Validating Heat-Transfer-Based Modeling Approach for Wax

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Validating Heat-Transfer-Based Modeling Approach for Wax Deposition from Paraffinic Mixtures: An Analogy with Ice Deposition Sina Ehsani, and Anil K Mehrotra Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b03777 • Publication Date (Web): 20 Feb 2019 Downloaded from http://pubs.acs.org on February 21, 2019

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Validating Heat-Transfer-Based Modeling Approach for Wax Deposition from Paraffinic Mixtures: An Analogy with Ice Deposition Sina Ehsani and Anil K. Mehrotra* Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada Abstract The process of solid deposition from wax–solvent mixtures was compared with that of ice deposition from liquid water by means of experiments using a cold ﬁnger apparatus and a transient mathematical model based on the Stefan ‘moving boundary problem’ formulation. Two agitation speeds of 250 rev min–1 (Reynolds number of 4100) and 500 rev min–1 (Reynolds number of 8200), with two coolant temperatures of Tf–4°C and Tf–7°C, at 4 water temperatures (from Tf+3°C to Tf+0.7°C), and for 11 deposition times between 30 s to 8 h were used in the ice deposition experiments. The wax deposition experiments were undertaken using a 10 mass% wax–solvent multicomponent mixture, at an agitation speed of 250 rev min–1 (Reynolds number of 1400), with a constant coolant temperature of WAT–12°C, at 4 mixture temperatures (from WAT+6°C to WAT), and for 17 deposition times ranging from 2 s to 48 h. Both the ice deposition and the wax deposition processes were remarkably similar. Both of these phasechange systems were extremely rapid during the first few minutes. A higher deposit mass was achieved with lowering the liquid water temperature, the coolant temperature, and the agitation speed. The experimental results from this investigation, supported by those from previous studies, indicated that a higher deposit mass is achieved with lowering of the liquid mixture temperature, the coolant temperature, and the agitation speed. The results of both sets of experiments were consistent with predictions from the Stefan moving boundary problem framework, which considers both of these phase-change processes to be governed only by the heat-transfer steps involved in the freezing of a liquid. The present study confirms that the solid deposition from wax–solvent mixtures is described adequately based entirely on heat-transfer considerations. Keywords: solid deposition; freezing; heat transfer; waxy crude oil; paraffin; wax appearance temperature; ice deposition; solidification; moving boundary problem *Corresponding Author. Tel.: 1–403–220–7406. Email: [email protected] 1 ACS Paragon Plus Environment

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1. Introduction Crude oils are complex mixtures containing hydrocarbons ranging from alkanes, naphthenes, aromatics and resins, to high molar mass waxes and asphaltenes. The high temperature and pressure in petroleum reservoirs cause the high molar mass paraffins (C18-C65) to stay dissolved in the crude oil. Crude oils could be exposed to substantial temperature differences in pipelines, especially in subsea conditions1, 2. The exposure of waxy (highly paraffinic) crude oil to cooler surfaces decreases their solubility, causing the precipitation of wax molecules from the liquid phase. The precipitated wax constituents could deposit on the pipeline wall, resulting in expensive flow assurance challenges3. Wax deposition occurs when waxy (paraffinic) solids separate from paraffinic crude oils while being subjected to temperatures lower than their wax appearance temperature (WAT). Upon cooling the crude oil, the highest temperature at which the initial crystalline wax solids begin to form in the crude oil is known as the WAT or cloud point temperature (CPT). The wax deposition process has been explained in terms of several different mechanisms. Shear dispersion4-10, Brownian diffusion5, 11, gravity settling7, thermophoresis5, 12, the Saffman effect1315,

molecular diffusion4, 16-19, and heat-transfer3, 20-23 are among the mechanisms that have been

proposed for the wax deposition process. The shear dispersion mechanism describes wax deposition process based on cross-stream transport of suspended solid particles4. In the Brownian diffusion mechanism, wax deposition is attributed to the concentration difference between the suspended solids particles5. Since the precipitated wax crystals are more dense than the liquid oil phase, gravity settling could be a feasible mechanism for wax deposition7. The thermophoresis deposition mechanism is based on the movement of micro-particles due to a temperature gradient between the hot and the cold regions5, 12. The particles under shear flow in a pipeline undergo a 2 ACS Paragon Plus Environment

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lift force (called the Saffman lift force) perpendicular to the direction of flow, which could result in the separation of particles from the pipe wall14. However, the effects of these mechanisms on wax deposition have been considered to be relatively minor or small5-7. The heat-transfer3, 20-24 and molecular diffusion4, 16-19 approaches have been proposed as the primary mechanisms for wax deposition. The molecular diffusion approach considers the crude oil flow in a pipeline with a wall temperature lower than the WAT such that it would result in a concentration gradient of the wax molecules in the radial direction4, 7, 16-18. The transient growth of the deposit is modelled based on the Fick’s diffusion equation. The molecular diffusion approach predicts an increase in the liquid–deposit interface temperature from the pipe-wall temperature to the wax appearance temperature at steady-state (at which point further solid deposition would cease)16. The heat-transfer modeling approach explains wax deposition as a liquid-to-solid phase transformation or (partial) freezing process. It is assumed, and also validated experimentally25-28, that, throughout the deposit growth process, the interface temperature between the liquid and the deposit is at a constant temperature, which is equal to the WAT. Furthermore, experimental investigations have reported a rapid deposit growth during the initial stages of the deposition process, which is observed typically in thermally-driven processes29. It should be noted that, despite many differences, the steady-state interface temperature is equal to the WAT in the molecular-diffusion and heat-transfer modeling approaches30. A number of studies in the literature have reported a variation in the deposit composition during the deposition process3, 22, 23, 27, 28, 31, 32. A mathematical model of the deposit, involving a cubical cage and its deformation, was proposed by Mehrotra and Bhat23, and validated with experimental deposit compositional data under laminar flow, in which the extent (or, the angle)

