Chapter 4 Two-Phase Flow

I. Basic Definitions

(1) Void Fractions

: the time-averaged volumetric fraction of vapor in a two-phase mixture.

[eqn: alpha ~ under TRIANGLE ~ int int int from V_g ~dV ``/`` int int int from V ~dV ~=~ V_g / ( V_l +V_g )]                    (4.1.1)

o Measurement

1) Needle-probe method

☞ electrical probe

☞ optical probe

2) X-ray absorption method (Jones and Zuber)

[eqn: alpha ~=~ { sum from i=1 to N_v DELTA``t_i } over T]

그림 4-1 Representation of void fraction.

For differential length element

[eqn: alpha ~=~ DELTA`` z int int from A_g ~dA``/(DELTA``z int int from A ~dA``)]

[eqn: =~A_g ``/(A_l +A_g )]

(4.1.2)

(2) Phase Velocity

: the volumetric flow rate of the phase through its cross-sectional flow area

[eqn: u_l ~==~ Q_l ````/A_l]

[eqn: u_g ~==~Q_g ````/A_g]                                  (4.1.3)

where [eqn: Q_i] is the volumetric flow rate ,[eqn: ft^3 /sec], of phase i

(3) Volumetric flux or superficial velocity

: the volumetric flow rate of the phase divided by the total cross-sectional flow

area.

[eqn: j _{l} ` == `Q _{l} ````/(A _{l} +A _{g} )]

[eqn: j_g ~==~ Q_g ```` /(A_l +A_g )]                             (4.2.4)

Now, using the relations given in Eqs.(4.1.2) and (4.1.3), we can write

[eqn: j_l ~=~ u_l ``(1-alpha)]

[eqn: j_g ~=~u_g alpha]                                      (4.1.5)

And the total volumetric flux of the mixture, j, is

[eqn: j~=~j_l ~+~ j_g]                                   (4.1.6)

(4) Relative velocity, [eqn: u_r]

: velocity difference between the vapor and the liquid phase

[eqn: u_r ~==~ u_g -u_l]                                   (4.1.7)

For homogeneous flow

[eqn: u_r ~=~0]

Thus, from Eq. (4.1.5)

[eqn: j_g`` /alpha `=~ j_l ``/(1-alpha)]

By solving for the void fraction

[eqn: alpha ~=~ j_g ``/(j_l +j_g )]                                (4.1.8)

(5) Slip ratio

: the ratio of the phase velocity

[eqn: S ~==~ u_g ``/u_l]                                               (4.1.9)

[eqn: =~ x over (1-x)``` (1-alpha) over alpha `` rho_l over rho_g]

For homogeneous flow, [eqn: S=1], by definition.

(6) Quality

① Thermal equilibrium quality : the flow fraction of vapor in the thermal

equilibrium condition

[eqn: x_e ~==~ (h-h_f )/h_fg]                                           (4.1.10)

② Flow quality : the true flow fraction of vapor regardless of whether thermal

equilibrium exists or not

[eqn: x~==~rho_g u_g A_g ``/ (rho_l u_l A_l + rho_g u_g A_g )]

[eqn: =~w_g ``/w]                                                (4.1.11)

③ Static quality : the mass fraction of vapor

[eqn: x_s ~==~ rho_g A_g ``/(rho_l A_l + rho_g A_g )]                                   (4.1.12)

From the definitions given in Eqs.(4.1.11) and (4.1.12), [eqn: x] and [eqn: x_g] can be related          through the following identity

[eqn: x``/(1-x) ~=~(u_g ``/u_l )``x_s ``/(1-x_s ) ~=~S x_s ``/(1-x_s )]              (4.1.13)

(7) Cross-sectional Average Notation, < >

[eqn: < zeta >~==~ 1 over A_x-s int int from A_x-s zeta ~dA]

[eqn: <zeta_l >_l ~==~ 1 over {A_x-s (1-<alpha>)} int int _A_x-s zeta_l ``(1-alpha)~dA ~=~{<(1-alpha)zeta_l `>} over <1-alpha>]

[eqn: <zeta_g >_g ~==~ 1 over {A_x-s <alpha>} int int from A_x-s zeta_g ``alpha dA ~=~{<alpha``zeta_g >}over<alpha>]                      (4.1.14)

Note : [eqn: <j^2 > != <j>^2]

