Mixtures in a Pipeline under Laminar Flow Conditions via Moving

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Ind. Eng. Chem. Res. 2006, 45, 8728-8737

Modeling of Deposition from “Waxy” Mixtures in a Pipeline under Laminar Flow Conditions via Moving Boundary Formulation Nitin V. Bhat and Anil K. Mehrotra* Department of Chemical and Petroleum Engineering, UniVersity of Calgary, Calgary, Alberta, Canada T2N 1N4

A mathematical model, based on the moving boundary approach, is developed for the growth of the deposit from paraffinic mixtures due to heat transfer in a pipeline. The model extends a recent study on the deposition from “waxy” mixtures by including unsteady-state energy balance and heat transfer in both radial and axial directions for hydrodynamically established laminar flow. Numerical solutions were obtained for the growth of deposit with time, both radially and axially, from a binary eutectic mixture of n-C16H34 and n-C28H58. Two different scenarios for the initiation of the deposition process were investigated. Even though the transient results for the temperature profile and deposit thickness differed significantly, identical steady-state predictions were obtained for the two scenarios. For a constant pipe-wall temperature, the steady-state deposit thickness was predicted to increase with the pipe length. The predicted deposit thickness was lower for a higher inlet mixture temperature, pipe-wall temperature, and Reynolds number. The trends in model predictions were compared with the published results from deposition experiments performed on similar waxy mixtures. Introduction Wax deposition is a serious concern in the production and transportation of crude oils.1 “Waxy” crude oils consist of a variety of heavier paraffinic hydrocarbons that precipitate and deposit on the cooler pipe wall as the crude oil temperature decreases during pipeline transportation.2 With time, the accumulated deposit on the pipe wall causes pipeline blockage, leading to higher pressure drop and/or decreased flow rate. The highest temperature, at which the first crystals of paraffins (waxes) appear upon cooling of waxy crude oils, is referred to as the wax appearance temperature (WAT). Prepared waxy mixtures showed non-Newtonian behavior only at temperatures below their WAT.3 The deposit formation from waxy mixtures on the pipe wall is attributed to the precipitation of solids at temperatures below the WAT. A recent study reported quantitative differences between the WAT and the wax disappearance temperature (WDT), recorded while heating.4 It was shown that, whereas WAT is the temperature of significance for solids deposition (which occurs due to cooling), the WDT represents the highest temperature for the coexistence of liquid and solid phases under thermodynamic equilibrium.4 The deposition from waxy mixtures is a complex phenomenon that has been interpreted and modeled using different approaches. The deposit is comprised of liquid and solid phases in a gel-like state, whose composition and relative proportions vary across the deposit thickness due to differences in temperature, concentration, and shear stress. Burger et al.5 proposed that the deposition occurs as a consequence of the transport of both dissolved and precipitated waxy crystals during the cooling of oil. The lateral transport of wax crystals was interpreted in terms of molecular diffusion, shear dispersion, and Brownian diffusion. It was proposed that molecular diffusion is the dominant mechanism at higher temperatures under high heat flux, while shear dispersion dominates at lower temperatures under low heat flux. The contribution of Brownian diffusion was considered relatively small as compared to the other * Corresponding author. Phone: (403)220-7406. Fax: (403)2844852. E-mail: [email protected].

mechanisms. Models based on the molecular diffusion approach have been proposed for predicting wax deposition.5-11 Predictions based on the molecular diffusion approach have been reported for different thermodynamic models for the solidliquid equilibrium.12-14 The important role of heat transfer in solids deposition from waxy crude oils has been identified in a number of studies.15-21 While earlier studies reported increased deposition with an increase in the overall temperature difference, recent studies have shown that the temperature difference across the deposit is more significant.6,19,21-24 It has been shown that the extent of deposition decreases with an increase in shear rate and insulation. Mehrotra and co-workers21-24 have presented further details of the heat-transfer approach for solids deposition. In a recent study,23 we presented a mathematical model, based on the moving boundary problem approach, that was based on the heat-transfer considerations for the deposition process. The model was used to explore the time-dependent deposition behavior and the deposit layer growth in the radial direction under both static and laminar flow conditions. It was shown that the phase transformation occurring at the liquid-deposit interface and the rate of heat transfer in the liquid and deposit regions would determine the deposit layer growth and the steady-state deposit thickness. This study extends the same modeling approach to a pipeline by incorporating energy balances in radial and axial directions together with the moving boundary framework in the radial direction to predict the variation and growth of the deposit thickness. Similar to our recent study,23 calculations were performed with a binary mixture, comprising n-C16H34 (denoted by C16) and n-C28H58 (denoted by C28) to represent the lighter (liquid) and the heavier (waxy) fractions of a crude oil, respectively. Lacking any experimental or field deposition data in long pipelines, the trends in the steady-state predictions are compared with the observations from two recent experimental investigations in our laboratory.22,24 In these experimental studies, the deposition data from two bench-scale setups, under thermal steady state, were reported for the mixtures of a wax (with C28 as the average constituent) and C12 or Norpar13 (a mixture of n-alkanes, with C13 as the average constituent).

10.1021/ie0601706 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/10/2006

Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006 8729

Figure 2. Schematic of pipeline showing the deposit profile.

Figure 1. Predictions for the eutectic phase behavior of C16-C28 mixture, and the phase regions encountered during the deposition process.

