HYDRATE FORMATION EFFECTS ON SLUG FLOW HYDRODYNAMICS AND HEAT TRANSFER: WALL DEPOSITION VS. DISPERSION
FORMATION Carlos L. Bassania, [email protected] Fausto A. A. Barbutoa, [email protected] Amadeu K. Sumb, [email protected] Rigoberto E. M. Moralesa, [email protected] aMultiphase Flow Research Center (NUEM), Federal University of Technology – Paraná (UTFPR), Rua Deputado Heitor Alencar Furtado 5000, Bloco N, CEP 81280-340, Curitiba/PR, Brazil. bHydrates Energy Innovation Laboratory, Chemical and Biological Engineering Department, Colorado School of Mines, 1500 Illinois St., Golden, CO 8040, USA. Abstract. Pipe blockage due to gas hydrate formation is a main concern to the oil and gas industry due to the costs of production interruptions. Hydrate formation scenarios are usually found in offshore production pipelines, where oil and gas flow along the pipeline as a mixture that may also contain sand and brine. The high pressure conditions and the heat transfer with the external medium – the ocean – may create the necessary conditions for hydrate formation. Hydrates may form: (i) as a deposit on the pipe inner wall, where the temperature gradient is higher and the wall imperfections trigger the nucleation process; or (ii) in the gas-water interface, where a more effective contact between the phases occurs, forming a hydrate-in-liquid dispersion. This work gathers three years of study from NUEM – Multiphase Flow Research Center on slug flow modeling using a mechanistic approach and shows the main equations, results and discussions of: (i) slug flows with heat transfer; (ii) mass transfer and heat generation during hydrate formation; (iii) hydrate formation as a layer deposited on the pipe wall; and (iv) hydrate formation as a dispersion, thus creating a new flowing phase. A theoretical discussion about the possible mechanisms of transition between dispersion and wall deposition is also included in the article, focusing on: (i) whether the hydrate dispersion deposits and (ii) whether hydrate particles may detach from the wall to suspend in the liquid as a dispersion. Keywords: flow assurance, hydrates, slug flow, heat and mass transfer.
NOMENCLATURE Roman letters A Cross sectional area [m2] c Specific heat [J/(kg.K)] D Pipe inner diameter [m] d Hydrate particle diameter [m] dm/dt Mass variation rate [kg/s] freq Slug flow frequency [Hz] h Heat transfer coefficient [W/(m2.K)]
Hh Hydrate enthalpy of formation [J/kg]
j Phase superficial velocity [m/s] J Mixture superficial velocity [m/s] k Thermal conductivity [W/(m.K)] K Head loss coefficient [-] L Length [m] m Mass flow rate [kg/s] M Molar mass [kg/kmol] P Pressure [Pa] R Phase volumetric fraction [-] S Wetted perimeter [m] t Time [s] T Temperature [K]
eqHT Hydrate formation equilibrium temperature [K]
U Real velocity [m/s] VD Critical deposition velocity [m/s] Vt Terminal velocity [m/s]
z Pipe axial coordinate [m] Z Compressibility factor [-] Greek letters Pipe inclination [rad]
Hydrate layer thickness [m]
H Hydration number [-]
Thermal scooping factor [-] Viscosity [Pa.s]
Density [kg/m3]
Shear stress [Pa] Phase ( ; ;L G H )
Slug region ( ;B S )
Indexes ( )L Hydrate in water dispersion
*( ) Parameter after the consideration of a hydrate
layer deposited in the pipe inner wall B Bubble region eq Equilibrium ext External medium i Gas-water interface in Pipe inlet
C.L. Bassani, F.A.A. Barbuto, A.K. Sum and R.E.M. Morales Hydrate Formation Effects on Slug Flow Hydrodynamics and Heat Transfer: Wall Deposition vs. Dispersion Formation
G Gas H Hydrate L Liquid m Mixture P Particle ov. Overall
S Slug region sub Subcooling T Unit cell translation U Unit cell W Wall
1. INTRODUCTION
Hydrates are crystals formed by gas molecules trapped into cages formed by hydrogen-bonded water molecules (Sloan and Koh, 2008). Whenever natural gas and water mixtures flow through pipelines at high pressures, hydrates may form: (i) on the pipe wall, where the temperature gradient is higher and the wall imperfections trigger the nucleation process; or (ii) on the gas-water interface, where a more effective contact between the phases occurs (Sloan et al., 2011). In either case, the hydrate formation may change the flow hydrodynamics and heat transfer: (i) by reducing the flow cross sectional area and creating an insulating hydrate layer, in the first case; or (ii) by precipitating solid particles and forming a dispersion, in the second case (Zerpa et al., 2013). In both situations, pipe blockage may occur if hydrate formation is out of control, causing either a partial or a complete pipe obstruction due to: (i) the reduction of the pipe cross sectional area by the formation of a hydrate layer or (ii) the formation of a massive plug due to particle agglomeration (Sloan et al., 2011).
