An eulerian-eulerian approach for oil&gas separator design conference Paper

Flare knock-out drums: 
flare knock-out (KO) drums are one type of gas-liquid separators 
that are used specifically for separation of liquids carried with gas streams flowing to the 
flares in an Oil and Gas plants. 

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3 
4. 
Filters separators:
gas-liquid filter separator (usually called filter separator) is used in 
separation of liquid and solid particles from a gas stream. A gas filter has a higher separation 
efficiency than the centrifugal separator, but is uses filter elements that must periodically be 
replaced. 
In the present paper, the focus is on a horizontal, three-phase conventional separator (Figure 1). It 
is worth saying that the mathematical model employed to describe the separation process and the 
numerical algorithm to solve the associated equations are general and can be rigorously applied to 
other configurations and geometry. 
 
Figure 1: Oil and gas separator 
MATHEMATICAL MODEL
In the field of the multiphase CFD analysis, the main classes of numerical techniques are usually 
divided in the following three groups: 
1. Discrete phase elements methods: the discrete phase elements method assumes that the 
topology of the two-phase is dispersed. The two phases are therefore referred as the 
continuous and the dispersed phase with the former treated from an Eulerian perspective 
(phase properties regarded as a field and described in an absolute reference frame), while 
the latter under the Lagrangian framework (phase properties described in frame moving with 
the particle). This approach is considered appropriate when the phase-fraction of the 
dispersed phase does not overcome the limit of 6-10 %; 
2. Multi-fluid methods: under the multi-fluid formulation, each phase is described with a set of 
mass, momentum and energy equation. These equations are derived from the instant local 
formulation with an appropriate averaging process that introduces additional terms for which 
a modelling formulation is required. This method is particularly appropriate for treating flow 
with an arbitrary number of phases that span the entire value of phase fractions, face 
separation and inversion phenomena without a clear interface among them;
3. Interface resolving methods or direct numerical simulation (DNS): unlike the previous classes 
that do all require a priori information on the flow regime and knowledge on the size of 
bubbles and drops, these methods are able to reconstruct the position and the shape of the 
interface and capture arbitrary topological changes in the flow. Unfortunately, the present 
computational capabilities restricts these methods to a limited range of applications. 
Moreover, a realistic reconstruction of the interface coupled with a model for the superficial 
tension requires finer grid with respect to the other model adding additional computational 
effort to simulations that involves this approach. 

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4 
For the aforementioned reasons combined with the available computational resources, the multi-
fluid or Eulerian-Eulerian model has been employed in the present work. For this purpose, only two 
equations are needed, i.e. mass and momentum equation (for each phase): 
𝜕(𝛼
𝑘
𝜌
𝑘
)
𝜕𝑡
+ ∇ ⋅ (𝛼
𝑘
𝜌
𝑘
𝒗
̅
𝑘
) = 0
𝜕(𝛼
𝑘
𝜌
𝑘
𝒗
̅
𝑘
)
𝜕𝑡
+ ∇ ⋅ (𝛼
𝑘
𝜌
𝑘
𝒗
̅
𝑘
𝒗
̅
𝑘
) + ∇ ⋅ (𝛼
𝑘
𝑹
̅
𝑘
𝑒𝑓𝑓
) = −𝛼
𝑘
∇𝑝̅ + 𝛼
𝑘
𝜌
𝑘
𝒈 + 𝑴
𝑰,𝒌𝒋
Two manipulations must be performed on the previous equations in order to provide a stable and 
robust discretization. Regarding the mass conservation, the convective term is expanded and 
expressed in terms of the total averaged velocity and the relative velocities of the generic phase k 
and the other phases. Assuming incompressible phases, the final expression reads as: 
𝜕𝛼
𝑘
𝜕𝑡
+ ∇ ⋅ (𝛼
𝑘
𝒗
̅) + ∇ ⋅ ( ∑ 𝛼
𝑗
𝛼
𝑘
𝒗
̅
𝑟,𝑘𝑗
3
𝑗=1,𝑗≠𝑘
) = 0
The analysis of the momentum equation shows that at the boundary, where a non-slip impermeable 
boundary condition is usually applied, it reduces to: 
∇𝑝̅ = 𝜌
𝑘
𝒈
In presence of a multiphase flow with potential large density variations among phases, the wall 
pressure gradient may assume different values. Since this possibility is unphysical, the pressure 
𝑝
is replaced with a corrected pressure 
𝑝̅
𝑟𝑔ℎ
defined as
𝑝
̅
𝑟𝑔ℎ
= 𝑝̅ − 𝜌𝑔ℎ.
As a result, the momentum equation becomes: 
𝜕(𝛼
𝑘
𝜌
𝑘
𝒗
̅
𝑘
)
𝜕𝑡
+ ∇ ⋅ (𝛼
𝑘
𝜌
𝑘
𝒗
̅
𝑘
𝒗
̅
𝑘
) + ∇ ⋅ (𝛼
𝑘
𝑹
̅
𝑘
𝑒𝑓𝑓
) = −𝛼
𝑘
∇𝑝̅
𝑟𝑔ℎ
− 𝛼
𝑘
𝑔ℎ∇𝜌 + 𝛼
𝑘
(𝜌
𝑘
− 𝜌)𝒈 + 𝑴
𝑰,𝒌𝒋
Where 
𝜌
is the mixture density. 
The momentum equation requires two additional closure to be solved: the momentum exchange 
source 
𝑴
𝑰,𝒌𝒋
and the turbulence term 
𝑹
̅
𝑘
𝑒𝑓𝑓
.