The developed model is used to explore the deposition behavior for two different scenarios for the initiation of the deposition process in a pipeline under hydrodynamically established laminar flow of the binary mixture. In the first scenario, the deposition process is commenced when the pipewall temperature is suddenly lowered below the WAT, while the mixture enters the pipeline at a temperature higher than the WAT. The second scenario mimics the pipeline “batching” of liquids, as it assumes the displacement of an imaginary “nondepositing” liquid in the pipeline by the waxy (depositing) mixture. The transient and steady-state predictions for both scenarios are compared for a set of base-case values of the mixture and pipe-wall temperatures, mixture composition, pipe diameter, and Reynolds number. The steady-state results are also presented for the mixture temperature and deposit thickness along the pipe length for different values of several important parameters that affect the deposition process. Problem Description: Deposition in a Cylindrical Pipe As mentioned previously, the calculations were performed with a binary mixture, comprising C16 (a pseudo-component to represent the solvent fraction) and C28 (a pseudo-component to represent the wax fraction). It is noted that the deposition modeling approach presented here could be extended to multicomponent waxy mixtures by incorporating a suitable thermodynamic framework for flash calculations. The C16-C28 binary mixture was predicted to have a eutectic composition of w28 ) 0.0042 at the eutectic temperature of TE ) 13.5 °C.23 As shown in the isobaric temperature-composition phase diagram in Figure 1, a mixture of C16 and C28 would exist as a liquid at Thi > Th > TL, a two-phase liquid-solid mixture at TL > Tδ > TE, and a mixture of two solid phases at TE > Tf > Ts. The liquidus temperature, TL, of a mixture would depend on its composition, w28. In the two-phase liquid-solid region, the liquid phase consists of both C16 and C28, in equilibrium with a solid phase, which could be either pure C16 or pure C28; however, at the concentrations of C28 considered in this study, the solid phase will be pure C28. For a C16-C28 mixture flowing in a pipeline at a temperature Th > TL, the deposition on pipe surface would occur once the pipe-wall temperature, Tc, is lower than TL. Note that, during the cooling of a waxy mixture, the first solid crystals would precipitate at the WAT of the mixture, which has been observed to be lower than TL.4 For several prepared mixtures of a wax

sample and different solvents, the average difference between the WAT and the WDT (which was shown to be close to TL) was reported to be about 3 °C.4 Furthermore, experimental results from recent deposition studies indicated the liquid-solid interface temperature, Td, can be approximated as the WAT of the mixture;22,24 that is, Td ≡ WAT = (TL - 3.0). For a pipeline with a constant wall temperature, Tc (such that Td > Tc > TE), there would be two phase regions at any axial location. The liquid-solid region near the pipe wall, at Td > Tδ > Tc, would consist of two phases with pure solid C28 in equilibrium with a liquid phase consisting of C16 and C28. This two-phase region, hereafter referred to as the deposit, is the one that is relevant for the deposition from waxy mixtures. The liquid core, at Thi > Th > Td, would consist of the C16-C28 mixture flowing in the pipeline. During the flow of waxy mixtures at Th > WAT or Td, the deposition is expected only if Tc < WAT. Since the deposition process is strongly influenced by the mixture temperature,23 heat transfer in the radial direction has a significant effect on the deposit thickness variation in the axial direction. Model Development A mathematical formulation for the deposition process, in both radial and axial directions, is developed in this section. Figure 2 shows schematically the two phase regions in the axial direction of a pipeline. The mixture enters the pipeline, of inside diameter D and length L, at an inlet mixture temperature, Thi (>WAT), while the cooler pipe wall is held at a constant temperature, Tc ( 0 (3) kδ 2 + r ∂r ∂r R′δ ∂t ∂z where Tδ denotes the deposit temperature. Note that R′δ represents a modified thermal diffusivity of the deposit:23

(4)

where f is the mass fraction of solid phase in the deposit, λ is the latent heat of fusion, and kδ is the thermal conductivity at Tδ. Preliminary calculations indicated that the rate of change of deposit layer thickness axially (∂δ/∂z) was very small. Furthermore, since T ) Td at r ) s and T ) Tc at r ) R, (∂Tδ/∂z) = 0; therefore, ∂2Tδ/∂z2 = 0. That is, the axial heat conduction in the deposit layer was neglected, particularly in comparison to that in the radial direction, and eq 3 was simplified as follows:

( )

1 ∂Tδ 1 ∂ ∂Tδ ; s < r < R, z > 0 r ) r ∂r ∂r R'δ ∂t

|

∂Tδ ds - h(Th - Td) ) Fλ fs ; r ) s, z > 0 kδ ∂r dt z

(5)

(6)

where fs is the equilibrium solid-phase fraction at the liquiddeposit interface (i.e., at r ) s) corresponding to the interface temperature, Td. Initial Conditions. Two initial conditions are necessary for the deposition problem: one for the liquid region (i.e., eq 1) and the other for the deposit region (i.e., eq 6). As mentioned previously, two different scenarios for the initiation of the deposition process were considered. In the first scenario, it is assumed that the flow of waxy mixture in the pipeline is hydrodynamically established under isothermal conditions at the pipeline inlet temperature, Thi. This implies that the pipe-wall temperature initially is also Thi. Thus, without any thermal driving force for heat transfer, the entire pipeline would be filled initially with the flowing mixture at Thi throughout. The deposition process is commenced, at t ) 0, by lowering the pipe-wall temperature from Thi to Tc ( r > 0, z > 0

m˘ C(Th-in - Th-out) ) hA(Th - Tc) + (πR2∆z)FC

1 Fλ df 1 ) R′δ Rδ kδ dTδ

Next, the energy balance at the liquid-deposit interface at the axial location z is

(7a)

