Energy Fuels 2010, 24, 2240–2248 Published on Web 10/14/2009
: DOI:10.1021/ef9008357
Deposition from “Waxy” Mixtures under Turbulent Flow in Pipelines: Inclusion of a Viscoplastic Deformation Model for Deposit Aging† Anil K. Mehrotra* and Nitin V. Bhat Department of Chemical & Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4 Received July 31, 2009. Revised Manuscript Received September 21, 2009
A mathematical model is described and used to predict solids deposition from multicomponent paraffinic “waxy” mixtures under turbulent flow in pipelines. The model is based on the moving boundary problem formulation, in which the deposit formation and growth is modeled primarily as a heat-transfer process with phase change. The effects of shear stress (or Reynolds number) and deposition time (or deposit aging) are incorporated via a viscoplastic model, which is based on one-dimensional deformation of a cubical cage that squeezes out a fraction of the liquid phase from the deposit. Numerical solutions were obtained for the radial and axial growth of the deposit with time at Reynolds numbers (Re) of 10 000-25 000. The predicted trends are in agreement with the experimental results from recent laboratory deposition studies. The steady-state deposit thickness under turbulent flow is predicted to be considerably smaller than that under laminar flow, and it decreased with an increase in Re. The average wax content of the deposit is predicted to also increase with Re and deposition time (or aging), causing the deposit to become enriched in heavier paraffins and depleted in lighter paraffins. The results indicate that, although an increase in Re and deposition time causes wax enrichment in the deposit, the deposit thickness is dependent on heat-transfer and thermodynamic phase equilibrium considerations.
history has been reported to affect the structure and the rheological properties of the formed gel in its final state significantly. Even small deformations and stresses can affect the aggregation of paraffin crystals during gel formation.1 The precipitation and deposition of wax crystals are two different but related processes; wax precipitation is governed primarily by thermodynamic considerations, whereas wax deposition is predominantly a transport process, which requires a thermal gradient. The deposit is comprised of liquid and solid phases in a gel-like state, whose composition and relative proportions vary across the deposit-layer thickness, because of changes in temperature, concentrations, and shear stress (caused by the flowing crude oil). The deposition of solids from “waxy” mixtures is a complex phenomenon, for which several different models or approaches have been proposed.4 Among these, the molecular diffusion approach has been used in several studies on solids deposition from “waxy” mixtures and crude oils.5-11 The molecular diffusion
Introduction The adverse effects of wax deposition are encountered in all sectors of the petroleum industry, ranging from the damage to oil reservoir formations to the blockage of pipelines and process equipment. The presence of deposited solid in pipelines leads to an increased pressure drop and/or a decreased flow rate and causes substantial expenditures for the control and remediation of solids deposition. Wax molecules crystallize out of the crude oil when its temperature is reduced, because of the lower solubility of n-alkanes heavier than C18H38 in the liquid phase, which may lead eventually to their deposition on the pipe wall. The highest temperature at which the first crystals of paraffin wax appear upon cooling of “waxy” crude oils is called the wax appearance temperature (WAT). The rheological behavior of prepared waxy mixtures has been shown to change with temperature, i. e., from Newtonian at temperatures above the WAT to shear thinning and apparently plastic at lower temperatures.1,2 At temperatures below the pour-point temperature (PPT), waxy mixtures have been noted to behave as weakly attractive colloidal gels; the structural buildup resulting from the crystal formation and aggregation is favored by lower temperatures and longer times.1,3 Furthermore, the mechanical or shearing
(4) Azevedo, L. F. A.; Teixeira, A. M. A Critical Review of the Modeling of Wax Deposition Mechanisms. Petrol. Sci. Technol. 2003, 21, 393. (5) Burger, E. D.; Perkins, T. K.; Striegler, J. H. Studies of Wax Deposition in the Trans Alaska Pipeline. J. Petrol. Technol. 1981, 33, 1075. (6) Svendsen, J. A. Mathematical Modeling of Wax Deposition in Oil Pipeline Systems. AIChE J. 1993, 39, 1377. (7) Creek, J. L.; Lund, H. J.; Brill, J. P.; Volk, M. Wax Deposition in Single Phase Flow. Fluid Phase Equilib. 1999, 158-160, 801. (8) Kok, M. V.; Saracoglu, R. O. Mathematical Modeling of Wax Deposition in Crude Oil Pipelines (Comparative Study). Petrol. Sci. Technol. 2000, 18 (9&10), 1121. (9) Singh, P.; Venkatesan, R.; Fogler, H. S.; Nagarajan, N. Formation and Aging of Incipient Thin Film Wax-Oil Gels. AIChE J. 2000, 46, 1059. (10) Singh, P.; Venkatesan, R.; Fogler, H. S.; Nagarajan, N. Morphological Evolution of Thick Wax Deposits during Aging. AIChE J. 2001, 47, 6. (11) Ramirez-Jaramillo, E.; Lira-Galeana, C.; Manero, O. Modeling Wax Deposition in Pipelines. Petrol. Sci. Technol. 2004, 22 (7&8), 821.