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of the cubical cage deformation was related to the influence of shear stress and deposition time on the compositional changes in the deposit. The predictions from the same ‘cubical cage deformation’ modeling were confirmed subsequently by Tiwary and Mehrotra28 with deposition experiments under the turbulent flow condition. It is noted that the freezing or solidification process involves phase change from the liquid phase to the solid phase. A pure liquid freezes completely when it is held at its freezing point temperature (Tf). In the absence of another diffusing species, the solidification of a pure liquid (i.e., a single component system) is described solely by heat-transfer considerations. That is, there is no molecular diffusion involved in the phase transformation of a single-component liquid or solid. Previous studies with prepared waxy mixtures have validated the heat-transfer based modeling of laboratory-scale solid deposition data21, 30. In addition, it has been used to describe the deposition behavior in pipelines, and also with deposit aging3, 21, 24, 30, 32-34. If the same modeling approach could also be used to explain the freezing of water, this analogy would serve as a further validation for the heat-transfer approach for wax deposition. A key objective of this study, therefore, is to demonstrate that the heat-transfer approach alone is capable of predicting solid deposition from waxy mixtures, similar to the case of ice deposition from pure water. Thus, the present experimental and modeling investigation was undertaken to investigate the heat-transfer approach for solid deposition from a pure liquid as well as waxy mixtures. All deposition experiments were performed using a cold-finger set-up, which has been shown to be satisfactory for describing the deposition behavior of ‘waxy’ mixtures29. The ice deposition experiments were performed using liquid water over varying deposition times, between 30 s and 8 h. A prepared wax–solvent mixture, consisting of multicomponent paraffin and solvent, was used for the wax deposition experiments, which is similar to previous studies from our


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laboratory3, 29, 35. The wax deposition experiments were performed under hot flow conditions (i.e., Th >WAT) over varying deposition times, between 2 s and 48 h. The experimental results for both systems were modeled based on a steady-state heat-transfer model3, 27-29, 31, 32, 35 as well as a transient model using the Stefan moving boundary problem framework33, 36, 37.

2. Mathematical models 2.1 Steady-state heat transfer model The steady-state heat transfer model for solid deposition involves a cylindrical cold finger put inside a well-mixed reservoir containing a liquid (water or waxy mixture) held at a constant temperature of Th (Th > Tf or Th > WAT), where Tf is the freezing point temperature. A coolant flows through the cold finger at an average temperature of Tc (Tc < Tf for ice deposition or Tc WAT, the 6 ACS Paragon Plus Environment

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solid–liquid interface temperature, Td, is equal to WAT3, 25, 32, 39. The experimental values of the deposit thickness (xd), the mixture temperature (Th ), the interface temperature (Td ) as well as the estimated deposit thermal conductivity (kd)22,33were used in eq 4 to estimate the steady state heat transfer coefficient (hh).

2.2 Transient heat-transfer model The time-dependant solid formation on the external surface of the cold finger was predicted using a transient (unsteady-state) heat-transfer mathematical model for solid deposition from liquid water, as well as a transient heat-transfer mathematical model for solids deposition from a waxy mixture, using the Stefan moving boundary problem approach22, 24, 37. In the Stefan problem dealing with transient phase transformation, the location of the moving phase boundary must be ascertained in terms of time and space36. The Stefan problem is associated with freezing and melting processes, which are encountered in many practical applications and industrial processes, such as ice formation, solidification of casts, thermal energy storage systems, hydrocarbon processing, etc36. In these processes, liquid-to-solid or solid-to-liquid phase change occurs such that the boundary separating the phases moves with time. The two phases can have different transport properties, resulting in differences in energy, mass and momentum transport. The location of the moving boundary is not known a priori, and it is specified as part of the numerical or analytical solution. In the one-phase Stefan problem for a pure substance, one of the phases is held at its freezing point, while the temperature change occurs in the other. In the two-phase Stefan problem, the temperature varies in both liquid and solid phases, while interface is held at the phase-change temperature. For the freezing of a pure substance, for example, the initial 7 ACS Paragon Plus Environment

