” conditions. The energy balance equation for two-dimensional unsteadystate heat transfer by conduction in the deposit layer is32,33 1 ∂ ⎛ ∂Tδ ⎞ 1 ∂Tδ ⎜r ⎟= , + ⎝ ⎠ r ∂r ∂r αδ′ ∂t

(7a)

where g(z) is the steady-state temperature profile for the nondepositing liquid in the pipeline.33 Without any deposit formation with the nondepositing liquid initially, the initial condition for eq 3 for the deposition process is given by

2

∂ 2Tδ

t = 0, 0 < r < R , z > 0

(6)

The available flow area in the pipe changes because of wax deposition. Thus, Nu and Re in eq 6 were calculated on the basis of the actual available local flow diameter (to account for the blockage). Initial and Boundary Conditions. The following initial and boundary conditions were used in solving eq 1 or 2 and eqs 3 and 5, which constitute the moving boundary formulation. As

WPT = a + bx + c ln y

(9)

The regressed values of a, b, and c were reported as 24.17 ± 0.21, −4.155 ± 0.49, and 6.684 ± 0.09, respectively. With a 6480


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for the transition from “hot flow” to “cold flow” considered in this study. Simulation Procedure. The input quantities for the numerical calculations were the mixture composition (w29), the inlet mixture temperature (Thi), the constant pipe wall temperature (Tw), the pipe diameter (D), inlet Reynolds number (Rei), the length of the pipeline (L), and the deposition time (t). The inlet mixture velocity was estimated from D and Rei together with the density and viscosity values obtained from the property estimation methods reported elsewhere.32,33 The solution methodology involved discretization in the radial direction for an axial element Δz. The equations in the moving boundary formulation were solved for each axial element over the length of the pipeline, which were then used to predict the axial variation of the liquid-phase temperature (Tl) and the deposit thickness (δ) with time and axial distance (z). These results yielded the profiles for the deposit thickness and temperature in the axial direction. It should be noted that the deposition and temperature profiles in the radial direction have been reported previously.32,33 As mentioned previously, at time t = 0 a nondepositing liquid is assumed to be flowing through the pipeline for all values of r and z (i.e., 0 < r < R and 0 < z < L) under a thermal steady state. Thus, there is no deposition initially anywhere in the pipeline. For times t > 0, while the inlet mixture temperature corresponds to Tl = Thi, the temperature of the inside pipe wall at r = R is held at Tw < WAT. This causes the liquid temperature near the pipe wall to decrease below the WAT, thereby initiating the deposition process. The first deposit layer adjacent to the pipe wall corresponds to a radial location R − Δr and an axial distance z = uΔt, all the way to the location of the imaginary disk separating the depositing and nondepositing liquids. The liquid region along the radial direction would be confined to 0 < r < (R − Δr), and the deposit region would exist over (R − Δr) < r < R. The movement of the liquid− deposit interface would cause changes in the boundary between the liquid region and the deposit region. As mentioned earlier, in the “hot flow” regime, the liquid−deposit interface temperature is held at the WAT of the mixture (i.e., Tl = Tδ = Td = WAT at r = s while Tl > WAT for r < s). Equation 1 (for the “hot flow” regime) or eq 2 (for the “cold flow” regime) along with eqs 3, 5, 7, and 8 were solved numerically to obtain the average liquid phase temperature, the deposit region temperature profile and the location of the liquid−deposit interface. To solve eq 1 or 2 to obtain the average liquid-phase temperature, a Newton−Raphson iterative procedure with a relative convergence criterion of 1 × 10−6 was used. The calculations were repeated by setting Tl‑out for the one axial element as Tl‑in for the immediately next axial element, and so on. For axial locations with Tl > WAT, the liquid− deposit interface temperature was always equal to the WAT. For each time step, the average cooling rate for the flowing depositing liquid was used to estimate the WPT at all axial locations using eq 9, which was compared with Tl to determine whether one or two phases would exist. When Tl approached the WPT, the new liquid−deposit interface temperature corresponding to the particular axial location z was estimated iteratively using eq 3 along with the boundary condition given as eq 8a. The average liquid temperature of the “waxy” mixture was observed to decrease steadily with time in the axial direction. Estimation of Properties and Thermodynamic Considerations. The methods used for estimating all of the liquid-

