Chapter 11: Two-Phase Flow and Heat Transfer Flow Models Flow Models

Transcription

1 11.3 Fow Modes Chapter 11: Two-Phase Fow 11.3 Fow Modes The homogeneous fow mode proides an easier approach to determining fow properties and behaiors, but it underestimates the pressure drop, particuary in a moderate pressure range. Furthermore, the homogeneous mode is ess accurate when eocity and fow conditions for both phases are more disparate. A separated fow mode, on the other hand, is somewhat more compex but tends to produce more accurate resuts. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 1

2 11.3 Fow Modes Chapter 11: Two-Phase Fow Homogeneous Fow Mode The eocities of the iquid and apor (gas) phases are assumed to be identica, i.e., u = u = u H. The density of the homogeneous mixture satisfies the reation ρ H 1 x 1 = + ρ ρ ρ = Mass fow rates of the iquid and apor phases H ρ ρ ρ x x + ρ (1 x) & = ρ wh A m = ρ w A H (11.30) (11.31) (11.32) (11.33) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 2

3 11.3 Fow Modes Chapter 11: Two-Phase Fow Substituting eqs. (11.32) and (11.33) into eq. (11.2) / ρ α = (11.34) / ρ + / ρ Considering the definition of quaity in eq. (11.17), the oid fraction becomes x α = (11.35) x + (1 x) ρ / ρ Tota mass fux in the channe becomes + ρ wa + ρ wa = = = A ρ w (11.36) Goerning equations for homogeneous mode incudes continuity, momentum and energy equations ρ (11.37) A + ( A) = 0 t z 2 (11.38) ( A/ ρ ) ( pa) A + = ρ g cosθ A τ wp t z z Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 3 A

4 11.3 Fow Modes Chapter 11: Two-Phase Fow 2 2 w 1 cos w cos P ρ h + + gz θ + ρ wa h + + gz θ = q p w + q + t 2 A z 2 A t Substituting eq. (11.36) into eq. (11.39), the energy equation becomes ( ρ h) 1 + ( Ah) t A z (11.39) 3 2 P 1 A p = qw + q cos 2 g θ + A A z 2ρ t 2ρ t (11.40) For steady-state two-phase fow in a circuar tube with constant cross-sectiona area, the momentum eq. (11.38) reduces to 2 dp 4 τ w ( / ρ ) = + + ρ g cosθ dz D z (11.41) which can be rewritten as dp dp dp dp dz dz dz dz F a = g (11.42) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 4

5 11.3 Fow Modes Chapter 11: Two-Phase Fow The energy equation for steady-state two-phase fow in a circuar tube with constant cross-sectiona area can be obtained by simpifying eq. (11.40), 2 dh 4q w q d 1 = + 2 dz D 2 dz ρ dx dz = 4q w Dh g cosθ (11.43) (11.44) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 5

6 11.3 Fow Modes Chapter 11: Two-Phase Fow Separated Fow Mode Compared with the homogeneous mode, the separated fow mode has been used more widey because it proides a better prediction of fow behaior with a manageabe ee of compexity. The separated fow mode assumes each phase to hae different properties and to fow at different eocities. It is a simper ersion of the two-fuid mode discussed in Chapter 4 because it is assumed that ony eocities differ between the two phases, whie the conseration equations are written ony for the combined fow. In addition, the pressure across any gien cross section of a channe carrying a mutiphase fow is assumed to be the same for both phases. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 6

7 11.3 Fow Modes Chapter 11: Two-Phase Fow Cross-sectiona area of the channe occupied by iquid and apor A = /( ρ w ) (11.45) A = /( ρ w ) (11.46) Void fraction of the two-phase fow /( ρ w ) α = (11.47) /( ρ w ) + /( ρ w ) Considering the definition of quaity in eq. (11.17), the oid fraction becomes x α = (11.48) ρ w x + (1 x) ρ w Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 7

8 11.3 Fow Modes Chapter 11: Two-Phase Fow The goerning equations for steady-state two-phase fow in a channe hae been gien in Section More generaized goerning equations that are appicabe to transient fow are presented here. The continuity equations for the apor and iquid phases are respectiey ( ρ α A) + ( ρ wα A) = t z (11.49) t [ ρ (1 α ) A] + [ ρ w (1 α ) A] = z (11.50) where m & and m & are the mass production rates of apor and iquid due to phase change in the two-phase fow system. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 8