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temperature of the liquid phase could be above the freezing temperature and that of the solid phase below the freezing temperature. Note that, in a multi-component system, the freezing point temperature is replaced by the liquidus temperature (at which the first crystal appears, upon cooling, or the last crystal disappears, upon heating) and the melting temperature is replaced by the solidus temperature (at which the last trace of liquid disappears, upon cooling, or the first trace of liquid appears, upon heating). At temperatures between the liquidus and solidus temperatures, the multi-component system would exist as a mixture of solid and liquid phases; it would be more liquid-like near the liquidus temperature and more solid-like near the solidus temperature. The Stefan problem framework has been used previously to predict solid deposition from waxy mixtures on the external surface of cold finger29 as well as inside a pipeline21. To simplify the calculations for wax deposition, the waxy mixture was approximated as a pseudo-binary mixture, with C14 and C30 representing the solvent and wax fractions, respectively. A comparison between three thermodynamic approaches for modeling the liquid–solid phase equilibrium was reported by Bhat and Mehrotra38 and it was shown that the pseudo-binary approach for modeling the multi-component wax–solvent mixture gave satisfactory predictions. The Stefan problem for the liquid−solid phase transformation of a binary mixture of n-alkanes may possibly include three phase regions: a liquid–phase region (𝑇 > 𝑇𝐿), a solid-phase region ( 𝑇 < 𝑇𝐸), and a liquid−solid mixed phase region (𝑇𝐸 < 𝑇 < 𝑇𝐿). Depending on the carbon number distribution, the solid phase of binary normal alkanes can be either miscible or immiscible. Similar to previous studies, the C14−C30 binary mixture was taken to completely immiscible in the solid phase29, 30.

2.2.1 Energy balance equation and heat-transfer considerations


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For the solidification of liquid water, the one-dimensional (radial) energy balance for the transient heat-transfer by conduction (through the ice layer forming on the external surface of the cold finger) is given by38: 1∂

∂𝑇𝛿

1 ∂𝑇𝛿

∂𝑟

𝛼𝛿 ∂𝑡

(𝑟 ) =

𝑟 ∂𝑟

rw < r < s

(5)

where s is the radial position of the growing liquid–ice interface and rw is the radius of the cold finger outer surface. The energy balance at the liquid–ice interface is given as38: ∂𝑇𝛿

𝑑𝑠

𝑘𝛿 ∂𝑟 ― ℎℎ(𝑇ℎ ― 𝑇𝑑) = 𝜌𝜆𝑑𝑡

r=s

(6)

For the solidification of a paraffinic mixture, the one-dimensional thermal energy balance equation for the transient heat-transfer by conduction (in the deposit layer forming radially on the external surface of the cold finger) is given by29, 37: 1∂

∂𝑇𝛿

1 ∂𝑇𝛿

∂𝑟

𝛼′𝛿 ∂𝑡

(𝑟 ) =

𝑟 ∂𝑟

rw < r < s

(7)

where s is the radial position of the growing liquid–deposit interface and rw is the radius of the cold finger outer surface. The solid phase modified thermal diffusivity, 𝛼′𝛿, is as follows: 1 𝛼′𝛿

=

1

𝜌𝜆∂𝑓𝛿

(8)

𝛼𝛿 ― 𝑘𝛿 ∂𝑇𝛿

The energy balance at the interface is: ∂𝑇𝛿

𝑑𝑠

𝑘𝛿 ∂𝑟 ― ℎℎ(𝑇ℎ ― 𝑇𝑑) = 𝜌𝜆𝑓𝑠𝑑𝑡

r=s

(9)

where fs is the equilibrium fraction of the solid phase at the interface, at r = s, at the liquid– deposit interface temperature, Td.

2.2.2 Boundary and initial conditions Eqs 5 and 6 for the ice deposition case were solved using the following boundary conditions: 𝑇𝛿 = 𝑇𝑤 = 𝑇𝑐,

r = 𝑟𝑤, t > 0 9

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(10a)

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𝑇𝛿 = 𝑇𝑑 = 𝑇𝑓 ,

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r = s, t > 0

(10b)

𝑇𝛿 = 𝑇𝑑 = 𝑇𝑓 ,

t=0

(10c)

s = rw,

t=0

(10d)

The initial conditions for eqs 5 and 6 are:

Eqs 7 to 9 for the wax deposition case were solved using the following boundary conditions: 𝑇𝛿 = 𝑇𝑤 = 𝑇𝑐,

r = 𝑟𝑤, t > 0

(11a)

𝑇𝛿 = 𝑇𝑑 = WAT,

r = s, t > 0

(11b)

𝑇𝛿 = 𝑇𝑑 = WAT,

t=0

(11c)

s = rw,

t=0

(11d)

The initial conditions for eqs 7 to 9 are:

2.2.3 Thermodynamic considerations for wax deposition As mentioned previously, the waxy mixture was approximated as a pseudo-binary mixture comprising C14 and C30, representing the solvent and wax fractions, respectively29. This pseudo-binary mixture was assumed to be completely immiscible (i.e., an ideal eutectic mixture) in the solid phase. The temperature−composition relationship for the ideal eutectic system was obtained from the following freezing point depression equation37: ln 𝑥𝑖 = ―

(∆𝐻𝑚)𝑖 𝑅

[(

1 𝑇𝐿)𝑖

1

]

― (𝑇𝑚) , i=1,2

(12)

𝑖

where (∆𝐻𝑚)𝑖, (𝑇𝑚)𝑖, (𝑇𝐿)𝑖 and (∆𝐻𝑚)𝑖 are the enthalpy of melting (or fusion), the melting point temperature, the liquidus temperature of component i, respectively. (𝑇𝑚) and (∆𝐻𝑚) of C30 and C14 have been reported previously29. In the equilibrium mixture at temperature Tδ, the solid phase mass fraction, f, was predicted using the lever rule, as follows: 𝑓=

∗ 𝑤30 ― 𝑤30

(13)