negative value of the parameter b, eq 9 indicates that the WPT of a waxy mixture increases linearly with a decrease in the cooling rate, x. It is worth pointing out that Kasumu et al.51 used the term “wax precipitation temperature” (WPT), which is measured at a constant and controlled cooling rate, to distinguish it from the “wax appearance temperature” (WAT), which is measured while cooling the sample in an uncontrolled stepwise manner. In this study, the underlying relationship between WPT, x, and y in eq 9 was used to achieve the transition from the “hot flow” regime to the “cold flow” regime. The significance of the cooling rate in transitioning from the “hot flow” regime to the “cold flow” regime is illustrated in Figure 2. The solid line in

Figure 2. Effect of the cooling rate on the wax precipitation temperature and liquid-to-solid phase transformation for w29 = 6 mass %.

Figure 2 represents the WPT predicted from eq 9 as the cooling rate is varied for a 6 mass % wax−solvent mixture. The area above the WPT line is the one-phase liquid region (corresponding to the “hot flow” regime), whereas the area below the WPT line is the two-phase liquid−solid region (corresponding to the “cold flow” regime). Line a−b−c in Figure 2 represents a decrease in the cooling rate for the 6 mass % mixture from 0.4 to 0.1 °C min−1 while the mixture is held at a constant temperature of 35 °C. From point a to b, the mixture would be a one-phase liquid; however, from point b to c, the mixture would exist in the two-phase (liquid + solid) state. That is, starting from point a, when the cooling rate decreases to approximately 0.27 °C min−1 (at point b), eq 9 predicts the transition from one phase (liquid) to two phases (liquid + solid). Line d−e−f in Figure 2 illustrates the cooling of the same mixture from 36.5 to 35.5 °C at a constant cooling rate of 0.1 °C min−1, for which the solid phase is predicted to precipitate at a temperature of 35.7 °C (at point e). Line g−h−i in Figure 2 corresponds to the cooling of the same mixture from 34.8 to 33.8 °C but at a higher constant cooling rate of 0.4 °C min−1, for which the solid phase is predicted to precipitate at a temperature of 34.5 °C (at point h). Thus, for the same 6 mass % wax−solvent mixture, an increase in the cooling rate from 0.1 to 0.4 °C min−1 is predicted to decrease the WPT from 35.7 to 34.5 °C. It is noted that the wax precipitation process illustrated by line a−b−c in Figure 2 is more relevant 6481


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deposit thickness profiles, especially when the deposit thickness became small, but the computational time increased significantly; however, the numerical results did not change appreciably.43 With 151 radial grids, one complete simulation typically required about 3 h of computation time on a desktop computer with 4 GB of RAM and a quad core processor with a processing speed of 2.66 GHz.

phase and deposit-layer properties were described previously.32,33 Also, as in previous studies,32,33,35,36,38,39 the density change caused by partial solidification of the liquid phase in the deposit layer was neglected. Deposition aging effects were not considered, although as mentioned previously, these could be incorporated by using the shear-induced deformation approach.35,37,38 To keep the thermodynamic calculations simple, the wax− solvent mixture was treated as a pseudobinary mixture comprising C13 (representing the liquid or solvent fraction) and C29 (representing the wax fraction). It was assumed that the C13−C29 binary mixture is an ideal eutectic mixture (i.e., it forms with no heat of mixing and no change in volume). The freezing-point depression for an ideal binary mixture was obtained from the following expression: ln xi = −