(8) Mass flux, G

[eqn: G~=~ w/A_x-s ~=~rho_l`` <u_l >_l``(1-<alpha>)/(1-<x>)~=~rho_g ``<u_g >_g <alpha>/<x>]     (4.1.15)

where the last two equalities come from the identities

[eqn: G(1-<x>)A_x-s ~=~rho_l ``<u_l >_l (1-<alpha>)A_x-s]

and

[eqn: G<x>A_x-s ~=~rho_g <u_g >_g <alpha>A_x-s]

which also imply

[eqn: <u_l >_l ~=~ G(1-<x>)/[rho_l ``(1-<alpha>)]]                         (4.1.16)

[eqn: <u_g >_g ~=~G<x> /(rho_g <alpha> )]                                  (4.1.17)

(9) The fundamental Void-Quality Relation

The fundamental void-quality relation is obtained as,

[eqn: <alpha>~=~ <x>/[<x> + S(rho_g /rho_l )(1-<x>)]]                     (4.1.18)

With respect to the static quality, Eq.(4.1.18) will be

[eqn: <alpha>~=~<x_s >rho_l ``/[<x_s >(rho_l -rho_g )+rho_g ]]                       (4.1.19)

and, from [eqn: <bar rho > ~=~(1-alpha) rho_l +alpha rho_g]

[eqn: < bar rho >~=~rho_l ``rho_g ``/[<x_s >rho_l +(1-<x_s >)rho_g ]~=~1/(v_l +<x_s >v_lg )]             (4.1.20)

For homogeneous flow condition, [eqn: S=1]

[eqn: <rho_H > ~==~ G over {<j>} ~=~ w over Q ~=~1/(v_l +<x>v_lg )]                   (4.1.21)

but for slip flow condition

[eqn: < bar rho >~ != ~ 1/(v_l +<x>v_gl )]

(10) Drift velocity, [eqn: V_gj]

: the void-weighted average velocity of vapor phase with respect to the velocity of        the center-of-volume of the mixture.

[eqn: V_gj ~==~<alpha(u_g - j )>/<alpha>]                                   (4.1.22)

II. Two-phase Model

o Number of unknowns for three dimensional flow

Single-Phase		Two-Phase
Parameter	No	Parameter	No
Velocity Vector Pressure Temperature Density	3 1 1 1	Void Fraction Velocity Vector of Liquid Velocity Vector of Vapor Density of Liquid Density of Vapor Pressure Temperature of each Phase	1 3 3 1 1 1 2
Total number of unknowns	6	Total number of unknowns	12

o Number of governing equations

Single-Phase		Two-Phase
Parameter	No	Parameter	No
Conservation of Mass Conservation of Momentum Conservation of Energy Equation of State	1 3 1 1	Mass Eq. for Liquid Mass Eq. for Vapor Momentum Eq. for Liquid Momentum Eq. for vapor Energy Eq. for each Phase Eq. of State for each Phase	1 1 3 3 2 2
Total number of equations	6	Total number of equations	12

o Constitutive Equations

☞ interfacial transfer (jump condition)

☞ various correlations dependent on the flow regime

o Various forms for two-phase flow modeling

☞ Local Instantaneous Formula

☞ Instantaneous Space- or Local Time-Averaged Equations

☞ Homogeneous Model

☞ Two-Fluid Model (Drift-Flux Model)

1. Local Instantaneous Formula

● balance equations for each phase

● interfacial condition(i.e. jump condition) as well as BC & IC

F = F(r, t)

■ Fundamental Importances in the Local Instantaneous Formula

a) direct application to study separated flow

; Δp calculation, instability analysis, bubble dynamics, etc

b) fundamental base of all two-phase models

- perform overall calculations using averaging techniques

- void fraction calculation, development of drift-flux model, etc

The general integral balance can be written as,

[eqn: {d} over {dt} int _{V _{m}} ^{} {} rho _{k} varphi _{k} dV=- oint _{A _{m}} ^{} {bar{n _{k}}} _{} CDOT bar{J} _{k} dA+ int _{V _{m}} ^{} {} rho _{k} phi _{k} dV]                (4.2.1)

Using the Renold transport theorem to obtain the differential form of the balance

equation, i.e

[eqn: {d over dt} int from V_m F_k dV = int from V_m partialF_k over partialt dV + int from A_m F_k {bar v}_k cdot bar n dA]                      (4.2.2)

the differential equation is,

[eqn: {Partial rho _{k} ` varphi _{k}} over {Partial t} + NABLA CDOT ( bar{v _{k}} rho _{k} varphi _{k} )=- NABLA CDOT bar{J _{k}} + rho _{k} phi _{k}]                           (4.2.3)

(1) Interfacial balance (Jump condition)

- Consider the interface as a singular surface across which the fluid density,               energy and velocity suffer from jump discontinuity, although mass, momentum           and energy must be conserved.