† Presented at the 10th International Conference on Petroleum Phase Behavior and Fouling. *Author to whom correspondence should be addressed. Phone: (403) 220-7406. Fax: (403) 284-4852. E-mail:
[email protected]. (1) Visintin, R. F. G.; Lapasin, R.; Vignati, E.; D’Antona, P.; Lockhart, T. P. Rheological Behavior and Structural Interpretation of Waxy Crude Oil Gels. Langmuir 2005, 21, 6240. (2) Tiwary, D.; Mehrotra, A. K. Phase Transformation and Rheological Behaviour of Highly Paraffinic “’Waxy” Mixtures. Can. J. Chem. Eng. 2004, 82, 162. (3) Vignati, E.; Piazza, R.; Visintin, R. F. G.; Lapasin, R.; D’Antona, P.; Lockhart, T. P. Wax Crystallization and Aggregation in a Model Crude Oil. J. Phys.: Condens. Matter 2005, 17, S3651.
r 2009 American Chemical Society
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: DOI:10.1021/ef9008357
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approach is based on the premise that the crude oil flow in a pipe with the wall temperature below the WAT provides a radial temperature gradient, which creates a radial concentration gradient for the diffusion of wax molecules. In the molecular diffusion approach, the liquid-deposit interface temperature is assumed to increase gradually from a value close to the pipe-wall temperature initially to the WAT as steady state is attained. The deposit layer is assumed to be formed on the cooler surface, because of the reduced solubility of paraffins at lower temperatures. The mass of deposited wax has been reported to decrease with increasing flow rate under both laminar and turbulent flows.10,12-16 Other related topics have also been addressed, such as the deposit removal or sloughing by shear, the deposit aging with time, and the effect of flow or shear rate.7,9,16-18 The deposit composition and “hardness” have been observed to change with time through a process that is called “deposit aging”.7,9,13,14,16,19,20 In addition to the role of molecular diffusion, the importance of heat transfer in the deposition process has also been identified in several studies.7,13-16,21-23 Recent laboratory investigations with prepared paraffinic mixtures have reported the deposition process to be a relatively fast process, with the thermal steady state being accomplished within 30 min.13-16 Beyond this time, the concentration of heavier paraffins in the deposit layer was observed to increase slightly. Although the overall temperature difference between the paraffinic mixture and the surroundings provides the driving force for solids deposition, it was shown that the temperature difference across the deposit layer is a more significant controlling parameter.13-16 A modeling approach for wax deposition, which is entirely different from the molecular diffusion approach, is based on the premise that the deposition process is thermally driven; i.e., the deposition of solids is controlled primarily by
heat-transfer considerations. In this thermally driven approach, the heat-transfer rate is dependent on the thermal driving force (between the bulk crude oil temperature and the cooler pipe-wall temperature), the latent heat released during the phase transformation of the crude oil or wax-solvent mixture into a solid at the interface, and the convective and/or conductive thermal resistances in series. This heat-transferbased approach has been used successfully to predict the trends in experimental results.12-16,21-26 In this approach, the solids deposition is treated essentially as a (partial) freezing or solidification process. The main assumption is that the liquid-deposit interface temperature remains constant at the WAT of the liquid phase (i.e., the crude oil or wax-solvent mixture).27 Bhat and Mehrotra24,25,28,29 presented a mathematical model, based on the moving boundary problem approach, for solids deposition from “waxy” mixtures under static and laminar flow conditions in a pipeline. It is emphasized that this model is based entirely on heat-transfer considerations. The effect of shear stress on the composition and growth of the deposit layer was incorporated via a simplified representation involving one-dimensional deformation of a cubical cage,28 leading to the release of a fraction of the liquid from the deposit.29 In this representation, the effect of shear stress is expressed in terms of an angle by which the cubical cage is deformed, and this deformation angle was shown to be dependent upon the fractional deposit thermal resistance (or the fractional temperature drop), deposit mass, and Reynolds number.28 Two recent studies30,31 reported that the liquiddeposit interface temperature, during the deposit-layer growth under static and laminar flow conditions, remained constant at the WAT of the deposit-forming liquid phase. These studies provided an important validation of the constant-interface-temperature assumption in the heat-transfer approach for modeling the deposition of solids from waxy mixtures. Recently, Tiwary and Mehrotra16 used a bench-scale flowloop apparatus, incorporating a double-pipe heat exchanger, for investigating the effects of shear rate and deposition time on the deposition of solids, under turbulent flow, from solutions of a multicomponent wax in a paraffinic solvent. They
(12) Wu, C.; Wang, K.-S.; Shuler, P. J.; Tand, Y.; Creek, J. L.; Carlson, R. M.; Cheung, S. Measurement of Wax Deposition in Paraffin Solutions. AIChE J. 2002, 48, 2107. (13) Bidmus, H. O.; Mehrotra, A. K. Heat-Transfer Analogy for Wax Deposition from Paraffinic Mixtures. Ind. Eng. Chem. Res. 2004, 43, 791. (14) Parthasarathi, P.; Mehrotra, A. K. Solids Deposition from Multicomponent Wax-Solvent Mixtures in a Benchscale Flow-Loop Apparatus with Heat Transfer. Energy Fuels 2005, 19, 1387. (15) Fong, N.; Mehrotra, A. K. Deposition under Turbulent Flow of Wax-Solvent Mixtures in a Bench-Scale Flow-Loop Apparatus with Heat Transfer. Energy Fuels 2007, 21, 1263. (16) Tiwary, R.; Mehrotra, A. K. Deposition from Wax-Solvent Mixtures under Turbulent Flow: Effects of Shear Rate and Time on Deposit Properties. Energy Fuels 2009, 23, 1299. (17) Singh, P.; Venkatesan, R.; Fogler, H. S.; Nagarajan, N. Prediction of the Wax Content of the Incipient Wax-Oil Gel in a Pipeline: An Application of the Controlled Stress Rheometer. J. Rheol. 1999, 43, 1437. (18) Bott, T. R.; Gudmunsson, J. S. Deposition of Paraffin Wax from Kerosene in Cooled Heat Exchanger Tubes. Can. J. Chem. Eng. 1977, 55, 381. (19) Singh, P.; Youyen, A; Fogler, H. S. Existence of a Critical Carbon Number in the Aging of a Wax-Oil Gel. AIChE J. 2001, 47, 2111. (20) Coutinho, J. A. P.; Lopes da Silva, J. A.; Ferreira, A.; Soares, M. R.; Daridon, J. L. Evidence for the Aging of Wax Deposits in Crude Oils by Ostwald Ripening. Petrol. Sci. Technol. 2003, 21, 381. (21) Mehrotra, A. K. Comments on: Wax Deposition of Bombay High Crude Oil under Flowing Conditions. Fuel 1990, 69, 1575. (22) Ghedamu, M.; Watkinson, A. P.; Epstein, N. Mitigation of Wax Buildup on Cooled Surfaces. In Fouling Mitigation of Industrial HeatExchange Equipment; Panchal, C. B., Bott, T. R., Somerscales, E. F. C., Toyama, S., Eds.; Begel House: New York, 1997; pp 473-489. (23) Cordoba, A. J.; Schall, C. A. Application of a Heat Transfer Method to Determine Wax Deposition in a Hydrocarbon Binary Mixture. Fuel 2001, 80, 1285.
