III Special Topics in Two-Phase Modelling

1. Interfacial Transport

(1) Differential balance equation for the [eqn: k^th] - phase

[eqn: {partial alpha_k rho_k vec {psi_k } } over partialt + del cdot (alpha_k rho_k vec psi_k `` vec v_k ``) ~=~ -del cdot [alpha_k (vec J_k + {vec j_k} ^T )] + alpha_k rho_k vec phi_k + I_k]     (3-1)

Mass Balance

[eqn: vec psi_k ~=~ 1 ~,~vec J_k ~=~0 ~,~vec phi_k ~=~0]

interfacial mass balance

[eqn: SUM from k=1 to 2 rho_k vec n_k cdot (vec v_k - vec v_i )~=~0]                                   (3-2)

Define

[eqn: GAMMA_k ~=~I_k ~=~ - 1 over DELTAt SUM from j left{ 1 over v_{n`i} vec n_k cdot rho_k (vec v_k - vec v_i ) right}]                   (3-3)

and     [eqn: I_m ~=~ SUM from k=1 to 2 I_k ~=~0]                                          (3-4)

[eqn: THEREFORE ~~{partial alpha_k `rho_k} over partialt + del cdot (alpha_k rho_k vec v_k ) = GAMMA_k]                              (3-5)

and           [eqn: SUM from k=1 to 2 GAMMA_k ~=~0" : conservation of mass at the interface "]       (3-6)

Momentum Balance

[eqn: vec{psi _{k}} `=~ vec{v _{k}} `,~ vec{J _{k}} `=~- vec{T _{k}} `=~P _{k} vec{I prime } - vec{tau _{k}} `,~ vec{phi _{k}} `=~ vec{g _{k}}]

interfacial momentum balance

[eqn: SUM from k=1 to 2 LEFT{ rho_k vec n_k cdot (vec v_k - vec v_i )` vec v_k - vec n_k cdot vec T_k RIGHT} + ( vec t_alpha ``a^alphabeta ``sigma)_beta ~=~0]             (3-7)

where, [eqn: vec t_k]   : hybrid tensor between the space and the suface

[eqn: a^alphabeta]  : metric tensor of interface

[eqn: (~)_beta] : surface covariant derivative

Define

[eqn: vec M_k ~=~ I_k ~=~ - 1 over {Delta t} sum_j LEFT { 1 over v_{n i} vec n_k LEFT [ rho_k ``(``vec v_k ``-``vec v_i ``)``vec v_k ``-``vec T_k RIGHT ] RIGHT }]          (3-8)

[eqn: vec M_m ~=~ I_m ~=~- 1 over {Delta t} sum_j LEFT { 1 over v_{n i} ``(``t_alpha `` a^{alpha`beta} sigma ``)``_beta RIGHT }]  [eqn: {vec tau_k}^T ~=~-``{vec J_k }^T ~=~- bar{ rho_k vec v_k vec v_k }]   [eqn: THEREFORE `` {Partial alpha _{k} rho _{k} vec{v _{k}}} over {Partial t} ``+`` NABLA CDOT ``(`` alpha _{k} rho _{k} vec{v _{k}} vec{v _{k}} ``)`=~ & - alpha _{k} NABLA P _{k} ``+`` NABLA CDOT alpha _{k} ( vec{tau _{k}} ``+`` vec{tau _{k}} ^{T} ``)``# # & +`` alpha _{k} rho _{k} g``+`` vec{M _{k}}]       and [eqn: sum_k=1 ^2 vec M_k ~=~ vec M_m] : mixture momentum source due to the surface tension                            effect                                        (3-12)

(3-9)

(3-10)

(3-11)

k-th phase momentum source from the interfacial transfort.

