Chapter 2.docx

Chapter 2 The Fundamentals of Two-Phase Flow 2.1 Flow pattern Two-phase flow is flow of pair immiscible fluid. The flow can be encountered external devices or internally. The pair immiscible fluid may content; liquid-liquid (refrigerant-oil in refrigeration system), liquid-solid (water-slag in engine cooling system), gas-solid (air-coal in pneumatic conveyor) or gas-liquid (steam-water in steam power cycle). Here, the study is emphasized on two-phase flow of gas-liquid. Mostly, in the authentic gas-liquid two-phase flow process sink or release heat. If the heat is sunk, called evaporation and another one is condensation, in which the heat is discharged. The evaporation and condensation is very complicated processes, since the fluid transform from liquid to gas or vice versa. Therefore, the basic fundamentals two-phase flow theory is derived from adiabatic two-phase flow. Air-water is the most pair immiscible fluid, used as experiment since decades, due to their unique. Air and water have contras in properties, table 2.1 and table 2.2. Therefore, this combination promotes stability two-phase form, resulting better investigation. Table 2.1 Air properties in atmospheric pressure, ThermopediaTM T



cp

k

x10-6

bx10-3

oC

kg.m-3

kJ.(kg.K)-1

w.(m.K-1)

m2.s-1

K-1

-150 -100 -50 0 20 40 60 80 100 120 140

2.793 1.980 1.534 1.293 1.205 1.127 1.067 1.000 0.946 0.898 0.854

1.026 1.009 1.005 1.005 1.005 1.005 1.009 1.009 1.009 1.013 1.013

0.0116 0.0160 0.0204 0.0243 0.0257 0.0271 0.0285 0.0299 0.0314 0.0328 0.0343

3.08 5.95 9.55 13.30 15.11 16.97 18.90 20.94 23.06 25.23 27.55

8.21 5.82 4.51 3.67 3.43 3.20 3.00 2.83 2.68 2.55 2.43

Pr

0.76 0.74 0.725 0.715 0.713 0.711 0.709 0.708 0.703 0.70 0.695

14 160 180 200

0.815 0.779 0.746

1.017 1.022 1.026

0.0358 0.0372 0.0386

29.85 32.29 34.63

2.32 2.21 2.11

0.69 0.69 0.685

Table 2.2 Water properties in atmospheric pressure, ThermopediaTM T o

C

0 0.01 4 5 10 20 30 40 50 60 70 80 90 100



Pa -2

kN.m

0.6 0.9 0.9 1.2 2.3 4.3 7.7 12.5 20.0 31.3 47.5 70.0 101.33

kg.m-3

cp

x10-6

h -1

-1

kJ.(kg.K)

kJ.kg

4.210

0

4.204 4.193 4.183 4.179 4.179 4.182 4.185 4.191 4.198 4.208 4.219

21.0 41.9 83.8 125.7 167.6 209.6 251.5 293.4 335.3 377.2 419.1

916.8 999.8 1000.0 1000.0 999.8 998.3 995.7 992.3 988 983 978 972 965 958

bx10-3

m2.s-1

K

1.792

-0.07

1.304 1.004 0.801 0.658 0.553 0.474 0.413 0.365 0.326 0.295

Pr

-1

0.160 0.088 0.207 0.303 0.385 0.457 0.523 0.585 0.643 0.665 0.752

13.67

9.47 7.01 5.43 4.34 3.56 2.99 2.56 2.23 1.96 1.75

Note: T: : cp: k: :

temperature density Specific heat Thermal conductivity kinematic viscosity

b: Pa: h: Pr:

expansion coefficient vapor pressure enthalpy Prandl’s number

If the air and water pass through a tube in different velocity, than they form unique gas-liquid configuration. It is termed; flow-pattern. It has been investigated since several decades.

2.2 Flow Patterns in Vertical Tubes For co-current up flow of gas and liquid in a vertical tube, the liquid and gas phases distribute themselves into several recognizable flow structures. These are referred to as flow patterns and they are depicted in Figure 2.1 and can be described as follows:

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 Bubbly flow. Numerous bubbles are observable as the gas is dispersed in the form of discrete bubbles in the continuous liquid phase. The bubbles may vary widely in size and shape but they are typically nearly spherical and are much smaller than the diameter of the tube itself.  Slug flow. With increasing gas void fraction, the proximity of the bubbles is very close such that bubbles collide and coalesce to form larger bubbles, which are similar in dimension to the tube diameter. These bubbles have a characteristic shape similar to a bullet with a hemispherical nose with a blunt tail end. They are commonly referred to as Taylor bubbles after the instability of that name. Taylor bubbles are separated from one another by slugs of liquid, which may include small bubbles. Taylor bubbles are surrounded by a thin liquid film between them and the tube wall, which may flow downward due to the force of gravity, even though the net flow of fluid is upward.  Churn flow. Increasing the velocity of the flow, the structure of the flow becomes unstable with the fluid traveling up and down in an oscillatory fashion but with a net upward flow. The instability is the result of the relative parity of the gravity and shear forces acting in opposing directions on the thin film of liquid of Taylor bubbles. This flow pattern is in fact an intermediate regime between the slug flow and annular flow regimes. In small diameter tubes, churn flow may not develop at all and the flow passes directly from slug flow to annular flow. Churn flow is typically a flow regime to be avoided in two-phase transfer lines, such as those from a reboiler back to a distillation column or in refrigerant piping networks, because the mass of the slugs may have a destructive consequence on the piping system.  Annular flow. Once the interfacial shear of the high velocity gas on the liquid film becomes dominant over gravity, the liquid is expelled from the center of the tube and flows as a thin film on the wall (forming an annular ring of liquid) while the gas flows as a continuous phase up the center of the tube. The interface is disturbed by high frequency waves and ripples. In addition, liquid may be entrained in the gas core as small droplets, so much so that the fraction of liquid

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entrained may become similar to that in the film. This flow regime is particularly stable and is the desired flow pattern for two-phase pipe flows.  Wispy annular flow. When the flow rate is further increased, the entrained droplets may form transient coherent structures as clouds or wisps of liquid in the central vapor core.  Mist flow. At very high gas flow rates, the annular film is thinned by the shear of the gas core on the interface until it becomes unstable and is destroyed, such that all the liquid in entrained as droplets in the continuous gas phase, analogous to the inverse of the bubbly flow regime. Impinging liquid droplets intermittently wet the tube wall locally. The droplets in the mist are often too small to be seen without special lighting and/or magnification.

Fig. 2.1 Two-phase flow pattern in vertical upward flow, ThermopediaTM

2.3 Flow Patterns in Horizontal Tubes Two-phase flow patterns in horizontal tubes are similar to those in vertical flows but the distribution of the liquid is influenced by gravity that acts to stratify the liquid to the bottom of the tube and the gas to the top. Flow patterns for co-current flow of gas and liquid in a horizontal tube are shown in Fig. 2.2 and are categorized as follows: 

Bubbly flow. The gas bubbles are dispersed in the liquid with a high concentration of bubbles in the upper half of the tube due to their buoyancy. When shear forces

17

are dominant, the bubbles tend to disperse uniformly in the tube. In horizontal flows, the regime typically only occurs at high mass flow rates. 

Stratified flow. At low liquid and gas velocities, complete separation of the two phases occurs. The gas goes to the top and the liquid to the bottom of the tube, separated by an undisturbed horizontal interface. Hence the liquid and gas are fully stratified in this regime.



Stratified-wavy flow. Increasing the gas velocity in a stratified flow, waves are formed on the interface and travel in the direction of flow. The amplitude of the waves is notable and depends on the relative velocity of the two phases; however, their crests do not reach the top of the tube. The waves climb up the sides of the tube, leaving thin films of liquid on the wall after the passage of the wave.



Intermittent flow. Further increasing the gas velocity, these interfacial waves become large enough to wash the top of the tube. This regime is characterized by large amplitude waves intermittently washing the top of the tube with smaller amplitude waves in between. Large amplitude waves often contain entrained bubbles. The top wall is nearly continuously wetted by the large amplitude waves and the thin liquid films left behind. Intermittent flow is also a composite of the plug and slug flow regimes. These subcategories are characterized as follows: o Plug flow. This flow regime has liquid plugs that are separated by elongated gas bubbles. The diameters of the elongated bubbles are smaller than the tube such that the liquid phase is continuous along the bottom of the tube below the elongated bubbles. Plug flow is also sometimes referred to as elongated bubble flow. o Slug flow. At higher gas velocities, the diameters of elongated bubbles become similar in size to the channel height. The liquid slugs separating such elongated bubbles can also be described as large amplitude waves.