(24) Bhat, N. V.; Mehrotra, A. K. Modeling of Deposit Formation from “Waxy” Mixtures via Moving Boundary Formulation: Radial Heat Transfer under Static and Laminar Flow Conditions. Ind. Eng. Chem. Res. 2005, 44, 6948. (25) Bhat, N. V.; Mehrotra, A. K. Modeling of Deposition from ‘Waxy’ Mixtures in a Pipeline under Laminar Flow Conditions via Moving Boundary Formulation. Ind. Eng. Chem. Res. 2006, 45, 8728. (26) Merino-Garcia, D.; Margarone, M.; Correra, S. Kinetics of Waxy Gel Formation from Batch Experiments. Energy Fuels 2007, 21, 1287. (27) Bhat, N. V. Modeling of Solids Deposition from Wax-Solvent Mixtures based on the Moving Boundary Problem Approach. Ph.D. Thesis, University of Calgary, Calgary, Canada, 2008. (28) Mehrotra, A. K.; Bhat, N. V. Modeling the Effect of Shear Stress on Deposition from “Waxy” Mixtures under Laminar Flow with Heat Transfer. Energy Fuels 2007, 21, 1277. (29) Bhat, N. V.; Mehrotra, A. K. Modeling the Effect of Shear Stress on the Composition and Growth of the Deposit Layer from “Waxy” Mixtures under Laminar Flow in a Pipeline. Energy Fuels 2008, 22 (5), 3237. (30) Bidmus, H.; Mehrotra, A. K. Measurement of the Liquid-Deposit Interface Temperature during Solids Deposition from Wax-Solvent Mixtures under Static Cooling Conditions. Energy Fuels 2008, 22, 1174. (31) Bidmus, H.; Mehrotra, A. K. Measurement of the Liquid-Deposit Interface Temperature during Solids Deposition from Wax-Solvent Mixtures under Sheared Cooling. Energy Fuels 2008, 22, 4039.
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: DOI:10.1021/ef9008357
Mehrotra and Bhat 24
layer, as follows:
extended the cubical-cage-deformation approach by expressing the time-dependent variations in the deposit composition with a viscoplastic model, in which Reynolds number (or shear rate) was assumed to influence the initial deposit properties. The proposed viscoplastic model afforded a new explanation for the deposit aging phenomenon, which, in the past, was explained as a counter-diffusion process.10,19 Our previous modeling studies, based on the moving boundary problem formulation, were focused on solids deposition under static and laminar flow conditions.24,25,28,29 This paper presents an extension of the mathematical model to incorporate solids deposition from hydrodynamically established turbulent flow of paraffinic mixtures in a pipeline. Also included are the effects of shear stress and deposition time on the depositing solid layer via the viscoplastic deformation model,16,28 which causes the release of a fraction of liquid from the deposit, leading to wax enrichment and a shift in the carbon number distribution in the deposit layer. Numerical simulations were performed using a modified UNIQUAC model32 for the liquid-solid phase equilibrium. The parameters considered are the temperature of wax-solvent mixture, the pipe-wall temperature, and the Reynolds number. Predictions are reported for the effect of shear stress on the steady-state deposit-layer thickness, the extent (angle) of deformation, and the deposit characteristics;namely, the deposit wax content and the wax enrichment. The trends in model predictions have been compared with experimental observations from bench-scale laboratory measurements in our laboratory, under both laminar and turbulent flow conditions.13-16,30,31
1 1 Fλ Df ¼ Rδ 0 Rδ kδ DTδ
ð3Þ
where f is the solid phase fraction in the deposit, λ is the latent heat of freezing, and Rδ, F, and kδ are the average deposit thermal diffusivity, density, and thermal conductivity, respectively. The energy balance at the liquid-deposit interface is25 DTδ ds ðr ¼ s, z > 0Þ ð4Þ -hðTh -Td Þ ¼ Fλfs kδ dt Dr where fs is the equilibrium solid-phase fraction at the liquiddeposit interface (i.e., at r = s) corresponding to the interface temperature, Td. The heat-transfer coefficient under turbulent flow was estimated from the Dittus-Boelter equation: Nu ¼ 0:023Re0:8 Pr0:33
ð5Þ