Energy Balance

[eqn: PSI _{k} `=~u _{k} `+~v _{k2} ^{2} `,`~~~J _{k} `=~q _{k} ``-`` vec{T _{k}} CDOT vec{v _{k}} `,~~` PHI _{k} `=~g _{k} CDOT v _{k} `+`Q _{k``} / rho_k]

Interfacial energy balance

[eqn: {d_s u_a } over dt + u_a TRIANGLED _s cdot vec v_i = sum_k=1 ^2 LEFT { rho_k vec n_k cdot ( vec v_k - vec v_i )LEFT ( u_k + {v_k ^2 } over 2 RIGHT )+ vec n_k cdot (- vec T_k cdot vec v_k + q_k ) RIGHT }+ (t_alpha a^{alpha beta} sigma ) cdot vec v_i]

(3-13)

Define

[eqn: rm E_k == I_k = - 1 over {Delta t} sum_j LEFT { 1 over v_{n i} vec n_k LEFT [ rho_k ``(``vec v_k ``-``vec v_i ``)``(`u_k `+` v_k^2 over 2 `)` ``-``vec T_k cdot vec v_k `+`q_k RIGHT ] RIGHT }]  [eqn: E _{m} == I _{m} = {1} over {DELTA t} sum _{j} ^{} LEFT { {1} over {v _{n i }} LEFT [ T _{i} {ds} over {dt} LEFT ( {d sigma } over {dT} RIGHT ) ``+~` LEFT ( {d sigma } over {dT} RIGHT ) ` NABLA _{s} CDOT v _{i} `+`(t _{alpha } a ^{alpha beta } `` sigma `) _{, beta } CDOT vec{v _{i}} RIGHT ] RIGHT }]  [eqn: q _{k} ^{T} ~~ == `J _{k} ^{T} ``-`` vec{T _{k}} CDOT vec{v _{k} ^{prime }} ``~~=~`` rho _{k} (u _{k} ``+`` {v _{k} ^{2}} over {2} ) ^{prime } `` vec{v _{k} ^{prime }} ``-`` vec{tau _{k}} CDOT vec{v _{k} ^{prime }} ``+``p _{k} vec{v _{k} ^{prime }}~~~~~~(3-16)]	and [eqn: {vec v_k }^'] = fluctuation component in turbulent flow

(3-14)

(3-15)

[eqn: epsilon _{A} `` == ``u _{k} `+~ {bar{``(``v _{k} ^{prime } ``)`` ^{2}}} over {2}] : thermal + turbulent kinetic energy

[eqn: Therefore & partial over partialt left[ alpha_k rho_k left( epsilon_k + {bar v_k }^2 over 2 right) right ] + TRIANGLED CDOT left[ alpha_k rho_k left( epsilon_k + {bar v_k }^2 over 2 right) {bar v_k }^2 right ] ### & = - Triangled CDOT left [ alpha_k left ( bar bar q_k + {bar bar q_k }^T right ) right ] + Triangled CDOT left [ alpha_k bar bar T_k CDOT bar v_k ] + alpha_k rho_k vec g_k CDOT vec v_k + E_k]

(3-17)

and [eqn: SUM_k=1^2 E_k = E_m]  : energy source for the mixture

With mechanical energy equation and introducing the enthalpy, Eq.(3-17)        will be

[eqn: { partial alpha_k rho_k h_k } over { partial t} + Triangled CDOT ( alpha_k rho_k h_k vec v_k ) &= - Triangled CDOT ( alpha_k bar bar q_k ) - Triangled Cdot alpha_k ( q_k^T + {vec tau_k }^T Cdot vec v_k ) # &+ D_k over Dt ( alpha_k bar rho_k ) + alpha_k ( tau_k + tau_k ^T )~ :~ Triangled vec v_k + ( v_k^2 over 2 Gamma_k - vec M _k Cdot vec v_k + E_k )]

(3-18)

(2) Interfacial transport

- hard and not practical to use [eqn: rm I_k] and [eqn: rm I_m] which are composed of local         instant values