Annular flow. At even larger gas flow rates, the liquid forms a continuous annular film around the perimeter of the tube, similar to that in vertical flow but the liquid film is thicker at the bottom than the top. The interface between the liquid annulus and the vapor core is disturbed by small amplitude waves and droplets may be dispersed in the gas core. At high gas fractions, the top of the tube with its thinner

18

film becomes dry first, so that the annular film covers only part of the tube perimeter and thus this is then classified as stratified-wavy flow. 

Mist flow. Similar to vertical flow, at very high gas velocities, all the liquid may be stripped from the wall and entrained as small droplets in the now continuous gas phase.

Fig. 2.2 Two-phase flow pattern in horizontal flow

2.4 Flow Patterns Map It is necessary to predict regimes as a basis for carrying out calculations on twophase flow, and the usual procedure is to plot the information in terms of a flow regime map. Many of these maps are plotted in terms of primary variables (superficial velocity of the phases or mass flux and quantity, for instance), but there has been a great deal of work aimed at generalizing the plots, so that they can be applied to a wide range of channel geometries and physical properties of the fluids. A generalized map for vertical flows is shown in Fig. 2.3 and is due to Hewitt and Roberts (1969) (see Hewitt, 1982). This map is plotted in terms of the superficial momentum fluxes of the two-phase

fUf2 and gUg2. A generalized flow pattern map for horizontal flow is that of Taitel and Dukler (1976) (see Dukler and Taitel, 1986), and is illustrated in Fig. 2.4. This is plotted in terms of the following parameters:

19

 dp   F  dz  f 2 X   dp   F  dz  g F

g  f  g

(2.1)

ug

(2.2)

D.g. cos 

 gug 2 D.u f K   f   g .D.g. cos   f 2

   T    f  

  dp    F  dz  f    g g. cos     

1

(2.3)

2

(2.4)

gug2

fuf2

Fig. 2.3 Flow pattern map obtained by Hewitt and Roberts (1969) for vertical two-phase co-current upwards flow in a vertical tube., ThermopediaTM

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Fig. 2.4 Flow pattern map for horizontal co-current flow obtained by Taitel and Dukler (1976). (See Dukler and Taitel, 1986), ThermopediaTM where (dpF/dz)f and (dpF/dz)g are the pressure gradients for the liquid phase and gas phase respectively, flowing alone in the channel, f and g are the phase densities, uf and ug are the superficial velocities of the phases, D the tube diameter, f the liquid kinematic viscosity, g the acceleration due to gravity, and  the angle of inclination of the channel. Taitel et al. (1980) also produced a flow pattern map for vertical flow, but this has met with less widespread use. Following similar approaches, Barnea (1987) has produced a unified model for flow pattern transitions for the whole range of pipe inclinations.

21

2.5 Two-phase flow model The complicated two-phase flow patterns can be approached as a simple model as illustrated on Fig 2.3. A gas and a liquid pass through channel having constant A cross section area. They form gas phase velocity ug normal to gas area Ag and liquid velocity uf normal to liquid area Af . Therefore total cross section area A=Ag+Af. If the point of view is in instant channel length dz at instant time dt.

Fig. 2.5 Two-phase flow model, Collier (1981) Thus the ug,uf , Ag and Af are approached to constant. Hence following set equations can be determined; 

The void fraction , ratio of gas cross section area Ag to total area A



Ag A

, so 1    

Af A

(2.5)

Void fraction is an essential dimensionless for two-phase flow parameters calculation. Since, in the real case Ag is not always constant along z, than the equation 2.5 is valid for very limited incident only. Therefore, most of void fraction is not defined

22

based on area, but based on volume, termed as volume void fraction. Later, several void fraction correlations, volume based, shall be presented soon. 

The mass quality x, ratio of gas mass flow rate Wg to total mass flow rate W x

Wg

, so (1  x) 

Wg  W f

Wf

(2.6)

Wg  W f

It is should be remarks, that the mass quality or some time called as ‘quality’ only is very different from void fraction. Because, quality is related to mass which strongly depend on density . However, both of quality and void fraction have particular proportionality, which will be discussed, later. 

The mass velocity/flux

G 

W u  u  A 

(2.7)

The mass flow rate Wg  GAx and W f  GA(1  x)



(2.8)

The phase velocity ug 

Wg

 g Ag

and u f 

Wf

(2.9)

 f Af

Where, the mass flow W is proportional to quantity, volume flow rate Q than; ug 

Qg Ag

and u f 

Qf

(2.10)

Af

Therefore, the phase velocity can be formed as function of void fraction and quality, u=f(, x);

ug  

Gx

 g

and u f 

G(1  x)  f (1   )

(2.11)

The volumetric quality



Qg Qg  Q f

so (1   ) 

Qf Qg  Q f

(2.12)

All of forgoing equations are based on phase area (Ag and Af), in which is vary along channel length z and time t. Accordingly, it is urgent to simply the equation based

23

on total cross section area A which is equal to tube cross section area, constant. This is superficial velocity parameter j. 