∗ 1.0 ― 𝑤30


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∗ where 𝑤30 is the mass fraction of the liquid–phase of C30 at Tδ. All physical, thermal and

thermodynamic properties were predicted from the methods described elsewhere30. The deposit mass per unit area, Ω, and the deposit thickness, xd, were related via the mass conservation equation to the density and deposit thickness, as follows29: 𝑥2𝑑 + 2𝑟𝑤𝑥𝑑

Ω = 𝜌𝑑

(14)

2𝑟𝑤

The following relationship was obtained by rearranging eq 14 to calculate 𝑥𝑑 from Ω. 𝑥𝑑 =

𝑟𝑤 2Ω𝜌 + 𝜌2𝑟𝑤 ― 𝜌𝑟𝑤

(15)

𝜌

In eqs 14 and 15, 𝜌𝑑 is the deposit density and 𝑟𝑖 is the inner tube radius.

2.2.4 Simulation procedure Eqs 5 and 6 for ice deposition and eqs 7, 8 and 9 for wax deposition were solved using Matlab software to estimate the solid-phase temperature profile and the radial position of the liquid–solid interface (s) with respect to the deposition time (t). The heat-transfer coefficient (hh), the cold finger external radius (rw), the constant coolant temperature (Tc), and the constant liquid temperature (Th) were the input parameters. In the explicit method used to discretize the equation, dependent variables were evaluated from the values at the previous time interval. The following stability criterion was used in the model37: 𝛼∆𝑡 2

∆𝑟

1

(16)

≤2

3. Experimental section 3.1 Materials The ice deposition (solidification) experiments were performed with distilled water. The coolant used was 50% ethylene glycol, with a freezing point temperature of –36°C. A 10 mass % wax–solvent mixture comprising of Bernardin Parowax (a multicomponent paraffin wax with 11 ACS Paragon Plus Environment

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carbon numbers between C21 and C58) dissolved in Linpar1416V (multicomponent solvent of nalkanes with carbon numbers between C10 and C20) was used for undertaking the deposition experiments. The compositional analyses and the physical properties for of the paraffin wax (Bernardin Parowax) and the solvent (Linpar1416V), as well as the WAT, PPT and the density of the 10 mass% wax–solvent mixture have been reported previously29.

3.2 Solid deposition experiments A cold finger assembly29, shown in Figure 1, was employed to perform ice deposition and wax deposition experiments at different deposition times and at various water and wax–solvent mixture temperatures. Briefly, the cold finger comprised a 146-mm long copper tube (of relatively low thermal resistance) with an outer dimeter of 9.5 mm. A thick layer of insulation was used to cover the bottom and top sections of the tube. With the circulation of the coolant inside the cold finger, the solid formation occurred on the external surface of the cold finger.

Figure 1: Details of the cold finger design.


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Liquid water or wax–solvent mixture was transferred into a container made of aluminum, whose temperature was maintained constant by placing it in a refrigerated bath (Haake DC1–V). A disc turbine stirrer (at an agitation speed of 250 rev min–1, corresponding to Re = 4100, or 500 rev min–1, corresponding to Re = 8200), with a diameter of 6 cm, a blade length of 2.5 cm and blade width of 1 cm, was used in the deposition experiments. Note that the Reynolds number, Re, for the cold finger was calculated for the case of flow across a cylinder, as follows: 𝑅𝑒 = 𝑉𝐷𝑜𝜌 𝜇 , where 𝑉 = rotational speed of agitator × r, where r is the center-to-center distance between the agitator and cold finger, and Do is the outside diameter of cold finger. Other details and dimensions are shown in Figure 2. The relative positions of the cold finger and stirrer were kept the same for all experiments. The coolant from another refrigerated bath, maintained at a constant temperature, was circulated through the cold finger. The temperatures (Th, Twb, Tci, and Tco) were recorded using calibrated T-type thermocouples.


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Figure 2: The stirrer and the cold finger placements in the aluminum container.

3.3 Procedure and design of experiments for the ice deposition experiments The coolant from the refrigerated bath (set at the desired temperature of Tc = Tf –4°C or Tc =Tf –7°C) was pumped through the cold finger at a constant volumetric flow rate of 2.8 mL s– 1. The

cold finger was inserted into the container to commence the deposition process. After a

pre-determined time interval (between 30 s and 8h), the deposition experiment was terminated by taking out the cold finger from the container. A Sartorius BP210S balance was used to obtain the ice mass to a precision of ±0.1 mg. Table 1 summarizes the conditions for ice-deposition experiments.

Table 1: Conditions of experiments for the ice deposition experiments. 14 ACS Paragon Plus Environment

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variable

number of levels

values of each variable

water temperature

4

Tf +3 °C, Tf +2 °C, Tf +1 °C, Tf +0.7 °C

coolant temperature

2

Tf – 4 °C, Tf −7 °C

impeller speed (and Reynolds number)

2

250 rev min–1 (Re = 4100), 500 rev min–1 (Re = 8200)

deposition time

11

30 s, 1 min, 2 min, 5 min, 10 min, 20 min, 30 min, 1 h, 2 h, 4 h, 8 h

3.4 Procedure and design of experiments for the wax deposition experiments The container placed in the cooling bath was filled with the wax–solvent mixture. The waxy mixture was heated to a temperature of 52 °C and stirred continuously at 250 rev min–1 (Re = 1400) for 1 h. The water from the refrigerated bath (set at Tc = WAT–12 °C) was pumped through the cold finger at a constant volumetric flow rate of 6.7 mL s–1. A uniform cooling rate (0.4 °C min–1) was used to lower the temperature of the wax–solvent mixture. The cold finger was inserted into the container to start the deposition process. After a pre-determined time interval, the deposition experiment was terminated by taking out the cold finger from the container. The deposit mass was obtained to a precision of ±0.1 mg. It should be noted that each deposition experiment was conducted using a fresh batch of the waxy mixture. Table 2 summarizes the conditions for wax-deposition experiments.

