Introduction to a Novel Approach for Modeling Wax Deposition in Fluid

Ind. Eng. Chem. Res. 2008, 47, 953-963

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Introduction to a Novel Approach for Modeling Wax Deposition in Fluid Flows. 1. Taylor-Couette System Kamran Akbarzadeh* and Mohammed Zougari† DBR Technology Center, Schlumberger, 9450, 17 AVenue, Edmonton, AB, Canada T6N 1M9

A novel approach for modeling wax deposition in fluid flows is developed. Particle diffusion and inertia, shear removal, molecular diffusion, shear dispersion, and aging are considered as possible mechanisms in the wax deposition process. Among these mechanisms, particle diffusion/deposition plays the most significant role in formation of the deposit at the realistic transport conditions. The model parameters are estimated using some experimental data obtained by the organic solids deposition cell (OSDC) that was designed based on the Taylor-Couette system. The model predictions are then compared with the deposition data at other conditions. The effect of each parameter on the model predictions is also discussed. The estimated model parameters can then be used in the pipeline deposition model to predict the deposition rates under actual flow conditions in production and transportation pipelines. Table 1. Single-Phase Wax Deposition Models in the Literaturea

Introduction Wax precipitation and deposition is one of the major challenges in deep-water petroleum production and transportation. It can significantly influence the economy of oil producers by increasing operational and remedial costs as well as decreasing and or halting production. Therefore, it is desirable to evaluate and predict wax deposition rates under various operating conditions in order to properly design and optimize the transported oil and gas productivity. This will also help the oil industry implement proper preventive and remediative strategies. Wax deposition is a complex process, which involves heat transfer, mass transfer, fluid and particle dynamics, kinetics/ crystal growth, and thermodynamics. Due to this theoretical complexity and also due to the lack of satisfactory experimental and production field data, the mechanisms responsible for wax deposition are not fully identified or understood. Molecular diffusion, shear dispersion, Brownian diffusion, shear removal (sloughing), internal diffusion (aging), settling, particle diffusion, and inertia are among possible mechanisms. Table 1 summarizes published wax deposition models along with their corresponding references1-18 and mechanisms involved. On the basis of this table, most of the available deposition models have considered the molecular diffusion mechanism as the predominant mechanism responsible for wax deposition. Molecular diffusion might be a dominant mechanism in laboratory scale flow loops where the flow of fluid is mostly laminar. However, in general, this may not be the case in pipelines where the flow of fluid is highly turbulent. At the realistic transport conditions of the production pipelines, other less-noticed mechanisms such as particle diffusion, the inertial effect, and shear removal (sloughing) are possibly the dominant mechanisms. Therefore, having proper laboratory scale equipment that could simulate the actual flow conditions and provide reliable experimental deposition data could play an important role in better identification of the solid deposition mechanisms and in developing more reliable deposition models. In this work, a novel approach for modeling wax deposition in which particle diffusion and the inertial effect play significant * To whom correspondence should be addressed. E-mail: [email protected]. Phone: 1-780-577-1341. Fax: 1-780-450-1668. † Currently with Gas Processing Center, Qatar University, Doha, Qatar.

no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

pub. year

mechanisms

ref no.

Burger et al. Majeed et al. Svendsen Brown et al. Hsu et al. Rygg et al. Creek et al. Singh et al. MSI

model

1981 1990 1993 1993 1995, 1998 1998 1999 2000, 2001 2000

1 2 3 4 5, 6 7 8 9, 10 11

Solaimany et al. Banki et al. Lindeloff Azevedo Fasano et al. Ramirez et al. Hernandez et al. this work

2001 2002 2002 2003 2004 2004 2004 2006

MD, SD MD MD MD, SD MD MD MD MD, ID, gelation MD, kinetic, crystallization MD, SR MD MD, SD MD, SD MD MD, SR, ID MD, SR, ID MD, SD, SR, ID, PD, and inertia

12 13 14 15 16 17 18

a MD molecular diffusion; SR shear removal (sloughing); SD shear dispersion; ID internal diffusion (aging); PD particle diffusion.

roles in the formation of the deposit at more realistic transport conditions is developed. The model parameters are estimated using experimental data obtained by running the organic solids deposition cell (OSDC).19,20 The developed model can be extended to the pipeline system. The obtained parameters can then be used in the pipe deposition model to determine the deposition rate in pipelines. Experimental Section Wax Precipitation and Quantification. Two waxy crudes, one from the Gulf of Mexico (GoM) and the other from Asia, were selected for this study. Table 2 provides some characteristics of these two oils. To obtain the wax precipitation curves for these fluids, the amounts of precipitated wax at different temperatures below the wax appearance temperature (WAT) of the selected crude oils were determined using a method similar to the one proposed by Burger et al.1 A description of the wax quantification procedure is provided in the Appendix. Organic Solids Deposition Cell (OSDC). The organic solids deposition cell19,20 (OSDC) is a novel device capable of generating individual and/or combined wax, asphaltene, and

10.1021/ie0711325 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/09/2008

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Ind. Eng. Chem. Res., Vol. 47, No. 3, 2008