Pipe blockage due to hydrate formation is one of the main flow assurance challenges faced by the oil and gas production companies, especially in offshore production operations in deep and cold waters (Cardoso et al., 2015). Offshore production is mainly composed by oil, gas, water (brine) and small solid particles such as sand, and the mixture is often considered to be flowing in the slug flow pattern due to the range of volumetric flow rate of the phases. Slug flows are characterized by the intermittent passage of elongated bubbles, flowing over a liquid film; and liquid slug bodies, which may contain dispersed bubbles in its interior (Shoham, 2006). Together, these structures form the so-called unit cell.
Several approaches have been used to model slug flows, namely: steady-state mechanistic models (Bassani et al., 2016b; Cook and Behnia, 2000; Shoham, 2006; Taitel and Barnea, 1990), Eulerian transient drift flux models (Danielson, 2011; Zerpa et al., 2013), Eulerian transient two-fluid models (Issa and Kempf, 2003; Simões et al., 2014), Lagrangian transient slug tracking models (Medina et al., 2015; Nydal and Banerjee, 1996; Taitel and Barnea, 1998) and hybrid models (Kjeldby et al., 2013). Fewer studies considering heat transfer exist (Bassani et al., 2016b; Medina et al., 2015; Simões et al., 2014; Zerpa et al., 2013), but the ones considering mass transfer during hydrate formation are still in development (Zerpa et al., 2013).
This work gathers studies on slug flow modeling using a steady-state mechanistic approach and shows the main equations, results and discussions about this topic over the last three years of work in NUEM – Multiphase Flow Research Center. The purpose is to understand the mathematical modeling and the main phenomena involved in: (i) slug flow hydrodynamics and heat transfer, (ii) mass transfer and heat generation when hydrates form; (iii) hydrate formation as a layer deposited on the pipe inner wall; and (iv) hydrate formation as a dispersion, which represents a new flowing phase. At the end of the article, a theoretical discussion based on literature from other groups is added to understand the main mechanisms of transition between dispersion and wall deposition, that is: (i) whether the hydrate dispersion deposits and (ii) whether hydrate particles detach from the wall. 2. SLUG FLOW HYDRODYNAMICS AND HEAT TRANSFER
The combined multiphase flow of the multicomponent mixtures from petroleum offshore production – that is, oil, gas, water (brine), sand and, circumstantially, hydrates – will be modeled as a combined liquid-gas two-phase slug flow. Figure 1a presents a depiction of a horizontal slug flow, which is characterized by the intermittent succession of unit cells constituted of two regions, herein designated by : the elongated bubble (B) and the slug (S). Each region may
contain both phases – that is, the gas (G) and the liquid (L), designated by . Each phase inside each region is called a
unit cell structure , being: the gas in the elongated bubble (GB), the liquid film that flows underneath the elongated
bubble (LB), the liquid in the slug body (LS) and the gas bubbles dispersed in the slug body (GS). Figure 1b presents the problem characterization for hydrodynamic and heat transfer calculations, as proposed by
Bassani et al. (2016b). The pipeline is divided in nodes spaced by z . The superficial velocities of the phases ( )j and
the mixture pressure (P) and temperature (T) are assumed to be known at the pipe inlet. The pipeline is cooled externally by an infinite medium of nearly constant temperature and heat transfer coefficient – which represents the ocean at offshore production scenarios. The pipeline is assumed to be horizontal.
Knowing the superficial velocities of the phases, the slug unit cell can be characterized by the Taitel and Barnea’s (1990) approach. Unit cell characterization is related to the volumetric fraction of the phases in each region of the unit cell, as well as the region lengths. The unit cell geometry is paramount for predicting the real velocities of the phases in
C.L. Bassani, F.A.A. Barbuto, A.K. Sum and R.E.M. Morales Hydrate Formation Effects on Slug Flow Hydrodynamics and Heat Transfer: Wall Deposition vs. Dispersion Formation
each region of the unit cell by the mass balance. The real velocities are used to find: (i) the shear stresses of the structures for estimating pressure in the next node by the momentum balance, eq. (1) (Bassani et al., 2016b; Cook and Behnia, 2000; Taitel and Barnea, 1990); and (ii) the heat transfer coefficient of the structures for estimating the mixture temperature in the next node by the energy balance, eq. (2) (Bassani et al., 2016b). The main phenomena involved in each conservation equation are indicated inside the equations.