All liquid properties were evaluated at the average temperature, (Th þ Td)/2.25 Initial and Boundary Conditions. With the flow of waxy mixture in the pipeline being under isothermal conditions, the initial temperature of the wax-solvent mixture (Th) is taken to be Thi throughout over 0 < r < R and 0 < z < L. The pipe-wall temperature initially is also held at Thi, which implies no deposit layer, initially, anywhere in the pipeline. At t = 0, the pipe-wall temperature is reduced from Thi to Tc, with Tc < WAT. This change causes the mixture temperature near the pipe wall to decrease below the WAT, thereby initiating the deposition process. The two initial conditions are ð6aÞ Th ¼ Thi , t ¼ 0, z g 0
Model Description
s ¼ R,
The details of a mathematical model;incorporating the moving boundary problem formulation;for the growth of the deposit layer from the cooling of a wax-solvent mixture, flowing through a long pipe under laminar flow, have been described elsewhere.24,25,27-29 A brief description of the mathematical model is presented below. Consider a wax-solvent mixture (or a “waxy” crude oil) at an inlet temperature Thi (>WAT), entering a pipeline of inside diameter D and length L, while the cooler pipe wall is held at a constant temperature, Tc (0
ð6bÞ
At t > 0, the deposit-layer temperature adjacent to the pipewall temperature is equal to Tc ( 0, z > 0 Tδ ¼ Tc ,
r ¼ R,
t > 0,
z>0
ð6dÞ
Thus, the moving boundary problem formulation for solids deposition in a pipeline is described by eqs 1, 2, and 4, together with the initial and boundary conditions given by eqs 6a, 6b, 6c, and 6d. Incorporating the Effect of Shear Stress via the Cubical Cage Representation. In a recent study,28 a modeling approach was proposed to account for the effect of shear stress, in which a simplified representation of the deposit layer was presented to account for the effect of shear stress on the relative amounts of the solid and liquid phases. The deposit layer formed under static conditions (i.e., without any shear stress) was considered to be comprised of a lattice-like structure with a unit cell, as shown in Figure 1a, which resembles a cubical cage of length a and volume V (note that V = a3). Each edge of the cubical cage is a rectangular member with a cross section of ξ ξ, and the ratio of the edge thickness to the side of cubical cage is Z = ξ/a. The solid phase volume fraction in the cubical cage is given as
where 2πsΔz is the liquid-deposit interfacial area for heat transfer, s the radial location of the interface or the flow radius of the liquid region (s = R - δ), and Th is the average liquid stream temperature. Neglecting any axial conduction within the deposit, the energy balance equation for the deposit layer is24,25,27 1D DTδ 1 DTδ ¼ 0 r ðs < r < R, z > 0Þ ð2Þ r Dr Rδ Dt Dr where Tδ denotes the deposit temperature. Note that the term Rδ0 represents a modified thermal diffusivity of the deposit (32) Coutinho, J. A. P. Predictive UNIQUAC: A New Model for the Description of Multiphase Solid-Liquid Equilibria in Complex Hydrocarbon Mixtures. Ind. Eng. Chem. Res. 1998, 36, 4870.
VS ¼ 12Z 2 -16Z 3 2242
ð7Þ
Energy Fuels 2010, 24, 2240–2248
: DOI:10.1021/ef9008357
Mehrotra and Bhat Table 1. Regression Constants in eq 12 for Parowax-Norpar13 Mixtures with 10 and 15 mass % Parowaxa Value constant b1 (°) b2 (°) b3 (h) b4 (h) b5 (°) b6 (dimensionless) b7 (dimensionless) b8 (°)
Figure 1. Schematics of the cubical-cage representation: (a) untilted cubical cage (under static conditions), and (b) tilted cubical cage (under shear at a deformation angle of β). (Taken from Mehrotra and Bhat.28)
a
10 mass % wax
15 mass % wax
0.0 79.394 -2.8163 4.5554 85.023 4.4113 0.22010 -77.767
-166.65 248.80 -5.6580 2.2148 257.13 4.3917 0.23757 -252.39
Data taken from ref 16.
The liquid phase volume fraction, by difference, is expressed as VL ¼ 1 -12Z 2 þ 16Z 3
ð8Þ
The application of a one-dimensional shear stress in the direction of flow was assumed to cause the cubical cage to tilt by an angle β, transforming it into a shape similar to the monoclinic structure, as shown in Figure 1b.28 The solidphase volume fraction in the tilted cage is given as VS V 12Z 2 -16Z 3 ð9Þ ¼ VSβ ¼ Vβ cos β and the liquid-phase volume fraction in the tilted cage is given as 12Z 2 -16Z 3 ð10Þ VLβ ¼ 1 cos β
Figure 2. Predictions from eq 12 for the variation of deformation angle (β) with Reynolds number (Re) and deposition or aging time (τ): (a) mixture with 10 mass % wax, and (b) mixture with 15 mass % wax.