Table 2. Some Characteristics of the Oils under Study

a

characteristic

GoM oil

Asian oil

API gravity wax content (wt %)a WAT (°C) pour point (°C) viscosity at 30 °C (cP)

36 1.4 34 -35 6.5

34 7.3 36 15 8.5

With the UOP 46-64 method.

scale deposit buildup under actual filed conditions. The OSDC, which is based on Taylor-Couette (TC) flow principals, was designed to simulate the hydrodynamic (e.g., pressure, turbulence, and shear) and thermal (temperature and heat transfer) characteristics encountered in typical oil production lines. The rotational movement of a spindle at the center of the device produces a fluid movement that creates a flow regime similar to pipe flow. A schematic of the OSDC device is shown in Figure 1. Deposit Measurement. To initiate the test, the OSDC cell is evacuated and either filled with helium in the case of live oils or kept empty in the case of dead oils. A certain amount (∼150 cm3) of sample is then charged into the OSDC cell at a temperature higher than the wax appearance temperature of the fluid (in the case of waxy crudes) and a pressure higher than the asphaltene onset pressure of the reservoir fluid. For wax deposition tendency tests, the fluid and wall temperatures are set to the predefined values where the wall temperature is usually less than the fluid temperature. Next, the fluid is depressurized to the specified test pressure and the spindle (inner cylinder) speed is set to a predefined shear rate to match the wall shear stress conditions of the flow line. Then, the test is run for a certain amount of time (Figure 2). After completion of the test, the production oil is displaced isobarically and isothermally from the OSDC device using helium. To remove the oil coating the deposits formed on the cell’s wall, the OSDC device is systematically depressurized to the ambient pressure and gently rinsed with cold dichloromethane. To recover the deposits, the cell is sealed again with new end caps and charged with hot toluene to wash off the deposited material on the cell wall under high shear. After the solvent is evaporated, the residue is weighted. The weight of this residue represents the total amount of the deposit including the entrapped oil. In order to determine the amount of occluded oil, a high-temperature gas chromatograph (HTGC) is used. By comparing the HTGC data of the original oil and the deposit,

Figure 1. Schematic representation of cylindrical Couette device.

the amount of entrapped oil in the deposit as well as the actual amount of solid wax in the whole deposit can be estimated. Theory Model Assumptions. The following is a list of assumptions that were made to develop this introductory deposition model: (1) Quasi-steady-state flow. (2) Newtonian fluid. (3) The average particle size and the distribution of particles in the fluid remains constant during the run. (4) Negligible heat is generated or consumed due to phase change. (5) Negligible kinetic effects on the wax particle size. Deposition Mechanisms. In this work, the deposition process involves six mechanisms including particle diffusion, the inertial effect, shear removal (sloughing), molecular diffusion, shear dispersion, and internal diffusion (aging). These mechanisms and their formulations are described below. Particle Diffusion. In a waxy crude, when the temperature of the bulk liquid reaches the wax appearance temperature, the wax crystals/particles appear in the liquid flow. The presence of wax particles in the fluid flow makes the deposition process more complicated. Due to such complexity and also the scarcity of experimental evidence, particulate deposition in flow conditions encountered in oil pipelines has been mostly neglected in deposition modeling. Although in laminar flow conditions the negligence of particle deposition is acceptable, in turbulent flow conditions, where large eddies and vortices containing wax particles have the ability to hit the walls and easily penetrate through the boundary layer, particle deposition may not be ignored. According to Beal,21 the deposition rate due to particle diffusion can be expressed as

Jpd ) kpdC/b

(1)

where kpd is the particle deposition coefficient and C/b is the wax particle concentration in the bulk fluid. The particle deposition coefficient was estimated using the following equation:

kpd ) kmPsIp

(2)

where km is the radial mass transfer coefficient, Ps is the sticking

Ind. Eng. Chem. Res., Vol. 47, No. 3, 2008 955

Figure 2. Schematic of deposition in Taylor-Couette system.

probability of the particles to the deposit, and Ip is the particle inertial coefficient. (A) Mass Transfer Coefficient. The radial mass transfer coefficient in Taylor-Couette flow can be estimated from the following correlation:22

km )

ADB c b Ta Sc 2d

(3)

where DB is the Brownian diffusion coefficient, d is the gap between the two cylinders in the OSDC (or Taylor-Couette device), Ta is the Taylor number, Sc is the Schmidt number, and A, b, and c are constants that have been determined empirically.22 The majority of the experiments conclude that A ) 0.4 - 0.9, b ) 1/3, and c ) 0.5. In this work, the proposed values for b and c and an average value of 0.65 for A were used. (B) Inertial Effect. Particles entrained in turbulent eddies are assumed to travel toward the wall by a combination of turbulent and Brownian diffusion to the relatively quiescent region adjacent to the wall.23 At this point, the turbulent eddies dissipate, but particles continue moving toward the wall and impact on the surface where they may deposit owing to their inertia. In the present work, the inertial effect on the particle deposition is determined by a particle inertial coefficient, Ip, which is defined as