For the second scenario, it was assumed that the pipeline initially is carrying the non-depositing liquid under hydrodynamic and thermal steady-state such that the inlet temperature is Thi and the pipe-wall temperature is Tc. Due to heat transfer in the radial direction, the steady-state temperature of the nondepositing liquid would vary in the axial direction. At t ) 0, the flow of the non-depositing liquid at the pipeline inlet is switched with the depositing (waxy) mixture. The initial condition for the second scenario is expressed as follows:

Th ) g(z); t ) 0, R > r > 0, z > 0

(7b)

where g(z) represents the steady-state temperature profile for the non-depositing liquid in the pipeline. However, g(z) would not affect the depositing mixture since it has been assumed to be physically and thermally isolated from the non-depositing liquid being displaced. Note that, for both scenarios, no deposit would exist initially at all locations in the pipe. For the first scenario, the deposit formation would commence simultaneously at all axial locations in the pipeline since the Tc is lower than the WAT at t > 0. For the second scenario, even though Tc is below the WAT of the depositing mixture, the deposit formation would commence from the pipe inlet as the depositing mixture displaces the nondepositing liquid. However, only the segment of the pipe length occupied by the displacing (depositing) mixture would have a deposit region, and its length would increase with time. For both scenarios, without any deposit initially, the initial condition for eq 6 is

s ) R; t ) 0, z > 0

(7c)

Boundary Conditions. Two boundary conditions are needed for eq 5. Note that, for both scenarios, Tc < WAT (or Td) at t > 0. For the first scenario, at t > 0, the pipe-wall temperature is Tc for all values of z. For the second scenario, however, the non-depositing liquid is displaced by the depositing mixture at its average velocity, uj, while the pipe-wall temperature is held


at Tc. For the second scenario, the region of interest is the one occupied by the depositing (waxy) mixture; hence, calculations are not needed for the portion of the pipeline that is filled with the non-depositing liquid. For the first scenario, the following boundary conditions specify the liquid-deposit interface temperature to be Td and the pipe-wall temperature as Tc ( 0, z > 0

(8a)

Tδ ) Tc; r ) R, t > 0, z > 0

(8b)

The first scenario of the deposit formation problem is described by eqs 1, 5, and 6 along with the initial and boundary conditions given by eqs 7a, 7c, 8a, and 8b. For the second scenario, the two boundary conditions are

Tδ ) Td; r ) s, t > 0,

ujt > z > 0

(8c)

Tδ ) Tc; r ) R, t > 0,

ujt > z > 0

(8d)

The second scenario of the deposit formation problem is described by eqs 1, 5, and 6 along with the initial and boundary conditions given by eqs 7a, 7c, 8c, and 8d. Estimation of Heat Transfer Coefficient, h. The following infinite series solution,25 for the local heat transfer coefficient of a Newtonian liquid under laminar flow through a circular pipe under constant wall-temperature, was used for estimating the local Nusselt number, Nuz, at a distance z from the pipe inlet: ∞

Nuz ) [

∞

∑ Gn exp(-An2z+)]/[2 n)0 ∑ (Gn/An2) exp(-An2z+)] n)0

(9)

where z+ ) (z/R)/RePr, and Gn and An are constants and eigenvalues, respectively, whose numerical values were provided by Sellars, Tribus, and Klein in ref 25. The deposit thickness, δ, and h are related as follows:

h ) [Nuz kl]/[2(R - δ)] ≡ Nuz kl/2s

(10)

Equation 10 indicates that, for a constant Nuz, h increases with an increase in δ. For constant flow rate in a pipe, Re ∝ (R - δ)-1; hence, an increase in δ will increase Re, which in turn will cause an increase in Nuz. Although not considered here, an increase in δ will also cause an increased pressure drop resulting from a decrease in the cross-sectional area for flow due to deposition. An increase in δ will also decrease the liquiddeposit interfacial area, Ad, in eq 1. Estimation of Liquid and Deposit Properties. All mixture and deposit properties were calculated by using the estimation methods described elsewhere.23 Depending on the temperature and composition of C16-C28 mixtures, Pr varied from 43 to 48. For the different Re and D used, the ranges of other parameters in eq 9 were as follows: 0 < z+ < 0.09 and 4 < Nuz < 75. Calculation Procedure. The input quantities for numerical calculations were the liquid- mixture composition (w28), the inlet mixture temperature (Thi), the constant pipe-wall temperature (Tc), the pipe diameter (D ) 2R), the inlet Reynolds number (Rei), and the length of pipeline (L). The numerical solution procedure for solving the radial deposition problem, based on the moving boundary approach, was described in our previous work.23 Briefly, the deposit was modeled as concentric layers, each of thickness ∆r, with each layer assumed to be under

