tentative criteria for the transition of the flow from one type to another. Equations to

*n 0.2 1.0 0.2 1.0 Cl 0.046 16 0.046 16 Cg 0.046 0.046 16 16 Rel > 2000 < 1000 > 2000 < 1000 Reg > 2000 > 2000 < 1000 < 1000*

*f l*

,

**turbulentturbulent**

Also, they presented the relationship of

laminar-turbulent

turbulent- laminar

laminar-laminar

mechanisms of liquid-gas (the third type) was not given.

given in Table 4.

*f g*

In addition, they expressed the two-phase frictional pressure drop in terms of factors, which multiplied single-phase drops. These multipliers were given by:

$$\left\|\phi\right\|\_{l}^{2} = \frac{\left(dp \;/\, dz\right)\_{f}}{\left(dp \;/\, dz\right)\_{f,l}}\tag{181}$$

$$\left\| \phi \right\|\_{\mathcal{S}}^2 = \frac{\left( dp \,/\, dz \right)\_f}{\left( dp \,/\, dz \right)\_{f, \mathcal{S}}} \tag{182}$$

Using the generalized Blasius form of the Fanning friction factor, the frictional component single-phase pressure gradient could be expressed as

Two-Phase Flow 307

$$\left(\frac{dp}{dz}\right)\_{f,l} = \frac{2C\_l \mu\_l^n \mathbf{U}\_l^{2-n} \rho\_l^{1-n}}{d^{1+n}}\tag{183}$$

$$
\left(\frac{dp}{dz}\right)\_{f,\mathcal{g}} = \frac{2C\_{\mathcal{g}}\mu\_{\mathcal{g}}^n \mathcal{U}\_{\mathcal{g}}^{2-n} \rho\_{\mathcal{g}}^{1-n}}{d^{1+n}}\tag{184}
$$

Values of the exponent (*n*) and the constants *Cl* and *Cg* for different flow conditions are given in Table 4.

306 An Overview of Heat Transfer Phenomena

*<IP_ADDRESS>. Lockhart-Martinelli model*

al. (2009).

For void fraction, the separated model is used by both analytical and semi-empirical methods. In the analytical theories, some quantities like the momentum or the kinetic energy is minimized to obtain the slip ratio *(S)*. The momentum flux model and the Zivi model (1964)

For two-phase flow modeling in microchannels and minichannels, it should be noted that the literature review on this topic can be found in tabular form in a number of textbooks such as Celata (2004), Kandlikar et al. (2006), Crowe (2006), Ghiaasiaan (2008), and Yarin et

For two-phase frictional pressure gradient, a number of models have been developed with varying the sophistication degrees. These models are all reviewed in this section in a

Lockhart and Martinelli (1949) presented data for the simultaneous flow of air and liquids including benzene, kerosene, water, and different types of oils in pipes varying in diameter from 0.0586 in. to 1.017 in. There were four types of isothermal two-phase, two-component flow. In the first type, flow of both the liquid and the gas were turbulent. In the second type, flow of the liquid was viscous and flow of the gas was turbulent. In the third type, flow of the liquid was turbulent and flow of the gas was viscous. In the fourth type, flow of both the liquid and the gas were viscous. The data used by Lockhart and Martinelli consisted of experimental results obtained from a number of sources as detailed in their original paper and covered 810 data sets including 191 data sets that are for inclined and vertical pipes and

Lockhart and Martinelli (1949) correlated the pressure drop resulting from these different flow mechanisms by means of the Lockhart-Martinelli parameter *(X)*. The Lockhart-

> 2 (/) (/)

In addition, they expressed the two-phase frictional pressure drop in terms of factors, which

(/) (/)

*dp dz dp dz*

(/) (/)

Using the generalized Blasius form of the Fanning friction factor, the frictional component

*dp dz dp dz*

*X*

2

*l*

2

*g*

multiplied single-phase drops. These multipliers were given by:

single-phase pressure gradient could be expressed as

*dp dz*

, ,

*dp dz* (180)

(181)

(182)

*f l f g*

,

*f l*

,

*f g*

*f*

*f*

*l/g)1/2* and *(*

*l/g)1/3*.

are two examples of this technique, where the slip ratio *(S)* equals *(*

chorological order starting from the oldest to the newest.