(ΔHm)i ⎡ 1 1 ⎤ − ⎢ ⎥, R ⎣ (TL)i (Tm)i ⎦

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RESULTS AND DISCUSSION Predicted Temperature and Deposit Thickness Profile for Base-Case Scenario. The predictions for the variation of the liquid mixture temperature, Tl, and the deposit thickness, δ/ R, along the axial distance, z, of the pipeline at various residence times, t, are shown in Figures 3 and 4, respectively. These predictions are for a 10 cm diameter, 4000 m long pipeline (corresponding to a dimensionless length L/D = 40 000), carrying a “waxy” mixture under turbulent flow conditions. The calculations were terminated at a residence time of 4.0 h since a thermal steady state was attained over the length of the pipeline well within 4.0 h for most of the simulations with varying parameters. In Figure 3, the predictions for the liquid phase temperature profile, expressed in terms of the difference between the mixture temperature, Tl, and its WAT, are shown as a function of z and t. The predicted liquid region temperature profiles show a steadily decreasing trend for both “hot flow” and “cold flow” regimes. For residence times less than 0.2 h, the liquid mixture temperature was higher than the WAT, representing the “hot flow” of the “waxy” mixture. At a residence time of about 0.4 h, the liquid temperature became less than the WAT at axial distances larger than z/D = 3710, indicating the transition from “hot flow” to “cold flow”. Below the WAT, the rate of cooling of the liquid mixture decreases with an increase in residence time, t, until the liquid mixture cools down to the pipe wall temperature in about 4 h, at the exit of the pipeline. The predicted deposit thickness profile at various axial locations and at different residence times for the base-case scenario is shown in Figure 4. At residence times of 0.2, 0.4, and 0.6 h, the deposit thickness is predicted to increase along the axial distance in the “hot flow” region (i.e., z/D < 3710), and for an increase in the residence time beyond 0.6 h, there was no change in the steady-state deposit thickness in the “hot flow” regime. The steady-state deposit thickness at residence times of 0.6 h and higher showed an increasing trend for axial locations with liquid temperature higher than the WAT and a decreasing trend for axial locations with liquid temperature lower than the WAT. The maximum deposit thickness was predicted to occur at z/D = 3710, when the flow transitions from “hot flow” to “cold flow” and the liquid temperature Tl is close to the WAT. At this axial location, the maximum deposit thickness is predicted to be δ/R ≈ 0.28, which would correspond to about 48% blockage of the pipe cross-sectional area, thereby causing a significant pressure drop due to solids deposition. From an operational point of view, this observation points out practical limitations in maintaining a high flow rate through pipelines carrying “waxy” crude oils under the “hot flow” regime. For axial locations under “cold flow”, the deposit thickness decreases axially with an increase in the residence time and as the wax−solvent mixture flows down the length of the pipeline. The results in Figures 3 and 4 confirm that as Tl approaches Tw (with a corresponding decrease in the thermal driving force), the deposit thickness decreases steadily.

i = 1, 2 (10)

where (Tm)i, (TL)i, and (ΔHm)i are the melting-point temperature, liquidus temperature, and enthalpy of melting (or fusion), respectively, for component i. The values of Tm and ΔHm for C13 and C29 were taken as 267.8 and 335.4 K and 28.5 and 106.6 MJ/kmol, respectively.43 It should be noted that the thermodynamic relationships for the flash calculations of a multicomponent “waxy” mixture would be different, requiring increased computation time. In the “cold flow” regime especially, the heavier alkanes would be expected to precipitate preferentially. Bhat and Mehrotra35,36 described the calculations for multicomponent wax−solvent mixtures with a UNIQUACbased phase equilibrium approach. Numerical Solution Methodology. Each set of equations was solved numerically using MATLAB to obtain the temperature profile over the pipeline length and the radial movement of the liquid−deposit interface with time for each axial element. Equations were discretized using an explicit scheme in which the dependent variables were estimated from the known values at the previous time interval. The pipe radius R was divided into n equally spaced concentric rings such that the radial increment is given by Δr = R/n. The time increment Δt and the radial increment Δr were chosen to satisfy the following stability criterion:32,33 αΔt 1 ≤ 2 2 Δr

(11)

The base-case values of the parameters used in the calculations and other values of parameters used in this study are provided in Table 1. For all of the calculations, the pipeline length was chosen as 4000 m. Several sets of preliminary calculations were used to select the number of radial grids as 151.43 A larger number of radial grids yielded “smoother” Table 1. List of Parameters and Their Base-Case and Other Values Used for Model Predictions parameter

base-case value

other values

pipe diameter, D (m) mixture composition, w29 (mass % C29) inlet mixture temperature, Thi (°C)

0.1 m 10.0 mass % WAT + 15 °C

pipe wall temperature, Tw (°C)

WAT − 15 °C

inlet Reynolds number, Rei

15000

− − WAT + 5 °C WAT + 25 °C WAT − 5 °C WAT − 25 °C 5000, 25000 6482


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Figure 3. Predicted variation of the average liquid mixture temperature along the axial direction of a pipeline at different residence times for the basecase scenario.