thermodynamic equilibrium at its average temperature for each time increment, ∆t. A numerical procedure was developed to include the axial variation of the mixture temperature (Th) and the deposit thickness (δ). For the first scenario, the entire pipeline length was divided into a number of equal divisions, each ∆z in length. For the second scenario, ∆z was obtained as the average distance traveled by the plate in one time increment, which was calculated from the average velocity of the depositing (displacing) mixture. At t ) 0, the initial mixture temperature was set to Thi (i.e., Th-in ) Thi for all values of r and z). For the first time increment, the temperature at r ) R was set equal to Tc (< Td) for all ∆z elements. The first deposit layer, adjacent to the pipe wall, corresponded to ∆r, which means that the liquid region existed between 0 and R-∆r and the deposit region existed between R-∆r and R for all axial elements. The liquid-deposit interface temperature, Td, was specified as the WAT (i.e., Th ) Tδ ) Td ) WAT at r ) s). For the first ∆z element, the initial guess for the outlet liquid temperature was Th-out ) Th-in ) Thi. The mixture and depositlayer properties were estimated at average temperatures. The heat transfer coefficient, h, was estimated from eqs 9 and 10. Next, eqs 5 and 6 along with eqs 7c, 8a, and 8b were solved, and the new liquid-deposit interface location was estimated. Equation 1 along with eq 7a or eq 7b were solved iteratively for Th using the Newton-Raphson method with a relative convergence criterion of 1 × 10-6. The calculations were repeated for the next axial element by setting Th-in equal to Th-out of the previous axial element. The above procedure was repeated for all axial elements up to z ) L, for the first scenario, or up to the plate (i.e., (z ) ujt)) for the second scenario. Note that, for the second scenario, the calculation procedure became identical to the first scenario after the non-depositing liquid was displaced completely from the pipeline. For each axial element, the converged value of Re was checked to ensure it was in the laminar regime (i.e., Re < 2100). If Re > 2100, the calculations were terminated beyond that axial location. The above calculations were repeated until the mixture temperature and the deposit thickness did not change with time for all axial elements, at which point the rates of heat transfer from the liquid region at the interface and across the deposit were equal (i.e., h(Th - Td) ) kδ(∂Tδ/∂r) in eq 6). Numerical Solution Methodology. Each set of equations was solved numerically to obtain the temperature profile over the pipeline length and the radial movement of the liquid-deposit interface with time for each axial element. All equations for the radial heat transfer were discretized using the explicit method, in which the dependent variables were estimated from their known values at the previous time interval. The domain, 0 < r < R, was divided into n equally spaced concentric rings (i.e., ∆r ) R/n). The sizes of ∆r and ∆t were selected such that they satisfied the following stability criterion:

(R∆t)/(∆r)2 e 1/2

(11)

All computations were performed on the WestGrid Network, available for high performance computing at universities in Western Canada. Typically, one complete run with n ) 150 took about 12-36 h of computation time on the Westgrid Network. With ∆t being proportional to 1/∆r2, a further increase in n, beyond n ) 150, involved a much smaller ∆t, which increased the computation time considerably. Therefore, n (≡ R/∆r) ) 150 was chosen as the optimum for all simulation runs;

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Table 1. List of Parameters with Their Base-Case and Other Values Used in Calculations parameter

base-case value

Thi

Td + 5 °C () 38.6 °C)

Tc

Td - 5 °C () 28.6 °C)

D

10.0 cm

w28

10.0 mass % (WAT ) TL - 3 ) 33.6 °C)

Rei

500

other values used Td + 2.5 °C () 36.1 °C) Td + 7.5 °C () 41.1 °C) Td + 10 °C () 43.6 °C) Td - 2.5 °C () 31.1 °C) Td - 7.5 °C () 26.1 °C) Td - 10 °C () 23.6 °C) 2.5 cm 5.0 cm 7.5 cm 5.0 mass % (WAT ) 28.2 °C) 15.0 mass % (WAT ) 37.0 °C) 20.0 mass % (WAT ) 39.4 °C) 250 750 1000

it was estimated to yield an average relative error of 80 m, such that the flow was no longer laminar; hence, the calculations beyond this axial location were discontinued. The results indicated that the transient temperature profiles for the depositing mixture were related to the deposit thickness profile. In Figure 4, the deposit thickness (δ) predictions at the same times of 400 s, 1000 s, 3000 s, and steady state are plotted against the axial location (z) as curves a, b, c, and d, respectively. The predictions show that δ increased with time at all axial locations until reaching the steady state. Notice that the steadystate curve is again truncated at z ≈ 80 m, due to Re > 2100, at which δ ) 4.0 cm. At larger z for the depositing mixture, a lower (Th - Td) in Figure 3 corresponds to a larger δ in Figure 4. This trend in Figures 3 and 4 is consistent with experimental results22,24 as well as the predictions reported in our previous study,23 which indicated that a lower (Th - Td) gave a higher steady-state deposit thickness. In other words, the deposit thickness increases with axial distance from the pipe inlet as the mixture temperature decreases. Another important observation made from these calculations was that the steady-state deposit thickness at all axial locations was achieved later than the steady-state temperature profile. The


Figure 4. Variation of the deposit thickness for depositing mixture in the axial direction. Base-case conditions for the first scenario at different times: 400 s (curve a), 1000 s (curve b), 3000 s (curve c), and steady state (curve d).

Figure 5. Variation of different parameters with time at axial locations of 5, 50, and 100 m. Base-case conditions for the first scenario: (a) deposit thickness, (b) mixture temperature, and (c) heat loss from pipe wall.

slower approach to steady state for the deposit thickness profile is attributed to a relatively slower rate of heat transfer by conduction across the deposit. Approach to Steady State at Different Axial Locations. The three plots in Figure 5 show the predicted results for the effects of time on δ, (Th - Td), and the heat loss at pipe wall, respectively, at selected z of 5, 50, and 100 m in the 10 cm diameter pipe. Figure 5a shows δ to increase with time until reaching the steady-state values of about 0.5 and 3.0 cm at z ) 5 and 50 m, respectively. At z ) 100 m, δ was predicted to continue increasing with time until Re > 2100. For all axial

Figure 6. Variation of temperature for the first and second scenarios in the axial direction. Base-case conditions at different times: 1000 s (curve a), 2000 s (curve b), 5000 s (curve c), and steady state (curve d).