The volumetric flux or the superficial velocity, j

j

Q Q Q , so j g  g and j f  f A A A

Gx

j g  u g  j  Gg  j g  g  Gx ,

g

(2.13)

and j f  u f (1   )  j (1   ) 

G(1  x)

f

G f  j f  f  G (1  x) in which G  G g  G f

(2.14) (2.15)

Superficial velocity is very important parameter for defining the phases velocity ug and uf , by condition of which void fraction is known. Also, superficial velocity is easy parameter to calculate, since the variables are easy to measure, as following measured variables, quantity Q. This work, quantity of each phase was measured carefully, by means, the phase is separated in separator tube, followed by quantifying liquid and gas volume Vg and Vf in certain time interval and the last phase quantity is determined by; Qg 

Vg tg

and Q f 

Vf

(2.16)

tf

Since the gas and liquid velocity is different. It is very important to define ratio in between gas velocity ug and liquid velocity uf, termed as the slip factor S 

The slip ratio S

ug uf



 x   f   W f  g Ag  1  x   g

Wg  f A f

 1        

(2.17)

Later, the control volume of instant two-phase flow in Fig. 2.5 can be solved analytically.

2.5.1 Conservation of Mass By assuming the system is adiabatic and the both of phase incompressible, then the total phase mass flow rate is the sum of gas flow rate and liquid flow rate

24

Wg  W f  W

(2.18)

Because the total mass flow rate is constant, then the differentiation of equation 2.18 results; dWg  dW f

(2.19)

Since; Wg  Ag  g u g  Wx

(2.20)

And; W f  A f  f u f  W (1  x) Then;

(2.21)

d dx dWg ( Ag  g u g )  W  dz dz dz

(2.22)

d dx dW f ( A f  f u f )  W  dz dz dz

(2.23)

2.5.2 Conservation of Momentum Also, from Fig. 2.5 the momentum conservation is; pAg  ( p  dp) Ag  dFg  S  Ag dz. g g sin   [(Wg  dWg )(u g  du g )  Wg u g  dWg u f ]

(2.24) S similar force exerted with respect to the gas-liquid interface, equation 2.24 can be simplified to  Ag dp  dFg  S  Ag dz g g sin   Wg du g  dWg u g  dWg u f

(2.25)

Relationship for liquid in which S is a force on the liquid  A f dp  dF f  S  A f dz f g sin   W f du f

(2.26)

Adding eq. 2.25, 2.26 and using eq. 2.19 yields  Adp  dFg  dF f  g sin dz[ A f  f  Ag  g ]  d (W f u f  Wg u g )

(2.27)

The net frictional force acting on each phase;  dp  dp   (dFg  S )   Ag  gF dz ; (dF f  S )   A f  fF dz  dz  dz  

 dp  (dFg  dF f )   A F dz  dz   dp  The term  F  represent frictional pressure drop,  dz 

(2.28) (2.29)

25

while total pressure drop (overall static pressure gradient) is  dp   dp   dp   dp  z     F    a    dz   dz   dz   dz 

(2. 30)

Where; 2 2 d  x  g 1  x   f   dp  1 d   a  (Wg u g  W f u f )  G 2    1     dz    dz  A dz

(2.31)

And

Af  Ag   dp    z   g sin    g   f   g sin  g  (1   )  f A  dz  A 





(2.32)

It should be emphasized at this point that the frictional component has been defined in terms of the force (dFg  dF f ) To solve all of forgoing equation is hard due to existing two unknown differential variable ug and uf. Therefore, approach solving is required. There are two approaching models i.e. homogeneous model and separated model. Homogeneous model assumes that both of phase gas and liquid pass in equal velocity, in its mean velocity. So as, the two phase flow problem considers as single phase flow and all properties are determined based on mean properties of both phases. Meanwhile, the separated model assumes that the phases is artificially segregated into stream; one of is liquid and another one is gas, and each phase velocity is the mean velocity of each phase, so that is constant. If both of phases have equal mean velocity, the equation reduces to those of homogeneous model.