9 11.3 Fow Modes Chapter 11: Two-Phase Fow Conseration of mass requires that the summation of m & and m & equa zero, so the continuity equation for the two-phase system can be obtained by adding eqs. and together, i.e., ( ρ TP A) + [ ρ wα A + ρ w (1 α ) A] = 0 t z (11.51) Considering the definition of mass fux in eqs. (11.20)-(11.22), the continuity equation can be rewritten as (11.52) ( ρ A) + ( A) = 0 t z The momentum equations for the apor and iquid phases are 2 (11.53) ( ρ wα A) + ( ρ w α A) t z p = α A g ρ α Acosθ τ w, Pw, + F, z Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 9

10 11.3 Fow Modes 2 [ ρ w (1 α ) A] ρ w (1 α ) A + t z p = (1 α ) A g ρ (1 α ) Acosθ τ w Pw + F z Chapter 11: Two-Phase Fow,,, Momentum equation of the two-phase system can be obtained by adding eqs. (11.53) and (11.54) (11.54) 2 2 p A + ρ w α A + ρ w (1 α ) A = A g ρ Acosθ τ wp t z z (11.55) Substituting eqs. (11.20) and (11.21) into eq. (11.55), the momentum equation becomes 2 2 & (11.56) m G G p A + A + = A g ρ Acosθ τ wp t z ρ α ρ (1 α ) z Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 10

11 11.3 Fow Modes Chapter 11: Two-Phase Fow The momentum equation in the separated fow mode becomes 2 2 & 1 (1 ) τ wp (11.57) m x x p + A + = g ρ cosθ t A z ρ α ρ (1 α ) z A Energy equations for the apor and iquid phases are ( ρ eα A) p + ( ρ ) (11.58) wα Ae = P q e + q α A + α A + q, t z t [ ρ e (1 α ) A] + t z p = P q e + q (1 α ) A + (1 α ) A + q t [ ρ w (1 α ) Ae ], (11.59) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 11

12 11.3 Fow Modes Chapter 11: Two-Phase Fow The energy equation for the two-phase mixture is then obtained by adding eqs. (11.61) and (11.62) A [ ρ eα + ρ e (1 α )] + { A[ ρ wα e + ρ w (1 α ) e ]} t z (11.60) p = Pq e + q A + A t Substituting eqs. (11.20) and (11.21) into eq. (11.60), the energy equation becomes p A [ ρ (1 )] { [ (1 ) ]} (11.61) eα + ρ e α GA xe x e Pq e q A A + + = + + t z t Eq. (11.61) can be modified as A [ ρ hα + ρ h (1 α )] + { A[ xh + (1 x) h ]} t z A x (1 x) = Pqw + q A + m Ag cosθ & z 2 ρ α ρ (1 α ) x (1 x) p A + + A t 2 2ρ α 2 ρ (1 α ) t (11.62) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 12

13 11.3 Fow Modes Chapter 11: Two-Phase Fow Substituting eqs. (11.20) and (11.21) into eq. (11.60) 2 2 G G A ρ α h + + gz cos θ ρ (1 α ) h gz cosθ t 2ρ α 2 ρ (1 α ) 2 2 G G + A G h + + gz cosθ G h gz cosθ z 2ρ α 2 ρ (1 α ) p = Pqw + q A + A t (11.61) Since G =xg and G =G-G =(1-x)G, eq. (11.61) can be modified as A [ ρ hα + ρ h (1 α )] + { GA[ xh + (1 x) h ]} (11.62) t z G A x (1 x) = Pq e + q A + GAg cosθ z 2 ρ α ρ (1 α ) A + + A t 2 2ρ α 2 ρ (1 α ) t G x (1 x) p Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 13

14 11.3 Fow Modes Chapter 11: Two-Phase Fow The momentum equation (11.51) for steady-state, twophase fow in a circuar tube reduces to 2 2 dp 4 τ w d x (1 x) = dz D dz ρ α ρ (1 α ) cosθ (11.63) The energy equation in the separated fow mode can be obtained by simpifying eq. (11.62) (11.64) d 4 qw q d x (1 x) [ xh + (1 x) h ] = + + g cosθ dz D 2 dz ρ α ρ (1 α ) g ρ Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 14

15 11.3 Fow Modes Chapter 11: Two-Phase Fow Frictiona Pressure Drop The pressure drop is ery important for the design of two-phase deices because it dictates the pump power that is required to drie the fow. As shown in eqs. (11.41) and (11.63), the pressure drop incudes three parts: friction, acceeration and graity pressure drops. The acceeration and graity pressure drop terms in eq. (11.41) and (11.63) can be cacuated based on physica properties and the oid fraction. The frictiona pressure drop for steady-state two-phase fow in a circuar tube can be cacuated by dp F = dz 4τ w D (11.65) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 15