Mass Jump

[eqn: rho_1 (vec v_1 - vec v_i )․vec n_1 ~+~ rho_2 (vec v_2 - vec v_i )․vec n_2 ~=~ 0]

or    [eqn: dot m_1 ~+~ dot m_2 ~=~ 0]                                                (4.2.4)

If no mass transfer,   [eqn: dot m_1 ~=~ dot m_2 ~=~ 0]

[eqn: therefore ~~(vec v_1 - vec v_2 )․vec n ~=~ 0 ~~~~~~or~~~ vec v_1 ~=~ vec v_2]

Linear Momentum Jump

[eqn: dot m_1 vec v_1 + dot m_2 vec v_2 - vec n_1 ․ vec T_1 - vec n_2 ․ vec T_2 ~=~0]       (4.2.5)           [eqn: vec n_1]

From Eq. (4.2.4),   [eqn: dot m_1 ( vec v_1 - vec v_2 ) - vec n_1 ․ vec T_1 - vec n_2 ․ vec T_2 ~=~0]

ⓐ for inviscid phase ( [eqn: vec T ~~-> ~~p] )

[eqn: dot m_1 ( vec v_1 - vec v_2 ) + vec n_1 (p_1 - p_2 ) ~=~0]

ⓑ for viscous flow (1D flow which interface perpendicular to flow direction

[eqn: dot m_1 ( vec v_1 - vec v_2 ) + vec n_1 ․ (p_1 - p_2 ) + vec n_1 ․ (tau_2 - tau_1 ) ~=~0]

For 2 dimensional case with surface tension

[eqn: dot m_1 vec v_1 + dot m_2 vec v_2 - vec n_1 ․ vec T_1 - vec n_2 ․ vec T_2 + {d sigma} over dl vec tau - sigma over R vec n_1 ~=~0]                (4.2.6)

If no mass transfer and the phase are inviscid,

[eqn: vec n_1 ․ ( p_1 - p_2 ) + {d sigma} over dl vec tau - sigma over R vec n_1 ~=~0]                                 (4.2.7)

If the surface tension is constant,

[eqn: p_1 ~-~ p_2 ~=~ sigma over R]                                       (4.2.8)

Total Energy Jump

[eqn: dot m_1 (u_1 + 1 over 2 v_1^2 ) + dot m_2 (u_2 + 1 over 2 v_2^2 ) + vec q_1 ․ vec n_1 + vec q_2 ․ vec n_2 - ( vec n_1 ․vec T_1 ) ․vec v_1 - ( vec n_2 ․vec T_2 ) ․vec v_2 ~=~ 0]

(4.2.9)

(2) Interfacial boundary conditions

; 各 phase 간 kinematical, dynamical 혹은 thermal relation을 制限할 수 있는 경계조건

ⓐ thermal boundary condition

[eqn: T_1 ~=~ T_2 ~=~ T_i]                                             (4.2.10)

ⓑ mechanical boundary condition (No slip condition)

[eqn: {vec v}_{t``1} ~=~ {vec v}_{t``2} ~=~ {vec v}_{t``i}]                                         (4.2.11)

ⓒ phase change boundary condition

[eqn: g_1 ~-~ g_2 ~=~ left { {| vec v_2 - vec v_i ``|^2} over 2 ~-~ {| vec v_1 - vec v_i ``|^2} over 2 right} - left{ 1 over rho_2 (vec tau_2 ․vec n_2 ``)․vec n_2 - 1 over rho_1 (vec tau_1 ․vec n_1 ``)․vec n_1 right}]

(4.2.12)

Note that in thermodynamic equilibrium,

[eqn: T_1 - T_2 ~=~ 0 ## p_1 - p_2 ~=~ 0 ## g_1 - g_2 ~=~ 0]