locations, the rate of deposit growth was higher initially, but it decreased with time due to a higher thermal resistance offered by the increased deposit thickness. Figure 5b shows the trends for the mixture temperature at the three axial locations. At all three locations, the mixture temperature was predicted to decrease rapidly from the initial temperature, which was equal to Thi. The predicted steady-state values of (Th - Td) were 3.2, 0.9, and 0.1°C at z ) 5, 50, and 100 m, respectively. These results also show that the steadystate mixture temperature decreases with an increase in z. A comparison of the results in plots a and b of Figure 5 shows that, at all axial locations, the mixture temperature attained steady state faster than the deposit thickness. The predictions for the rate of heat loss per unit pipe length, at the same three axial locations, are shown in Figure 5c. The rate of heat loss decreased considerably from about 720 W/m at z ) 0 to 49 W/m and 6 W/m at z ) 5 and 50 m, respectively. Whereas the thermal driving force across the deposit (Td - Tc) was constant, the thermal driving force for the liquid region (Th - Td) decreased as Th decreased steadily in the axial direction. The results also indicated that the deposit generally offered the controlling resistance to heat transfer. At early times, a small deposit thickness resulted in a higher rate of heat loss. As time progressed, an increase in the deposit thickness increased its thermal resistance, which resulted in a gradual decrease in the rate of heat loss at each axial location until reaching the steady state. (B) Unsteady-State Results for the Base Case: The Second Scenario. As mentioned previously, the second scenario considers the displacement of the non-depositing liquid with the depositing mixture. The model equations were solved numerically to obtain predictions for the profiles of Th and δ as a function of z and t. For Rei ) 500, the average velocity at the pipe inlet was calculated to be 0.02 m/s, and ∆t from eq 11 was 0.5 s corresponding to n ) 150. The average distance traveled by the depositing mixture (or the imaginary plate) in one time step was calculated to be 0.01 m. Figure 6 compares the axial temperature profiles for the depositing mixture, shown as (Th - Td), for both scenarios. Note that the solid curves are for the second scenario and the dotted curves are for first scenario. The temperature profiles are shown for 1000 s, 2000 s, 5000 s, and the steady state as curves a, b, c, and d, respectively. Each curve for the second scenario extends up to a certain value of z that corresponds to the location of the plate separating the two liquids at that time. For example, the plate was at z ≈ 19 m for 1000 s and at z ≈ 97 m for 5000 s.

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Figure 7. Variation of the deposit thickness for the first and second scenarios in the axial direction; Base-case conditions at different times: 1000 s (curve a), 2000 s (curve b), 5000 s (curve c), and steady state (curve d).

It is noted that, despite the significant differences between the two scenarios for the initiation of the deposition process, the steady-state temperature profiles in Figure 6 are identical for the base-case set of conditions. This indicates that the steadystate results shown in Figure 6 are independent of the path taken to attain them. Figure 7 shows a comparison of the predictions, from both scenarios, for the deposit thickness (δ) at 1000 s, 2000 s, 5000 s, and steady state as curves a, b, c, and d, respectively. The deposit thickness profiles for the second scenario, at different times, display peculiar characteristics, which are quite different from those for the first scenario. Until the complete displacement of the non-depositing liquid, the profiles show an increase in δ followed by a decrease. With time, the location of the maximum δ moves away from the pipeline inlet. For example, the maximum δ for 1000 s occurred at z ≈ 11 m, which shifted to z ≈ 72 m for 5000 s. In Figure 7, the predicted steady-state deposit profiles for both scenarios are identical. This is despite the fact that the results in Figure 7 were obtained with ∆z ) 0.5 m for the first scenario and ∆z ) 0.01 m for the second scenario. Due mainly to this large difference in the magnitudes of ∆z, the calculations with the second scenario took much longer computation times. Finally, the observation that both scenarios gave identical steadystate temperature and deposit thickness profiles indicates consistency in the modeling approach presented in this study. (C) Steady-State Results: Effects of Important Parameters. Additional calculations were undertaken to investigate the effects of important parameters on the deposition process. Only the steady-state temperature and deposit thickness profiles are presented here. Even though both scenarios were shown to give identical steady-state results, the first scenario was chosen for these calculations as it required less computation time. The parameters considered for these calculations were as follows: the inlet mixture temperature (Thi), the pipe-wall temperature (Tc), the pipe diameter (D), the liquid composition (w28), and the inlet Reynolds number (Rei). The different values of each parameter used in these calculations are listed in Table 1. In each of these calculations, only one parameter was varied while the others were held at their respective base-case values. The five plots in Figure 8 show the effects of different parameters on the steady-state mixture temperature, plotted as the thermal driving force for convection in the liquid region (i.e., (Th - Td)) as a function of the relative axial location, z/D. The range for the ordinate in Figure 8a is from 0 to 10 °C, whereas it is from 0 to 5 °C in all others. For the same five

Figure 8. Axial variation of the steady-state mixture temperature for different values of (a) inlet mixture temperature, Thi; (b) pipe-wall temperature, Tc; (c) pipe diameter, D; (d) mixture composition, w28; and (e) inlet Reynolds number, Rei.