Table 2: Conditions of experiments for the wax deposition experiments. variable

number of levels

values of each variable

wax concentration, xi

1

10 mass%

wax–solvent mixture temperature

4

WAT+6°C, WAT+4°C, WAT+2°C, WAT


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coolant temperature

1

WAT−12°C

impeller speed (and Reynolds number)

1

250 rev min–1 (Re = 1400)

experiment time

17

2 s, 5 s, 10 s, 20 s, 30 s, 10 min, 20 min, 30 min, 1 h, 2 h, 4 h, 8 h, 12 h, 18 h, 24 h, 36 h, 48 h

4. Results and discussion In this section, the data from ice deposition and the wax deposition experiments are presented as a function of time and the liquid phase temperature. These are compared to the predictions obtained from the Stefan moving boundary problem formulation.

4.1 Data and predictions for ice deposition from water Experiments to investigate the effects of liquid temperature (Th), the coolant temperature (Tc) and the agitation speed on the amount of ice deposition over different deposition times were performed. The experimental results are compared with predictions from the Stefan moving boundary problem approach for the phase transformation of pure substance, given by eqs 5 and 6 in addition to the initial and the boundary conditions given by eqs 10a–10d. The average hh for each experiment was estimated by using the steady-state deposit mass results (i.e., at 2 h, 4 h and 8 h) from eq 4. For these experiments, the estimated hh ranged from 3500 W m–2 K–1 at Re = 4100 to 6500 W m–2 K–1 at Re = 8200. The estimated hh was used in the transient model and the predictions are compared with experimental results in Figures 3 to 5. Two agitation speeds of 250 (Re = 4100) and 500 rev min–1 (Re = 8200), with two coolant temperatures of Tf–4 °C and Tf–7 °C, at 4 water temperatures (from Tf+3 °C to Tf+0.7 °C), and 11 deposition times (30 s to 8 16 ACS Paragon Plus Environment

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h) were used in the ice deposition experiments. Table 3 summarizes the results these experiments in terms of the deposit thickness (δ) and deposit mass per unit area (Ω). Figure 3 compares the experimental results and the predictions from the transient model with respect to deposition time at various liquid temperatures. A decrease in Th yielded an increase in the deposit mass, with the highest solid mass at Tf +0.7 °C and the lowest at Tf +3 °C. At each liquid temperature, the deposit mass increased very rapidly during the first few minutes of each experiment. As shown in Table 3, about 10% and 65% of the final deposit mass were achieved within the initial 30 s and 30 min, respectively. The solid mass reached steady-state after about 2 h. In Figure 3, a satisfactory agreement is observed between the predictions and the experimental results.

Figure 3: Data and predictions for the variation of the ice deposit mass with time at the indicated experimental liquid temperatures (symbols represent experimental results and smooth curves represent model predictions). 17 ACS Paragon Plus Environment

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Table 3: Experimental ice deposit thickness and ice deposit mass per unit area at different liquid water temperatures. Th = Tf +0.7 ˚C deposition

Th = Tf +1 ˚C

Th = Tf +2 ˚C

Th = Tf +3 ˚C

δ

Ω

δ

Ω

δ

Ω

δ

Ω

(mm)

(kg m-2)

(mm)

(kg m-2)

(mm)

(kg m-2)

(mm)

(kg m-2)

30 s

0.69

0.74

0.54

0.57

0.43

0.45

0.31

0.32

1 min

0.93

1.02

0.81

0.88

0.67

0.72

0.39

0.41

2 min

1.38

1.58

1.14

1.27

0.96

1.05

0.59

0.63

5 min

2.10

2.56

1.72

2.02

1.32

1.50

0.67

0.72

10 min

3.05

4.02

2.39

2.98

1.47

1.69

0.85

0.93

20 min

3.86

5.42

3.12

4.14

1.73

2.04

0.95

1.05

30 min

4.39

6.41

3.47

4.74

1.85

2.21

1.04

1.15

1h

5.17

7.98

3.88

5.47

1.94

2.33

1.14

1.27

2h

5.79

9.32

4.07

5.81

2.07

2.52

1.22

1.38

4h

6.11

10.0

4.03

5.73

2.05

2.49

1.28

1.45

8h

6.10

10.0

4.06

5.80

2.07

2.52

1.29

1.47

time

The ice deposition experiments were undertaken at two coolant temperatures of Tc =Tf−7 °C and Tc =Tf−4 °C, at a constant water temperature of Th = Tf+0.7 °C, and an agitation speed of 250 rev min–1 (Re = 4100). Table 4 summarizes the results of these experiments in terms of the deposit thickness (δ) and deposit mass per unit area (Ω). Figure 4 compares the experimental results and the predictions with respect to the deposition time at two coolant temperatures. As Tc was decreased, the ice deposit mass increased. The predictions from the transient model, in Figure 4, are in agreement with the experimental results.