619 data sets for horizontal flow (Cui and Chen (2010)).

Martinelli parameter *(X)* was defined as:

**Table 4.** Values of the Exponent (*n*) and the Constants *Cl* and *Cg* for Different Flow Conditions.

Also, they presented the relationship of *l* and *g* to *X* in graphical forms. They proposed tentative criteria for the transition of the flow from one type to another. Equations to calculate the parameter (*X*) under different flow conditions are given in Table 5.

**Table 5.** Equations to Calculate the Parameter (*X*) under Different Flow Conditions.

It should be noted that Lockhart and Martinelli (1949) only presented the graphs for *g* versus *X* for the t-t, l-t and l-l flow mechanisms of liquid-gas, and the graph of t-l flow mechanisms of liquid-gas (the third type) was not given.

Recently, Cui and Chen (2010) used 619 data sets for horizontal flow to recalculate the original data of Lockhart-Martinelli following the procedures of Lockhart-Martinelli. Once the researchers separated the data into the four flow mechanisms based on the superficial Reynolds number of the gas phase (*Reg*) and liquid phase (*Rel*) respectively, the corresponding values of *X*, *g* and *l* were calculated, and the data points were plotted on the *g*-*X* diagram. They compared these data points with the four Lockhart-Martinelli correlation curves respectively. They commented that there was no mention of how the correlation curves were developed from the data points and there was also no evidence of any statistical analysis in the original Lockhart-Martinelli paper. It appeared that the curves were drawn by following the general trend of the data points. Furthermore, from the original graph of the correlation curves given in the original Lockhart-Martinelli paper, it was noted that the middle and some of the right-hand portions of the curves were shown as "solid lines" while the left-hand portion of the curves were drawn as "dashed lines". It was obvious that the "dashed lines" were not supported by data points and were extrapolations. They mentioned that computers and numerical analysis were not so readily accessible when the Lockhart-Martinelli paper was published in 1949. With the help of modern computers, the goodness of fit of data to empirical correlations could be analyzed and new empirical curves that better fit the existing data points might be obtained using the non-linear least squares method. The t-l curve had a percentage error significantly lower than for the other curves. However, this did not necessarily mean that the t-l curve was the best-fitted correlation because there were only nine data points associated with this curve. Also, these data points were in a very narrow range of 10 < *X* < 100 while the empirical correlation given was for the range 0.01 < *X* < 100. The t-t, l-t and l-l curves had similar but larger values of percentage error compared with the t-l curve.

Two-Phase Flow 309

*g*-*X* curves

(185)

Martinelli curves, while the stratified data was best represented by the l-l curve with an average error of 14.3%. As a result, when the data were categorized according to flow patterns, none of the four curves (t-t, t-l, l-t, and l-l) provided improved correlation, but with the exception of the bubbly flow data that showed an averaged error of 9.7%. It should be

Although the Lockhart-Martinelli correlation related to the adiabatic flow of low pressure air-liquid mixtures, they purposely presented the information in a generalized form to enable the application of the model to single component systems, and, in particular, to steam-water mixtures. Their empirical correlations were shown to be as reliable as any annular flow pressure drop correlation (Collier and Thome, 1994). The Lockhart-Martinelli model (1949) is probably the most well known method, commonly used in refrigeration and wet steam calculations. The disadvantage of this method was its limit to small-diameter pipes and low pressures because many applications of two-phase flow fell beyond these

Since Lockhart and Martinelli published their paper on two-phase or two-component flows in 1949 to define the methodology for presenting two-phase flow data in non-boiling and boiling flows, their paper has received nearly 1000 citations in journal papers alone is a

In his Ph. D. thesis, Turner (1966) developed the separate-cylinder model by assuming that the two-phase flow, without interaction, in two horizontal separate cylinders and that that the areas of the cross sections of these cylinders added up to the cross-sectional area of the actual pipe. The liquid and gas phases flow at the same flow rate through separate cylinders. The pressure gradient in each of the imagined cylinders was assumed to be equal, and its value was taken to be equal to the two-phase frictional pressure gradient in the actual flow. For this reason, the separate-cylinder model was not valid for gas-liquid slug flow, which gave rise to large pressure fluctuations. The pressure gradient was due to frictional effects only, and was calculated from single-phase flow theory. The separatecylinder model resembled Lockhart and Martinelli correlation (1949) but had the advantage

that it could be pursued to an analytical conclusion. The results of his analysis were