The actual volume of the liquid released from the cubical cage, because of its tilting by a deformation angle of β, is equal to the overall volume change, i.e., VL V -VLβ Vβ V -Vβ ¼ ð1 -cos βÞa
3
the Reynolds number Re and deposition time (or deposit aging) τ:
ð11Þ
"
" #-1 #-1 b3 -τ b6 -log Re β ¼ b1 þ b2 1 þ exp þ b5 1 þ exp b4 b7
Note that VLβ = 0, at β = cos-1 (12Z2 - 16Z3), corresponds to the complete release of all of the liquid from the tilted cage. That is, β e cos-1 (12Z2 - 16Z3). The aforementioned relationships for the cubical cage representation involve only two parameters: Z and β. Parameter Z in eqs 7 and 8 corresponds to the original or undeformed cubical cage, i. e., without any shear stress, which is estimated directly from the liquid-solid phase equilibrium calculations.28 To simplify the numerical calculations, it was assumed that all of the shear-induced deformation of the cubical cage occurs when a layer of deposit is formed initially. In other words, the effect of shear stress was assumed to be only on the newly formed layer and not on the previously formed layers buried beneath it. However, the liquid-solid phase equilibrium calculations were repeated as the deposit temperature profile changed with time, because of additional deposition, which caused a variation in the liquid-to-solid ratio in each layer. For the constant flow rate in the pipeline, the increasing deposit-layer thickness axially would increase the local Reynolds number (Re) and, consequently, the shear stress on the deposit, because of the decreased cross-sectional area for flow. In this study, β was estimated from the following correlation proposed by Tiwary and Mehrotra,16 for the effects of
"
b3 -τ þ b8 1 þ exp b4
#-1 "
#-1 b6 -log Re 1 þ exp ð12Þ b7
Table 1 includes the regression constants for b1-b8 in eq 12 for the two mixture compositions (10 mass % wax and 15 mass % wax).16 Note that an interpolation of eq 12 for other wax contents should be avoided. Figure 2 shows predictions for the effects of Re and τ on the deformation angle (β). Characterization of Norpar 13-Wax Mixtures. The gas chromatography (GC) analyses of Norpar 13 (the solvent) and Parowax (the wax), which have been used in previous experimental investigations15,16 and are used this study, are shown in Figure 3. Norpar 13 is a paraffinic solvent with carbon numbers ranging from C8 to C16 and a mean carbon number of C13. Parowax is a paraffinic wax with carbon numbers ranging from C20 to C50 and a mean carbon number of C29. Because of their low concentrations, C41 to C50 were neglected, and the wax composition was normalized to a C20-C40 range. The methods used to estimate all liquid-phase and deposit-layer properties have been described previously.24,25 Also, as in the previous studies,24,25,28,29 the density change caused by a partial solidification of the liquid phase in the deposit layer was neglected. 2243
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: DOI:10.1021/ef9008357
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The simulation results were obtained as a function of both radial and axial locations and the deposition time. The sequence of events in a typical simulation run is summarized as follows. With Tc < Td < Th, the deposition started at the pipe wall at all axial locations and, with time, the liquiddeposit interface moved radially inward. Thermodynamic and mass balance calculations were performed for each new (incremental) layer of deposit to obtain the volume fractions of liquid and solid phases (i.e., VS and VL), along with their compositions. Equation 7 (or 8) was used to obtain parameter Z from VS (or VL). For the specified deposition time τ, the deformation angle β was estimated from eq 12. The portion of the liquid phase retained in the deposit layer was estimated from eq 11 to account for the effect of shear stress (or Re) on the newly formed, incremental deposit layer. The composition of the tilted cage was recalculated by accounting for the amount of released liquid phase. With the growth of the deposit layer, the thermal resistance in the liquid region decreased, because of an increased h (mainly because of an increased Re), while the thermal resistance of the deposit layer increased primarily because of its increased thickness. With an increase in the thickness of the deposit layer, the rate of heat transfer decreased gradually, causing a corresponding decline in the rate of interface movement (ds/dt). When the rate of forced-convection heat transfer in the liquid region became equal to the conductive rate of heat transfer across the deposit layer, the interface movement ceased altogether. At steady state, the equality of the rate of heat transfer for the two thermal resistances in series is given as follows: kd ðTd -Tc Þ ð14Þ hðR -δÞðTh -Td Þ ¼ ln½R=ðR -δÞ
Figure 3. Analysis of Norpar13 (solvent) and Parowax (wax).
Liquid-Solid Phase Equilibrium Considerations. As mentioned previously, a modified UNIQUAC model that was proposed by Coutinho,32 was used to predict the liquid-solid phase equilibrium in the deposition process. Details of the steps involved in the phase transformation calculations have been reported elsewhere.27,33 Numerical Solution Methodology. Equations 1, 2, and 4, along with the initial and boundary conditions given by eqs 6a-6d, were solved numerically to obtain the time-dependent variation of the average liquid temperature over the pipeline length and the radial movement of the liquid-deposit interface with time for each axial element. All equations were discretized using the explicit method, in which the dependent variables are estimated from the known values of parameters at the previous time interval. The domain 0 < r < R was divided into n equally spaced concentric rings with a thickness of Δr (i.e., Δr = R/n). The values of Δr and Δt were selected to satisfy the following stability criterion: RΔt 1 ð13Þ e 2 ðΔrÞ2
According to eq 14, an increase in the average thermal conductivity of the deposit (kd) (e.g., because of an increased solid-phase content) corresponds to an increased deposit thickness. Results and Discussion
Preliminary calculations indicated that the numerical results did not change appreciably for n > 500 (note that n R/Δr) and RΔt/(Δr)2 < 0.4. The values of Δr and Δt were chosen accordingly to obtain the simulation results presented in this study. Further details of the numerical solution procedure have been provided elsewhere.24,25,27,29 Simulation Procedure. The input quantities for numerical calculations were as follows: the mixture composition, the inlet mixture temperature (Thi), the (constant) pipe-wall temperature (Tc), the inlet Reynolds number (Rei), and the deposition time (τ). The movement of the liquid-deposit interface caused changes in the boundaries of the liquidphase and deposit regions. As the deposition progressed radially inward, the deposit thickness increased and the liquid-phase region decreased, which also caused an increase in Re with a corresponding increase in h. For the pipe lengths considered in this study, the pipe was not filled completely with the deposit at steady state, because of the constant liquid flow rate. Also, the deposit growth caused a sharp decline in the rate of heat transfer (or loss) from the solvent-wax mixture to the pipe wall, because of the relatively low thermal conductivity of the deposit.