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Figure 4: Data and predictions for the variation of the ice deposit with time at the indicated experimental coolant temperatures (symbols represent experimental results and smooth curves represent model predictions).

Table 4: Experimental ice deposit thickness and ice deposit mass per unit area at different coolant temperatures. Tc = Tf – 7 ˚C

Tc = Tf – 4 ˚C

deposition time

δ

Ω

δ

Ω

(mm)

(kg m-2)

(mm)

(kg m-2)

30 s

0.69

0.74

0.40

0.42

1 min

0.93

1.02

0.71

0.76

2 min

1.38

1.58

0.97

1.06

5 min

2.10

2.56

1.59

1.85

10 min

3.05

4.02

2.14

2.62


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20 min

3.86

5.42

2.58

3.28

30 min

4.39

6.41

2.74

3.53

1h

5.17

7.98

2.97

3.90

2h

5.79

9.32

3.18

4.25

4h

6.11

10.03

3.25

4.35

8h

6.10

10.02

3.23

4.32

Two agitation speeds of 250 rev min–1 (Re = 4100) and 500 rev min–1 (Re = 8200), a constant water temperature of Th = Tf+0.7°C and a constant coolant temperature of Tc = Tf −7°C were used in the ice deposition experiments. Table 5 summarizes the results of these experiments. Figure 5 compares the experimental results with the predictions from the transient model for ice deposition with respect to deposition time at the two agitation speeds. A higher agitation speed (Re) resulted in a lower deposit mass, which is explained as follows. An increase in the Re would result in a higher hh, which would lead to a lower thermal conductive resistance of the deposit and a lower deposit mass or thickness34. In Figure 5, the predictions are in agreement with the experimental results.


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Figure 5: Data and predictions for the variation of the ice deposit mass with time at the indicated agitation speeds (symbols represent experimental results and smooth curves represent model predictions).

Table 5: Experimental ice deposit thickness and ice deposit mass per unit area at different agitation speeds. 250 rev min–1

500 rev min–1

deposition

δ

Ω

δ

Ω

time

(mm)

(kg m-2)

(mm)

(kg m-2)

30 s

0.69

0.74

0.58

0.61

1 min

0.93

1.02

0.78

0.84

2 min

1.38

1.58

1.11

1.23

5 min

2.10

2.56

1.70

2.00

10 min

3.05

4.02

1.99

2.41


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20 min

3.86

5.42

2.30

2.86

30 min

4.39

6.41

2.70

3.46

1h

5.17

7.98

2.84

3.69

2h

5.79

9.32

2.95

3.86

4h

6.11

10.03

3.01

3.95

8h

6.10

10.02

3.00

3.95

It is emphasized that the Stefan moving boundary problem modeling framework, used in this study, did not involve any “tuning” parameters, other than the physical, transport and thermodynamic properties of liquid and solid phases; all of these were measured or estimated from established methods. As mentioned previously, an average hh was estimated from the three steady-state deposit mass results of each deposition experiment, which was used to predict the deposit thickness as a function of time. The effect of hh on the variation of the predicted deposit mass with time was investigated using a sensitivity analysis. The transient calculations were repeated with hh±5% (i.e., with 1.05 hh and 0.95 hh) at Th = Tf+1.0 C. The results of this sensitivity analysis are shown in Figure 6, where the experimental data are seen to be within the hh±5% region. The results in Figure 6 also validate the relationship given by eq 2; that is, under steady state, an increase or decrease in Rh (as a result of a decrease or an increase in hh, respectively) corresponds with an increase or decrease in Rd (resulting an increase or a decrease in the deposit mass, respectively) to ensure the equality of the heat transfer rates35.


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Figure 6: Sensitivity analysis for hh at Th = Tf+1.0C (symbols represent experimental results and smooth curves represent model predictions).

4.2 Data and predictions for wax deposition from wax–solvent mixtures The effects of wax–solvent mixture temperature (Th), the coolant temperature (Tc) and the agitation speed on the amount of wax deposit are investigated in this section. It is pointed out that several previous experimental investigations in our laboratory have investigated the result of variation of Th and Tc on solid deposition from wax–solvent mixtures3, 27, 29, 31, 32. Similarly, the variation in shear stress on solid deposition has been investigated either at different flow rates (in the case of flow loop) or at different agitation speeds (in the case of cold finger)3, 29, 31. Hence, the deposition experiments were not repeated in this study at different Tc and agitation speeds.


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The deposition experiments were performed with the 10 mass% wax–solvent mixture at 4 mixture temperatures, between WAT+6 C and WAT. A constant coolant temperature of Tc = WAT−12 °C was used in all experiments. Seventeen deposition times, varying between 2s and 48 h were tested in the wax deposition experiments. The mass of wax–solvent mixture in the container was much larger than the deposit mass; hence, it was assumed that a relatively small amount of solid depositing on the cold finger did not cause a significant difference in the composition of the liquid phase. Table 6 summarizes the results of these experiments, in terms of the deposit thickness (δ) and deposit mass per unit area (Ω). It should be noted that eq 14 was utilized to obtain the wax deposit mass per unit area. Figure 7 presents the experimental results along with the predictions based on the Stefan problem approach, given by eqs 7–9 in addition to the initial and boundary conditions given by eqs 11a–11d. Figure 7 shows that a higher deposit mass is achieved with lowering the mixture temperature. The deposit mass reaches a high value at WAT (which actually represents a maximum before entering the cold flow regime, as shown in previous studies21, 30). It could be seen that a thermal steady state was achieved within 24 h of each experiment. The average hh for each experiment was estimated by using the steady-state deposit mass results (i.e., at 24 h, 36 h and 48 h) from eq 4. The estimated hh values varied between 120 and 440 W m–2 K–1. These estimated hh values were subsequently used in the transient model to predict the changes in the deposit mass per unit area with time, shown in Figure 7. Figure 7 shows satisfactory agreement between the predictions from the transient model and the experimental results.