1/ 1/

The values of *n* were dependent on whether the liquid and gas phases were laminar or

In Table 6, it should be noted the following for turbulent flow (analyzed on a basis of

2 2 1 1 1 *n n*

*l g*

0.25 and *fg* = 0.079/*Reg*0.25.

turbulent flow. The different values of *n* are given in Table 6.

testament to its contribution to the field of two-phase flow.

noted that the bubbly data points were located at large *X* values where the four

tended to merge.

limits.

*<IP_ADDRESS>. Turner model*

friction factor):

i. *n* = 2.375 for *fl* = 0.079/*Rel*

Moreover, Cui and Chen (2010) re-categorized the Lockhart-Martinelli data according to flow pattern. In order to re-categorize the Lockhart-Martinelli data according to flow pattern, the researchers needed to make use of the Mandhane-Gregory-Aziz (1974) flow pattern map because the original Lockhart-Martinelli data had no information on flow patterns. Having calculated the superficial velocities of the gas phase (*Ug*) and liquid phase (*Ul*) respectively, the Lockhart-Martinelli data were plotted as scatter points on the Mandhane-Gregory-Aziz (1974) flow pattern map with the X-axis "*Ug*" was the superficial velocity of the gas phase, while the Y-axis "*Ul*" was the superficial velocity of the liquid phase. It was clear that the data used by Lockhart-Martinelli fell into five categories in terms of flow patterns: A, Annular flow; B, Bubbly flow; W, Wave flow; S, Slug flow and Str, Stratified flow. There were no data in the D, Dispersed flow region. They observed that the majority of the data fell within the annular, slug and wavy flow patterns. A few points fell within the stratified flow and the bubbly flow patterns. Also, in every flow pattern, the distribution of data points based on the four flow mechanisms of t-t, t-l, l-t and l-l flow was presented. After all the data had been re-categorized according to flow pattern, Cui and Chen (2010) compared the new data groups with the Lockhart-Martinelli curves. Again, the "Mean Absolute Percentage Error" , which referred to the vertical distance between the data point and the curve expressed as a percentage deviation from the curve, was used for making the comparison. The t-t curve was the best correlation for the annular (13.4% error), bubbly (9.0% error) and slug (15.8% error) data used by Lockhart-Martinelli. The wavy data showed an error greater than about 20% when compared with any one of the LockhartMartinelli curves, while the stratified data was best represented by the l-l curve with an average error of 14.3%. As a result, when the data were categorized according to flow patterns, none of the four curves (t-t, t-l, l-t, and l-l) provided improved correlation, but with the exception of the bubbly flow data that showed an averaged error of 9.7%. It should be noted that the bubbly data points were located at large *X* values where the four *g*-*X* curves tended to merge.

Although the Lockhart-Martinelli correlation related to the adiabatic flow of low pressure air-liquid mixtures, they purposely presented the information in a generalized form to enable the application of the model to single component systems, and, in particular, to steam-water mixtures. Their empirical correlations were shown to be as reliable as any annular flow pressure drop correlation (Collier and Thome, 1994). The Lockhart-Martinelli model (1949) is probably the most well known method, commonly used in refrigeration and wet steam calculations. The disadvantage of this method was its limit to small-diameter pipes and low pressures because many applications of two-phase flow fell beyond these limits.

Since Lockhart and Martinelli published their paper on two-phase or two-component flows in 1949 to define the methodology for presenting two-phase flow data in non-boiling and boiling flows, their paper has received nearly 1000 citations in journal papers alone is a testament to its contribution to the field of two-phase flow.