VB + VI Ip ) VB

(4)

where VB is the radial velocity of the particles due to Brownian motion and VI is the velocity of the particles due to inertia. For small particles, the Brownian velocity dominates and, therefore,

Ip approaches unity, while for bigger particles the inertial effect becomes noticeable resulting in Ip values higher than unity and greater particle deposition rates. Brownian motion velocity, which is usually caused by molecules of the fluid striking the particles constantly, can be obtained from21

VB )

( ) KBT 2πmp

1/2

(5)

where KB is the Boltzman constant, T is temperature, and mp is the particle mass. According to Beal,21 in Brownian motion, the particle behaves like a molecule except that, because of its higher mass, it moves more slowly. The particle velocity due to fluid motion (inertia) can be determined from the following:24

VI )

S+ S + + 10

(6)

where S+ is the dimensionless stopping distance. The stopping distance, S, of a particle is the characteristic distance that the particle, given an initial velocity, will travel through stagnant fluid before coming to rest.23 This distance originally defined by Friedlander and Johnstone25 is calculated from

S)

0.05Fpdp2u/ dp + µo 2

(7)

where Fp is the particle density, u/ is the shear velocity, and dp is the particle diameter. (C) Sticking Probability. Once the particles reach the deposit wall, they may or may not stick to the wall depending on various factors such as shear, particle size, and deposit bond strength.

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The probability of a particle sticking to the deposit upon impact has not been studied theoretically. However, its relation with the above-mentioned factors has been investigated empirically.21,24,27,28 In the current study, the sticking probability is considered as a tuning parameter. A sensitivity analysis will be performed on this parameter to examine its effect on the deposition rate. It is planned to carry out some specifically designed deposition tests for investigating the dependency of the sticking probability on the aforementioned factors. Shear Removal. This mechanism also known as sloughing is based on the idea that the shear stress exerted by fluid flow at the deposit interface may be sufficiently high to mechanically remove some of the deposit. The shear removal mechanism has been mostly used in fouling models that are mainly based on the Kern-Seaton model.29 According to this model, the removal flux, Jsr, due to shear can be expressed by

Jsr ) ksrτwmd

(8)

where ksr is the shear removal coefficient, τw is the wall shear stress, and md is the mass of deposit per surface area. The shear removal coefficient is a parameter that represents the deposit strength and depends on various factors such as temperature, pressure, and physiochemical characteristics of both the fluid and deposit. Molecular Diffusion. The idea of molecular diffusion comes from the transport of dissolved waxy constituents due to the temperature gradient across the sublaminar layer adjacent to the wall on which deposition occurs. When the oil is being cooled, if the fluid temperature drops below the so-called wax appearance temperature (WAT) or cloud point, wax crystals start to come out of solution. This will cause a concentration gradient across the sublaminar layer where dissolved wax material is transported toward the wall by molecular diffusion and deposited as soon as it reaches the solid-liquid interface. By assuming an identical boundary layer for heat transfer and mass transfer, Burger et al.1 used Fick’s law and the chain rule to estimate the deposit mass flux due to molecular diffusion:

Jmd ) -Dwo

|

dC dT dT dr r)ri

(9)

where Dwo is the diffusivity of wax molecules in oil, C is the dissolved wax molecule concentration, r is the radius, and T is temperature. The diffusivity, Dwo, can be estimated from a correlation provided by Hayduk and Minhas:30 1.47 10.2/Vw-0.791

Dwo ) kmdTi

µ

Vw

-0.71

(10)

where kmd is the molecular diffusion constant, Ti is the interface temperature in kelvin, µ is the oil viscosity in centipoise, and Vw is the wax molar volume in cubic centimeters per mole. Shear Dispersion. Burger et al.1 recognized this mechanism of particle transport as the dominant mechanism at lower temperatures and low heat fluxes. Shear dispersion is a mechanism tied to the shearing of the fluid. It is well-known that when a dilute suspension of noncolloidal particles is sheared under laminar flow conditions, the particles will experience some displacement away from their original streams. These displacements will lead to random walk motion, which can be characterized by the shear-induced coefficient of self-diffusion or shear dispersion coefficient.31 At low volumetric fractions of precipitated solids in liquid, this coefficient can be estimated from

DSD )

dp2γφ/w 10

(11)

where dp is the particle diameter, γ is the shear rate at the wall, and φ/w is volume fraction of solid wax particles in fluid at the wall/interface. The mass flux due to shear dispersion, Jsd, can then be represented by

Jsd ) -DSD

|

|

dC* dC* dT ) -DSD dr r)ri dT dr r)ri

(12)

where C* is the particle concentration and ri is the interface radius. The concentration gradient of wax crystals in eq 12, dC*/ dT, can be obtained from the precipitation curve. The concentration of dissolved wax molecules in eq 9 is then equal to -dC*/ dT. Burger et al. replaced eq 12 with the following chemical reaction type relation