Figure 4. Predicted deposit layer thickness profiles along the axial direction of a pipeline at different residence times for the base-case scenario.

for a 6 mass % wax solution at similar flow rates and Reynolds numbers. The abscissa of Figure 5 is actually the difference between the liquid temperature and the corresponding WAT, so the results to the left of the WAT are for the “hot flow” regime and those to the right are for the “cold flow” regime. The trends in these experimental results indicate that the deposit mass (or deposit thickness) increases as Tl decreases to approach the WAT (i.e., in the “hot flow” regime), reaches a maximum when Tl becomes equal to WAT, and then decreases thereafter as Tl decreases to approach Tw (i.e., in the “cold flow” regime). The trends in the experimental deposition results in Figure 5 can be compared with the predictions in Figure 6, which was

The predictions for the dissolved wax content in the liquid mixture showed a constant value of 10 mass % C29 in the “hot flow” regime (up to the maximum in Figure 4 at z/D = 3710) followed by a steady decline in the “cold flow” regime to about 1 mass % C29 at z/D = 40 000. This indicates that the solidphase concentration in the solid−liquid suspension in the “cold flow” regime gradually increased to about 9 mass % at z/D = 40 000. The experimental deposition results reported by Bidmus and Mehrotra14 for a bench-scale flow-loop apparatus in both “hot flow” and “cold flow” regimes have been replotted in Figure 5. Figure 5a presents the deposition results for a 3 mass % wax solution at two flow rates, and Figure 5b presents similar results 6483


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increasing deposit thickness profile axially in the “hot flow” regime and a decreasing deposit thickness profile axially in the “cold flow” regime, which is confirmed by the trends in the experimental results in Figure 5. In fact, in the “cold flow” regime, the deposit thickness is expected to approach zero as the oil temperature approaches Tw because there would not be any thermal driving force for the deposition to take place. Also, the oil flow would be in the form of a two-phase suspension or slurry. The agreement between the trends in the experimental and predicted results is taken as an important validation of the modeling approach presented in this study. Effect of the Pipe Wall Temperature, Tw. For these simulations, only the pipe wall temperature was varied while all of the other parameters were held at their base values. In Figure 7, the steady-state predictions for the average liquid temperature and deposit thickness profiles along the pipe length are shown for three pipe wall temperatures, WAT − 5 °C, WAT − 15 °C, and WAT − 25 °C. In Figure 7a, for a residence time of 4.0 h, the liquid mixture temperature Tl decreases from the inlet mixture temperature Thi to the pipe wall temperature Tw at the outlet of the pipeline, representing a fully developed flow in the pipeline, for all the three values of the pipe wall temperature. In Figure 7b, the predicted deposit layer thickness decreases with an increase in the pipe wall temperature under both “hot flow” and “cold flow” of the “waxy” mixture. It can be seen that the maximum value of the deposit thickness predicted during the transition from the “hot flow” regime to the “cold flow” regime decreased by about 62% because of an increase in the pipe wall temperature, Tw, from WAT − 25 °C to WAT − 5 °C. The maximum deposit thickness at the transition from “hot flow” to “cold flow” shifts farther down the pipeline by z/D = 4500 when the pipe wall temperature is increased from WAT − 25 °C to WAT − 5 °C. The predictions in Figure 7 indicate that solids deposition under both “hot flow” and “cold flow” of “waxy” mixtures can be substantially decreased by increasing the pipe wall temperature Tw. The predicted trends in Figure 7 are consistent with the experimental results for both “hot flow” and “cold flow” regimes.14,29,31,34,37,42 Effect of Inlet Mixture Temperature, Thi. Figure 8 shows the predicted effects of the inlet mixture temperature, Thi, on the average liquid mixture temperature and the deposit layer thickness profiles along the length of the pipeline for a residence time of 4.0 h. The calculations were performed at three Thi values, WAT + 5 °C, WAT + 15 °C, and WAT + 25 °C, while all of the other parameters were held at their base values. The predictions in Figure 8 for the three values of Thi are very similar. In Figure 8a, an increase in Thi is predicted to shift the axial location for the transition from the “hot flow” regime to the “cold flow” regime about 450 m further down the 4 km pipeline. A higher Thi implies an increased sensible heat in the flowing mixture, which would be expected to increase its cooling rate in the “hot flow” regime.14,29,31,34,37 Moreover, since the deposit thickness is known to decrease with an increase in Thi, the mixture with a higher Thi would have a smaller deposit thickness (consequently, offering a decreased thermal resistance), and therefore, it would cool more rapidly near the pipeline entrance region. As shown in Figure 8b, the deposit thickness profiles are nearly identical for the three cases. At a fixed axial location, the mixture with a higher Thi has a higher rate of heat loss in both the “hot flow” and “cold flow” regimes. The main impact of an