parameters, the plots in Figure 9 show the corresponding variations in the steady-state relative deposit thickness (i.e., δf/ R). Effect of the Inlet Mixture Temperature, Thi. Figures 8a and 9a show the effects of the inlet mixture temperature, Thi, on the mixture temperature (Th) and deposit thickness (δf/R) up to z/D ) 1000, respectively. A higher Thi gives rise to a larger thermal driving force, which corresponds to a relatively faster cooling of the mixture near the pipeline entrance. The rate of decline in Th decreases as it approaches Td, which as shown in Figure 8a causes a flattening of all curves at larger values of z/D. Figure 9a shows the corresponding values of the deposit thickness at steady state. At the same axial location, the steadystate deposit thickness is predicted to be higher for a lower Thi. This observation is supported by the experimental results.22,24 For example, at z/D ) 100, the predicted δf/R increases from 0.10 for (Td + 10 °C) to 0.29 for (Td + 2.5 °C). The predicted δf/R is much higher at z/D ) 500, as it increases from 0.39 for (Td + 10 °C) to 0.77 for (Td + 2.5 °C). Thus, the higher the mixture temperature (relative to Td or WAT), the smaller is the predicted steady-state deposit thickness at all axial locations. Effect of the Pipe-Wall Temperature, Tc. Figures 8b and 9b show the effects of the pipe-wall temperature on Th and δf/R at different axial locations, respectively. For the four values of


Figure 9. Axial variation of the steady-state deposit thickness for different values of (a) inlet mixture temperature, Thi; (b) pipe-wall temperature, Tc; (c) pipe diameter, D; (d) mixture composition, w28; and (e) inlet Reynolds number, Rei.

Tc in Figure 8b, the predicted profiles for (Th - Td) differ slightly. At larger z/D, the curve for the highest Tc is only slightly higher than that for the lowest Tc. In other words, the pipe-wall temperature (relative to Td or WAT) does not significantly affect the mixture temperature, at least for the pipeline length considered here. In Figure 9b, the variation of Tc has a very pronounced effect on the deposit thickness profile. At the same axial location, the steady-state deposit thickness is predicted to be higher for a lower Tc. This observation is also supported by the previously reported experimental results.22,24 For example, at z/D ) 100, δf/R is predicted to increase from 0.09 for (Td - 2.5 °C) to 0.29 for (Td - 10.0 °C). The predicted δf/R is much higher at z/D ) 500, as it increases from 0.38 for (Td - 2.5 °C) to 0.78 for (Td - 10.0 °C). Thus, the lower the pipe-wall temperature, the larger is the predicted steady-state deposit thickness at all axial locations. Effect of the Pipe Diameter, D. The effects of varying the pipe diameter on Th and δf/R at different axial locations are shown in Figures 8c and 9c, respectively. Interestingly, the mixture temperature profiles for all four diameters, ranging from 2.5 to 10.0 cm, are identical. Since the variable z/D, used for the abscissa in Figure 8, is also dependent on the pipe diameter, the results in Figure 8c imply that the same extent of mixture cooling would occur over a shorter pipe length of a smaller

diameter pipeline. Moreover, for these calculations, the inlet Reynolds number was kept the same (i.e., Rei ) 500); hence, the average velocity of the mixture was 4 times higher for D ) 2.5 cm than that for D ) 10.0 cm. It was also observed that a smaller diameter pipeline required less time for attaining the steady state. The curves in Figure 9c for the steady-state deposit thickness also show an identical variation between δf/R and z/D for the four diameters considered. That is, regardless of the pipe diameter, the deposit is predicted to occupy the same fractional cross-section area of the pipe at the same relative axial location, z/D. This observation is similar to that made in our previous study on deposition in the radial direction.23 In terms of the absolute values, a larger diameter pipe will have a larger deposit thickness at steady state, which would occur at a greater axial distance from the pipeline inlet. Effect of the Mixture Composition, w28. Figures 8d and 9d show the effects of mixture composition on Th and δf/R at different axial locations, respectively. It is pointed out that, aside from relatively small changes in all of the mixture and deposit properties, a variation in w28 caused a corresponding change in the predicted TL of the mixture (as shown in Figure 1) and, therefore, the liquid-deposit interface temperature, Td (or WAT). In Figure 8d, therefore, the predicted trends for Th are nearly the same for different mixture compositions, ranging from 5 to 20 mass % at all axial locations. Similar to the trends seen in Figure 8d, the δf/R profiles in Figure 9d are also nearly the same for all mixture compositions. Note that the inlet mixture temperature and the pipe-wall temperature for all mixtures were selected relative to the respective Td (i.e., Thi ) (Td + 5 °C) and Tc ) (Td - 5 °C)). Even with different numerical values of Thi and Tc for each mixture composition, the thermal driving forces for the resistances offered by the liquid region and the deposit were the same. The minor differences between the δf/R profiles, particularly at larger z/D, can again be attributed to small changes in mixture and deposit properties. Recent experimental studies have identified the roles of the liquid-deposit interface temperature, Td (or WAT), and the ratio of the thermal resistances present in the deposition process for waxy mixtures of different compositions.22,24 The predicted trends in Figure 9d, together with those from our previous study,23 are consistent with the experimental observations. Effect of the Inlet Reynolds Number, Rei. Finally, the effects of the inlet Reynolds number on Th and δf/R at different axial locations are shown in Figures 8e and 9e, respectively. Four values of Rei, between 250 and 1000, were used in the calculations. A lower inlet Reynolds number (e.g., Rei ) 250) implies a proportionally lower velocity or flow rate of the mixture; hence, the heat lost from the relatively low flow rate caused a rapid decrease in the mixture temperature. This led to a faster cooling of the mixture at lower Rei. Conversely, for Rei ) 1000, (Th - Td) was predicted to be higher, by about 1 °C, at z/D ) 1000. In Figure 9e, δf/R is the largest corresponding to the lowest Rei. Thus, the predictions in Figures 8e and 9e indicate that, at all axial locations, a lower Rei corresponds to a lower Th and a higher δf/R. An increase in Rei leads to an increase in Th while decreasing δf/R. It is also interesting to note that a larger Rei allows the pipeline to remain under laminar conditions over a longer length. That is, an increase in the Reynolds number of the mixture helps in keeping it relatively “warmer” over a longer pipe length, in achieving a lower deposit thickness, and in maintaining the flow under laminar conditions over an extended

