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A Computational Fluid Dynamic (CFD) model is presented to analyze the flow dynamics of a two phase incompressible flow in the application sectors of different industries. The Finite Element method (FEM) which is based on the Galerkin approximation, has been implemented for this two phase flow model. Generally, two-phase flows can occur in different forms like gas-liquid, liquid-liquid and solid-liquid forms. The Oil and Water two-phase flow is an important phenomenon in petroleum industry for crude oil production and transportation. In our study, a laminar flow of liquid-liquid phase is considered to simulate the flow dynamics where the liquid phases are water and oil. The COMSOL Multiphysics software is used to perform the simulation including velocity profile, volume fraction, shear rate, pressure distributions and interfacial thicknesses at different times. A typical circular tube domain with radius 0.05 m and length 8 m is assumed for our simulation.

The two-phase flows are found widely in the nature and in a whole range of industrial applications. The two-phase material is often called the mixture [

A variety of two-phase flows can occur depending on the combination of two-phases such as gas-liquid, liquid-solid, liquid-liquid and gas-solid. Generally, we see these two phase flows in transport of oil-gas mixtures in the pipeline system, air conditioning and refrigeration system, crystallization system, clay extraction, oil water mixture flow in pipelines and fuel combustion etc.

There are various studies which have been done on two phase flow. Aparallel streamline Upwind Petrov-Galerkin model using the Finite Element method for two-phase flow in a T-junction showed that the primary phase accumulates at the middle of the pipe and top of the wall while the secondary phase accumulates near the wall after the junction [

Nowadays CFD has become an essential tool in simulation studies. In our study, a two-phase flow model is solved by using the Finite Element approximation based on the Galerkin method. The two-phases are considered as oil and water respectively. The Finite Element method is a numerical technique to obtain an approximate solution to boundary value problems and the method has a distinct advantage that it allows naturally for handling complex arbitrary geometries so that it can be applied using irregular grid of various shapes. It also provides a set of functions which give the variation of differential equations between the grid points [

The oil-water flow is very common in the petroleum industries. Generally the oil phase is transported in a multiphase flow condition as water and oil are normally produced together. The presence of water has a significant effect during the transportation of oil. The complex interfacial structure of oil water flow makes it difficult to predict the hydrodynamics of the fluid flow [

In literature, we found different models for two-phase flows including the Petrov-Galerkin method, the projection level set method, the volume of fluid method, and the population balanced model.

In our study, we considered a FEM model for two-phase flow using the Galerkin method and an oil-water mixture flow is taken to simulate according to our model. To investigate the velocity profile, the volume fraction and the pressure distribution of the oil-water two-phase flow through a circular tube we applied boundary and initial condition in the inlet, outlet and wall of a typical computational domain.

In this section, a mathematical model will be developed to discuss the two-phase flow. A computational domain will also be constructed to simulate the fluid flow. A suitable mesh design is considered as the computational domain. The initial and boundary conditions are for the oil-water two-phase flow in a phase field platform.

In this study, we consider a two-phase incompressible Newtonian flow in which the phases are liquid and the flow mixture is laminar. Therefore, the governing equation of the problem becomes as follows along with Boundary Conditions (BC) and Initial conditions (IC) in the computational domain Ω ,

∇ ⋅ u = 0 (1)

ρ ( ∂ u ∂ t + u ⋅ ∇ u ) = − ∇ p + ∇ ⋅ ( μ ( ∇ u + ∇ u T ) ) + ρ g + F s t , (2)

where u denotes the velocity of the mixture, ρ and μ are its density and viscosity respectively, g is the gravity and F s t is the surface tension force. The phase field variable [

∂ φ ∂ t + u ⋅ ∇ φ = ∇ ⋅ γ ∇ G , (3)

where G is the chemical potential.