Figure 5. Experimental data for the variation of deposit thickness with liquid temperature in the “hot flow” and “cold flow” regimes: (a) 3 mass % wax; (b) 6 mass % wax. (Data taken from ref 14).

Figure 6. Predicted variation of deposit thickness with liquid temperature in a pipeline under turbulent flow (replot of the predictions in Figures 3 and 4).

obtained by replotting the predictions in Figures 3 and 4. In both cases, the deposit mass (or thickness) increases in the “hot flow” regime while the liquid temperature decreases, goes through a maximum as the liquid temperature approaches the WAT, and then decreases in the “cold flow” regime while the liquid temperature decreases steadily because of heat transfer to the pipe wall held at Tw. The predictions in Figure 6 show an 6484


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Figure 7. Predictions of the effect of the pipe wall temperature in the axial direction of a pipeline: (a) average liquid temperature profile; (b) deposit thickness profile.

at a lower Rei is dissipated more quickly to the pipe wall held at Tw. The predictions also indicate that the axial location for the transition from “hot flow” to “cold flow” shifts away from the pipe entrance with an increase in Rei. In Figure 9b, the predicted deposit thickness during “hot flow” decreases with an increase in Rei. This effect has been explained in previous experimental and modeling studies by Mehrotra and co-workers14,15,26,29−34,37,42 by equating the steady-state heat flux across the convective thermal resistance (in the liquid region) and across the conductive thermal resistance (offered by the deposit layer). Thus, the predictions in Figure 9b are in agreement with the trends obtained from previous experimental studies.14,31,34,37,42 The maximum deposit thickness at the transition from “hot flow” to “cold flow” is also predicted to decrease with an increase in Rei. In the “cold flow” regime, the predictions indicate that when Rei is 5000, negligible deposit thickness is achieved at z/D > 16 000, whereas δ/R ≈ 0.04 at z/D = 40 000

increase in Thi is that the deposit thickness profile is delayed or shifted down the pipeline, but the maximum deposit thickness at the transition from “hot flow” to “cold flow” is predicted to be essentially the same. Effect of (Inlet) Reynolds Number, Rei. The results in Figure 9 show the predicted average liquid mixture temperature profile and the deposit thickness profile along the pipeline at steady state with a variation in the inlet Reynolds number, Rei. It should be noted that the Reynolds number varies axially because of changes in liquid properties as well as the velocity and diameter (due to changes in the cross-sectional area available for flow). The inlet Reynolds numbers used in these calculations were 5000, 15 000, and 25 000, while all of the other parameters were held at their base values. These results were obtained for a residence time of 4.0 h. In Figure 9a, the average liquid temperature is predicted to decrease more rapidly and over a shorter pipe length at a lower Rei than at a higher Rei. This is partly because the sensible heat 6485


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Figure 8. Predictions of the effect of inlet mixture temperature in the axial direction of a pipeline: (a) average liquid temperature profile; (b) deposit thickness profile.