2
(n 1) 2LB LB GB GB i i LB TLS LS S B
LnU U U
friction in the friction in the elongated buble region head loss in theslug region elongated bubble rear
S S S U US L LP P K z
A L A L L
(1)
1 exp
heat transfer heat transfer scoopingin the film in the slug phenomenon
LB LB B LS LS S z LW Wn n
L LU T U L
liquid heat capacity
h S L h S L m cT T T T z
R AU L c
(2)
Figure 1 – (a) Unit cell with its respective regions and flow structures, b) problem characterization for slug flow
hydrodynamics and heat transfer calculations. being L the region lengths, A the cross sectional area occupied by each phase in each unit cell region, their
shear stresses, S their wetted perimeters, R their volumetric fractions, U their real velocities and h their heat
transfer coefficients. The indexes (n) and (n+1) refer to the nodes of the pipe. Details on the nomenclature are presented as a separated section in the beginning of this article. The wall temperature WT relates to the external one by means of
the overall heat transfer coefficient .ovh in each unit cell structure, which by its turn depends on the internal and
external convection and on the wall conduction (Bassani et al., 2016b):
Yet from Fig. 1b, the superficial velocities of the gas phase can be corrected due to pressure and temperature
variations along the pipeline. The liquid superficial velocity is considered constant when assuming an incompressible fluid hypothesis:
j ; jT ; P
L G
T ext ; hext
LB LS
Mass balance Velocities
Momentum balance P(n+1)
Energy balance T(n+1)
Gas superficial velocity correction
RLSRLB
Unit cell geometry
(b)(a)
Elongated bubble region Slug region
Gas in theelongated bubble
Liquid filmLiquid inthe slug
Gas bubbles dispersed in the slug
fST
rBT
;U UT P
f rB ST T
Unit cell
IV Journeys in Multiphase Flows (JEM 2017)
1 1
1 11
;n n n
G n G n L n L nn n n
Z P Tj j j j
Z P T
(4)
being Z the gas compressibility factor. Equations (1) to (4) demonstrate how to estimate pressure, temperature and superficial velocities node after node for a case of two-phase gas-liquid slug flow. Since these are the required input parameters, the routine can be repeated until the end of the pipeline is reached, following an upwind-wise logic, as shown in Fig. 1b. 3. HYDRATE FORMATION
Hydrate formation implies mainly two phenomena: (i) water and gas consumption and (ii) heat generation, since hydrate formation has an exothermic nature. Turner (2005) proposes that the gas mass consumption rate dmG/dt due to hydrate formation depends on the gas-water interfacial surface Ai and on the subcooling of the system subT , eq. (5).
This latter is defined as the difference between the hydrate formation equilibrium temperature and the mixture temperature, eq
sub HT T T (Zerpa et al., 2013). Hydrates are assumed to form when ,sub sub critT T , where ,sub critT
is the critical subcooling that activates the first particles nucleation, here assumed as , 3.6sub critT K (Matthews et al.,
2000). The water consumption and the hydrate formation rates, eq. (6), are calculated with respect to the molar masses of the phases M and the hydration number H , i.e., an averaged stoichiometry between the amounts of water and gas
to form the hydrate (Sloan and Koh, 2008).
21 expG
i sub
dm kk A T
dt T
(5)
; 1G GL L H HH H
G G
dm dmdm M dm M
dt M dt dt M dt (6)
where k1 and k2 are experimental constants. The liquid and gas superficial velocities are recalculated, node by node, due to the mass transfer when hydrates form (Bassani et al., 2016a). The gas superficial velocity must also take the pressure and temperature variations into account.
1 1
1 11
1 1;
n n n G LG n G n L n L n
G U L Un n n
Z P T dm dmz zj j j j
Z P T A dt L A dt L
(7)
From the energy balance standpoint, hydrate formation generates heat. The heat generated can be expressed by the
hydrate enthalpy of formation, in [kJ/kg of gas] (Sloan and Koh, 2008), eq. (8). Therefore, the mixture temperature estimation between two consecutive nodes should be calculated as shown in eqs. (9) and (10) (Bassani et al., 2016a).
GH H
dmQ h
dt (8)
1 expn n
p p nT T z
n n m
(9)
21
1
; ; ; ; expeqL L LU T U W H LB LB B LS LS S z L H i
n
km c AR U L n q r p qT rT q h S L h S L m c r h k A
T
(10)
Figure 2 shows the distribution of temperature (a) and of the mixture superficial velocity (b), evaluated for two
distinct cases: (i) when hydrates do not form (blue line) and (ii) when hydrates form (red line). The model evaluation is made for a methane-water mixture flowing along a 1.5-km length, 26-mm ID pipeline. The input data for the model evaluation is specified in Table 1. The following works were used to evaluate the properties of the phases: Haar et al. (1984) for water, Setzmann and Wagner (1991) for methane and Jung et al. (2010) for hydrates.
The temperature distribution (Fig. 2a) presents a decaying exponential trend, characteristic of the constant external temperature boundary condition. When hydrate formation is not taken into account (blue line), the mixture temperature has an asymptote at the external medium temperature (green line). However, the temperature distribution changes when hydrates are allowed to form. When the decreasing mixture temperature crosses the hydrate equilibrium temperature – indicated by the red dashed line and evaluated using CSMGem software (Ballard and Sloan, 2004) – the mixture
C.L. Bassani, F.A.A. Barbuto, A.K. Sum and R.E.M. Morales Hydrate Formation Effects on Slug Flow Hydrodynamics and Heat Transfer: Wall Deposition vs. Dispersion Formation
becomes metastable and hydrate formation may occur at any moment. Here, hydrate formation is assumed to form right after the critical subcooling is reached. This subcooling yields the necessary energy for triggering the nucleation of the first hydrate particles. Energy is released when hydrate forms,, thus the mixture temperature increases. The mixture reheats towards the equilibrium temperature, but never to reach it. Actually, a nearly constant subcooling is reached after the hydrate nucleation. This subcooling is the result of the competition between: (i) the energy released by the exothermic nature of hydrate formation and (ii) the heat exchange between the mixture and the external medium.