- thus need alternative physical mechanism to control them based on their

exact physical meaning

∘distinguish the dominant transfer mechanism

∘eliminate some of complicated terms which have insignificant effects in

the macroscopic field

1) interfacial mass transfer

Define the interfacial mass efflux from kth - phase as,

[eqn: rm Dot m_k = rho_k vec n_k Cdot ( vec v_k - vec v_i )]                 (3-19)

and IAC (interfacial area concentration), [eqn: rm a _{j}] ,

[eqn: a_j = 1 over{ TRIANGLE t} lim_delta->0 2epsilon_i over delta = 1 over { Triangle t } left( 1 over v _{n i} right ) _j] [cm-1], for only one interface.           (3-20)

※ [eqn: 2epsilon = delta over v _{n i}] : time interval associated with each interface

IAC represents the interfacial area per volume and the total area                concentration is given as,

[eqn: rm a=`{ 1} over {triangle t}sum from {j } ( 1 over v_{n i} )_j = sum a_i]                                   (3-21)

Next, we assign state density function for phase 1, phase 2 and interface        as,

[eqn: rm M _{k} (x,t)```=1,~````M _{s} (x,`t)=0```] : a point occupied [eqn: rm k^th]-phase

[eqn: rm M_s (x,t)=1,~M_k =0]  : a point occupied by interface            (3-22)

Then the time-averaged phase density function will be,

[eqn: rm alpha _{k} (x _{o} ,`t _{o} )= lim _{delta -> 0} {} {1} over {TRIANGLE t} int _{[ TRIANGLE t]} ^{} {} M _{k} (x _{0} ,t)dt=0~````=```` {TRIANGLE t _{k}} over {TRIANGLE t} ``~~~~~~]    (3-23)

For [eqn: rm v_{n i} != 0],

[eqn: alpha _{1} ``+``` alpha _{2} ````=```1]                                           (3-24)

which is the consequence that the average interfacial density function is 0

[eqn: rm alpha _{s} = lim _{delta -> 0} {} {1} over {TRIANGLE t} int _{TRIANGLE t} ^{} {} M _{s} (x _{0} ,t)dt=0~~~~~~~]                   (3-25)

Time-averaged mean value

[eqn: rm bar{F} `(x _{o} ,t _{o} )``= lim _{delta -> 0} {} {1} over {TRIANGLE t} int _{[TRIANGLE t] _{T}} ^{} {} F(x _{0} ,t)dt~~~~~~~]

[eqn: rm bar{F _{k}} `(x _{o} ,t _{o} )``= lim _{delta -> 0} {} {1} over {TRIANGLE t} int _{[ TRIANGLE t] _{T}} ^{} {} F _{k} (x _{0} ,t)dt~~~~~~~]

[eqn: alpha _{k} ````=`` bar{M _{k}}]                                               (3-26)

If [eqn: rm v_{n i}=0] (Quasi-stationary interface) ,

[eqn: rm alpha_s ={triangle t_s} over {triangle t}>=0]

[eqn: rm THEREFORE``` triangle t =triangle t_s+sum from{k=1} to {2} triangle t_k~~~or ~~alpha_s + alpha_1 + alpha_2 =1]

Thus two distinct cases of singularities

1) [eqn: rm v_{n i}=0], and [eqn: rm alpha_s =0]

2) [eqn: rm v_{n i}=0], and [eqn: rm alpha_s` >`0]

Case 1)

- does not bring discontinuities in[eqn: rm alpha_k], but only in its derivative in the             microscopic sense

- This microscopic singularities should be neglected in macroscopic

problem (fundamental hypothesis on smoothness of mean value)

Case 2)

- Only for this case, the singular interfacial surface appers in macroscopic          formulation.

- However, highly fluctuating two-phase flow systems rarely have such a          stationary interface.

As the time-averaged macroscopic formulation is intended to be applied for two-phase flows, these interfacial singularities can be neglected in most application.

