Investigation of flow through centrifugal pump impellers 3

3.1.1 Governing Equation of Liquid Phase Water flow liquid phase within the pump impellers can be described by the three-dimensional Reynolds-averaged Navier-Stokes RANS equations with

THEORETICAL MODELS OF TWO-PHASE FLOW

3.1 Governing Equations

In formulating the governing equations for air-water two phase flow in centrifugal pump impeller, the following assumptions are made:

(1) The mixture is a homogeneous bubbly flow entraining fine bubbles The bubble size is small compared to a characteristic length of the impeller channel

(2) In this case, air bubble is treated as the dispersed phase and water is treated as the continuous phase

(3) The bubbles maintain their spherical shape Neither fragmentation nor coalescence of bubble occurs The liquid phase is incompressible

(4) The drag coefficient of a bubble is the same as that of a solid particle The influence of interactions between bubbles is negligible

(5) The mixture flow is steady in a relative frame or reference, which rotates around

an axis with a constant velocity Neither mass nor heat transfer takes place between the two phases

3.1.1 Governing Equation of Liquid Phase

Water flow (liquid phase) within the pump impellers can be described by the three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations with a suitable turbulence model Assuming a constant property flow, the RANS equations for liquid phase in a relative frame of reference are as follows:

[(1− ) ]+∇•[(1− ) ]=0

∂

∂

l l

l

T l l

leff

l l

l l l l

l

M U

U p

r U

U U U

t

+

∇ +

∇

−

•

∇ +

∇

−

−

=

× +

×

− +

×

−

•

∇ +

−

∂

∂

]}

) ( [

) 1 {(

) 1 (

) 2

( ) 1 ( ] )

1 [(

] )

1

[(

µ α α

ω ω

ρ α ρ

α ρ

α

(3.2)

where “×” is a vector cross-product, U is the velocity of liquid phase, l ρl the density

of water, α is the void fraction,ω is angular velocity of impeller, r is radial

vector, µleff is the effective viscosity of liquid phase accounting for turbulence (µleff =µl +µlt ; µl and µlt are the molecular and turbulent viscosities of liquid

phase respectively), p is the pressure, M is interfacial forces acting on liquid phase l

due to the presence of gas phase

3.1.2 Governing Equation of Gas Phase

The governing equations for bubbly flow (gas phase) in a relative frame of

reference are written as follows under the assumption

( )+∇•( )=0

∂

∂

g g

g

T g g

geff

g g

g g g g

g

M U

U p

r U

U U U

t

+

∇ +

∇

•

∇ +

∇

−

=

× +

× +

×

•

∇ +

∂

∂

]}

) ( [

{

) 2

( ]

( ) (

αµ α

ω ω

αρ αρ

αρ

(3.4)

where U is the velocity of gas phase, g ρg the density of air, µgeff is the effective viscosity of gas phase accounting for turbulence (µgeff =µg +µgt; µg and µgt are the molecular and turbulent viscosities of gas phase respectively), M describes g

interfacial forces acting on gas phase due to the presence of liquid phase

The density of air ρg in Eqs (3.3) and (3.4) may be described as a function of temperature, pressure and one or more additional variables:

)

, , ,

g ρ p T AV AV AV

For an ideal gas, the density is defined by the Ideal Gas Law:

T R

wp

g

0

=

where w is the molecular weight of gas, and R is the Universal Gas constant 0

Eqs (3.1), (3.2), (3.3), (3.4) and (3.5) form a closed set of non-linear partial differential equations to solve the unknowns α, U , l U , g ρg and p M and l M in g

Eqs (3.2) and (3.4) are interfacial forces which can be expressed by using suitable models The effective viscosities of liquid phase and gas phase µleff and µgeff can also be solved by using suitable turbulence models

3.2 Interfacial Forces

Generally, the total interfacial forces acting between two phases may arise from several independent physical effects If two phases are labelled using Greek indices α and β respectively and the total force on phase α due to interaction with phase β is denoted M , then we will have the following relation for α M : α

+ + +

+ +

=M D M p M L M VM M TD

where the term MαβD is the interphase drag force, Mαβp the force due to the pressure gradient, MαβL the lift force, MαβVM the force due to virtual mass, and MαβTD the turbulence dissipation force

In the present stage of simulation, the fluid motion in the centrifugal impeller

is assumed as a steady turbulent flow Therefore, the transient term MαβVM makes a negligible contribution to the intergrated results, because its value changes sign frequently and its overall integrated effect becomes very small The lift force and pressure force can also be negligible because of assumed small bubble size Based on the above analyses and assumptions, only interphase drag force and turbulence dissipation force are considered in our current two-phase fluid simulation Thus the interphase forces for liquid and gas phase in Eqs (3.2) and (3.4) can be written as follows:

TD D g

3.2.1 Interphase Drag Force

The following general form can be used to model interphase drag force acting

on liquid phase due to gas phase:

) (

lg

lgD C U g U l

where C is the drag coefficient lg

For spherical particle, the coefficient C can be derived analytically The area lg

of a single particle projected in the flow direction, A , and the volume of a single p

particle V are given by p

4

2

d

6

3

d

where d is the mean diameter

6

d V

n

p

α

α =

=

The drag exerted by a single particle on the continuous liquid phase can be written as