#### *<IP_ADDRESS>. Turner model*

308 An Overview of Heat Transfer Phenomena

corresponding values of *X*,

the  *g* and

percentage error compared with the t-l curve.

the researchers separated the data into the four flow mechanisms based on the superficial Reynolds number of the gas phase (*Reg*) and liquid phase (*Rel*) respectively, the

Moreover, Cui and Chen (2010) re-categorized the Lockhart-Martinelli data according to flow pattern. In order to re-categorize the Lockhart-Martinelli data according to flow pattern, the researchers needed to make use of the Mandhane-Gregory-Aziz (1974) flow pattern map because the original Lockhart-Martinelli data had no information on flow patterns. Having calculated the superficial velocities of the gas phase (*Ug*) and liquid phase (*Ul*) respectively, the Lockhart-Martinelli data were plotted as scatter points on the Mandhane-Gregory-Aziz (1974) flow pattern map with the X-axis "*Ug*" was the superficial velocity of the gas phase, while the Y-axis "*Ul*" was the superficial velocity of the liquid phase. It was clear that the data used by Lockhart-Martinelli fell into five categories in terms of flow patterns: A, Annular flow; B, Bubbly flow; W, Wave flow; S, Slug flow and Str, Stratified flow. There were no data in the D, Dispersed flow region. They observed that the majority of the data fell within the annular, slug and wavy flow patterns. A few points fell within the stratified flow and the bubbly flow patterns. Also, in every flow pattern, the distribution of data points based on the four flow mechanisms of t-t, t-l, l-t and l-l flow was presented. After all the data had been re-categorized according to flow pattern, Cui and Chen (2010) compared the new data groups with the Lockhart-Martinelli curves. Again, the "Mean Absolute Percentage Error" , which referred to the vertical distance between the data point and the curve expressed as a percentage deviation from the curve, was used for making the comparison. The t-t curve was the best correlation for the annular (13.4% error), bubbly (9.0% error) and slug (15.8% error) data used by Lockhart-Martinelli. The wavy data showed an error greater than about 20% when compared with any one of the Lockhart-

*g*-*X* diagram. They compared these data points with the four Lockhart-Martinelli correlation curves respectively. They commented that there was no mention of how the correlation curves were developed from the data points and there was also no evidence of any statistical analysis in the original Lockhart-Martinelli paper. It appeared that the curves were drawn by following the general trend of the data points. Furthermore, from the original graph of the correlation curves given in the original Lockhart-Martinelli paper, it was noted that the middle and some of the right-hand portions of the curves were shown as "solid lines" while the left-hand portion of the curves were drawn as "dashed lines". It was obvious that the "dashed lines" were not supported by data points and were extrapolations. They mentioned that computers and numerical analysis were not so readily accessible when the Lockhart-Martinelli paper was published in 1949. With the help of modern computers, the goodness of fit of data to empirical correlations could be analyzed and new empirical curves that better fit the existing data points might be obtained using the non-linear least squares method. The t-l curve had a percentage error significantly lower than for the other curves. However, this did not necessarily mean that the t-l curve was the best-fitted correlation because there were only nine data points associated with this curve. Also, these data points were in a very narrow range of 10 < *X* < 100 while the empirical correlation given was for the range 0.01 < *X* < 100. The t-t, l-t and l-l curves had similar but larger values of

*l* were calculated, and the data points were plotted on

In his Ph. D. thesis, Turner (1966) developed the separate-cylinder model by assuming that the two-phase flow, without interaction, in two horizontal separate cylinders and that that the areas of the cross sections of these cylinders added up to the cross-sectional area of the actual pipe. The liquid and gas phases flow at the same flow rate through separate cylinders. The pressure gradient in each of the imagined cylinders was assumed to be equal, and its value was taken to be equal to the two-phase frictional pressure gradient in the actual flow. For this reason, the separate-cylinder model was not valid for gas-liquid slug flow, which gave rise to large pressure fluctuations. The pressure gradient was due to frictional effects only, and was calculated from single-phase flow theory. The separatecylinder model resembled Lockhart and Martinelli correlation (1949) but had the advantage that it could be pursued to an analytical conclusion. The results of his analysis were

$$\left(\frac{1}{\phi\_l^2}\right)^{1/n} + \left(\frac{1}{\phi\_\mathcal{g}^2}\right)^{1/n} = 1\tag{185}$$

The values of *n* were dependent on whether the liquid and gas phases were laminar or turbulent flow. The different values of *n* are given in Table 6.