☆ Shock Discontinuities

- 1Φ 혹은 2Φ flow에서 When various properties are abruptly changed in          single- or two-phase flow

․compressibility effect

․concentration shock in mixture

- use microscopic jump conditions which stand as balance eq. at a                 surface of discontinuities in the averaged field

Interfacial mass transfer term, [eqn: GAMMA _{k}]

[eqn: GAMMA _{k} `=~- sum _{j} ^{} {1} over {DELTA t} {1} over {v _{n`i}} LEFT { vec{n _{k}} CDOT rho _{k} ( vec{v _{k}} - vec{v _{i}} ) RIGHT } ````=``-` sum _{j} ^{} a _{j} dot{m} _{k}]      (3-27)

where [eqn: dot{m} _{k}] is the rate of mass loss per interfacial area in unit time from the     kth-phase.

Mean mass transfer per unit interfacial area

[eqn: bar{dot{m}} ` _{k} ```````=`` {sum _{j} ^{} a _{j} dot{m _{k}}} over {a _{s}} ```````]                                     (3-28)

Then, the interfacial mass transfer condition can be rewritten as,

[eqn: rm SUM from {k=1} to 2 GAMMA_k ~=~ 0] ,  [eqn: rm GAMMA _{k} `=~-a _{s} `` bar{dot{m}} _{k} ````]                    (3-29)

2) Interfacial momentum transfer

From Eq.(3-8), the interfacial momentum transport term is,

[eqn: rm vec{M_k} ~=~ - SUM from { j }` a_j "{"][eqn: rm dot m_k vec{v_k} + p_k vec{n_k} - vec{n_k} ․vec{tau_k}][eqn: rm "}"]                        (3-30)

rate of the interfacial mom. loss per area from k-th phase

And the mixture mom. source from the interface, Eq.(3-9), is given as,

[eqn: rm vec{M_m} ~=~ - SUM from { j }` a_j` "{"][eqn: rm a^{alpha beta} (t_alpha ),_beta` sigma] + [eqn: rm a^{alpha beta}` t_alpha` sigma,_beta`][eqn: "}"]

effect of mean curvature,   effect of surface tension gradient

The vectorial form of kth-phase interfacial momentum gain

[eqn: rm & vec{M _{k}} ``=```` vec{M _{k} ^{GAMMA }} ```+` vec{M _{k} ^{n}} ```+``` bar{p} _{ki} ` bar{NABLA } alpha _{k} ````+``` vec{M _{k} ^{t}} ````]

(3-31)

[eqn: rm and,~& {vec M_k }^Gamma = Gamma_k vec v_ki ##&rm {vec M_k }^n = SUM _j a_j (bar P_ki -P_k ) n_k ##&rm {vecM_k } ^t = sum_j a_j tau_{tk} ~:~"tangential component"]

On th other hand,

[eqn: rm &vecM_m = 2barH_ 21 sigma vec del alpha_2 + under {Sum_j {a_j `2(H_21 - barH_21 )sigma`` vec n}} + sum_j a_j ``a^alphabeta ``t_alpha ``sigma ,_beta]

change of mean curvature       0  ∵ [eqn: TRIANGLED sigma] is small

[eqn: THEREFORE ~ vec M_m = 2bar H_21 sigma vec del alpha_2 + {vec M_m }^H]

(3-32)

where   H21 ; mean curvature

[eqn: vec{M} _{m} ^{H}] ; the effect of the change of the mean curvature on the

mixture momentum source

From Eqs. (3-31) and (3-32)

[eqn: rm SUM_k=1 ^2 left{ left( {Gamma_k ^2 L_s ^2 } over rho_ki + P_ki right) vec del alpha_k + {vec M_k }^n right}-2H_21 sigma vec del alpha_2 - M_m ^H = 0 " : for normal component"]

[eqn: rm sum_k=1 ^2 {vec M_k }^t ~=~0 " : for tangential component"]

As a summary of interfacial momentum transfer condition,

[eqn: rm&vec M_k = {vec M_k} ^Gamma + P_ki del alpha_k + M_k ^d]

[eqn: rm vec M_m = 2bar H_21 sigma vec del alpha_2 + {vec M_m }^H]

[eqn: rm sum_k=1 ^2 M_k ~=~M_m]

where [eqn: vec{M _{k} ^{d}} `````=` vec{M _{k} ^{n}} ```````+```` vec{M _{k} ^{t}} `````] ;  total drag force of each phase

3) Interfacial energy transfer

Consider interfacial thermal energy transfer term.