Figure 2b shows the distribution of the mixture superficial velocity. Since the liquid (water) phase is assumed to be incompressible, variations in the mixture superficial velocity are related to gas expansion/contraction or to the consumption/formation rates of the phases. When hydrates are not allowed to form (blue line), the gas: (i) contracts due to temperature drop, thus decreasing the gas (and mixture) superficial velocity; and (ii) expands due to pressure drop, thus increasing the gas (and mixture) superficial velocity. The first mechanism prevails at the pipeline inlet section, where the temperature gradient between the mixture and the external medium is the highest. Nearly 300 m downstream, the mixture superficial velocity presents a minimum, indicating that both mechanisms cancel out themselves. After this point, the pressure drop mechanism prevails and the mixture velocity increases. When mass transfer due to hydrate formation is taken into account (red line), the mixture undergoes a slowdown due to gas consumption (a phase with high specific volume) to form the hydrate (a phase with low specific volume). That is, a mixture volume contraction takes place, herein related to the mixture slowdown since the system is open (there would be a pressure drop related to this volume decrease if the system were closed).
Table 1. Input parameters for model evaluation. Pipe length / ID / width 1.5 km / 26 mm / 1 mm Pipe inclination Horizontal Pipe conductivity 30 W/(m·K) Gas superficial velocity 1 m/s Liquid superficial velocity 1 m/s Fluids CH4 / H2O Pressure at the inlet 10 MPa Temperature at the inlet 298 K (25oC) External medium temperature 277 K (4 oC) External medium heat transfer coefficient 100 W/(m2·K)
Figure 2. Distribution along the pipeline of: (a) mixture temperature and (b) mixture superficial velocity, comparing
cases with and without hydrate formation.
(a) (b)
Figure 3. Two possible cases of hydrate nucleation: (a) at the pipe inner wall as a deposit and (b) at the gas-water interface, as a homogeneous dispersion.
0 0.3 0.6 0.9 1.2 1.5Position [km]
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Without hydratesWith hydratesExternal mediumHydrate equilibrium
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z
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Hydratelayer
Elongated bubble region Slug region
Gas in theelongated bubble
Liquid film
Hydrates dispersed in the liquid film
Hydrates dispersed in the slug
Liquid in the slug Gas bubblesdispersed in the slug
IV Journeys in Multiphase Flows (JEM 2017)
The aforementioned phenomena involving hydrate formation – that is, mass transfer and heat generation – are independent on how hydrates form. However, hydrates may form: (i) as a deposit on the pipe wall, where the temperature gradient is higher and the tiny wall imperfections trigger the nucleation process; or (ii) as a dispersion in the gas-water interface, where a more effective contact between the phases exists. Figure 3 illustrates both cases. Next, their effects on slug flow hydrodynamics and heat transfer will be discussed. 3.1 Wall deposition
Hydrates may form on the pipe inner wall, where: (i) the temperature is lower than in the mixture bulk and (ii) the wall imperfections trigger the nucleation process. Figure 3a shows a hydrate layer t deposited symmetrically around
the pipe inner wall, which can be expressed in terms of the hydrate formation rate as:
1 1 Ht
Ht
dmdt
D dt
(11)
The open flow diameter reduces due to the hydrate wall deposition, eq. (12) (Bassani et al., 2017). The cross
sectional area of the flow is therefore reduced, eq. (13), thus accelerating the phases, eq. (14).
0 2t tD D (12)
2
00
21
t
tA AD
(13)
2
,00
21
t
tj jD
(14)
being 0D and 0A the pipe inner diameter and cross sectional areas when no hydrate layer is deposited, with related
superficial velocities of the phases ,0j . Equations (12) to (14) will affect the pressure gradient, eq. (1), in the following
manners (Bassani et al., 2017): (i) The friction terms are inversely proportional to the cross sectional area reduction, thus increasing pressure
drop. (ii) The shear stresses of the unit cell structures are directly proportional to the increase of the superficial
velocities, thus increasing pressure drop. (iii) The wetted perimeters of the unit cell structures are directly proportional to the diameter reduction, thus
reducing the pressure gradient. Figure 4 presents the model evaluation for three different constant thicknesses of hydrate layer deposited on the
wall, from 1 to 3 mm. The other input parameters for the model evaluation follow Table 1. The wall deposition is assumed as already formed – that is, mass transfer and heat generation have already ceased. A case with no hydrate layer is also plotted for comparison.