In the following sections, predictions are reported and compared for the effects of the deposition time (τ), the axial location in pipeline (z/D), the inlet temperature of the Norpar13-Parowax mixture (Thi), and the pipe-wall temperature (Tc) on the steady-state deposit thickness (δ/R), deformation angle (β), two deposit wax contents, and the compositional changes in the deposit [(Wd)j - (Wh)j]. Note that (Wd)j and (Wh)j denote normalized concentrations (expressed in units of mass %) of the jth carbon number, with 20 e j e 40, in the deposit and Norpar13-Parowax mixture, respectively.28,29 Basecase Conditions. For undertaking the simulation calculations, a basecase set of conditions was defined as follows. We considered a 10 mass % Norpar 13-wax mixture entering a circular pipe, with R = 5 cm, at Rei = 15 000 and Thi = WAT þ 5 °C, for which the deposit-layer formation was predicted over 0 < z/D < 4000. The constant pipe-wall temperature (Tc) was taken to be equal to WAT - 5 °C. In a previous study,33 we reported the wax disappearance temperature (WDT), measured during heating, and the wax appearance temperature (WAT), measured during cooling, for several prepared wax-solvent mixtures. It was shown that WDT is closer to the thermodynamic liquidus temperature (TL), and, for the mixtures studied, the average difference between WDT and WAT was determined to be 3 °C.33 It was
(33) Bhat, N. V.; Mehrotra, A. K. Measurement and Prediction of the Phase Behavior of Wax-Solvent Mixtures: Significance of the Wax Disappearance Temperature. Ind. Eng. Chem. Res. 2004, 43, 3451.
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Table 2. List of Parameters along with their Basecase and Other Values Used in Calculations parameter
base-case value
inlet mixture temperature, Thi
WAT þ 5 °C
pipe-wall temperature, Tc
WAT - 5 °C
Parowax-Norpar13 composition, w inlet Reynolds number, Rei
10 mass % wax
pipeline diameter, D
5 cm
15 000
other values used WAT þ 10 °C WAT þ 15 °C WAT - 10 °C WAT - 20 °C 15 mass % wax 10 000 25 000
Figure 5. Predictions for the effect of mixture inlet temperature (Thi) at different deposition times or aging under basecase conditions: (a) deposit thickness (δ/R) and (b) average wax content.
deposition time or aging. Also note that these deposit thickness values in Figure 4a, under turbulent flow, are much smaller compared to those reported previously under laminar flow conditions.25,29 The results in Figure 4b show that h increases with axial distance from ∼400 W m-2 K-1 at the pipeline inlet to ∼1000 W m-2 K-1 at z/D = 4000, and the trend is similar to that for δ/R in Figure 4a. As mentioned previously, h is dependent on Re, which increases with the deposit thickness (because of a decreased area for flow). Also, h is predicted to not vary significantly with the deposition time (or aging). In Figure 4c, the predictions indicate that the deformation angle β increases with both the axial distance and the deposition time. As indicated by eq 11, an increase in β implies a larger fraction of the liquid phase released from the deposit. An increase in β with z/D is attributed to a corresponding increase in the Reynolds number. The predictions show that the deposition time affects β significantly. For example, at shorter axial distances, β is predicted to increase from ∼60° initially to ∼82° ultimately after 24 h of deposition time. The predictions in Figure 4d show the wax content in the deposit to be essentially constant over the entire range of axial distance. However, the wax content is predicted to increase with the deposition time; that is, with aging, the deposit is predicted to become more enriched in wax from ∼10 mass % initially to ∼17 mass %. A slight minimum in the wax content is predicted at z/D ≈ 2500, which is due to the axial variations in Re, h, and β. Effect of the Mixture Inlet Temperature (Thi). Plotted in Figure 5 are the predictions for the effect of wax-solvent mixture inlet temperature (Thi) on δ/R and wax content at the deposition times of τ = 0, 8, and 24 h. The predictions are for three values of Thi, namely, WAT þ 5 °C, WAT þ 10 °C, and WAT þ 15 °C, while all other parameters were held at the basecase conditions. The predictions in Figure 5a show that the deposit thickness decreases with an increase in the wax-solvent mixture temperature; these trends are similar to those reported previously for laminar flow.25,29 An increase in the wax-solvent mixture temperature by 10 °C, from WAT þ 5 °C to WAT þ 15 °C, causes the deposit thickness to decrease by more than 50%. Similar to the results in Figure 4a, the deposit thickness increases slightly with deposit aging.
Figure 4. Predicted axial variations of deposit properties at different deposition times or aging under basecase conditions: (a) deposit thickness (δ/R), (b) heat-transfer coefficient (h, W m-2 K-1), (c) average deformation angle (β), and (d) average wax content.
assumed that the liquid-deposit interface temperature (Td) is held at WAT, which was taken to be equal to TL 3 °C,33 with TL obtained from the thermodynamic model.32 Table 2 lists the basecase set of conditions and other values of parameters used in this study. Effect of Deposit Aging for Basecase Conditions. Figure 4 shows the predicted effect of deposit aging on the axial variation of δ/R, h, β, and wax content at selected deposition or aging times of τ = 0, 8, 16, and 24 h. These results were obtained with all of the parameters held at their basecase values listed in Table 2. Note that the curves for τ = 0 represent the predictions at the beginning of the aging process; that is, they correspond to β at τ = 0 in eq 12. Figure 4a shows that the deposit thickness increases with the axial distance, which is attributed to a decrease in the wax-solvent mixture temperature, because of cooling. However, the deposit thickness increases only slightly with the 2245
Energy Fuels 2010, 24, 2240–2248
: DOI:10.1021/ef9008357
Mehrotra and Bhat
Figure 7. Predictions for the effect of Reynolds number (Re) at different deposition times or aging under basecase conditions: (a) deposit thickness (δ/R) and (b) average wax content.
Figure 6. Predictions for the effect of pipe-wall temperature (Tc) at different deposition times or aging under basecase conditions: (a) deposit thickness (δ/R) and (b) average wax content.