) k*C*γ JBurger sd

(13)

where k* represents the reaction rate constant and its average value by fitting the experimental data was 7.92 × 10-4 g‚s/ cm2‚day. In this work, eq 12 will be directly used to calculate the deposition rate due to shear dispersion. It will be shown that the shear dispersion contribution is negligible compared to other mechanisms. Aging/Internal Diffusion. The aging mechanism, which is responsible for hardening the deposit with time, was introduced and mostly used by Fogler’s research group at the University of Michigan.9,10 The idea comes from the fact that the deposited wax on the cold surface traps a large quantity of oil in its network. The trapped liquid acts as a medium for further diffusion of the heavier molecules into the deposited gel. This diffusion of heavier molecules into the deposit is accompanied by the counterdiffusion of the trapped oil out of the deposit. The result of this process is an increase in the fraction of solid wax inside the deposit as well as an increase in the hardness of the deposited gel over time. The internal diffusion mass flux of wax molecules inside the deposit, Jid, can be expressed in a similar way described for the molecular diffusion mechanism:9,10,18

Jid ) -De

|

dC dT dT dr r)ri

(14)

where De is the effective diffusivity of wax molecules in the deposit and can be estimated from eq 15 proposed by Cussler et al.:32

De )

Dwo R2Fw2 1+ 1 - Fw

(15)

where R is the average aspect ratio of wax crystals and Fw is the weight fraction of solid wax inside the deposit that increases with time. The mass flux due to internal diffusion is subtracted from the total mass flux due to molecular diffusion in order to determine the overall growth rate of the deposit resulting from a combination of molecular diffusion and aging mechanisms. Mass Balance. Figure 5 shows a sketch of a deposition process in TC flow. By writing a total mass balance on the


whole deposit and a mass balance on the solid wax inside the deposit and involving all the contributing mechanisms for each mass balance, one can derive formulas to determine the deposition rate and the rate of change in the wax fraction of deposit, respectively. Mass Balance on the Whole Deposit. Among the mechanisms previously discussed, particle diffusion, molecular diffusion, and shear dispersion help the deposit grow while shear removal and aging (internal diffusion) do the opposite. Therefore, a total mass balance on the whole deposit can be written as follows:

Jt )

d [π(rw2 - ri2)]FdH ) (Jpd + Jmd + Jsd - Jsr - Jid + dt Jo)(2πriH) (16)

where t is time, rw is the inner radius of the outer cell (without deposit), H is the OSDC (TC device) height, Jt is the accumulation of total mass in the deposit, Jpd is the mass flux due to particle diffusion and inertia, Jmd is the mass flux due to molecular diffusion, Jsd is the mass flux due to shear dispersion, Jsr is the mass flux due to shear removal or sloughing, Jid is the mass flux due to internal diffusion (aging), and Jo is the net gain or loss of oil due to new deposit volume and is obtained from the following:18

Jo ) (Jpd + Jmd + Jsd - Jsr - Jid)

( ) 1 - Fw Fw

(17)

Combining eqs 16 and 17 will provide the following term for determining changes in deposit thickness, δ, with time:

Jg dδ ) dt FdFw

(18)

q0 ) Uo(Tf - Tw) ) -ko

δ ) rw - ri

(19)

Jg ) Jpd + Jmd + Jsd - Jsr - Jid

(20)

and

The mass fluxes in eq 20 can be obtained from the relations provided in previous sections. Mass Balance on the Wax in the Deposit. In order to find out the change in the wax fraction of the deposit, Fw, with time, a mass balance can be written on the solid wax in the deposit: 18

d [π(rw2 - ri2)FdFw] ) (Jpd + Jmd + Jsd - Jsr)(2πriH) dt (21)

where Jw is the rate of change of wax in the deposit and the right-hand term includes the mechanisms that are involved in adding wax to or removing wax from the deposit. Combining, manipulating, and rearranging eq 21, the change in the wax fraction of deposit is expressed as

dFw 2(rw - δ)Jid ) dt Fdδ(2rw - δ)

(22)

By assuming a quasi-steady-state system, one can solve eqs 18 and 22 and determine deposit thickness, deposit mass, and change in the solid wax fraction in the deposit over time in the

|

dT dr r)ri

(23)

where q0 is the total heat flux in radial direction, Uo is the overall heat transfer coefficient, Tf is the average fluid temperature, Tw is the wall temperature in the OSDC, and ko is the oil thermal conductivity and assumed to be equal to 0.1 W/m‚K for the studied oils. In the OSDC, the temperatures of the bulk fluid and inner wall of the cell are controlled at constant values. Therefore, the overall heat transfer coefficient can be obtained from the following equation:

Uo )

where

Jw )

TC flow or OSDC. A similar method can be used for pipeline deposition modeling which is the subject of another paper. Hydrodynamics and Heat Transfer. The deposition model is not complete without having information on fluid hydrodynamics and heat transfer. For instance, wall shear stress due to fluid flow needs to be determined prior to calculating the shear removal mass flux, or a heat balance must be performed on the system, in order to find out the interface temperature or the temperature gradient at the interface. The hydrodynamics module in the developed deposition model consists of correlations/equations for estimating the Reynolds number, friction factor, wall shear stress, and shear rate at the interface. Some of these variables/correlations are defined in the nomenclature section. Others can be easily found in the literature. A heat balance on the OSDC can help us determine the oilwax interface temperature, the temperature gradient at the interface, total heat flux, etc. By assuming a one-dimensional heat transfer in radial direction, a total heat balance gives us the following:

[

]

ln(rbl/ri) ln(rw/ri) 1 1 + + ri riho ko kd

-1

(24)

where ho is the oil heat transfer coefficient and kd is the deposit thermal conductivity. In this work, the deposit thermal conductivity is assumed to be equal to the oil thermal conductivity. For the heat transfer coefficient, the correlation developed by Ball et al.33 was adapted to our own applications. The correlation is the following:

14.54 + 0.74Re0.53 h o ) ko 2(ri - rs)

(25)

Once q0 is known, the temperature gradient at the interface can be calculated from eq 23:

|

q0 dT )dr r)ri ko

(26)

Wax Concentration. Solid wax and dissolved wax concentrations and their gradients are required for estimating the deposition mass flux in the previously discussed mechanisms. These values can be determined using different methods including the wax precipitation and quantification method (an experimental method), crystallization kinetic modeling method,34 and/or thermodynamic precipitation models. The thermodynamic compositional models need the composition of fluid and wax in addition to their need for tuning. The kinetic model will be applied in this deposition model in the future. At the current stage, however, we employ the experimental wax precipitation and quantification technique.

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Figure 5. Experimental data and model calculations for the Asian fluid at Tf ) 43 °C, Tw ) 10 °C, and zero shear (ω ) 0 Hz).

parameters are determined using some experimental data. The effect of the model parameters on the deposition rate will be investigated through a sensitivity analysis. Figure 3. Wax precipitation curve for the GoM fluid.

Figure 4. Wax precipitation curve for the Russian fluid.

The experimental wax quantification technique is used to obtain a precipitation curve for an individual oil by separating and quantifying the amount of precipitated wax at different temperatures. A description of the wax quantification procedure is provided in the Appendix. Figures 3 and 4 show the measured points for the studied oils. The concentration gradient of wax particles with regard to temperature, dC*/dT, for each oil can be easily obtained from the precipitation curve for that oil. As mentioned earlier, the dissolved wax concentration gradient, dC/ dT, is equal to -dC*/dT. Model Parameters. The key model parameters include the molecular diffusion factor (kmd), the average size of wax particles (dhp), the shear removal (sloughing) coefficient (ksr), the particle sticking probability (Ps), and the initial solid wax content of the deposit (F0w). Since all the experimental data in this work are based on the solid wax deposit not the whole deposit (solid wax + entrapped oil), F0w in the model calculations equals unity, which means that the aging mechanism has no contribution in our calculations. Despite this, the influence of aging will be discussed later in a separate example. The rest of the

Results and Discussion Zero Shear TestsDetermining Molecular Diffusion Factor. At static conditions, the only mechanism that is involved in deposition is molecular diffusion. Upon fitting the experimental deposition data for the Asian oil demonstrated in Figure 5, a value of 1.2 × 10-10 was obtained for the molecular diffusion factor, kmd. In order to validate this factor, Figure 6 compares the model predictions at zero shear conditions with the experimental deposition data for the GoM oil with kmd ) 1.2 × 10-10. As it can be seen from the plot, there is a fair agreement between the model prediction and the experimental data meaning that the proposed molecular diffusion factor can be used for other oils with reasonable accuracy. Once the molecular diffusion factor is estimated, it is fixed and used for model calculations at other conditions. Determining the Average Particle Size. The appearance of wax crystals due to precipitation in the waxy crude creates a polydispersed system. The wax particles are considered to have a size distribution with an average particle size. In the current study, a gamma distribution function35 was used to describe the size distribution:

f(dp) )

[

β 1 Γ(β) (dhp - dmin p )

]

β

×

β-1 exp (dp - dmin p )

[

]

β(dp - dmin p ) (dhp - dmin p )

(27)

where dhp and dmin are the average and minimum wax particle p sizes, respectively, and β is a parameter which determines the shape of the distribution. The recommended value for β in the polydispersed systems is 2 that will be used throughout this work. The average particle size of wax particles can be a function of temperature, shear, fluid composition, and time. The wax appearance temperature (WAT) or cloud point can be an indicator of the biggest particle size depending on what the detection limit of the equipment for measuring the onset point is. For instance, for the GoM fluid under study, the measured WAT using cross polar microscopy (CPM) is 34 °C and the detection limit of CPM is 2 µm. Therefore, if the fluid


Figure 6. Experimental data and model calculations for the GoM fluid at Tf ) 30 °C, Tw ) 25 °C, and zero shear (ω ) 0 Hz).

Figure 8. Particle size distribution with an average size of 1 µm. Deposition data and modeling results are for GoM fluid at Tf ) 30 °C, Tw ) 27 °C, and ω ) 60 Hz.