The following equation is used to track the interface in the Phase field method obtained from the Cahn-Hilliard equation [

∂ φ ∂ t + u ⋅ ∇ φ = ∇ ⋅ γ λ ε 2 ∇ ψ , (4)

ψ = − ∇ ⋅ ε 2 ∇ φ + ( φ 2 − 1 ) φ + ( ε 2 λ ) ∂ f ∂ φ , (5)

where φ is the dimensionless phase field variable, γ is the mobility, ε is a

controlling interface parameter, λ is the mixing energy density, the term ∂ f ∂ φ

denotes the φ derivative of external free energy and χ is the mobility tuning parameter. The density and viscosity of the mixture are the function of volume

fraction of water V w . The volume fraction of water and oil are V w = 1 + φ 2 and V o = 1 − φ 2 respectively. The density and dynamic viscosity of the two-phase

model are calculated over the interface according to

ρ = ρ o + ( ρ w − ρ o ) V w (6)

μ = μ o + ( μ w − μ o ) V w , (7)

where the subscript w and o are used for water and oil respectively. The surface tension force for the method is applied as a body force

F s t = G ∇ φ . (8)

The chemical potential G is given by the following equation related to the phase field variable and the interface controlling parameter.

G = λ [ − ∇ 2 φ + φ ( φ 2 − 1 ) ε 2 ] . (9)

As the mixture of oil and water is incompressible, the flow is assumed to be laminar two-phase flow where the volume fraction of water and oil are considered 0.91 and 0.09 respectively [

u = 0 (10)

T = σ ⋅ n = T ¯ , where σ = − p I + μ ( ∇ u + ∇ u T ) (11)

Atypical circular tube domain as in ^{3} and 2.518 m^{2} respectively. As the mesh design plays a significant role in accuracy of numerical results, we considered a suitable mesh design for our simulation with 81,143 elements and 619,748 degrees of freedom according to

The variational statement of a boundary value problem is the integral representation of the problem. We can obtain the variational statement by setting the total weighted residual error to zero. When the local residual error is multiplied by a weighting function and then integrating over the domain, the weighted residual error is obtained [

r ( x , t ) = ρ ( ∂ u ∂ t + u ⋅ ∇ u ) + ∇ p − ∇ ⋅ ( μ ( ∇ u + ∇ u T ) ) − ρ g − F s t (12)

According to weighted residual method [

( r , v ) = 0 ∀ v ∈ V and v = 0 on Γ u (13)

( ∇ ⋅ u , q ) = 0 , ∀ q ∈ Q (14)

u ( x , 0 ) = u 0 in Ω (15)

u = u ¯ in Γ u (16)

where V and Q are velocity and pressure spaces and the inner product is defined by

( a , b ) = ∫ Ω a ⋅ b d Ω (17)

The integral representation of (13) is presented as follows

∫ Ω ρ D u D t ⋅ v d Ω + ∫ Ω ( − p ∇ ⋅ v + μ ( ∇ u + ∇ u T ) ∇ v ) d Ω − ∫ Ω ρ g ⋅ v d Ω − ∫ Ω F s t ⋅ v d Ω = ∫ ∂ Ω T ¯ ⋅ v d s (18)

Since ∂ Ω = Γ u ∪ Γ t and u is specified on Γ u and v is chosen to be zero on Γ u .

The variational statement of the problem is,

This variational statement is also known as the weak formulation of a two-phase Newtonian incompressible flow in a computational domain Ω .

To find the Finite Element approximation, let V h ⊂ V be a N-Dimensional subspace of V with basis functions { ϕ 1 , ϕ 2 , ⋯ , ϕ N } . Approximating v and q in (19) by

v h = ∑ k = 1 N ϕ k v k and q = ∑ p = 1 M θ p q p

∑ k = 1 N { ( ρ D u D t , ϕ k ) − ( p , ∇ ⋅ ϕ k ) + ( μ ( ∇ u + ∇ u T ) , ∇ ϕ k ) − ( ρ g , ϕ k ) − ( F s t , ϕ k ) − b ( T ¯ , ϕ k ) } v k = 0 (20)