2.6 The Homogeneous Model 2.6.1 Derivation of Model and Assumption; a. Equal vapor and liquid velocities b. The attainment of thermodynamic equilibrium between the phases c. The use of a suitably defined single-phase friction factor for two-phase flow Continuity, W  A u

(2.33)

26

Momentum

 Adp  dF  Ag sin dz  Wdu Where  



(2.34)

 



Q j 1  x g  (1  x) f   f  x fg   W G 

From the assumption of a; u f  u g  u

(2.36)

u  G  j

So that

(2.35)

(2.37)

And



x g



,

1    

1  x  f 

dF   W Pdz

 1   

(2.38) (2.39)

Where P is wall perimeter of circular inner tube

 u 2    W  fTP  2  

(2.40)

2  dp  1 dF  W P fTP P  u     F    A A  2   dz  A dz

(2.41)

Where

For circular channel (P/A=4/D), P; Perimeter, so 2  dp  2 f G  2 fTP Gj   F   TP  D D  dz 

(2.42)

From eq. (2.31) d u  d    dp    a  G  G2 dz dz  dz 

(2.43)

Neglecting the compressibility of the liquid phase

d g  dp  d   dx   fg x   dz dz dp  dz 

(2.44)

From eq (2.32) and eq (2.38) g sin   dp   z    g sin     dz 

Eqs (2.42), (2.43), (2.45), (2.30) become

(2.45)

27

  fg 2 f TP G 2 f  1  x D  f  dp      dz 

    G 2 f  fg     f  d g 1  G x  dp 2

 dx    dz 

g sin  

  fg      f 

 f 1  x 

  

(2.46)

=0, g closed to constant

Accordingly, general equation of pressure gradient along z for homogeneous modeling is; 2  dp  2 fTPG  f    D  dz 

   fg    g sin    G 2 f  fg  dx  1  x     dz        f   f   f 1  x fg     f 

(2.47)

2.6.2 The Two-Phase Friction Factor All the terms in eq. (2.47) are definable, except one (fTP); (a) fTP with assumption all the fluid is liquid, an denote as ffo as function of Reynolds number (GD/f) and the pipe relative roughness (/D). so Equation (2,42) becomes 2    fg    dp  2 f foG  f     dp F  1  x fg  F  1  x   D  dz   dz  fo     f   f

   

(2.48)

 dp  Where   F  is frictional pressure gradient calculated from the Fanning equation for  dz  fo total flow (liquid plus vapour) assumed to flow as liquid, so

2 f foG 2 f  dp   F  D  dz  fo

(2.49)

(b) the viscosity using mean viscosity  of liquid and gas, where x=0,    f ; and x=1,    g

(2.50)

and the  correlation by McAdam, et.al.

1





x

g



1 x

f

(2.51)

28

Cicchitti, et.al.   x g  1  x  f



Dukler et.al.

Akers et al.

tp 

(2.52)

x g  g  1  x  f  f

(2.53)

x g  1  x  f

f   1  x   x g    f 

   

0.5

(2.54)

   

 tp   f

Owens

(2.55)

Beattie and Whalley tp   g  1   1  2.5  f , where  

 tp 

Lin et al

x g

(2.56) (2.57)

 f  x fg

 f g  g  x1.4  f   g 

(2.58)

Assuming that the friction factor may be expressed in term of the Reynolds number by Blasius equation f TP

 GD    0.079  TP 



1 4

 GD   0.079    



1 4

(2.59)

For equation (2.51) the         dp   dp     F    F  1  x fg  1  x fg       dz   dz  fo   f    g 

(2.60)

In general equation;

 dp   dp  2   F    F   fo  dz   dz  fo

(2.61)

 fo 2 , known as the two-phase frictional multiplier;

 fo 2

   fg  1  x    f

    1  x fg       g

   



1 4

(2.62)

29

2.7 The Separated Flow Model 2.7.1 Derivation of Model and Assumption; a. Each phase velocity is constant b. The attainment of thermodynamic equilibrium between the phases c. The use of empirical correlations or simplified concepts to relate  2 and  to independent variable of the flow The momentum equation  x 2 g (1  x) 2 f   dp   dp  2 d      FG     g sin   g  (1   )  f dz   (1   )   dz   dz 





 2 f foG 2 f  2  dp   dp  2  F    F   fo    fo D  dz   dz  fo  

(2.63)

(2.64)

May be expressed as liquid phase  2 f f G 2 (1  x) 2  f   dp   dp  2  F    F  f    D  dz   dz  f  

(2.65)

Using the Blasius equation (2.54) 1

ff f fo

 1 4    (1  x) 