16 11.3 Fow Modes Chapter 11: Two-Phase Fow Correations Based on the Homogeneous Mode Frictiona pressure gradient 2 dpf 4τ w 2 ftpg = = (11.66) dz D Dρ H where f TP is the two-phase friction factor, which can be determined by empirica correation for singe phase fow. To determine the homogeneous friction factor, f TP 1 2κ 9.35 = og 10 + f D ( ) (11.67) TP ReTP f TP Reynods number can be cacuated based on a homogeneous fow GD = (11.68) Re TP µ The homogeneous iscosity of the two-phase mixture is obtained by 1 x 1 x (11.69) = + µ µ µ H H Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 16

17 11.3 Fow Modes Chapter 11: Two-Phase Fow Correations Based on the Separated Fow Mode The frictiona pressure gradient of two-phase fow can be reated to that of either the apor or iquid phase fowing aone in the channe. The frictiona pressure gradients of the apor or iquid phase fow in the channe, with their actua fow rate and properties, can be defined as dp dz F = 2 f G x Dρ 2 2 (11.70) dpf 2 fg (1 x) = dz Dρ 2 2 (11.71) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 17

18 11.3 Fow Modes Chapter 11: Two-Phase Fow Simiary, the frictiona pressure gradient in the channe, with the same tota mass fow rate of the two-phase fow, but the properties of the apor or iquid phase, can be defined as dp dz F = o 2 0 f G (11.72) 2 dpf 2 f0g (11.73) = dz o Dρ Through the standard equations and charts fir the singe-phase fow, the friction factors defined in eqs. (11.70)-(11.73) can be reated to the respectie Reynods numbers: xd Re (11.74) = µ (1 x) D (11.75) Re = µ Dρ 2 Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 18

19 11.3 Fow Modes Chapter 11: Two-Phase Fow Re o = D µ (11.76) Re o = D µ (11.77) The reationships between the frictiona factor and the Reynods number are different for aminar and turbuent fow. f 16 Re < 2000 = Re > Re Re 2000 (11.78) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 19

20 11.3 Fow Modes Chapter 11: Two-Phase Fow Figure 11.6 Lockhart- Martinei correations for pressure drop Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 20

21 11.3 Fow Modes Chapter 11: Two-Phase Fow The frictiona pressure gradient of the two-phase fow can be reated to those defined in eqs. (11.70) (11.73) through pressure drop mutipiers defined as 2 dpf / dz φ = (11.79) dp / dz ( ) F 2 dpf / F φ = dz ( dp / dz ) 2 dpf / o F φ = dz ( dp / dz ) 2 dpf / o F φ = dz ( dp / dz ) o o (11.80) (11.81) (11.82) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 21

22 11.3 Fow Modes Chapter 11: Two-Phase Fow Two commony-used parameters in two-phase fow inestigations are the Martinei parameter, X, which was defined in eq. (11.26), and the Chishom parameter, Y, ( dpf / dz) o Y = (11.83) ( dpf / dz) o Parameter X, the Martinei parameter, is a ratio of pressure drops of singe-phase fow terms. As can be seen from eqs. (11.79) (11.82), the pressure drop in two-phase fow can be determined if any one of the four mutipiers is known. 1 1 C X X 2 φ = / 2 (11.84) φ = 1 + CX + X 2 2 (11.85) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 22

23 11.3 Fow Modes Chapter 11: Two-Phase Fow Tabe 11.1 Vaue of C in eqs. (11.84) and (11.85). Liquid Vapor Subscripts C Turbuent Turbuent tt 20 Viscous Turbuent t 12 Turbuent Viscous t 10 Viscous Viscous 5 Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 23

24 11.3 Fow Modes Chapter 11: Two-Phase Fow According to eq. (11.78), n equas 1 for aminar fow and 0.25 for turbuent fow. The parameter B is gien by 55 0 < Y < B = 9.5 < Y < 28 Y Y > 28 2 Y (11.87) For cases where µ / µ < 1000, the foowing correation points can proide a better prediction: C2 φ 0 = C1 + (11.88) Fr We where 2 2 ρ f0 C1 = (1 x) + X ρ f0 (11.89) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 24