Figure 7. Particle size distribution with an average size of 1 µm.

temperature is under the WAT (e.g., 30 °C), some of the wax crystals have particle sizes bigger than 2 µm while the rest are under 2 µm. On the basis of this, an average particle size of 1 µm for the GoM wax crystals at 30 °C might be reasonable. Figure 7 shows the size distribution for the wax crystals in the GoM fluid at 30 °C with a minimum particle size of 1 nm and an average particle size of 1 µm. In order to validate the demonstrated crystal size distribution, particle size measurement techniques need to be applied or developed. Determining the Sticking Probability and Shear Removal Coefficient. To estimate the rest of the model parameters (i.e., Ps, and ksr), the experimental deposition data obtained for the GoM fluid at Tf ) 30 °C, Tw ) 27 °C, and ω ) 60 Hz (τw ) 16.94 Pa) were fitted using the developed model. For simplicity, a repeatability of (30% symbolized by error bars on all the deposition data points in this work was considered. Our recent deposition data with various oils show an average repeatability of (15%. As demonstrated in Figure 8, a value of 1 × 10-7 for the shear removal coefficient, ksr, and a value of 0.029 for the sticking probability, Ps, matched the experimental data well. This means that only 2.9% of the wax crystals that reach the wall stick there and become part of the deposit. It should be noted that no individual test could be run to independently determine the contribution of either the particle diffusion mechanism or shear removal mechanism in the deposition process. However, the impact of each parameter on the calculated deposition tendency through a sensitivity analysis will be discussed later. Model Validation. Once the model parameters were determined, they were fixed and used to predict the deposited mass

Figure 9. Deposition data and modeling results for GoM fluid at Tf ) 30 °C, Tw ) 25 °C, and ω ) 100 Hz.

of solid wax at other conditions. With this approach we can investigate the impact of each parameter on the deposition through changing the parameters and matching the predicted results with the experimental data. This can be helpful in determining the dependency of the key parameters on the fluid and flow characteristics/conditions. Figure 9 compares the model predictions with the experimental data for the OSDC test with the GoM fluid at Tf ) 30 °C, Tw ) 25 °C, and ω ) 100 Hz (τw ) 40.38 Pa). As it can be seen, the model with the fixed parameters overpredicts the deposition data. This may suggest the shear dependency of the parameters. Since the effect of shear was included in the sloughing equation, eq 8, the shear removal coefficient may not be considered as a shear dependent coefficient. The average size of the wax crystals, however, may be reduced at higher shears resulting from higher rotational speeds. Figure 13 shows that a decrease in the average size from 1 to 0.5 µm (500 nm) is not very effective in matching the data points while a decrease in the particle sticking probability from 0.029 to 0.017 matches the data very well. This may mean that, at higher shears, particles with an average size of 1 µm either have less chance for sticking to the wall or do not possess enough energy and/or momentum to reach the wall. Figure 10 demonstrates a benchmark between the model predictions and the experimental deposition data for the OSDC test with the GoM fluid at Tf ) 30 °C, Tw ) 25 °C, and ω ) 30 Hz (τw ) 5.97 Pa). There is a good agreement between the

960




data and the model predictions for the 1-h run time. However, the model underpredicts the deposit amount for the 2-h run time. If the average particle size increases from 1 to 2 µm, then a better match is obtained which may show the effect of shear on the crystal size. A very similar match can also be obtained by increasing the sticking probability from 0.029 to 0.04. This indicates that at lower shear the particles in the 1-µm range may have a greater chance to stick to the wall upon impact, which is consistent with what was concluded for the 100-Hz run. To investigate the effect of temperature on the model predictions and the parameters, a test was conducted at Tf ) 25 °C, Tw ) 12 °C, and ω ) 60 Hz (τw ) 18.64 Pa). In Figure 11, although the model predicts the experimental deposition data accurately, having a fluid temperature lower than 30 °C (in this case 25 °C), may amplify the average size of the wax particles. Consequently, if, for instance, the average particle size increases from 1 to 1.5 µm, the model predictions error will increase as well (Figure 11). By keeping the average particle size at 1.5 µm, if the sticking probability drops from 0.029 down to 0.010, a very good match is obtained which may suggest that at lower wall temperatures particles would have less possibility to stick to the wall upon impact. On the basis of the results and discussions in this section, if the model parameters are fixed, the developed model, in general, does a fairly good job in predicting the deposition data at various conditions. To get a match between the data and model

Figure 12. Contribution of each mechanism for GoM fluid at Tf ) 30 °C, Tw ) 27 °C, and ω ) 60 Hz.

Figure 13. Experimental data and modeling results for the total amount of deposit (wax + entrapped oil) by contributing the aging effect for the GoM fluid at Tf ) 30 °C, Tw ) 27 °C, and ω ) 60 Hz.

