Behavior of a Two-Phase Gas–Liquid Flow at the Inlet into a Catalytic Reactor

Abstract

The behavior of a two-phase flow inside the inlet pipeline of a catalytic reactor is investigated. In addition to the classical approach using familiar flow diagrams, means of computational fluid dynamics are used for three-dimensional modeling of the spatial distribution of phases in the pipeline during operation. Results show a nonuniform distribution of the liquid phase over the over the pipeline outlet cross section surface, plus a mass flow of the liquid phase that is not stable over time. The maximum peak flow rates exceed the average values by ~300%. Compared to data from flow diagrams, CFD modeling shows that a change in the gas flow in the investigated range does not alter the nature of a two-phase flow, but an increase in the gas flow reduces the irregularity of the distribution of the liquid phase over the pipeline outlet cross section. Data on the behavior of a flow are needed to design catalytic reactor structures that ensure the uniform distribution of a two-phase flow to the catalyst bed for, e.g., hydrotreating reactors in the oil refining industry.

Appendices

Calculation Model

To solve problems with multiphase flows (in our case, a two-phase gas–liquid system), Euler’s approachis used in computational fluid dynamics, which treats all phases as interpenetrating continua [12]. In each calculated cell there can be any phase, the amount of which is determined through the volume fraction of each phase, which varies in the range of 0 to 1.

To solve problems with stratified phases and the free surface of liquid, a version of the Euler approach is used that is labeled the Volume of Fluid (VOF) model in the literature [12].

The VOF model can simulate two or more phases, solving one set of momentum equations and tracking the volume fraction of each phase across the entire calculated domain. The VOF model is based on each control volume having a sum of the volume fractions of all phases that is equal to one. In other words, if the volume fraction of the qth fluid in a cell is denoted as α_q, three conditions are possible:

• α_q = 0: the cell is empty (for the qth liquid),

• α_q = 1: the cell is filled (for the qth liquid),

• 0 < α_q < 1: the cell contains a surface (interface) between qth liquid and one or more other liquids.

Based on the local value of the volume fraction α_q, the corresponding properties of flow and individual phases and hydrodynamic variables are calculated in each control volume inside the calculated domain.

Continuity Equation

The surface between the phases is tracked by solving the continuity equation for the volume fraction of one phase (e.g., the liquid). For the qth phase, this equation generally has the form

$$\left[ {\frac{\partial }{{\partial t}}\left( {{{\alpha }_{q}}{{\rho }_{q}}} \right) + \nabla \left( {{{\alpha }_{q}}{{\rho }_{q}}{{{\vec {V}}}_{q}}} \right) = {{S}_{{{{\alpha }_{q}}}}} + \sum\limits_{{\text{p}} = 1}^n {\left( {{{{\dot {m}}}_{{pq}}} - {{{\dot {m}}}_{{qp}}}} \right)} } \right],$$

((D1))

where ${{\dot {m}}_{{qp}}}$ is the mass transfer from phase to phase and represents the mass transfer from the qth phase to the pth phase; ${{\vec {V}}_{q}}$ is the velocity of qth phase. In our case, the right part of the equation is equal to zero, since ${{S}_{{{{\alpha }_{q}}}}} = {{\dot {m}}_{{qp}}} = {{\dot {m}}_{{qp}}} = 0$.

The volume fraction of the other phase (e.g., gas) is determined using the relation

$$\sum\limits_{q = 1}^2 {{{\alpha }_{q}}} = 1.$$

((D2))

Preliminary analysis of the behavior of a two-phase flow in the considered pipeline showed that a nonstationary type of flow occurs. An explicit calculation scheme was therefore used in our calculations, i.e., a time step solution diagram:

$$\frac{{\alpha _{q}^{{n + 1}}{\rho }_{q}^{{n + 1}} - \alpha _{q}^{n}{\rho }_{q}^{n}}}{{\Delta t}}{v} + \sum\limits_f {\left( {{{\rho }_{q}}U_{f}^{n}{\alpha }_{f}^{n}f} \right)} = 0,$$

((D3))

where n + 1 is the index for a new (current) time step; n is the index for the previous time step; α_{q, f} is the nominal value of volume fraction of the qth phase, calculated in the first or second order against the flow; v is the volume of control cell; and U_f is the volume of the flow through the face of control cell, calculated from the velocity normal to the face.

In an explicit approach, standard finite difference interpolation schemes are applied to volume fraction values that were calculated at the previous time step. This formulation does not require an iterative solution to the transport equation at each time step, as is required for an implicit scheme.

The flow properties used in the transport equations are determined by the presence of phases in each control volume. For example, in a two-phase system, if the phases are represented by subscripts g and l, and if the volume fraction of the second one is monitored, the density in each cell is determined using the relation

$$\rho = {{\alpha }_{l}}{{\rho }_{l}} + \left( {1 - {{\alpha }_{l}}} \right){{\rho }_{q}}.$$

((D4))

Equation for the Conservation of Momentum

The equation for the conservation of angular momentum is solved throughout the computational domain, and the resulting velocity field is distributed between the phases. The pulse equation given below depends on the volume fractions of all phases through the properties of ρ and μ:

$$\begin{gathered} \frac{\partial }{{\partial t}}\left( {\rho \vec {V}} \right) + \nabla \left( {\rho \vec {V}\vec {V}} \right) \\ = - \nabla \rho + \nabla \left[ {\mu \left( {\nabla \vec {V} + \nabla {{{\vec {V}}}^{T}}} \right)} \right] + \rho g + \vec {F}, \\ \end{gathered} $$

((D5))

where g is the acceleration of gravity, and $\vec {F}$ is the external bulk forces. In our case, $\vec {F}$ = 0.

Equation for the Conservation of Energy

The equation for the conservation of energy is also solved throughout the computational domain:

$$\frac{\partial }{{\partial t}}\left( {\rho E} \right) + \nabla \left( {\vec {V}\left( {\rho E + {\text{p}}} \right)} \right) = \nabla \left( {{{k}_{{{\text{eff}}}}}\nabla T} \right) + {{S}_{h}}.$$

((D6))

The VOF model considers energy E and temperature T to be average-weight variables:

$$E = \frac{{\sum\limits_{q = 1}^n {{{\alpha }_{q}}{{\rho }_{q}}{{E}_{q}}} }}{{\sum\limits_{q = 1}^n {{{\alpha }_{q}}{{\rho }_{q}}} }} = \frac{{{{\alpha }_{g}}{{\rho }_{g}}{{E}_{g}} + {{\alpha }_{l}}{{\rho }_{l}}{{E}_{l}}}}{{{{\alpha }_{g}}{{\rho }_{g}} + {{\alpha }_{l}}{{\rho }_{l}}}},$$

((D7))

where E_q for each qth phase is based on the specific thermal capacity of each phase and the total temperature.

External sources of energy are considered on the right-hand side of the equation by member S_h = 0, which can contain contributions from radiation and any other volumetric source of heat. In our case, S_h = 0.