temperature (i.e., δ/R → 0.0 when Tl ≫ Tw).30 That is, no deposition could occur with a very high oil temperature even though it would lead to a very large thermal driving force. On the other hand, as mentioned previously, no deposition could occur in the absence of a thermal driving force (i.e., δ/R → 0.0 when Tl → Tw). The latter observation has been confirmed experimentally even in the “cold flow” regime with precipitated wax crystals suspended in the liquid phase.14 In Figure 10, two sets of predictions for the deposit thickness profile are compared. Figure 10a shows two different deposit thickness profiles predicted for the same overall thermal driving force, (Thi − Tw) = 20 °C. When Thi = (WAT + 15 °C) and Tw = (WAT − 5 °C), the deposit thickness is significantly less than that with Thi = (WAT + 5 °C) and Tw = (WAT − 15 °C). Between these two cases, the deposit thickness is increased when either or both Thi and Tw are decreased. Similarly, in Figure 10b two different deposit thickness profiles are shown for the same overall thermal driving force, (Thi − Tw) = 40 °C. When Thi = (WAT + 25 °C) and Tw = (WAT − 15 °C), the

when Rei = 25 000. That is, the deposit thickness is predicted to decrease over shorter pipe lengths at lower Reynolds numbers. These predictions suggest that a higher Reynolds number might be preferred in the “hot flow” regime (which would yield a lower deposit thickness although over a longer pipe length), but a lower Reynolds number should be preferred in the “cold flow” regime (which would yield a lower deposit thickness over a shorter pipe length). This is an interesting finding that calls for further investigation and confirmation in future studies. Effect of the Overall Temperature Difference, (Tl − Tw). Recently, the available experimental observations were used to explain the relationship between the temperature difference (or the thermal driving force) and the wax deposition tendency.15 It was pointed out that it is incorrect to assume that the deposit thickness would increase with an increase in the overall thermal driving force or that the deposit thickness would decrease when the overall thermal driving force is decreased. In fact, it has been shown that wax deposition can be prevented if the waxy crude oil can be maintained at a very high 6486


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Figure 9. Predictions for the effect of inlet Reynolds number in the axial direction of a pipeline: (a) average liquid temperature profile; (b) deposit thickness profile.

■

deposit thickness is significantly less than that with Thi = (WAT + 15 °C) and Tw = (WAT − 25 °C). Again, for these two cases, the deposit thickness increases when either or both Thi and Tw are decreased. As mentioned previously, experimental studies have established that even though a thermal driving force is needed for wax deposition, the overall thermal driving force does not influence the deposit thickness directly. Rather than the overall thermal driving force (Tl − Tw), the deposition tendency can be explained in terms of two temperature difference components (i.e., across the two thermal resistances), namely, (Tl − Td) across the convective thermal resistance and (Td − Tw) across the conductive thermal resistance, with Td = WAT.14,15,29,30 The predictions in Figure 10 clearly support the experimental observation that the overall temperature difference is not sufficient to determine the extent of wax deposition.

CONCLUSIONS

A mathematical model based on the moving boundary formulation was modified to predict wax deposition under both “hot flow” and “cold flow” regimes. To simplify the numerical calculations, the “waxy” mixture was approximated as a pseudobinary mixture comprising C13H28 (C13, representing the liquid solvent fraction) and C29H60 (C29, representing the wax fraction). In the model, the transition from “hot flow” to “cold flow” was accomplished by incorporating the effect of the cooling rate on the wax precipitation temperature (WPT). A recently published correlation for the effect of the cooling rate and wax concentration on the WPT was used to account for the crystallization kinetics in an empirical manner. Predictions were provided for the variation in liquid temperature and deposit thickness in a pipeline under turbulent flow conditions. The predicted trends for the deposit thickness profiles for both the “hot flow” and “cold flow” regimes were explained in terms of 6487


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Figure 10. Predicted effect of the overall thermal driving force on the deposit thickness profiles: (a) (Thi − Tw) = 20 °C; (b) (Thi − Tw) = 40 °C.