∑ p = 1 M ( ∇ ⋅ u , θ p ) q p = 0 (21)

which can be written as

( ρ ∂ u ∂ t , ϕ k ) + ( ρ u ⋅ ∇ u , ϕ k ) − ( p , ∇ ⋅ ϕ k ) + ( μ ( ∇ u + ∇ u T ) , ∇ ϕ k ) = ( ρ g , ϕ k ) − ( F s t , φ k ) − b ( T ¯ , ϕ k ) (22)

( ∇ ⋅ u , θ p ) = 0 (23)

Now approximating u and p respectively by

u h = ∑ l = 1 N ϕ l u l and p h = ∑ p = 1 M ω p p p

Therefore, we get from (22) and (23) that

∑ l = 1 N { ( ρ ϕ l , ϕ k ) u ˙ l + ( ρ ϕ l ⋅ ∇ ϕ l , ϕ k ) u l + ( μ ∇ ϕ l , ∇ ϕ k ) u l + ( μ ∇ ϕ l T , ∇ ϕ k ) u l T } − ∑ p = 1 M ( ω p , ∇ ⋅ ϕ k ) p p = ( ρ g , ϕ k ) + ( F s t , ϕ k ) + b ( T ¯ , ϕ k ) (24)

∑ k = 1 N ( ∇ ⋅ ϕ k , ω p ) u k = 0 (25)

which can be represented by

M U ˙ + A U − C P = F − C 1 T U 1 − C 2 T U 2 − C 3 T U 3 = 0

using M = ( m k l ) with m k l = ( ρ ϕ k , ϕ l )

D = ( D k l ) with D k l = ( ρ ϕ l ⋅ ∇ ϕ l , ϕ k )

K i j = K i j k l = ( μ ∇ ϕ k , ∇ ϕ l )

A = K i j + D

C = C i k p with C i k p = ( ω p , ∇ ⋅ ϕ k )

F = F i k with F i k = ( ρ g , ϕ k ) + ( F s t , ϕ k ) + b ( T ¯ , ϕ k )

k , l = ( 1 , 2 , ⋯ , N ) and i , j = 1 , 2 , 3

In our study, COMSOL Multiphysics version 4.2 has been used to simulate the oil-water two-phase flows. Properties of the fluid phases and parameters values are given in

Property | Symbol | Water phase | Oil phase |
---|---|---|---|

Density | 998.2 kg/m^{3} | 780 kg/m^{3} | |

Dynamic viscosity | 0.001003 Pa∙s | 0.00157 Pa∙s | |

Oil-water Interfacial Tension | 0.17 N/m at 20˚C |

The

Symbol | Quantity | Values |
---|---|---|

Inlet initial velocity | 0.024 m/s | |

Phi-derivative of external free energy | 0.01 J/m | |

Parameter controlling interface thickness | 0.01 m | |

Mobility tuning parameter | 1 m∙s/kg | |

Gravity | 9.8 m/s^{2} |

In case of fluid flow inside a tube domain the shear stress can be found on the wall of the domain. We investigated the shear rate distribution for the entire domain which is shown in

One of the important parameters used to characterize the two-phase flow is the volume fraction of different phases.

The

except for a small change around zero of the value of the phase field variable. The phases have started to separate (

A CFD model is proposed based on the Galerkin Finite Element approximation to simulate the two-phase flows. According to the Galerkin FEM, we obtained the variational statement of the fluid flow problem by setting the total weighted residual error to zero. A computational domain with 8 m length and 0.05 m radius respectively is constructed to simulate an oil-water two-phase flow. The velocity magnitude has been observed higher at the middle of the tube and the lowest at the boundary of the tube. The shear rate decreased with small fluctuation along the domain. The Oil phase volume fraction increased consequently the water phase decreased at the same ratio. From the inlet to the outlet, a linear pressure drop is observed. It is also notable that the fluids tend to separate into two distinct phases.

The authors gratefully acknowledge for the technical support to The Centre of excellence in Mathematics, Department of Mathematics, Mahidol University, Bangkok, Thailand.

Akter, F. and Deb, U.K. (2017) Computational Modeling and Dynamics of the Oil and Water Flow Using the Galerkin Approximation. American Journal of Computational Mathematics, 7, 58-69. https://doi.org/10.4236/ajcm.2017.71005