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length of the pipeline. In other words, a pipeline carrying waxy crude oils at low Reynolds numbers would be susceptible to increased deposition over a shorter pipe length. The above observations for the effect of Reynolds number are consistent with experimental results,22,24 which showed an inverse relationship between Re and the deposit mass. Applications of Steady-State Predictions. As mentioned before, the model presented here for solids deposition from waxy mixtures is based on heat-transfer considerations. The phase transformation was assumed instantaneous, governed only by thermodynamic considerations and not constrained by molecular diffusion. The model also did not consider any shear-induced changes in the deposit properties. The predicted trends were shown to be consistent with the experimental results from recent laboratory studies22,24 on solids deposition from similar waxy mixtures. Furthermore, two different scenarios for the commencement of the deposition process in a pipeline were shown to give identical steady-state results. For the base-case conditions, a steady-state deposit thickness of about 2.5 cm (i.e., δf/R ≈ 0.5) was predicted at an axial distance of about 40 m (i.e., z/D ≈ 400). This implies that at the end of a relatively short 40 m pipeline, maintained under steady state at the base-case set of conditions, the deposit would occupy approximately 75% of the cross-sectional area of the 10 cm diameter pipe. This extent of deposition in the pipeline would obviously be unacceptable. The predicted results in Figure 9 suggest that a lower δf/R in the pipeline could be achieved for a higher Thi, a higher Tc, and a higher Rei. That is, the steady-state deposit thickness could be decreased but not avoided entirely by preheating the waxy mixture to higher temperatures (relative to Td or WAT), by insulating the pipeline to achieve higher pipe-wall temperatures (relative to Td or WAT), and/or by maintaining higher values of the Reynolds number. Although not presented in this study, our preliminary calculations indicated that an insulation layer could delay and/or lessen the extent of deposition in the pipeline, but it would not avoid the deposition from taking place unless Tc > WAT. The predictions in Figure 9e show a lower δf at a higher Rei for a given pipe diameter. As shown in Figure 9c, the steadystate deposit thickness on a relative basis (i.e., δf/R) does not depend on the pipe diameter. Furthermore, the mass rate (m˘ ), pipe diameter (D), and Reynolds number (Re) for a pipe are related as m˘ ∝ D Re. Thus, for a given m˘ , a smaller diameter pipeline corresponds to a higher Reynolds number. As shown in Figures 8e and 9e, an increase in the Reynolds number would maintain the mixture relatively “warm” and the deposit thickness relatively small over an extended pipeline length. The above analysis indicates that the liquid-deposit interface temperature, Td, shown to be equal to the WAT under steady state,22,24 is an important parameter for the deposition process. The analysis also suggests that the smallest possible pipeline diameter would be preferable as it would yield a smaller deposit thickness due to a higher Reynolds number. Obviously, the selected pipeline diameter should satisfy any pressure drop constraint for the specified flow rate. If feasible, the average deposit thickness in the pipeline could be decreased further by diluting the mixture to lower its WAT, by preheating the mixture to increase (Th - WAT), and/or by increasing the pipe-wall temperature (e.g., by insulating the pipeline) to decrease (WAT - Tc). Conclusions A recently proposed mathematical model, involving the moving boundary problem framework, was extended to the

formation and growth of deposit from waxy mixtures in a pipeline under laminar flow conditions. The model was used to study the deposition process for a binary mixture of C16 and C28. Two scenarios for the commencement of the deposition process were considered. For both scenarios, the partial differential equations for unsteady-state heat transfer in the liquid and deposit regions, both radially and axially, were solved numerically to predict variations in the mixture temperature and the deposit thickness as a function of time and axial distance. Even though the predictions for the two scenarios for the initiation of the deposition process differed significantly in the approach to steady state, identical steady-state results were obtained for the variation of the mixture temperature and deposit thickness in the axial direction. For the base-case set of conditions considered in this study, the approach to steady state was slower for the depositing (waxy) mixture than for the non-depositing liquid under similar conditions, due mainly to the additional thermal resistance offered by the deposit. The steady-state deposit thickness was predicted to increase in the axial direction while the temperature of waxy mixture was predicted to decrease. The model was used to explore the effects of the mixture temperature, pipe-wall temperature, pipe diameter, mixture composition, and Reynolds number. At steady state, a smaller deposit thickness along the pipe length was predicted for higher mixture temperature, pipe-wall temperature, and inlet Reynolds number. The trends in the predicted steady-state results were satisfactory as these did not contradict the experimental deposition results reported on similar waxy mixtures.22,24 A comparative analysis of the parameters that affect the deposition process suggested that, for transporting a certain mass rate of waxy mixture under laminar flow conditions, the smallest possible pipeline diameter, selected within the pressure drop constraint, should be used as it would result in a smaller deposit thickness due to an increase in the Reynolds number. The results also suggested that an increase in the difference between the mixture temperature and WAT and/or a decrease in the difference between the WAT and pipe-wall temperature could further decrease the average deposit thickness in the pipeline. Acknowledgment Financial support was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Department of Chemical and Petroleum Engineering, University of Calgary. We acknowledge the member universities of Westgrid Network, Canada, for allowing the use of their computation facilities. Nomenclature A ) pipe-wall area for heat transfer (non-depositing liquid), m2 Ad ) liquid-deposit interface area for heat transfer, m2 An ) eigenvalues in eq 9 C ) specific heat capacity, J/kg‚K Cn ) n-alkane (≡ normal CnH2n+2) D ) pipe diameter, m f ) mass fraction of solid-phase in deposit fs ) mass fraction of solid-phase at liquid-deposit interface g(z) ) steady-state temperature profile for non-depositing liquid Gn ) constants in eq 9 h ) heat transfer coefficient, W/m2‚K kδ ) thermal conductivity of deposit, W/m‚K