The predictions indicated that β decreases only slightly with an increase in Thi, especially at larger axial distances. On a relative basis, the deposition time or aging was observed to have a much stronger effect on β, which increased considerably during the early stages of deposit aging. In Figure 5b, the deposit wax content is predicted to increase with deposit aging and with an increase in Thi, but only for shorter axial distances (z/D) of ∼2000. For longer axial distances, variations in Thi do not affect the deposit wax content significantly. Effect of Pipe-Wall Temperature (Tc). The results in Figure 6 show the effect of the pipe-wall temperature (Tc) on the axial variation of δ/R and wax content at the deposition times of τ = 0, 8, and 24 h. The calculations were performed at three Tc values: WAT - 5 °C, WAT - 10 °C, and WAT - 15 °C. The predictions in Figure 6a show an increase in the deposit thickness at lower Tc values, which is supported by experimental results under turbulent flow.15,16 Again, similar to the results in Figures 4a and 5a, the deposit thickness is predicted to increase slightly with the deposition time for all pipe-wall temperatures. The results in Figure 6b do not show a significant effect of Tc on the deposit wax content at all deposition times. However, the deposit wax content does increase with the deposit time, which is attributed to the deposit aging effect. Effect of Reynolds Number (Re). Figure 7 shows the predicted effects of Re on the deposit characteristics at three deposition times. As listed in Table 2, the Re values used were Re = 10 000, 15 000, and 25 000, while all other parameters were maintained at the basecase conditions. The effect of Re on the axial variation of δ/R is shown in Figure 7a. The deposit thickness is predicted to decrease with increasing Re, which is supported by experimental results under turbulent flow.15,16 This trend is also similar to that observed under laminar flow in experimental studies,13,14 as well as model predictions.24,25,29 Again, the deposit thickness is predicted to increase slightly with the deposition time for all Re. Figure 7b shows the effect of Re on the deposit wax content. Following a small decrease initially, the deposit wax content is predicted to remain essentially unchanged along the pipeline length. The wax content is predicted to increase with Re for all deposition times, which suggests a greater “squeezing out” effect with an increase in shear stress
Figure 8. Predictions for the effect of mixture wax content at different deposition times or aging under basecase conditions: (a) deposit thickness (δ/R) and (b) average wax content.
or Re. For all Re values, the deposit wax content is predicted to increase with the deposition time or aging. Effect of Wax Concentration. Figure 8 shows the predictions for the effect of wax concentration in the Norpar13Parowax mixture on the deposit characteristics at deposition times of τ = 0, 8, and 24 h. Calculations were performed at two wax concentrations of 10 and 15 mass %, which were the same as those in the deposition experiments.16 Figure 8a compares the predictions for the deposit thickness in the axial direction at different deposition times. The deposit thickness is predicted to be only slightly higher for the 10 mass % mixture. Because the wax deposition is modeled mainly as a heat-transfer process with Th and Tc for each mixture selected in relation to the respective WAT, the deposit thicknesses for the two mixtures are expected to be almost the same. The deposit thickness is predicted to increase slightly with the deposition time for both mixtures. Figure 8b presents the predictions for the deposit wax content for both mixtures at different deposition times. The deposit wax content for both cases is predicted to be much higher than the corresponding wax concentration in the mixture. For 10 mass % and 15 mass % wax mixtures, the 2246
Energy Fuels 2010, 24, 2240–2248
: DOI:10.1021/ef9008357
Mehrotra and Bhat
Figure 9. Predicted variation of the deposit wax content with deposition times or aging for 10 mass % and 15 mass % mixtures at z/D = 2000. Figure 11. Predicted effects of deposition times or aging on the change in average deposit composition at z/D = 2000 under basecase conditions: (a) 10 mass % wax and (b) 15 mass % wax.
distribution such that the deposit is predicted to become depleted in lighter n-alkanes (i.e., Cj < C32) and enriched in heavier n-alkanes (i.e., Cj > C32). Similar results were observed at other Re values for both 10 mass % and 15 mass % mixtures. The predictions in Figures 9 and 10 indicate that the combined effect of shear stress and aging is to cause the deposit wax content to increase, together with an enrichment of heavier n-alkanes (and a corresponding depletion of lighter n-alkanes). These changes would be expected to cause a “hardening” of the deposit with aging, which has been reported in several studies.4,7,10,15,16,19,20 Figure 11 presents the effect of deposit aging on changes in the carbon number distribution, from that of Parowax, for both 10 mass % and 15 mass % mixtures at Re = 15 000 and z/D = 2000. In Figure 11, a positive Δmass % in the normalized concentration of an n-alkane implies its enrichment in the deposit, and vice versa. For both 10 mass % and 15 mass % mixtures, at all deposition times in Figures 11a and 11b, respectively, Δmass % is negative for Cj < C32 (which implies a depletion for these constituents) and is positive for Cj > C32 (which implies an enrichment for these constituents). Note that the changes in the carbon number distribution, similar to those predicted in Figure 11, have been reported in previous experimental deposition studies under both laminar and turbulent flow conditions.9,10,13-16,19,20 Furthermore, the extent of this depletion-enrichment characteristic in Figure 11 increases with the deposition time. There is a considerable change in the profiles at deposition times of τ = 0, 8, and 24 h for the 10 mass % mixture in Figure 11a; however, for the 15 mass % mixture in Figure 11b, the profiles are essentially identical at τ = 8 and 24 h. Thus, as previously noted, the deposit aging is predicted to be faster for the 15 mass % mixture. Similar results were also obtained at the other two Re values. Finally, in Figures 11a and 11b, a value of Δmass % ≈ 0 is predicted for a carbon number of 32 at all deposition times; that is, neither depletion nor enrichment is predicted for C32 for both mixtures with 10 mass % wax and 15 mass % wax. Note that the carbon number, for which the concentration in the deposit does not change with deposition time, has been
Figure 10. Predicted effects of deposition times or aging on the average deposit composition for the 10 mass % mixture at z/D = 2000 under basecase conditions.