[eqn: LAMBDA_k` =~{{barv}_k}^2 over 2`GAMMA _k - VEC M_k ․vecv_k + E_k]                                       (3-21)

[eqn: = sum _{j} ^{} {1} over {TRIANGLE t} {1} over {v _{n`i}} ``[- dot{m _{k}} ``(u _{k} `+` {v _{k} ^{2}} over {2} ````-v _{k} ․ bar{v _{k}} +` {bar{v _{k} ^{2}}} over {2} `)``# ``````````````````````````````````````````````````````+ vec{n _{k}} ․ vec{T _{k}} ․(v _{k} - bar{v _{k}} )-n _{k} ․ vec{q _{k}} ```]]        (3-22)

Define the virtual internal energy of the interface

[eqn: bar{epsilon _{ki}} `` == `` {sum _{j} ^{} ``[a _{j} dot{m _{k}} (u _{k} + {|v _{k} - bar{v _{k}} ``| ^{2}} over {2} )]} over {sum _{j} ^{} ``a _{j} dot{m _{k}}}]

And the heat input per unit interfacial area is defined by

[eqn: q_ki prime prime =- LEFT {`` SUM from { j }`` a_j`` vec n_k ․vec q_k ``RIGHT }`` L_s]

Then Eq. (3-22) can be rewritten as,

[eqn: LAMBDA _{k} = LEFT { GAMMA _{k} `` bar{epsilon _{ki}} + {q _{ki} prime prime } over {L _{s}} RIGHT } + sum _{j} ^{} `` LEFT { a _{j} vec{n _{k}} ․ vec{T _{k}} ․(v _{k} - bar{v _{k}} ) RIGHT}]

(3-23)

From some mathematical manipulation(Chapter VIII in Ishii's book) and          introducing the virtual enthalpy of kth-phase at interface as,

[eqn: h _{ki} ```` == `` bar{epsilon } _{ki} ```+`` {p _{ki}} over {rho _{ki}}]

Eqs. (3-23) will be,

[eqn: LAMBDA _{k} =( GAMMA _{k} h _{ki} +aq _{ki} prime prime )-p _{ki} `` {D _{k} alpha _{k}} over {Dt} +M _{k} ^{d} ` BULLET ( vec{v} _{ki} `-` vec{v} _{k} `)`+`W _{ki} ^{T}]   ( 3-24)

where [eqn: W _{ki} ^{T} ````]is the turbulent flux of work due to drag

And from Eq.(2-21)

[eqn: E _{k} = GAMMA _{k} (h _{ki} + vec{v _{ki}} ․ vec{v _{k}} - {bar{v _{k}} ^{2}} over {2} )+aq _{ki} prime prime -p _{ki} {Partial alpha _{k}} over {Partial t} +M _{k} ^{d} ` BULLET vec{v} _{ki} `+`W _{ki} ^{T}]

(3-25)

Next, assuming [eqn: {d sigma } over {dT } ~ APPROX~ constant] , Eq. (3-15) will be

[eqn: E_ m= SUM from {j} a_j left{ T_i left({d sigma} over dT right) triangled_s cdot v_i + (t_alpha a^alphabeta sigma ),_beta cdot v_i right}]

Since [eqn: 1 over da ds over dt (da) = triangled_s cdot v_i]

∴ [eqn: E_m = T_i left( {d sigma} over dT right) {D_i a} over Dt + 2H_21 sigma partialalpha_1 over partialt +E_m^H]                 (3-26)

3) Two-Fluid Model

★ For two-fluid model it is necessary to specify [eqn: Gamma_1 , M_1 , E_1 ,M_m , E_m] by            constitutive equations, whereas for diffusion (mixture) model, it is               sufficient to supply only [eqn: Gamma_1 , M_m , E_m] .