kl ) thermal conductivity of liquid, W/m‚K L ) length of pipeline, m m˘ ) mass flow rate, kg/s n ) number of radial divisions () R/∆r) Nuz ) local Nusselt number Pr ) Prandtl number r ) radial distance, m R ) pipe radius, m Re ) Reynolds number Rei ) Reynolds number at the pipe inlet s ) radial location of liquid-deposit interface, m t ) time, s Tc ) pipe-wall (cold) temperature, °C Td ) liquid-deposit interface temperature () WAT), °C Tδ ) deposit-region temperature, °C TE ) eutectic temperature, °C Tf ) solid-region temperature, °C Th ) mixture (hot) temperature, °C Thi ) pipeline initial or inlet temperature, °C Th-in ) mixture or liquid temperature at inlet of an axial element, °C Th-out ) mixture or liquid temperature at outlet of an axial element, °C TL ) liquidus (or saturation) temperature, °C w28 ) mass fraction of C28 (in C16-C28 mixture) z ) axial distance, m z+ ) dimensionless axial distance (≡ (z/R)/RePr) Greek Letters R ) thermal diffusivity, m2/s R′δ ) apparent thermal diffusivity of deposit, m2/s F ) density, kg/m3 δ ) deposit thickness, m δf ) steady-state deposit thickness, m λ ) latent heat of fusion, J/kg Acronyms WAT ) wax appearance temperature, °C WDT ) wax disappearance temperature, °C Literature Cited (1) Carnahan, N. F. Paraffin deposition in petroleum production. J. Pet. Technol. 1989, 41, 1024. (2) 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. (3) Tiwary, D.; Mehrotra, A. K. Phase transformation and rheological behaviour of highly paraffinic ‘waxy’ mixtures. Can. J. Chem. Eng. 2004, 82, 162. (4) Bhat, N. V.; Mehrotra, A. K. Measurement and prediction of the phase behaviour of wax-solvent mixtures: significance of the wax disappearance temperature. Ind. Eng. Chem. Res. 2004, 43, 3451.

(5) Burger, E. D.; Perkins, T. K.; Streigler, J. H. Studies of wax deposition in the Trans Alaska Pipeline. J. Pet. Technol. 1981, 33, 1075. (6) Singh, P.; Venkatesan, R.; Fogler, H. S.; Nagarajan, N. Formation and aging of incipient thin film wax-oil gels. AIChE J. 2000, 46, 1059. (7) Svendsen, J. A. Mathematical modeling of wax deposition in oil pipeline systems. AIChE J. 1993, 39, 1377. (8) Majeed, A.; Bringedai, B.; Overa, S. Model calculates wax deposition for North Sea oils. Oil Gas J. 1990, 18, 63. (9) Rebiero, F. S.; Souza Mendez, P. R.; Braga, S. L. Obstruction of pipelines due to paraffin deposition during the flow of crude oils. Int. J. Heat Mass Transfer 1997, 40, 4319. (10) Singh, P.; Venkatesan, R.; Fogler, H. S.; Nagarajan, N. Morphological evolution of thick wax deposits during aging. AIChE J. 2001, 47, 6. (11) Lindeloff, N.; Krejbjerg, K. A compositional model simulating wax deposition in pipeline systems. Energy Fuels 2002, 16, 887. (12) Ramirez-Jaramillo, E.; Lira-Galeana, C.; Brito, O. M. Numerical model for wax deposition in oil wells. Pet. Sci. Technol. 2001, 19, 587. (13) Kok, M. V.; Saracoglu, R. O. Mathematical modeling of wax deposition in crude oil pipelines (comparative study). Pet. Sci. Technol. 2000, 18, 1121. (14) Ramirez-Jaramillo, E.; Lira-Galeana, C.; Manero, O. Modeling wax deposition in pipelines. Pet. Sci. Technol. 2004, 22, 821. (15) Creek, J. L.; Lund, H. J.; Brill, J. P.; Volk, M. Wax deposition in single phase flow. Fluid Phase Equilib. 1999, 158-160, 801. (16) Cole, R. J.; Jennsen, F. W. Paraffin deposition. Oil Gas J. 1960, 58, 87. (17) Patton, C. C.; Casad, B. M. Paraffin deposition from refined waxsolvent systems. Soc. Pet. Eng. J. 1970, 10 (1), 17. (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) 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.; Begell House: New York, 1997; pp 473-489. (20) 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. (21) Mehrotra, A. K. Comments on: wax deposition of Bombay high crude oil under flowing conditions. Fuel 1990, 69, 1575. (22) Bidmus, H. O.; Mehrotra, A. K. Heat-transfer analogy for wax deposition from paraffinic mixtures. Ind. Eng. Chem. Res. 2004, 43, 791. (23) 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. (24) 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. (25) Kays, W. M.; Crawford, M. E. ConVectiVe Heat and Mass Transfer, 3rd ed.; McGraw-Hill: New York, 1993.

ReceiVed for reView February 12, 2006 ReVised manuscript receiVed September 21, 2006 Accepted October 2, 2006 IE0601706

Mixtures in a Pipeline under Laminar Flow Conditions via Moving

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