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Figure 7: Data and predictions for the variations in the wax deposit mass with deposition time at the indicated experimental wax–solvent mixture temperatures (symbols represent experimental results and smooth curves represent model predictions).

Table 6: Experimental wax deposit thickness and wax deposit mass per unit area. Th = WAT+6 ºC deposition time

δ (mm)

Ω (kg

m-2)

Th = WAT+4 ºC δ (mm)

Th = WAT+2 ºC

Ω (kg

m-2)

δ (mm)

Ω (kg

m-2)

Th= WAT δ

Ω

(mm)

(kg m-2)

2s

0.22

0.17

0.48

0.39

0.59

0.48

0.83

0.70

5s

0.34

0.27

0.65

0.53

0.69

0.57

1.19

1.03

10 s

0.48

0.39

0.76

0.63

0.78

0.65

1.61

1.45

20 s

0.53

0.43

0.83

0.69

0.86

0.72

1.91

1.77

30 s

0.56

0.45

0.87

0.73

0.91

0.76

2.55

2.49

5 min

0.92

0.78

1.07

0.91

1.20

1.04

4.05

4.45

20 min

0.97

0.82

1.17

1.02

1.34

1.18

4.67

5.36


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30 min

1.02

0.87

1.21

1.05

1.40

1.23

5.02

5.90

1h

1.13

0.98

1.42

1.26

1.53

1.36

5.53

6.73

2h

1.44

1.27

1.65

1.49

1.97

1.83

6.29

8.04

4h

1.70

1.54

2.02

1.88

2.38

2.29

6.92

9.19

8h

2.00

1.87

2.33

2.23

2.67

2.63

7.86

11.05

12 h

2.14

2.02

2.44

2.36

2.92

2.94

7.93

11.19

18 h

2.22

2.10

2.59

2.53

3.03

3.07

7.99

11.32

24 h

2.26

2.15

2.69

2.66

3.15

3.23

8.12

11.58

36 h

2.28

2.18

2.67

2.63

3.06

3.12

8.16

11.67

48 h

2.28

2.18

2.67

2.63

3.13

3.20

8.16

11.66

Previous studies on wax deposition have shown an increase in the deposit mass or thickness increases with a lowering of Tc 3, 31, 32, which is supported by the predictions from steady-state and transient models21, 23, 30, 33, 37. Similarly, previous studies using the cold finger apparatus have shown that a decrease in the agitation speed results in a higher amount of deposit mass29. A lower agitation speed results in a lower hh, or a higher convective thermal resistance (Rh), which signifies a higher deposit thermal resistance (Rd), and accordingly a higher deposit mass (Ω)34.

4.3 Comparison of the results between ice and wax deposition experiments As mentioned previously, the approximate 2 h deposition time required to attain a thermal steady-state the ice deposition experiments was much faster than about 24 h for the wax deposition experiments. This is attributed to much higher hh values for the ice deposition experiments, which were higher, by more than one order of magnitude at the same agitation speed, than for the wax deposition experiments. In addition, the properties of water and ice are


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different from those of liquid and solid paraffins and their mixtures. The thermal conductivity, density, physical properties and specific heat capacity of water (liquid) and ice (solid) along with the latent heat of fusion and freezing point temperature of water were obtained from NIST Chemistry WebBook40. The thermal conductivity of ice is about much higher than that of paraffins41. For example, the thermal diffusivity of ice at its freezing point of 0oC is about 1.1 × 10–6 m2 s–1 compared to about 0.1 × 10–6 m2 s–1 for n-pentacosane (C25) at its freezing point of 53.3oC41. Moreover, whereas the ice deposit was completely solid, the wax deposit comprised of solid and liquid phases. It is pointed out that all of these differences in the properties and characteristics of ice and wax deposits were included in obtaining the predictions from the moving boundary problem formulation. The data for the mass of ice deposit and wax deposit at steady state, are given in Tables 3 and 6, respectively. The variations in the steady state solid (ice or wax) deposit mass with a change in the temperature difference between Th and Td are plotted in Figure 8. Again, in Figure 8, the markers are for the experimental results and the smooth curves show the predictions. For the ice deposition experiments, as the water temperature was lowered, the mass of ice deposit increased, reaching a high value at (Th – Td) = 0.7 C. Note that ice deposition experiment at Th = Tf, could not be performed because pure water freezes at Th = Td = 0 C. For the wax deposition experiments, the mass of the wax deposit increased with lowering the wax–solvent mixture temperature, reaching a high value at (Th – Td) = 0 C. The two sets of experimental results and predictions in Figure 8 follow a similar pattern. In both cases, the deposit mass increases gradually as the liquid-phase temperature is lowered, but it increases more sharply as the liquid temperature becomes closer to the WAT (for the waxy mixture) or the freezing point (for water). In Figure 8, the predicted masses of ice deposit and


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wax deposit agree with the experimental results. It was pointed out previously that the Stefan moving boundary problem framework is based entirely on heat-transfer considerations. Therefore, the wax deposition process in the hot flow regime (i.e., Th ≥ WAT) is also described completely using heat-transfer considerations; that is, without the need to invoke any other proposed mechanism for wax deposition.