(2.66)

Eq (2.47) and 2.48

 fo 2   f 2 (1  x) 2

ff f fo

  f (1  x)1.75 2

2 d  x  g 1  x   f  dp    a  G2   1    dz    dz  2

(2.67)   

(2.68)

Compressibility negligible and expansion theorem

d  x  g 1  x   f   1    dz   2

2

 dx  2 x g 21  x  f     1     dz  

2 2       1  x   f x  g          2 2     1      x  p   

2 2 dp  x 2 d g      1  x   f x  g           2 2  dz   dp  p  x     1       

(2.69)

30

2 f foG 2 f D

 dp      dz 

 fo 2  G 2

2 2 dx  2 x g 21  x  f  d  1  x   f x  g         g sin   g   f 1    1     dx  1   2 dz    2     x 2  d g  d  1  x 2 f x 2 g    1  G 2    2   2      dp  dp  1    0, compressibility of gaseous phase isnegligible





(2.70)

So that; 2 2 2  2 x g 21  x  f  d   1  x   f x  g    dp  2 f foG  f 2 2 dx       fo  G   2   g sin   g   f 1       2 1     dx  1    D dz      dz   





(2.71) 2.7.2 The evaluation of the Two-phase Multiplier  fo and void fraction  2

2.7.2.1 The Lockhart-Martinelli correlation 

flow regime were defined on the basis of the behavior of the flow (viscous or turbulent) when the phases were considered to pass alone through the channel



The liquid and gas phase pressure drop were considered equal irrespective of the detail of the particular flow pattern.

  dp   dp   dp fF    F  gF       dz   dz   dz

(2.72)

The frictional pressure drop for liquid and gas

 dp  2 ff fuf  fF   Df  dz 

2

 dp  2 f g  gug   gF   Dg  dz 

2

Using Blasius

(2.73)

(2.74)  2 Af    D f  4  

(2.75)

 2 Ag    Dg  4 

(2.76)

31

  f u f Df  ff  Kf     f 

f   2

(2.59, 2.67, 2.69, 2.71)

g  2

n2

 D  D  f

 dp   F  dz  f

   

n

(2.77)

5 n

(2.78)

 dp   F  dz  g



n 2

 D  D  g

   

5n

(2.79)

Lockhart-Martinelli assumption to the case of annular flow, Dividing eq 2.69 by A=D2/4     1    D D  f  

 f  (1   ) 2

n2

2

 D  D  f

(2.80)    

n 1

(2.81)

For annular flow with liquid film of thickness (),

Df 

4D 4 4D   4 ; 1     D 2 D D

Thus

D D 1   D f 4 1    So that;  f  1    2

n 2

1   n1  1   3

(2.82)

Result is incorrect, the correct result  f  1    2

2

(2.83)

Empirical multiplier as function of X  dp   F dz  f 2 X 

 dp   F  dz  g

(2.84)

Where;

 f 2  1

C 1  2 X X

 g 2  1  CX  X 2

(2.85)

32

Liquid

Gas

C

Turbulent

Turbulent

tt

20

Viscous

Turbulent

vt

12

Turbulent

Viscous

tv

10

Viscous

Viscous

vv

5

2.7.2.2 Universal approach to predicting two-phase frictional pressure drop, Kim, S.M and Mudawar, I. (2012)

Where  f

2

 dp   dp  2      f  dz  F  dz  f

(2.86)

 dp   dp  2      g  dz  F  dz  g

(2.87)

 dp   F C 1  dz  f 1  2 , X 2  X X  dp   F  dz  g

(2.88)

2 f f  f G 2 (1  x) 2 2 f g g G 2 x 2  dp   dp   F  ;  F  Dh Dh  dz  f  dz  g

(2.89)

1

f k  16 Re k , for Rek 2000 f k  0.079 Re k

0.25

f k  0.046 Re k

0.2

, for 2000≤ Rek 20,000

, for Rek 20,000

For laminar flow in rectangular channel f k Re k  24(1  1.3553  1.9467  2  1.7012 3  0.9564 4  0.2537  5 )

(2.90)

Where subscript k denotes f or g for liquid and vapor phases, respectively

Re f 

G1  x Dh

f

, Re g 

GxDh

g

, Re fo 

GDh

f

, Suratman number Su go 

 gDh g 2

(2.91)

33

Liquid

Gas(Vapor)