In Table 6, it should be noted the following for turbulent flow (analyzed on a basis of friction factor):

i. *n* = 2.375 for *fl* = 0.079/*Rel* 0.25 and *fg* = 0.079/*Reg*0.25.

**Table 6.** Values of Exponent *(n)* for Different Flow Types.

In the case of the two mixed flow regimes, Awad (2007a) mentioned in his Ph. D. thesis that the generalization of the Turner method could lead to the following implicit expressions:

$$\boldsymbol{\phi}\_{l}^{2} = \left[1 + (\boldsymbol{\phi}\_{l}^{2})^{(3/38)} \left(\frac{1}{X^{2}}\right)^{1/2.375}\right]^{2} \tag{186}$$

Two-Phase Flow 311

groups for the friction pressure drop during the flow of gas-liquid or vapor-liquid mixtures in pipes. His theoretical development was different from previous treatments in the method of allowing for the interfacial shear force between the phases. Also, he avoided some of the anomalies occurring in previous "lumped flow". He gave simplified equations for use in

> 1 1 *l C X X*

2 2 1

The values of *C* were dependent on whether the liquid and gas phases were laminar or turbulent flow. The values of *C* were restricted to mixtures with gas-liquid density ratios corresponding to air-water mixtures at atmospheric pressure. The different values of *C* are

> Liquid **Gas** *C*  Turbulent Turbulent 20 Laminar Turbulent 12 Turbulent Laminar 10 Laminar Laminar 5

He compared his predicted values using these values of *C* and his equation with the Lockhart-Martinelli values. He obtained good agreement with the Lockhart-Martinelli

The meaning of the Chisholm constant (*C*) can be easily seen if we multiply both sides of Eq.

*dp dp dp dp dp C dz dz dz dz dz*

, , ,, , int *f tp f l fl fg f g*

The physical meaning of Eq. (190) is that the two-phase frictional pressure gradient is the sum of three components: the frictional pressure of liquid-phase alone, the interfacial contribution to the total two-phase frictional pressure gradient, and the frictional pressure of

> , ,, int *f i fl fg*

*dp dp dp C dz dz dz*

*erfacial*

*erfacial*

0.5

0.5

(190)

(191)

2

(188)

*CX X* (189)

2

*g*

**Table 7.** Values of Chisholm Constant *(C)* for Different Flow Types.

(188) by (*dp/dz*)*f,l* or both sides of Eq. (189) by (*dp/dz*)*f,g* to obtain:

gas-phase alone. As a result, we may now write

engineering design. His equations were

given in Table 7.

empirical curves.

for the laminar liquid-turbulent gas case (*fl* = 16/*Rel* and *fg* = 0.079/*Reg*0.25), and

$$\boldsymbol{\phi}\_l^2 = \left[ 1 + (\boldsymbol{\phi}\_l^2)^{(-3/38)} \left( \frac{1}{X^2} \right)^{0.5} \right]^{2.375} \tag{187}$$

for the turbulent liquid-laminar gas case (*fl* = 0.079/*Rel* 0.25 and *fg* = 16/*Reg*). Equations (186) and (187) can be solved numerically.

Also, Muzychka and Awad (2010) mentioned that the values of *n* in Eq. (185) for the case of the two mixed flow regimes were *n* = 2.05 for the turbulent liquid-laminar gas case and *n* = 2.10 for the laminar liquid-turbulent gas case.

Wallis (1969) mentioned in his book that there is no rationale for the good agreement between the analytical results the separate-cylinder model and the empirical results of Lockhart and Martinelli (1949). In spite of this statement, the method is still widely accepted because of its simplicity.

#### *<IP_ADDRESS>. Chisholm model*

In the following year after Turner (1966) proposed the separate cylinders model in his Ph. D. thesis, Chisholm (1967) proposed a more rigorous analysis that was an extension of the Lockhart-Martinelli model, except that a semi-empirical closure was adopted. Chisholm's rationale for his study was the fact that the Lockhart-Martinelli model failed to produce suitable equations for predicting the two-phase frictional pressure gradient, given that the empirical curves were only presented in graphical and tabular form. In spite of Chisholm's claims, he developed his approach in much the same manner as the Lockhart-Martinelli model. The researcher developed equations in terms of the Lockhart-Martinelli correlating groups for the friction pressure drop during the flow of gas-liquid or vapor-liquid mixtures in pipes. His theoretical development was different from previous treatments in the method of allowing for the interfacial shear force between the phases. Also, he avoided some of the anomalies occurring in previous "lumped flow". He gave simplified equations for use in engineering design. His equations were