Figure 4 – Distribution along the pipeline of: (a) mixture superficial velocity and (b) pressure, for three different
hydrate wall deposition thicknesses and for a case without the hydrate layer.
0 0.3 0.6 0.9 1.2 1.5Position [km]
0
1
2
3
4
5
Mix
ture
supe
rfic
ialv
eloc
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[m/s
]
No hydrate layer = 1 mm = 2 mm = 3 mm
0 0.3 0.6 0.9 1.2 1.5Position [km]
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a]
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C.L. Bassani, F.A.A. Barbuto, A.K. Sum and R.E.M. Morales Hydrate Formation Effects on Slug Flow Hydrodynamics and Heat Transfer: Wall Deposition vs. Dispersion Formation
Figure 4a shows the mixture superficial velocity distribution. After the point where the hydrate layer is present – that is, after the critical subcooling is reached, near 700 m downstream the pipeline inlet – the mixture undergoes a sudden acceleration. This acceleration is from ~2 to ~3.5 m/s (~75% augmentation) when a 3 mm hydrate layer is present (2 23%)D , and does not take the mixture deceleration due to mass transfer into account, since hydrates
were assumed as already deposited in this simulation. This intensifies the shear stresses, with a consequent pressure drop increase, as shown in Fig. 4b. The pressure gradient – that is, the slope of Fig. 4b – increases from ~0.9 to ~3.1 kPa/m for the 3 mm layer, which represents ~244% of pressure drop augmentation. This illustrates how sensitive pressure drop is on the deposited hydrate layer thickness and the high risk related to flow stoppage. The cross sectional area restriction is probably the hydrate formation phenomenon that influences pressure drops the most.
From the heat transfer standpoint, the hydrate wall deposition represents a conduction thermal resistance (Bassani et al., 2017). Neglecting contact thermal resistances, the equivalent thermal conductivity between the pipe wall and the hydrate layer eq. (3)should be used (evaluation of the overall heat transfer coefficient), instead of the wall thermal conductivity Wk :
1
0 0 0ln / ln / ln /
2 2 2
extt t t t exteq
H W
D D D D D D D D Dk
k k
(15)
On the other hand, the mixture acceleration intensifies the heat exchange between the unit cell structures and the
wall. Since the Reynolds and Nusselt numbers are directly proportional to the velocity of the phases, then an increase in the mixture heat transfer coefficient with a consequent decrease in the internal convection thermal resistance is expected. Therefore, the impact on the mixture temperature distribution along the pipeline will be dictated by the competition between: (i) the increase in the equivalent conductive thermal resistance and (ii) the reduction on the internal convection thermal resistance.
Figure 5 – Distribution along the pipeline of: (a) mixture convection heat transfer coefficient, (b) overall heat transfer coefficient and (c) mixture temperature, for three different hydrate wall deposition thicknesses and the case where no
hydrate layer is deposited.
Figure 5 presents the heat transfer model behavior with hydrate wall deposition, following the input parameters from Table 1 and the three different hydrate layer thicknesses as presented before. As discussed, the mixture heat transfer coefficient (Fig. 5a) increases due to the mixture acceleration, from ~3500 to ~6000 W/m2K (~71.4% increase for 3mm ). However, the equivalent thermal conductivity of the pipe reduces so drastically due to the deposited insulating layer of hydrate, passing from 30 W/mK (wall conductivity) to ~0.9 W/mK (~97% reduction), that the overall heat transfer coefficient (Fig. 5b) decreases from ~105 to ~80 W/m2K (~24% reduction). Therefore, the temperature gradient along the pipeline is reduced when the hydrate layer is present. It is important to notice that a heat generation case is not shown in Fig. 5, since hydrate formation is assumed to have already ceased. 3.2 Dispersion formation
Hydrates may form in the water-gas interface, such as in the elongated bubble border with the liquid film, or in the interface between the dispersed bubbles and the liquid slug body. When this occurs, a third phase is introduced in the flow, as illustrated in Figure 3b, and two new unit cell structures appear: (i) the hydrates dispersed in the slug body and (ii) the hydrates dispersed in the film. For instance, the hydrate particles are assumed to be homogeneously dispersed
0 0.5 1 1.5Position [km]
0
25
50
75
100
125
h m[W
/(m
2 .K)]
(a)1 500
3 000
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h m[W
/(m
2 .K)]
(b)
ov.
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]
No hydrate layer = 1 mm = 2 mm = 3 mmHydrate equilibrium
(c)
IV Journeys in Multiphase Flows (JEM 2017)
along the unit cell, a valid hypothesis for small particles and low concentrations – that is, at the beginning of hydrate formation (Joshi, 2012). Thus, the three-phase solid-liquid-gas flow can be treated as a two-phase dispersion-gas flow (Bassani et al., 2016a). The dispersion properties vary along the pipeline as hydrates forms. The dispersion density is calculated in terms of the volumetric fraction of water and hydrate, whereas the dispersion viscosity is estimated by means of the Krieger and Dougherty (1959) correlation:
1.575
; 10.63
L H H LL L H L L
L L
R R R R
R R
(16)
being ( )L the notation used for the dispersion. The properties of the liquid phase in eqs. (1) and (9) should be
substituted by the dispersion one. A new superficial velocity is also introduced in the model – the hydrate one, estimated via eq. (17) (Bassani et al., 2016a). The dispersion superficial velocity is defined as the sum of the liquid and hydrate ones, as presented in eq. (18).