25 11.3 Fow Modes Chapter 11: Two-Phase Fow ρ µ µ C2 = x (1 x) 1 ρ µ µ Fr = gdρ 2 2 (11.90) (11.91) We = 2 D ρ σ (11.92) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 25

26 11.3 Fow Modes Bounds on Two-Phase Fow Chapter 11: Two-Phase Fow Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 26

27 11.3 Fow Modes The ower bound of the friction pressure drop is Chapter 11: Two-Phase Fow dp (1 x) µ x ρ µ = dz F, ower D ρ 1 x ρ µ (11.93) The upper bound of the friction pressure drop is dp (1 x) µ x ρ µ = dz F, upper D ρ 1 x ρ µ 4 (11.94) dp dz An acceptabe prediction of pressure drop can be obtained by aeraging the maximum and minimum aues, i.e., F, ae = 0.79 (1 ) x µ 1.25 D ρ ρ µ x ρ µ x x ρ µ 1 x ρ µ (11.95) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 27

28 11.3 Fow Modes Exampe 11.2 Chapter 11: Two-Phase Fow Use the Taite-Duker fow map to determine the fow regime for a fow of 8 kg/s of water-steam at 15% quaity and 180 C in a 0.1 m ID horizonta tube. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 28

29 11.3 Fow Modes Soution: Chapter 11: Two-Phase Fow The thermophysica properties of water at p=10 bar can be found from Tabe B.48 in Appendix B, i.e., ρ = kg/m 3 ρ = kg/m 3, = 1493x10-7 N-s/m 2, and µ = 149x10-7 N-s/m 2. µ The mass fux of the two-phase fow in the horizonta tube is = = = = kg/m -s 2 2 A π D π 0.1 The superficia eocities of the apor and iquid are x j = = = 29.61m/s ρ (1 x) (1 0.15) j = = = 0.976m/s ρ Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 29

30 11.3 Fow Modes Chapter 11: Two-Phase Fow The Reynods numbers of the apor and iquid phases are obtained from eqs. (11.74) and (11.75), i.e., xd Re = = = µ (1 x) D (1 0.15) 0.1 Re = = = µ The fraction coefficients for the apor and iquid phases are determined from eq. (11.78), i.e., 0.25 f = Re = 2.51 f = Re = 2.18 The frictiona pressure gradients of the apor or iquid phase fow in the channe, with its actua fow rate and properties, can be found from eqs. (11.70) and (11.71), i.e., dpf 2 f x Pa/m dz = = = Dρ Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 30

31 11.3 Fow Modes dpf 2 f (1 x) = dz Dρ = = Pa/m Chapter 11: Two-Phase Fow Therefore, the Martinei parameter can be obtained from eq. (11.26), i.e., 1/ 2 4 1/ 2 ( dpf / dz) X = = = ( dpf / dz) It can be seen from the Taite and Duker (1976) fow map that actua fow regime depends on the aues of F and K, which can be found from eqs. and, i.e., F = ρ j ρ ρ Dg = = 2.29 Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 31

32 11.3 Fow Modes 2 ρ j ρ Dj K = ( ρ ρ ) Dg µ 1/ 2 Chapter 11: Two-Phase Fow = = ( ) / 2 The fow regime is annuar-dispersed iquid (AD), as indicated by Fig Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 32

33 11.3 Fow Modes Exampe 11.3 Chapter 11: Two-Phase Fow Determine the pressure gradient due to friction in Exampe Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 33

34 11.3 Fow Modes Soution: Since the ratio of the iscosity is Chapter 11: Two-Phase Fow µ = = 10 < µ eq. (11.88) deeoped by Friede (1979), shoud be used. In addition to the properties found in Exampe 11.2, the surface tension is σ = 42.19x10-3 N/m. The Reynods numbers of the apor or iquid phases obtained from eqs. (11.76) and (11.77) are D Re0 = = = µ D Re = = = µ Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 34

35 11.3 Fow Modes Chapter 11: Two-Phase Fow The fraction coefficients for the apor and iquid phases determined from eq. (11.78) are 0.25 = Re = f 0 0 f0 = Re0 = The frictiona pressure gradients when the iquid fows in the channe with the tota mass fow rate of the two-phase fow, found from eq. (11.73), is The homogeneous density of the two-phase system is obtained from eq. (11.31): ρ ρ ρ = = = 33.30kg/m ρ x + ρ (1 x) dp F 2 f0 (1 x) = dz Dρ = = Pa/m Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 35