$$\left|\phi\right|^2 = 1 + \frac{\mathcal{C}}{\mathcal{X}} + \frac{1}{\mathcal{X}^2} \tag{188}$$

$$\left\|\boldsymbol{\phi}\right\|\_{\mathcal{S}}^2 = 1 + \mathbf{C}\mathbf{X} + \mathbf{X}^2\tag{189}$$

The values of *C* were dependent on whether the liquid and gas phases were laminar or turbulent flow. The values of *C* were restricted to mixtures with gas-liquid density ratios corresponding to air-water mixtures at atmospheric pressure. The different values of *C* are given in Table 7.

**Table 7.** Values of Chisholm Constant *(C)* for Different Flow Types.

310 An Overview of Heat Transfer Phenomena

0.2 and *fg* = 0.046/*Reg*0.2.

**Table 6.** Values of Exponent *(n)* for Different Flow Types.

iii. *n* = 2.5 for *fl* = constant (i.e. not function of *Rel*) and *fg* = constant (i.e. not function of *Reg*). Flow Type *n*  Laminar Flow 2 Turbulent Flow (analyzed on a basis of friction factor) 2.375-2.5 Turbulent Flows (calculated on a mixing-length basis) 2.5-3.5 Turbulent-Turbulent Regime 4 All Flow Regimes 3.5

In the case of the two mixed flow regimes, Awad (2007a) mentioned in his Ph. D. thesis that the generalization of the Turner method could lead to the following implicit expressions:

>

2 2 (3/38)

2 2 ( 3/38)

1 1( ) *l l X*

Also, Muzychka and Awad (2010) mentioned that the values of *n* in Eq. (185) for the case of the two mixed flow regimes were *n* = 2.05 for the turbulent liquid-laminar gas case and *n* =

Wallis (1969) mentioned in his book that there is no rationale for the good agreement between the analytical results the separate-cylinder model and the empirical results of Lockhart and Martinelli (1949). In spite of this statement, the method is still widely accepted

In the following year after Turner (1966) proposed the separate cylinders model in his Ph. D. thesis, Chisholm (1967) proposed a more rigorous analysis that was an extension of the Lockhart-Martinelli model, except that a semi-empirical closure was adopted. Chisholm's rationale for his study was the fact that the Lockhart-Martinelli model failed to produce suitable equations for predicting the two-phase frictional pressure gradient, given that the empirical curves were only presented in graphical and tabular form. In spite of Chisholm's claims, he developed his approach in much the same manner as the Lockhart-Martinelli model. The researcher developed equations in terms of the Lockhart-Martinelli correlating

for the laminar liquid-turbulent gas case (*fl* = 16/*Rel* and *fg* = 0.079/*Reg*0.25), and

for the turbulent liquid-laminar gas case (*fl* = 0.079/*Rel*

2.10 for the laminar liquid-turbulent gas case.

(187) can be solved numerically.

because of its simplicity.

*<IP_ADDRESS>. Chisholm model*

1 1( ) *l l X*

2 1/2.375

2.375 0.5

(186)

(187)

0.25 and *fg* = 16/*Reg*). Equations (186) and

2

2

ii. *n* = 2.4 for *fl* = 0.046/*Rel*

He compared his predicted values using these values of *C* and his equation with the Lockhart-Martinelli values. He obtained good agreement with the Lockhart-Martinelli empirical curves.