1
1 HH n H n
H U
dm zj j
A dt L
(17)
1
1 1 1L HL L H L n L n
L H U
dm dm zj j j j j
dt dt A L
(18)
Figure 6a presents the dispersion viscosity along the pipelines – evaluated using input parameters of Table 1. When
there is not hydrate formation (dashed line), the liquid viscosity increases along the pipeline due to temperature drop, from ~0.8 to ~1.5 mPa.s (~87% augmentation) along the 1.5 km-length of the pipeline. When hydrates form, the liquid viscosification due to the dispersion formation prevails, going from ~1.3 to 2.7 mPa.s (~107% augmentation) from z ≈ 700 m to the end of the pipeline. However, this viscosification is not sufficient to influence the pressure distribution along the pipeline, which remains almost the same, as shown in Fig. 6b. The dispersion influence on pressure drop will be accentuated for higher hydrate volumetric fractions, when the particles distribution may change from homogeneous to heterogeneous or even forming a moving or stationary bed. However, the model is not yet prepared to handle those cases, since a homogeneous dispersion hypothesis is used, valid for up to 13% of hydrate volumetric fraction (Joshi, 2012). A theoretical discussion upon the settling of hydrate slurries is made in section 4.
Figure 6 – (a) Liquid viscosification due to hydrate dispersion formation and (b) influence on pressure drop when a
homogeneous dispersion is present.
Figure 7a presents the liquid/mixture superficial velocity ratio along the pipeline. Without hydrate formation, jL/J changes due to gas expansion and contraction. At the inlet section of the pipe (up to z ≈ 300 m), temperature drop is predominant and the gas contracts, thus jL/J increases. Beyond this point, pressure drop dominates and the gas expands, thus jL/J decreases. When hydrates form, gas is consumed at a far higher volumetric rate than water, since G L .
Furthermore, H L and then the volume of water consumed nearly equals the volume of hydrate formed, thus Lj
remains constant. Therefore, the pseudo-liquid/mixture ratio Lj J increases. This increases the slug flow frequency, as
shown in Fig. 7b, since Lfreq j J (Gregory and Scott, 1969; Schulkes, 2011). The slug flow frequency – important
in the design of pipelines and phase separators, and especially in predicting corrosion – increases in approximately 16.3% along ~800 m after the hydrates onset point.
0 0.3 0.6 0.9 1.2 1.5Position [km]
7
7.5
8
8.5
9
9.5
10
Pre
ssur
e [M
Pa]
1.4 1.45 1.57.5
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0 0.3 0.6 0.9 1.2 1.5Position [km]
0
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cosi
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mP
a.s]
With hydratesWithout hydrates
(a) (b)
C.L. Bassani, F.A.A. Barbuto, A.K. Sum and R.E.M. Morales Hydrate Formation Effects on Slug Flow Hydrodynamics and Heat Transfer: Wall Deposition vs. Dispersion Formation
Figure 7 – (a) Liquid/mixture ratio of superficial velocities and (b) slug flow frequency evaluated as proposed by
Schulkes (2011). 4. TRANSITIONS BETWEEN DEPOSITION AND DISPERSION
The technical literature has yet to present a well-established criterion to predict when hydrates may form as a dispersion or as a wall deposition. Hydrates may also change from dispersion to deposition or vice-versa depending on the flow conditions. This section brings forward some literature works to theoretically discuss transitions: (i) from dispersion to deposition, due to decantation of the hydrate particles – also known as settling of slurries; and (ii) from deposition to dispersion, due to particles detachment from the wall. 4.1 Dispersion-to-deposition transition: settling of slurries
In slurry flows, the critical deposition velocity is defined as the minimum velocity the continuum phase should have to avoid particle decantation (Peker and Helvaci, 2007). That is, below this velocity, the slurry flow is susceptible to settling and to forming a stationary bed. The critical deposition velocity depends mainly on the particle diameter, its drag coefficient and the particle/fluid density ratio. Wilson and Judge (1976) propose:
0.5
2.0 0.3log 2 1P HD
D L
dV gD
DC
(19)
being VD the critical deposition velocity, dP the particle diameter, D the inner wall diameter, CD the drag coefficient of the particle and g the gravity acceleration. Equation (19) was modified from the original one by the use of the absolute value of ( 1)H L , since the original work focus on particles that are heavier than the carrying fluid, which is not the
case of a hydrate-in-water dispersion. The drag coefficient is dependent on the Reynolds number of the particle, and can be calculated as proposed by Khan and Richardson (1987) assuming spherical particles and 50.01 3 10PRe :
3.450.31 0.062.25 0.36 L t PD P P P
L
V dC Re Re with Re
(20)
being Vt the terminal velocity of the particle in the continuum fluid, which is expressed in its dimensionless form as (Zigrang and Sylvester, 1981):
2* * 1.5
*
114.51 1.83( ) 3.81V d
d
(21)
2* *
2; H
t PH L
gV V d d
g
(22)
If the liquid velocity in one of the regions is below the critical deposition velocity, then the particles are susceptible
to form a deposit:
LB DU V hydrate deposition in the film region (23)
LS DU V hydrate deposition in the slug region (24)
0 0.3 0.6 0.9 1.2 1.5Position [km]
2
2.2
2.4
2.6
Slug
flo
w f
requ
ency
[H
z]
0 0.3 0.6 0.9 1.2 1.5Position [km]
0.4
0.5
0.6
0.7j L
/J [
-]
With hydratesWithout hydrates
(a) (b)
IV Journeys in Multiphase Flows (JEM 2017)
Figure 8a shows the critical deposition velocity in terms of the hydrate particle diameter. The deposition velocity increases for bigger particles – that is, the liquid velocity should be higher for the slurry not to settle when its particles
are bigger. Figure 8b shows the mean velocity of the liquid and film regions. When the hydrates nucleate, 2LSU m s
1.7LBU m s , while the deposition velocity is 1.8DV m s (very small particles). That is, in the film region the
particles might form a stationary bed, but in the slug region they will most probably flow as a suspension in the liquid continuous phase.