36 11.3 Fow Modes Chapter 11: Two-Phase Fow A the parameters necessary to use eq. (11.88) are obtained from eqs. (11.89) (11.92): 2 2 ρ f0 C1 = (1 x) + X ρ f = (1 0.15) = ρ µ µ C2 = x (1 x) 1 ρ µ µ = 0.15 (1 0.15) = Fr = = = gdρ D We = = = ρ σ Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 36

37 11.3 Fow Modes Chapter 11: Two-Phase Fow Therefore, the pressure mutipier is C φ 0 = C1 + = Fr We ( ) = The pressure gradient due to friction is obtained from eq. (11.82), i.e., dpf 2 dpf 4 6 = φ o = = Pa/m dz dz o Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 37

38 11.3 Fow Modes Void Fraction Chapter 11: Two-Phase Fow Correations for the frictiona pressure drop were discussed in the preceding subsection. To obtain the tota pressure drop, acceerationa and graitationa pressure drops must aso be determined. As can be seen from eqs. (11.44) and (11.69), knowedge of the oid fraction is required for determination of the acceerationa and graitationa pressure drops for both the homogeneous and separated fow modes. In the case of a horizonta circuar tube, the graitationa pressure drop term becomes zero but the acceerationa pressure drop terms are sti present. Therefore, correations for the oid fraction in the two-phase fow wi be discussed. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 38

39 11.3 Fow Modes Chapter 11: Two-Phase Fow Substituting eqs. (11.20) and (11.21) into eq. (11.13), one obtains G ρ (1 α ) S = G ρ α (11.96) which can be simpified by using eq. (11.23), i.e., ρ x(1 α ) S = ρ (1 x ) α (11.97) The sip ratio can aso be reated to the oumetric fow rate by substituting eqs. (11.5) and (11.6) into eq. (11.13), i.e., Q A S = Q A (11.98) Substituting eq. (11.2) into eq. (11.98), one obtains Q (1 α ) S = Q α (11.99) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 39

40 11.3 Fow Modes Chapter 11: Two-Phase Fow The reationships between the oid fraction and sip ratio can be obtained by rearranging eqs. (11.97) and (11.99): 1 α = 1 x ρ 1 + S x ρ Q α = SQ + Q (11.100) (11.101) Reationship between oid fraction and Martinei parameter φ, tt 1 (11.102) α = Where Φ,tt is the frictiona mutipier for turbuent fow φ (11.103), tt = X X Butterworth recommended a simper correation α = [ X ] (11.104) φ, tt Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 40

41 11.3 Fow Modes Chapter 11: Two-Phase Fow The correation of Premoi et a. gien in terms of the sip 1 ratio 2 y S = 1 + E1 ye2 1 ye (11.105) + 2 where β y = (11.106) ρ E1 = Re0 ρ 0.51 ρ E2 = We Re0 ρ The Weber number is defined as We = β 2 G D ρ σ (11.107) (11.108) (11.109) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 41

42 11.3 Fow Modes Chapter 11: Two-Phase Fow The oid fractions can be expressed in the foowing form (11.110) where the aues of the constant and exponents are gien in Tabe Chishom (1973b) presented a simpe correation in terms of 0.5 sip-ratio α = x ρ µ 1 + c 1 x ρ µ The ower bound of the oid fraction is x ρ µ α ower = x ρ µ q 1 ρ S = 1 x 1 ρ r s /19 1 (11.111) (11.112) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 42

43 11.3 Fow Modes Chapter 11: Two-Phase Fow Tabe 11.2 Constants and exponents for different oid fraction modes. Modes c q r s Homogeneous mode Zii (1964) Turner (1966) Lockhart-Martinei (1949) Thome (1964) Baroczy (1965) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 43

44 11.3 Fow Modes Chapter 11: Two-Phase Fow Figure 11.8 Comparison of oid fraction modes for two-phase fow (Awad and Muzychka, 2005b). Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 44

45 11.3 Fow Modes The upper bound of the oid fraction is α upper x ρ µ = x ρ µ Chapter 11: Two-Phase Fow (11.113) By aeraging the ower and upper bounds, an empirica correation for oid fraction in two-phase fow can be obtained. α ae x ρ 1 µ = x ρ µ x ρ µ x ρ µ 16/ (11.114) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 45

46 11.3 Fow Modes Chapter 11: Two-Phase Fow Figure 11.9 Void fraction ersus quaity (Awad and Muzychka, 2005b). Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 46