Figure 8: Data and predictions for variation of the steady-state deposit mass per unit area (Ω) with (Th – Td) for wax deposition and ice deposition (symbols represent experimental results and smooth curves represent model predictions).


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5. Conclusions The time-dependent growth of solid (ice or wax) layer for ice deposition and wax deposition experiments, over a range of temperatures, was investigated using a cold finger setup. The ice deposition experiments indicated a higher deposit mass with a decrease in Th, a decrease in Tc, and a decrease in the agitation speed (Re). The trends were observed to be the same for the wax deposition case as well; that is, a higher deposit mass was achieved with a lowering of Th, a lowering of Tc, and a lowering of the agitation speed (Re). The solidification process was observed to be fast during the initial period in all of the experiments. The ice deposit thickness ceased to grow and reached steady-state in about 2 h; however, the process was slower for the wax deposition case. The heat-transfer and phase-transformation processes involved in ice deposition (i.e., freezing of water) and wax deposition were modeled based on the Stefan moving boundary problem approach. This unsteady-state mathematical modeling approach considered the deposition process, during the complete or partial freezing of the liquid phase at the interface, as a heat-transfer process. For both cases, a satisfactory agreement was observed between the predictions from the mathematical model and the experimental measurements. Unlike other approaches described in the literature for predicting the time-dependent wax deposition process, the moving boundary problem modeling approach described and used in this study did not involve any adjustable or empirical “tuning” parameter(s). Overall, the experimental results obtained in this study showed many similarities between the ice deposition and wax deposition processes. For both cases, predictions from the moving boundary problem framework, which is based solely on heat-transfer analysis, were shown to match the experimental results. The only parameters in the mathematical models were the solidphase thermal conductivity (kd) and the heat transfer coefficient (hh), both of which could be 29 ACS Paragon Plus Environment

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measured or estimated independently. Finally, this study successfully demonstrated an analogy between ice deposition from pure water and wax deposition from wax–solvent mixtures. Since, being a single-component system, the ice deposition from pure water is an entirely thermallydriven process. The wax deposition process was described analogously by heat-transfer considerations alone.

Acknowledgments Financial support from the Department of Chemical and Petroleum Engineering, University of Calgary, and the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. We thank Ms. Samira Haj-Shafiei for her assistance.

Nomenclature Ai = Inside surface area of cold finger (m2) Aw = Surface area of cold finger (m2) Cc = Specific heat capacity of coolant (J kg–1 K–1) Do = Outside diameter of cold finger (m) 𝑓𝑠 = Solid phase mass fraction g = Mole fraction ∆𝐻𝑚 = Enthalpy of fusion (J kmol–1) hc = Heat transfer coefficient of coolant (W m–2 K–1) hh = Heat transfer coefficient of liquid (W m–2 K–1) kd = Thermal conductivity of deposit (W m–1 K–1) km = Thermal conductivity of cold finger tube (W m–1 K–1)


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L = Length of cold finger deposition surface (m) 𝑚𝑐 = Mass flow rate of coolant (kg s–1) q = Rate of heat transfer (W) r = center-to-center distance between the agitator and cold finger (m) Re = Reynolds number Rc = Thermal resistance of coolant (K W–1) Rd = Thermal resistance of deposit layer (K W–1) Rh = Thermal resistance of wax solution (K W–1) Rm = Thermal resistance of cold finger tube (K W–1) ri = Inside tube radius (m) ro = Outside tube radius (m) rw = Tube wall radius (m) s = Radial location of liquid–solid interface (m) t = Deposition time (s) Tc = Average temperature of coolant (C) Tci = Inlet temperature of coolant (C) Tco = Outlet temperature of coolant (C) Td = Average temperature at liquid–solid interface (C) Tf = Freezing point temperature (C) Th = Average temperature of liquid (C) Tm = Melting point temperature (C) TL = Liquidus temperature (C) TE = Solidus or eutectic temperature (C) 31 ACS Paragon Plus Environment

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Tw = Average wall temperature of cold finger (C) Twi = Temperature at the inside wall of cold finger (C) Two = Temperature at the outside wall of cold finger (C) T = Average deposit temperature (C) Ui = Overall heat transfer coefficient (W m–2 K–1) V = velocity (m s–1) Vs = Solid phase volume fraction in deposit Vl = Liquid-phase volume fraction in deposit xd = Deposit thickness (m) w = Mass fraction (mass%) Greek Letters  = Thermal diffusivity (m2 s–1)  = Deposit thickness (m)  = Latent heat of fusion (J kg–1) µ = viscosity (Pa s) ρ = density (kg m–3)  = Mass of deposit per unit area (kg m–2) 𝜙 = Superficial volume fraction Acronyms CPT = Cloud point temperature (C) PPT = Pour point temperature (C) WAT = Wax appearance temperature (C)


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