deposit aging is predicted to increase the deposit wax content to ∼16 mass % and ∼26 mass %, respectively. Figure 9 presents the changes in deposit wax content for both mixtures, at z/D = 2000, with deposition time or aging. The predictions show a more gradual change in the deposit wax content for the 10 mass % mixture. For the 15 mass % mixture, the increase in the deposit wax content from 15 mass % to ∼26 mass % is accomplished in less than ∼12 h. For the 10 mass % mixture, ∼20 h is required for the deposit wax content to increase from 10 mass % to ∼16 mass %. Effect of Aging on Deposit Composition. Figure 10 shows the predicted changes in the deposit wax composition, at z/D = 2000, for different deposition times for the 10 mass % mixture. These results represent the normalized concentrations of n-alkanes, C20-C40, in the deposit on a solvent-free basis. Also included in Figure 10 for comparison is the curve for Parowax (shown in Figure 3), which also represents the deposit composition without any deformation (i.e., for β = 0) or any effect of shear stress on the deposition process. For all cases with β > 0, the carbon number distribution is different from the original Parowax, and this difference becomes more pronounced with an increase in the deposition time or aging. The predictions for all three deposition times show a progressive shift in the deposit carbon number 2247
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: DOI:10.1021/ef9008357
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C = specific heat capacity of Parowax-Norpar13 mixture (J kg-1 K-1) D = pipe diameter (m) f = mass fraction of solid phase in the deposit fs = mass fraction of solid phase at the liquid-deposit interface h = (local) heat-transfer coefficient (W m-2 K-1) kd = average thermal conductivity of the deposit (W m-1 K-1) kδ = local thermal conductivity of the deposit (W m-1 K-1) L = pipe length (m) m_ h = mass flow rate of Parowax-Norpar13 mixture (kg/s) n = number of radial divisions; n = R/Δr r = radial distance (m) R = pipe radius (m) Re = Reynolds number s = radial location of the liquid-deposit interface (m) t = time (s) Tc = pipe-wall temperature (°C) Td = liquid-deposit interface temperature (°C) Tδ = deposit temperature (°C) Th = average temperature of Parowax-Norpar13 mixture (°C) Thi = temperature of Parowax-Norpar13 mixture at the pipeline inlet (°C) Th-in = temperature of Parowax-Norpar13 mixture at the inlet of the axial element (°C) Th-out = temperature of Parowax-Norpar13 mixture at the outlet of the axial element (°C) TL = liquidus temperature (°C) TS = solidus temperature (°C) Vh = volume of axial element (m3) V = volume of cubical cage; V a3 (m3) VL = volume fraction of liquid phase in cubical cage VS = volume fraction of solid phase in cubical cage Vβ = volume of tilted cage (deformation angle = β) (m3) VLβ = volume fraction of liquid phase in tilted cage VSβ = volume fraction of solid phase in tilted cage Wd = normalized mass fraction of C20þ n-alkanes in the deposit Wh = normalized mass fraction of C20þ n-alkanes in Parowax-Norpar13 mixture ww = mass fraction of C20þ n-alkane in Parowax-Norpar13 mixture z = axial distance (m) Z = ratio of the edge-thickness to the side of cubical cage; Z ξ/a
called the critical carbon number and explained in terms of a time-dependent, counter-diffusion process that has been proposed to occur within the deposit.19 It is emphasized that predictions shown in Figure 11 were obtained from a shearinduced viscoplastic deformation approach, in which heat transfer and thermodynamic considerations play important roles. The modeling approach, described in this study, involves the moving boundary problem formulation for wax deposition, together with a viscoplastic deformation model for shear-induced changes in deposit composition. It provides an alternate mechanism, based on momentum and heat transfer but not molecular diffusion, for the formation, growth, and aging of waxy deposits. Conclusions A mathematical model based on the moving boundary problem approach for solids deposition from wax-solvent mixtures was described to simulate turbulent flow conditions in a pipeline. The model is based primarily on heat-transfer considerations, and the effect of shear stress on deposit characteristics was incorporated using a cubical cage deformation approach. Specifically, the effect of shear stress was incorporated using a recently proposed viscoplastic model to account for the time-dependent variations in the deposit composition. Numerical solutions were presented for the axial variations of the deposit thickness and composition at different deposition times. Predictions were reported and discussed in relation to previously reported experimental trends for the effect of the initial mixture temperature, the coolant temperature, the Reynolds number (Re), and the Norpar13-Parowax mixture composition. The predictions showed that the deposit thickness increases with lower Th, lower Tc, and lower Re, but only slightly with the deposition time. The cubical-cage deformation angle was shown to influence the deposit composition directly. The deposit aging process was predicted to slow asymptotically with the deposition time. The predictions indicated that the combined effect of shear stress and aging caused the deposit wax content to increase, together with an enrichment of heavier n-alkanes and a corresponding depletion of lighter n-alkanes, which would cause “hardening” of the deposit with aging. The deposit aging was predicted to be faster for the 15 mass % mixture than for the 10 mass % mixture. The deposit wax content was predicted to increase with Th, Tc, Re, and the mixture wax concentration. The approach presented in this study provides an alternate interpretation for the critical carbon number.
Greek Symbols Acknowledgment. Financial support was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC). We wish to acknowledge the member universities of Westgrid Network, Canada, for allowing the use of their computation facilities.
R = thermal diffusivity of the deposit (m2/s) Rδ0 = apparent thermal diffusivity of deposit (m2/s) β = deformation angle of the tilted cage (°) ξ = edge thickness of the cubical cage (m) δ = deposit thickness (m) F = density (kg/m3) τ = deposition time (h)
Nomenclature Parameters
Acronyms
a = side of the cubical cage (m) b1-b8 = constants in eq 12 Cj = n-alkane with carbon number j; Cj CjH2jþ2)
WAT = wax appearance temperature (°C) WDT = wax disappearance temperature (°C)
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