- more complicated not only in terms of the number of governing                  equations(6), but also in terms of the number of the necessary above            constitute equations.

- Accurate equations for [eqn: Gamma_1 , M_m , E_m] and [eqn: E_k] govern the usefulness of the          two-fluid model.

Mass Continuity Equation

[eqn: rm {partial alpha_k rho_k} over {partial t} + del cdot (alpha_k rho_k {vec v}_k )=Gamma_k]

(3-27)

Momentum Equation

[eqn: rm {partial alpha_k rho_k {vec v}_k} over {partial t} + del cdot (alpha_k rho_k {vec v}_k {vec v}_k )=& -alpha_k del p_k + del cdot alpha_k ( {bar bar tau}_k +{{bar bar tau}_k}^t )#rm&+ alpha_k rho_k g + {vec v}_ki Gamma_k + {vec M}_ki -del alpha_k cdot tau_i` ~~~]

(3-28)

Energy(Enthalpy) Equation

[eqn: rm {partial alpha_k rho_k H_k} over {partial t} + del cdot (alpha_k rho_k {vec v}_k H_k )=& -del cdot alpha_k ( {bar bar q}_k +{{bar bar q}_k}^t )#rm&+ alpha_k D_k over Dt p_k + H_ki Gamma_k + q_ki^'' over L_s +PHI_k ~~~~~~]

(3-29)

Jump condition

[eqn: rm Sum_k Gamma_k ~=~0]

(3-30)

[eqn: rm Sum_k {vec M}_ki ~=~0]

(3-31)

[eqn: rm sum _{k} ^{} a _{i} (m _{k} h _{ki} +q _{ki} ^{'' } )~=~0]

(3-32)

When the lift force and the diffusion due to density gradient is negligible,       Zuber(1964), Ishii & Mishima(1981) and Ishii & Zuber(1979) suggested the       interfacial momentum transfer term as,

[eqn: rm vec{M} _{ki} = & alpha _{d} vec{F} _{D} /B _{d} + alpha _{d} vec{F} _{v} /B _{d}# rm & + {9} over {2} {alpha _{d}} over {r _{sm}} sqrt {{rho _{c} mu _{m}} over {pi }} int _{t} ^{} {} {D _{d}} over {D xi } ( vec{v} _{c} - vec{v} _{d} ) {d xi } over {sqrt {t- xi }}]

(3-33)

In steady-state, the drag force on the particle is

[eqn: rm {vec F}_D = - 1 over 2 C_D rho_c {vec v}_r |v_r | A_d]

(3-34)

Or,

[eqn: rm alpha_d {vec F}_D /B_d =- LEFT ( alpha_d A_d over B_d RIGHT ) C_D over 2 rho_c {vec v}_r |v_r |]

(3-35)

Define

[eqn: rm Sauter`~diameter`:~~r _{sm} `=~ {3B _{d}} over {A _{i}} ~~~~~~~~~~~]

(3-36)

[eqn: rm Drag~`radius~~:~~r _{D} `=~ {3B _{d}} over {4A _{d}} ~~~~~~]

(3-37)

Bubble number density and interfacial area concentration are given as,

[eqn: rm N_d = alpha_d over B_d]

(3-38)

[eqn: rm a_i = N_d A_i]

(3-39)

Then the momentum transfer by drag can be obtained as,

[eqn: rm alpha_d {vec F}_D /B_d =- a_i LEFT [ C_D over 4 LEFT ( r_sm over r_D ) RIGHT ) {rho_c {vec v}_r |v_r |}over 2 RIGHT ]]

Interfacial transfer term = Interfacial area concentration

× Driving force