Turbulen

Turbulent

C 0.39 Re fo

t Turbulen

Laminar

0.03

Su go

4

8.7 x10 Re fo

t Laminar

 f    g

Su go

0.50

Turbulent 0.0015 Re fo

Laminar

0.17

0.10

Laminar

0.59

5

3.5 x10 Re fo

Su go

0.44

0.19

Su go

   

0.35

 f    g

 f    g

0.50

(2.92)

   

 f    g

   

0.14

(2.93)

0.36

(2.94)    

0.48

(2.95)

Other proper correlations 1. Friedel (D>4mm, air-water,air-oil,R-12)

 dp   dp  2       fo  dz  F  dz  fo

 fo

2

  1  x   x  g   f 2

Frtp 

2

 f go      3.24 x 0.78 1  x 0.224  g  f    fo   f

(2.96)

   

0.91

 g    f

   

0.19

0.7

   1  g  Frtp  0.045Wetp  0.035    f   (2.97)

GDh G 2 Dh 1 G2 We  , , H  , Re go  tp 2  H x g  1  x  f g gDh  H

(2.98)

2. Muller-Steinhagen and Heck (D=4-392 mm, air-water, water, hydrocarbon, refrigerant)   dp   dp    dp   dp     dp  3 1/ 3       2      x 1  x     x  dz  F   dz  go  dz  go  dz  fo    dz  fo 

(2.99)

34

3. Lee and Lee (Dh=0.78-6.67mm, Air-water)

 dp   dp  2      f  dz  F  dz  f

(2.100)

f  j C 1 1  2 ,   f f ,   X X  f Dh  2

f

2

Cvv  6.833  108 1.317 0.719 Re fo Cvt  6.185  102 Re fo

0.726

0.557

, Ctv  3.627 Re fo

, Ctt  0.048 Re fo

(2.101) 0.174

0.451

(2.102) (2.103)

4. Chen (D=1.02-9mm, adiabatic, air-water, R410A, Ammonia) 2  Dh / 2  dp   dp      , Bo  g  f   g       dz  F  dz  fo,Friedel

For Bo*<2.5  

For Bo*2.5  

0.0333 Re fo Re g

0.09

(2.104)

0.45

1  0.4e 

(2.105)

 Bo*

Wetp

0.2

(2.106)

2.5  0.06 Bo 

5. Sun and Mishima (Dh=0.506-12mm, Air-water, Refrigerant, CO2)  Re g C 1  dp   dp  2 2       f ,  f  1   2 , C  1.79 X X  dz  F  dz  f  Re f

Re f 

G1  x Dh

f

, Re g 

GxDh

g

0.4

  1  x  0.5   (2.107)   x  

(2.108)

2.8 Void Fraction 2.8.1 Homogeneous Model



1 1 x  g 1    x  f

(2.109)

35

2.8.2 Zivi Void fraction



1  1  x   g 1    x   l

  

2

(2.110) 3

2.8.3 Smith Void fraction

  l 1 x   0.4    1  x  g x   1  g   0.4  0.6 1 x  l  x  1  0.4   x  

      

1

(2.111)

Homogenous model suitable for Bubbly and Disperse models

2.8.4. Local Void Fraction Using Drift Model x   x 1  x  U GU      C   G  o  G  L  m 

1

(2.112)

Valuable only if U GU  0.05 U At elevated pressure, Zuber (1967);

Co  1.13 With

U GU

g  L   G   1.41  L2  

1

4

(2.113)

Regardless flow regime. This also can be implemented for bubbly flow, vertical up flow, with particular value of Co Geometry

In-dim (mm)

Co

Tube

 50

Co=1-0.5Pr

Tube

 50

Co=1.2 for Pr

Pr ,Reduced pressure Except for Pr0.5, where Co=1.2

36

 50

Tube

Co=1.2-0.4(Pr-0.5)

for Pr >0.5

Co=1.4-0.4(Pr-0.5)

Rectangular

For bubbly flow, vertical up flow, Wallis (1969);

Co  1.0 U GU

g  L   G   1.53  L2  

1

4

(2.114)

2 2 Amc  1 d mc , Ac  1 d c , Wg   g Qg , W f   f Q f 4 4

2.9 Uniformity Distribution Uniformity distribution in between two outlet channels is equated Rg 

Qgcu  Qgcd Qgcu  Qgcd

, and R f 

Q fcu  Q fcd

(2.115)

Q fcu  Q fcd

Rg and Rf is dimensionless. If Rg >0, gas phase tend to go to upper channel and vice versa. If Rg=0, the gas is uniform. Similarly, for Rf is. If Rf >0, liquid phase tend to go to upper channel and vice versa. If Rf=0, the liquid is uniform

2.10 Pressure Losses through Merged Pipe Distributor Merge pipe distributor have inlet diameter 8 mm and 2 pair 5 mm outlet diameter, as

5mm

r  4 8

5mm

8 mm

shown in Fig. 2.6 (a)

(a)

(c)

37

(b) Fig. 2.6 Merged pipe distributor; (a) 2d sketch and Simplified as Incline straight tube, (b) simplified as converging diverging nozzle and (c) 3d sketch

Because there are no correlations, related to merge pipe distributor.