As the hydrates continue to form and grow (and agglomerate) along the pipeline, the mixture decelerates due to gas and water consumption to form a solid (as already discussed in Fig. 2b). This flow deceleration also affects the local film and slug velocities, as shown in Fig. 8b. Therefore, the hydrate slurry may settle due to two mechanisms: (i) the deposition velocity increases due to the diameter increase of the particles (or of the agglomerated cluster of particles);
and (ii) the carrying fluid velocity ( LSU and LBU ) decrease due to gas and water consumption to form the hydrate. For
example, LSU ≈ 1.87 m/s at z ≈ 1 km, which implies that only particles smaller than dP ≈ 0.8 mm will flow as a
dispersion in the liquid phase.
Figure 8 – (a) Critical deposition velocity against the particle diameter and (b) distribution along the pipeline of the
mean liquid velocities at the slug and film. 4.2 Deposition-to-dispersion transition: particle detachment from the wall
The hydrates that are deposited on the wall are submitted to forces due to the mixture flow. Whenever those forces are strong enough to break the adhesion forces between the hydrate and the wall and to lift the particles, the said particles will flow with the mixture. The evaluation of those forces is, however, extremely dependent on: (i) the shape of the hydrate layer, (ii) the liquid velocity profile at each region of the unit cell and (iii) the adhesion between the hydrate and the wall material. Regrettably, these three topics are still open in literature. For instance, a particle with spherical shape will be assumed to be deposited in the pipe wall, which may be representative of the hydrate nucleation onset, but is not representative after the hydrate continues to grow, forming a layer as depicted in Fig. 3a.
Figure 9a shows a force balance over a spherical particle deposited on the pipe wall, following the study of Nicholas et al. (2009). The particle is attached to the wall due to an adhesion force:
*
, / , /a H W a H WF F R (25)
where the index H/W stands for the hydrate to wall interaction, R* is the harmonic mean radius between the two surfaces that are in contact ( *
PR d for a sphere in contact with a plane; Yang et al., 2004) and , /a H WF is the adhesion
force normalized by R*. Nicholas et al. (2009) suggest that a value of , / 0.002a H WF N m can be used as a conservative
estimation for the normalized adhesion force (valid for interactions between cyclopentane hydrates and carbon steel). The particle of Fig. 9a is also submitted to lift (FL), gravity (FG) and buoyancy (FB). Forces in the horizontal
direction will not be discussed in this article, since the interest here is to evaluate when the particles will detach and incorporate into the mixture and assuming that this movement will be entirely vertical. For a spherical particle submitted to laminar flow – that is, assuming the particle is inside the laminar sublayer – those forces can be estimated as:
2
22
1.615P
P
L LL L P L d
dL
dUF d U
dr
(Saffman apud Nicholas et al., 2009) (26)
0 0.3 0.6 0.9 1.2 1.5Position [km]
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
Liq
uid
mea
n ve
loci
ty [
m/s
]
0 1 2 3 4Particle diameter [mm]
1.8
1.85
1.9
1.95
2
Cri
tical
dep
osit
ion
velo
city
[m
/s]
(a) (b)
Slug (ULS)
Film (ULB)
Without hydrateformation
With hydrateformation
C.L. Bassani, F.A.A. Barbuto, A.K. Sum and R.E.M. Morales Hydrate Formation Effects on Slug Flow Hydrodynamics and Heat Transfer: Wall Deposition vs. Dispersion Formation
3
6B G L H PF F g d (27)
where
2PL d
U and /2P
L ddU dr are the liquid velocity and its derivative evaluated at the center of the particle. To
evaluate these parameters, it is necessary to know the liquid velocity profile ( )L rU , whose estimation under two-phase
slug flow is not easily obtained from an analytical standpoint. For instance, a Poiseuille flow is assumed in the slug region, being the maximum velocity at the centerline of the pipeline the double of the mean slug velocity, 2L,max LSU U
(Fox et al., 2011). Thence, the liquid velocity and its derivative can be evaluated as:
22
21 2 1
P
PL,max L LSL r d
drU U U U
R D
(28)
,
/2
2
P
L r L r LS PL max
d
dU dU U drU
dr R dr D (29)
Finally, the particle is going to detach and to lift towards the center of the pipeline whenever the following force
criterion is reached (Nicholas et al., 2009):
, /L B G a H WF F F F (30)
which is valid for particles attached in the bottom inner wall – that is, buoyancy and gravity have opposite directions.