The meaning of the Chisholm constant (*C*) can be easily seen if we multiply both sides of Eq. (188) by (*dp/dz*)*f,l* or both sides of Eq. (189) by (*dp/dz*)*f,g* to obtain:

$$
\left(\frac{dp}{dz}\right)\_{f,tp} = \left(\frac{dp}{dz}\right)\_{f,l} + \underbrace{\mathcal{C}\left[\left(\frac{dp}{dz}\right)\_{f,l}\left(\frac{dp}{dz}\right)\_{f,g}\right]^{0.5}}\_{\text{int}\,\text{erf}\,\text{cal}^{-1}} + \left(\frac{dp}{dz}\right)\_{f,g} \tag{190}
$$

The physical meaning of Eq. (190) is that the two-phase frictional pressure gradient is the sum of three components: the frictional pressure of liquid-phase alone, the interfacial contribution to the total two-phase frictional pressure gradient, and the frictional pressure of gas-phase alone. As a result, we may now write

$$\mathbf{C} \left( \frac{dp}{dz} \right)\_{f,l} = \mathbf{C} \underbrace{\left[ \left( \frac{dp}{dz} \right)\_{f,l} \left( \frac{dp}{dz} \right)\_{f,g} \right]^{0.5}}\_{\text{interfacid}} \tag{191}$$

The means that the constant *C* in Chisholm's model can be viewed as a weighting factor for the geometric mean (*GM*) of the single-phase liquid and gas only pressure gradients.

The Chisholm parameter (*C*) is a measure of two-phase interactions. The larger the value, the greater the interaction, hence the Lockhart-Martinelli parameter (*X*) can involve ll, tl, lt, and tt regimes. It just causes the data to shift outwards on the Lockhart-Martinelli plot.

The Chisholm constant (*C*) can be derived analytically for a number of special cases. For instance, Whalley (1996) obtained for a homogeneous flow having constant friction factor:

$$\mathbf{C} = \left[ \left( \frac{\rho\_l}{\rho\_g} \right)^{0.5} + \left( \frac{\rho\_g}{\rho\_l} \right)^{0.5} \right] \tag{192}$$

Two-Phase Flow 313

*<IP_ADDRESS>. Hemeida-Sumait model*

Lockhart and Martinelli parameters

using SAS software, their equation was

according to the type of fluid and flow mechanisms (Table 8).

**Table 8.** Values of *a*, *b*, and *c* for Different Flow Mechanisms.

and standard deviation of 9.6%.

*<IP_ADDRESS>. Modified Turner model*

two-phase frictional pressure gradient.

The Lockhart-Martinelli (1949) correlation in its present form cannot be used to study a large set of data because it requires the use of charts and hence cannot be simulated numerically. As a result, Hemeida and Sumait (1988) developed a correlation between

2 exp 2.303 ( ) ( ) 2.30

 *a bLn X LnX*

Where *a*, *b*, and *c* were constants. They selected the values of the constants *a*, *b*, and *c*

Parameter *a b c*

*g,ll* 0.4625 0.5058 0.1551

*g,lt* 0.5673 0.4874 0.1312

*g,tl* 0.5694 0.4982 0.1255

*g,tt* 0.6354 0.4810 0.1135

*l,ll* 0.4048 0.4269 0.1841

*l,lt* 0.5532 -0.4754 0.1481

*l,tl* 0.5665 -0.4586 0.1413

*l,tt* 0.6162 -0.5063 0.124

In Table 8, the first subscript refers to whether the liquid is laminar or turbulent while the second subscript refers to whether the gas is laminar or turbulent. Equation (194) enabled the development of a computer program for the analysis of data using the Lockhart-Martinelli (1949) correlation. Using this program, they analyzed field data from Saudi flow lines. The results showed that the improved Lockhart-Martinelli correlation predicted accurately the downstream pressure in flow lines with an average percent difference of 5.1

It should be noted that the Hemeida-Sumait (1988) model is not famous in the literature like other models such as the Chisholm (1967) model although it gave an accurate prediction of

Awad and Muzychka (2004b) arrived at the same simple form as the empirical Turner (1966) model, but with a different physical approach. Rather than model the fluid as two distinct fluid streams flowing in separate pipes, the researchers proposed that the two- phase frictional pressure gradient could be predicted using a nonlinear superposition of the component pressure gradient that would arise from every stream flowing alone in the same

*c*

and *X* for a two-phase pressure drop in pipelines

as a function of *X*

(194)

using the Statistical Analysis System (SAS). To calculate the parameter

that for an air-water combination gives C 28.6 that is in good agreement with Chisholm's value for turbulent-turbulent flows. Also, Whalley (1996) shows that for laminar and turbulent flows with no interaction between phases the values of C 2 and C 3.66 are obtained, respectively.