Three approach

correlations are tested; area changes by Tapucu, 1989, straight incline pipe and Teejunction.

2.10.1 Pressure Losses Caused by Area Changes in a Single-Channel Flow, by Tapucu 1989.

38

Fig. 2.7 Sharp insert Assumption: Contraction and expansion cannot be separated in the case of short insert. 

Janssen & Kervinen (1964), assuming that the contraction losses are small compared to the expansion losses.

2 2 2  1  2  C  2  G1  1   L 2  1   C   2      1  x  1      pSI   x         2  L   2 C 2   G  3    4   1   3    1   4           

  1 C   1 C     1  x 2     2C  L x 2     1   3 1   4    G   3  4  

(2.116)

Where;

  1 2  3   4  

A A2 A2  and C  3 A2 A1 A4

Assumed as constant void fraction 1 Gi  1  pSI    1  2    C  2



2

(2.117)

If use Momentum Energy Equation of Hewitt & Hall Taylor (1970) based on Jansen assumption 2 1  G  2 1  1     1 H pSI  G1  2 C  3  4 

 1  1 1   2  2   2    C   3 4 

(2.118)

Where;



A A2 , and C  3 A2 A1

Assumed as constant void fraction G pSI  1 2

2

  2  1     1  1  H2   1     2       C     C  

(2.119)

39

Where   momentum specific volume and is defined as

 

x2

G



1  x 2 1    L

(2.120)

and 3  1  x   2 G 2 1   2  L 2 x 1  x   

x3

 2  1

H

G

(2.121)

(2.122)

L

   1  , is the momentum density Assuming a constant void fraction along the duct,

2.10.2 Pressure drop by Energy equation as T-Junction, Hwang et al. (1988)

p1i ,TP

2 2 2  Hom,i  Gi   G1        K1i ,TP G1   2   E ,i    E ,1   2 L



(2.123)



Where E is the energy density, defined as  1  x 3 x3  E   2  2 2   L 1     G  

 12

(2.124)

And K1-i,TP is a two-phase pressure loss coefficient formulated as K1i ,TP  K1i ,SP

   For Annular and churn flows, Rectangular Channel   1.60 L2 Hom,i    Hom,1     For plug and bubbly flow, Rectangular Channel   2.57 L2 Hom,i    Hom,1 

  L  Hom,i For circular channel,    2   Hom,1 For Re  5000, Rectangular channel

  

0.586

(2.125)

0.146

(2.126)

(2.127)

40

k1i

W  W   0.477  0.21 i   0.744 i   W1   W1 

2

(2.128)

For low Re, round channel; k1i

W  W   1  0.8285 i   0.6924 i   W1   W1 

2

(2.129)

.i:2,3: Upper chennel, lower channel

 Hom

 x 1  x        L   G

1

(2.130)

References Collier, J.G., (1981), Convective boiling and condensation, Second Edition, McGraw-Hill, ISBN: 0070117985. Thome, J.R., (2010), Engineering data book III, Wolverine Tube, Inc. ThermopediaTM , A to z guide to thermodynamics, Heat and mass transfer, and fluid engineering, http://www.thermopedia.com/ Hwang, S.T. and Lahey, R.T., (1988), A study on single- and two-phase pressure drop in branching conduits, Exp. Therm. Fluid Sci. 1, p. 111–125 Tapucu, A., Teyssedou, A., Troche, N. and Merilo, M., (1989), Pressure losses caused by changes in a single channel flow under two-phase flow, Int. J. Multiphase Flow 151, p. 51-64 Kim, S.M. and Mudawar, I., (2012), Universal approach to predicting two-phase frictional

41

pressure drop for adiabatic and condensing mini/micro-channel flows, Int. J. Heat and Mass Transfer 55, p. 3246–3261 Hewitt, G. F. & Hall Taylor, N. S. (1970), Annular Two-phase Flow, Pergamon Press, Oxford.