number (Rei). A decrease in the inlet liquid temperature (Thi) shifted the deposit thickness profile by about z/D = 4500 toward the pipe inlet. The predicted results confirmed that the overall thermal driving force, (Tl − Tw), does not influence wax deposition directly. Instead, the deposition tendency can be explained in terms of two temperature differences, namely, (Tl − Td) and (Td − Tw), with Td = WAT. A reduction in the deposit thickness could be accomplished by increasing either or both Thi and Tw. This study provides an important extension to the unsteady-state heat-transfer-based modeling approach by providing a method to include the transition from the “hot flow” regime to the “cold flow” regime of flowing “waxy” mixtures in pipelines. This important advancement has been made possible by the inclusion of the effect of the cooling rate on WPT of “waxy” mixtures. It thus provides a methodology for the unsteady-state modeling of wax deposition in the “cold flow” regime. This study also provides further confirmation that

the observations and trends in experimental data reported by Mehrotra and co-workers.14,29,31,34,37,42 For all cases, the deposit thickness along the pipe length was predicted to increase in the “hot flow” regime and to decrease in the “cold flow” regime. The maximum deposit thickness was predicted to be found at the location where the transition from “hot flow” to “cold flow” occurred. The axial location for the transition from “hot flow” to “cold flow” moved further down the pipeline with an increase in the inlet temperature (Thi), an increase in the pipe wall temperature (Tw), and an increase in the (inlet) Reynolds number (Rei). The predictions for wax deposition in the “hot flow” regime indicated that the deposit thickness could be decreased by increasing the pipe wall temperature (Tw) and/or by increasing the inlet Reynolds number (Rei). The predictions for wax deposition in the “cold flow” regime indicated that the deposit thickness could be decreased by increasing the pipe wall temperature (Tw) and/or by decreasing the inlet Reynolds 6488


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solids deposition from “waxy” mixtures or crude oils in both the “hot flow” and “cold flow” regimes can be modeled satisfactorily as a thermally driven process.

■

z = axial distance (m) Greek Letters

α1 = liquid-phase thermal diffusivity (m2 s−1) αδ = deposit-phase thermal diffusivity (m2 s−1) α′δ = deposit-phase modified thermal diffusivity (m2 s−1) δ = deposit layer thickness (m) λ = latent heat of solid−liquid phase transformation (J kmol−1) ρ = density at liquid−deposit interface (kg m−3) ρδ = density of deposit (kg m−3) ρl = density of liquid (kg m−3)

AUTHOR INFORMATION

Corresponding Author

*Phone: (403) 220−7406. Fax: (403) 284−4852. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Department of Chemical and Petroleum Engineering of the University of Calgary is gratefully acknowledged. We thank Dr. Nitin V. Bhat for his contributions to this study.

Acronyms

■

■

WAT = wax appearance temperature (°C) WPT = wax precipitation temperature (°C)

REFERENCES

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NOMENCLATURE a, b, c = regression constants in eq 9 Ad = interfacial area (m2) Cp = heat capacity (J K−1 kmol−1) Cp,l = paraffin liquid heat capacity (J K−1 kmol−1) D = pipe diameter (m) fδ = solid-phase mass fraction in the deposit f l = solid-phase mass fraction suspended in the liquid fs = mass fraction of solid phase at liquid−deposit interface h = heat-transfer coefficient (W m−2 K−1) k = paraffin thermal conductivity (W m−1 K−1) kl = liquid-phase thermal conductivity (W m−1 K−1) kδ = deposit thermal conductivity (W m−1 K−1) L = length of the pipeline (m) ṁ = mass flow rate (kg s−1) ṁ h = mass flow rate of wax−solvent mixture (kg s−1) n = number of grids Nu = Nusselt number Pr = Prandtl number r = radius of the pipe with deposition (m) R = radius of the pipe without deposition (m) Re = Reynolds number Rei = inlet Reynolds number s = effective radius of the pipe with deposition (m) t = time (s) Td = liquid−deposit interface temperature (°C) Tδ = average deposit layer temperature (°C) Th = average temperature of the wax−solvent mixture = 0.5Thi + 0.5Tho (°C) Thi = inlet temperature of wax−solvent mixture (°C) Tho = mixture or liquid temperature at the outlet of an axial element (°C) Tl = liquid region temperature (°C) TL = liquidus temperature (°C) Tl‑in = mixture or liquid temperature at the inlet of an axial element (°C) Tl‑out = mixture or liquid temperature at the outlet of an axial element (°C) Tw = pipe wall temperature (°C) u = average velocity of the liquid mixture (m s−1) Vl = volume of liquid across a differential element (m3) w29 = mass fraction of C29 in the liquid phase x = cooling rate (°C min−1) y = wax concentration (mass %) 6489


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