calculations with minimum error, the model parameters need to be adjusted. On the basis of our limited experimental deposition data, the sticking probability factor is a function of wall shear stress and wall temperature, has the biggest impact on the deposition tendency, and needs to be evaluated in more details. Some statistical methods such as Monte Carlo simulation may help in determining the sticking probability of the particles that hit the wall/deposit at various flow and fluid conditions. Contribution of Mechanisms. Figure 12 shows the contribution of each mechanism in the single-phase wax deposition study of the GoM fluid at Tf ) 30 °C, Tw ) 27 °C, and ω ) 60 Hz (τw ) 16.94 Pa) for which the proposed parameters are kmd ) 1.2 × 10-10, dhp ) 1 µm, Ps ) 0.029, and ksr ) 1 × 10-7. As it can be seen, molecular diffusion has little impact compared to particle diffusion while the shear dispersion contribution can be neglected. The shear removal contribution is also small in this case. To see the effect of aging mechanism on the deposition, the total amount of deposit, that is, the amount of solid wax plus the amount of entrapped oil in the deposit matrix, can be used for model calculations. Figure 13 shows the data for the above case. Assuming a spherical shape for wax crystals (R ) 1 in eq 15), a value of 0.1 for the initial solid wax fraction in the deposit, F0w, could match the experimental data well. According to the model calculations, after 8 h the percentage of the solid wax in the whole deposit increases from 30% to about 47%. This example shows that the model is capable of taking the effect of aging into account in the deposition process.


Figure 14. Effect of average wax crystal size on the model predictions for deposition tendency.

Figure 16. Effect of shear removal coefficient on the model predictions for deposition tendency.

Shear Removal Coefficient. Figure 14 demonstrates the sensitivity analysis performed on the shear removal coefficient, ksr. Depending on the strength of the deposit, this coefficient may be different from one fluid to another. The temperature and flow conditions may also have an impact on this coefficient and therefore on the deposition tendency. As it can be seen from Figure 16, at higher shear removal coefficients, the effect of sloughing is bigger and less deposit is obtained. Due to this high shear removal effect, the deposit amount and its thickness flatten over time. More experimental data at different flow conditions and with different fluids is required to investigate the sloughing mechanism more deeply. Conclusion Figure 15. Particle sticking probability factor on the model predictions for deposition tendency.

Sensitivity Analysis. In this section, the influence of each parameter on the amount of deposit calculated by the developed model for the test at Tf ) 30 °C, Tw ) 27 °C, and ω ) 60 Hz (τw ) 16.94 Pa) is examined. Average Particle Size. Figure 14 displays the effect of the average particle size on the wax deposition tendency. Assuming the same sticking probability for all the particles, when the size of wax crystals is smaller than 0.7 µm, the amount of deposit increases with a decrease in the average size of particles because the Brownian diffusivity and therefore the mass transfer coefficient are bigger for smaller particles. On the other hand, for particles larger than 0.7 µm, due to inertial effects, the amount of deposit increases with a rise in the average size. Sticking Probability. All the model calculations in this study are based on assuming the same sticking probability for particles with any size. If the sticking probability of smaller particles is different from that of bigger particles, the amounts of deposit will be different as well. Figure 15 shows the sensitivity of the model to the changes in the sticking probability. The higher the probability, the higher the deposition rate would be. As discussed earlier, the sticking probability factor has the most impact on the deposition tendency and needs to be evaluated in more details. Some statistical methods such as Monte Carlo simulation may help in estimating the sticking probability of the particles with different sizes that hit the wall/deposit at various flow and fluid conditions.

A new approach for modeling single-phase wax deposition in Taylor-Couette system was developed. Among the involved mechanisms, particle diffusion/deposition mechanism made the biggest contribution in formation of the deposit at the realistic transport conditions. The model parameterssmolecular diffusion factor, average particle size, shear removal coefficient, and sticking probabilityswere estimated using some experimental data obtained by the organic solids and deposition control (OSDC) device that was designed based on the Taylor-Couette system. The model predictions were compared with the deposition data at other conditions. The model, in general, did a fairly good job in predicting the deposition data at various conditions. To match the model calculations with the experimental deposition data with minimum error, the model parameters were adjusted and the effect of each parameter on the model predictions was discussed. On the basis of the limited experimental deposition data, the sticking probability factor and average particle size showed dependencies on temperature and shear. More experimental data are required to investigate the impact of each parameter in more details. Some statistical methods such as Monte Carlo simulation may help in determining the sticking probability of the particles that hit the wall/deposit at various flow and fluid conditions. Also, finding a more effective way to measure and determine the size distribution of wax crystals and their kinetics at flowing conditions is of great importance. The estimated model parameters can be used in the pipeline deposition model to predict the deposition rates under actual flow conditions in production and transportation pipelines. This will be the subject of the second part of this paper.