Figure 9b shows the evaluation of those forces against the particle diameter deposited on the wall. The adhesion force grows linearly with the particle diameter, whereas lift, buoyancy and gravity forces grow with 3
Pd . The lift force
(green line) represents the major contribution for the particle detachment, since buoyancy and gravity forces (red line) almost cancel out themselves (hydrates have a density similar to that of the water, thus resulting in
373 )L H kg m . The purple line represents the sum of lift, buoyancy and gravity forces – that is, the RHS of
eq. (30). For dP ≈ 1 mm, this combined forces equal the adhesion force (blue line). After that point, the particle will probably detach from the wall.
Figure 9 – (a) Force balance over a spherical hydrate particle that nucleated on the pipe wall (simplified from Nicholas
et al., 2009). (b) Evaluation of adhesion, buoyancy, gravity and lift forces against the particle diameter. 4.3 Final considerations
Conclusions drawn in sections 4.1 and 4.2 may cause confusion, due to the following two statements: (i) Hydrate particles with dP > 0.8 mm flowing in the slug body are most likely to settle. (ii) Hydrate particles with dP > 1 mm are most likely to detach from the wall. It is important to notice that statement (ii) is based on the fact that the lift force over a particle surpasses the
adhesion force between the hydrate and the wall. However, this lift force is not necessarily sufficient to push the
BF
, /a H WF
GF
(a) (b)
LF (lift)
(buoyancy)
(weight)
(adhesion force)
0.5 1 1.5Particle diameter [mm]
0
2
4
6
Forc
e[
N]
Criterion for particledetachment from the wall
FL + FB - FG > Fa,H/W
IV Journeys in Multiphase Flows (JEM 2017)
particle towards the centerline of the flow (since this was not analyzed in the present work). Further analysis to understand this situation should be carried out in a future work, as well as to verify the validity of the correlations and assumptions adopted. The analysis made in this work should be faced as an initiative to theoretically understand the mechanisms of transition between deposition and dispersion of hydrate particles. 5. CONCLUSIONS
The present study used a two-phase gas-liquid slug flow mechanistic model to investigate hydrate formation effects on hydrodynamics and heat transfer. Hydrate formation is related to mass transfer and heat generation. The gas and water consumption that lead to the formation of a solid – the hydrate – represents a mixture slowdown. By its turn, heat generation is competitive to heat transfer with the external medium, and thus the system converges to a nearly constant subcooling when hydrates form.
Other effects of hydrate formation depend on how it nucleates. Two scenarios exist, wall deposition and dispersion formation. Wall deposition is related to a cross sectional area restriction, which tends to accelerate the mixture and increase pressure drop. By its turn, hydrate dispersion formation is related to a third phase in the flow. The three-phase flow can be assumed as a two-phase flow between the dispersion and the gas if the dispersion is treated as homogeneous. The dispersion formation is related: (i) to a viscosification of the liquid phase, which however did not influence pressure drop, probably due to the consideration of a homogeneous dispersion (for up to 13% of hydrate volumetric fraction); and (ii) to an increase in the pseudo-liquid/mixture superficial velocity ratio, which causes the slug frequency to increase.
Finally, transitions between dispersion and deposition were theoretically analyzed, being: (i) the settling of slurries, using the critical deposition velocity in terms of the hydrate particle; and (ii) the particles wall detachment, comparing the lift, buoyancy and gravity forces with the adhesion force between the hydrate and the wall. 6. ACKNOWLEDGEMENTS
The authors acknowledge the financial support of ANP and FINEP through the Human Resources Program for Oil and Gas Segment PRH-ANP (PRH 10-UTFPR), TE/CENPES/PETROBRAS (0050.0068718.11.9) and the National Council for Scientific and Technological Development (CNPq). AKS thanks PETROBRAS for sponsoring his sabbatical leave at UTFPR during the time part of this study was performed. 7. REFERENCES Ballard, A.L., Sloan, E.D., 2004. The next generation of hydrate prediction IV: A comparison of available hydrate
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8. RESPONSIBILITY NOTICE
The authors are the only responsible for the printed material included in this paper.