) (

2

1

l g l g p l D

Hence, the total drag force per unit volume on the continuous liquid phase is

) (

4

3

lg

d

C D

n D

Compare Eq (3.9) with Eq (3.8), we get

l g l

d

C

4

3

where C D is non-dimensional drag coefficient

For a particle of simple shape, immersed in a Newtonian fluid and which is not rotating relative to the surrounding free stream, the drag coefficient, C , depends D

only on the particle Reynolds number Rep The function C D(Rep) may be determined experimentally, and is known as the drag curve The particle Reynolds number Re is defined using the particle mean diameter, and the continuous liquid p phase properties, as follows:

l

l g p l p

U U d

µ

=

where µl is the viscosity of the continuous liquid phase, d is the diameter of the p

particle

In the present simulation, the Schiller Naumann drag model is employed for solving C D because of its good agreement with experimental data for solid spherical particles, or for fluid particles that are sufficiently small that they may be considered spherical The empirical relation for C D is written as

) Re 15 0 1 ( Re

p p

D

To ensure the correct limiting behavior in the inertial regime, the above Schiller Naumann model drag model is modified as

















+

= (1 0.15Re ),0.44

Re

24

p p

D

3.2.2 Interphase Turbulent Dispersion Force

In Eq (3.7), MlgTD represents turbulent dispersion force acting on the continuous liquid phase due to the dispersed gas phase In such case, Lopez de Bertodano Turbulent Dispersion Model (Lopez de Bertodano, 1991) is implemented and written as

l l l TD

TD C k r

where k l is the turbulent kinetic energy of liquid phase; r l is the volume fraction of liquid phase which is equal to (1−α ); C TD is the non-dimensional turbulent dispersion coefficient, its values of 0.1-0.5 have been used successfully for bubbly flow with bubble diameters of order a few millimetres See Lopez de Bertodano (1998) for a general discussion on recommended values of C TD

3.3 Turbulence Modelling in Multiphase Flow

This section describes the extension of the single-phase turbulence models to multiphase simulation According to our assumption (1) and (2), the water flow through the impeller is treated as continuous phase, and dilute fine bubbly flow is treated as dispersed phase In such case, it is possible to mix algebraic and k−ε

models between two phases for simplicity A recommended model for dilute dispersed two-phase flow uses a k−ε model for the continuous phase, and an algebraic eddy viscosity model for the dispersed phase, which simply sets the dispersed phase viscosity propotional to the continuous phase eddy viscosity

3.3.1 k−ε Turbulence Model for the Continuous Phase

The eddy viscosity hypothesis is assumed to hold for the continuous turbulence phase Diffusion of momentum is governed by an effective viscosityµleff, which is equal to molecular viscosity coefficient of liquid phase µl plus turbulent eddy viscosity coefficient of liquid phase µlt:

lt l leff µ µ

Similar to the derivation in Chapter 2 for single-phase flow, the turbulence viscosity of liquid phase, µlt, can be linked to the turbulence kinetic energy and dissipation of liquid phase via the relation:









=

l

l l lt

k C

ε ρ

where C is a constant, the value is 0.09, µ k is the turbulent kinetic energy of liquid l

phase and ε is the turbulent dissipation rate of liquid phase

The transport equations for k and l εl are assumed to take a similar form to the single-phase transport equations:

) lg

) )(

1

(

]}

) (

)[

1 {(

] ) 1

[(

k l l k

l k

lt l l l l l

l

T P

k k

U k

t

+

−

−

=

∇ +

−

−

•

∇ +

−

∂

∂

ε ρ α

σ

µ µ ρ

α ρ

α

(3.15)

) ( lg 2

( ) 1

(

]}

) (

)[

1 {(

] ) 1

[(

ε ε

ε

ε

ε ρ

ε α

ε σ

µ µ ε ρ α ε

ρ α

T C

P C k

U t

l l k

l l

l

lt l l l l l

l

+

−

−

=

∇ +

−

−

•

∇ +

−

∂

∂

(3.16)

where P is the turbulent kinetic energy production term, its definition is given in Eq k

(2.11), C , ε1 C , ε2 σk, σε are empirical constants for k −ε turbulence model, their values are also given in Chapter 2, the additional terms ( )

lgk

T and ( )

lg ε

T represent interphase transfer for k and l εl respectively, but not considered in the current simulation

3.3.2 Algebraic Turbulence Model for the Dispersed Phase

The algebraic equation model is only available for the dispersed fluid when the continuous phase is set to use a turbulence model

σ

µ ρ

ρ

µ lt

l

g

where µgt and µlt are turbulent viscosities of gas phase and liquid phase respectively; the parameter σ is a turbulent Prandtl number

In situations where the particle relaxation time is short compared to turbulence dissipation time scales, e.g bubbles or very small particles, we can safely use the

3.4 Boundary Conditions

The implementation of boundary conditions for multiphase flow is very

similar to that for single-phase flow The main differences are:

• Boundary conditions need to be specified for both fluids for all variables except the shared pressure field

• Volume fractions of both phases must be specified on inlet boundary condition These must sum to unity

• For multifluid flow, pressure boundary conditions at inlet and outlet are always defined in terms of static pressure

Considering these differences, the boundary conditions for two-phase flow are specified as follows:

Inlet boundary: The inlet velocity of both phases and particle inlet volume

fraction are given; the inlet turbulent kinetic energy is given according to 3.7 percent

of turbulence intensity; the inlet dissipation rate of turbulent kinetic energy is given as the same as that for single-phase flow

Outlet boundary: The outlet boundary for multiphase flow is specified as the

same as that for single-phase flow However, the static pressure is assumed zero at the outlet

Wall boundary: A no-slip condition is imposed for the continuous liquid

(water) phase, and a slip condition is applied to dispersed particle phase Wall functions are used to model near-wall flow