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Acknowledgment The authors would like to thank Joshua Genereux and Amanda Prefontaine for performing the OSDC tests and analyzing the deposits, respectively. Appendix: Quantification of Precipitated Wax at Different Temperatures. (1) Preheat the samples to about 70 °C prior to any kind of process or subsampling and shake the sample jar manually during conditioning to dissolve wax crystals back to the solution, if any. (2) Weigh approximately 2 g of sample and transfer it quantitatively into a 1-L Erlenmeyer flask. Add 500 mL of petroleum ether and stir until the sample is completely dissolved. Add approximately 15 g of fuller’s earth and continue to stir for 30 min. (3) Filter the solution through an ashless filter paper pulp in a Millipore filter apparatus attached to a 1-L vacuum flask. Note the color of the filtrate. If after first filtration, the petroleum ether mixture has a significant color, then the asphaltenes have not been completely removed. Repeat the fuller’s earth treatment. (4) Once the filtrate is clear or very lightly colored, evaporate the petroleum ether on a heating plate down to a volume of 5-10 mL. (5) Add 200 mL of the acetone/petroleum ether mixture (volume ratio of 3 to 1) to the oil/wax mixture while it is still warm. Agitate it until the sample is dissolved. Place a rubber stopper on flask before placing it in the temperature control air bath. Chill the resulting solution for 4 h. Chill some additional acetone/petroleum ether mix for rinsing the filter and Millipore apparatus. (6) Filter the chilled solution through the chilled Millipore apparatus into a clean vacuum flask. Rinse with chilled acetone/ petroleum ether mixture. Attach the Millipore apparatus with wax into original vacuum flask and let the wax warm up to the room temperature. (7) Use dichloromethane or toluene to dissolve wax on the apparatus and rinse the vacuum flask with Dichloromethane or toluene. Place the solution into a preweighed aluminum dish and let the solvent evaporate. Cool the dish in desiccator, weigh the dish, and record the weight. Nomenclature All the parameters/variables are in SI units unless otherwise noted in the text. A, b, c ) constants in eq 3 C ) dissolved wax molecules concentration C* ) solid wax concentration C/b ) solid wax concentration in the bulk Cp ) oil heat capacity d ) gap between the cylinders in the TC device or OSDC dp ) particle size dhp ) average particle size DB ) Brownian diffusivity ) (1.38 × 10-23T)/(3πdpµo) De ) effective diffusivity Dsd ) shear dispersion coefficient Dwo ) wax molecular diffusivity f ) friction factor Fw ) weight fraction of solid wax in the deposit G ) dimensionless torque ) [0.23(rs/ri)1.5/(1 - rs/ri)1.75]Re1.7 when Re > 10 000 and ) [1.45(rs/ri)1.5/(1 - rs/ri)1.75]Re1.5 when Re e 10 000

H ) OSDC height ho ) oil heat transfer coefficient Ip ) particle inertia coefficient Jid ) internal diffusion (aging) mass flux Jmd ) molecular diffusion mass flux Jpd ) particle diffusion (deposition) mass flux Jsd ) shear dispersion mass flux Jsr ) shear removal mass flux Jt ) total deposition mass flux kd ) deposit thermal conductivity ) (2kw + ko + (kw - ko)Fw)/(2kw + ko - 2(kw - ko)Fw)ko km ) particle mass transfer coefficient kmd ) molecular diffusion parameter ko ) oil thermal conductivity kpd ) particle diffusion (deposition) coefficient ksr ) shear removal coefficient kw ) wax thermal conductivity KB ) Boltzman constant ) 1.38 × 10-23 md ) Mass of deposit per surface area ) (rw2 - ri2)Fd/2rw mp ) mass of wax particle ) πdp3Fw/6 Nu ) Nusslet number ) 2riho/ko Ps ) sticking probability Pr ) Prandtl number ) Cpµo/ko r ) radius rbl ) boundary layer radius ) ri - δbl ri ) interface radius ) rw - δ rs ) inner cylinder (spindle) radius rw ) OSDC cell inner radius Re ) Reynolds number ) 2πrsωFo(ri - rs)/µo S ) particle stopping distance S+ ) dimensionless particle stopping distance ) Su*Fo/µo Sc ) Schmidt number ) µo/FoDB t ) time T ) temperature Tf ) fluid temperature Ti ) interface temperature Tw ) wall temperature Ta ) Taylor number ) 4ω2Fo2d4/µo2 u* ) shear velocity ) xτw/Fo Uo ) overall heat transfer coefficient VB ) Brownian velocity VI ) particle velocity due to fluid motion and inertia VP ) total particle velocity R ) aspect ratio of wax crystals β ) gamma distribution function parameter γ ) shear rate at the wall ) τw/µo δ ) deposit thickness δbl ) boundary layer thickness µo ) oil viscosity Fd ) deposit density ) (Fw/Fw + (1 - Fw)/Fo)-1 Fo ) oil density Fp or Fw ) wax (particle) density τw ) wall shear stress ) Gµo2/2πri2Fo ω ) rotational speed Literature Cited (1) Burger, E.; Perkins, T.; Striegler, J. Studies of Wax Deposition in the Trans Alaska Pipeline. J. Pet. Technol. 1981, 33, 1075. (2) Majeed, A.; Bringedal, B.; Overa, S. Model Calculates Wax Deposition for North Sea Oils. Oil Gas J. 1990, 88, 63. (3) Svendsen, J. A. Mathematical Modeling of Wax Deposition in Oil Pipeline Systems. AIChE J. 1993, 39, 1377. (4) Brown, T. S.; Niesen, V. G.; Erickson, D. D. Measurement and Prediction of the Kinetics of Paraffin Deposition. In SPE Technical Conference and Exhibition, Houston, TX, 1993; SPE 26548, p 353.

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ReceiVed for reView August 17, 2007 ReVised manuscript receiVed October 16, 2007 Accepted October 19, 2007 IE0711325

Introduction to a Novel Approach for Modeling Wax Deposition in Fluid

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