In addition, Awad and Muzychka (2007, 2010b) mentioned that a value of *C* = 0 can be used as a lower bound for two-phase frictional pressure gradient in minichannels and microchannels. The physical meaning of the lower bound (*C* = 0) is that the two-phase frictional pressure gradient is merely the sum of the frictional pressure of liquid phase alone and the frictional pressure of gas phase alone:

$$
\left(\frac{dp}{dz}\right)\_{f,tp} = \left(\frac{dp}{dz}\right)\_{f,l} + \left(\frac{dp}{dz}\right)\_{f,g} \tag{193}
$$

This means there is no contribution to the pressure gradient through phase interaction. The above result can also be obtained using the asymptotic model for two-phase frictional pressure gradient (Awad and Muzychka (2004b)) with linear superposition. Further, using the homogeneous model with the Dukler et al. (1964) definition of two-phase viscosity for laminar-laminar flow leads to the same result as Eq. (193).The value of *C* = 0 is also in agreement with recent models in microchannel flows such as (Mishima and Hibiki correlation (1996) and English and Kandlikar correlation (2006)) that implies that as *dh* → 0, *C* → 0. The only disadvantage in these mentioned correlations is the dimensional specification of *dh*, as it is easy to miscalculate *C* if the proper dimensions are not used for *dh*. Other researchers such as Zhang et al. (2010) overcame this disadvantage by representing the hydraulic diameter (*dh*) in a dimensionless form using the Laplace number (*La*).

Moreover, if a laminar plug flow is assumed, a value of *C* = 0 can be easily derived that implies that the total pressure gradient is just the sum of the component pressure gradients based on plug length and component flow rate. This is a reasonable approximation provided that plug lengths are longer than fifteen diameters (Walsh et al., 2009).

In his Ph. D. thesis, Awad (2007a) reviewed additional extended Chisholm type models.

#### *<IP_ADDRESS>. Hemeida-Sumait model*

312 An Overview of Heat Transfer Phenomena

obtained, respectively.

and the frictional pressure of gas phase alone:

The means that the constant *C* in Chisholm's model can be viewed as a weighting factor for

The Chisholm parameter (*C*) is a measure of two-phase interactions. The larger the value, the greater the interaction, hence the Lockhart-Martinelli parameter (*X*) can involve ll, tl, lt, and tt regimes. It just causes the data to shift outwards on the Lockhart-Martinelli plot.

The Chisholm constant (*C*) can be derived analytically for a number of special cases. For instance, Whalley (1996) obtained for a homogeneous flow having constant friction factor:

that for an air-water combination gives C 28.6 that is in good agreement with Chisholm's value for turbulent-turbulent flows. Also, Whalley (1996) shows that for laminar and turbulent flows with no interaction between phases the values of C 2 and C 3.66 are

In addition, Awad and Muzychka (2007, 2010b) mentioned that a value of *C* = 0 can be used as a lower bound for two-phase frictional pressure gradient in minichannels and microchannels. The physical meaning of the lower bound (*C* = 0) is that the two-phase frictional pressure gradient is merely the sum of the frictional pressure of liquid phase alone

*f* ,*tp f l f g* , ,

This means there is no contribution to the pressure gradient through phase interaction. The above result can also be obtained using the asymptotic model for two-phase frictional pressure gradient (Awad and Muzychka (2004b)) with linear superposition. Further, using the homogeneous model with the Dukler et al. (1964) definition of two-phase viscosity for laminar-laminar flow leads to the same result as Eq. (193).The value of *C* = 0 is also in agreement with recent models in microchannel flows such as (Mishima and Hibiki correlation (1996) and English and Kandlikar correlation (2006)) that implies that as *dh* → 0, *C* → 0. The only disadvantage in these mentioned correlations is the dimensional specification of *dh*, as it is easy to miscalculate *C* if the proper dimensions are not used for *dh*. Other researchers such as Zhang et al. (2010) overcame this disadvantage by representing

the hydraulic diameter (*dh*) in a dimensionless form using the Laplace number (*La*).

provided that plug lengths are longer than fifteen diameters (Walsh et al., 2009).

In his Ph. D. thesis, Awad (2007a) reviewed additional extended Chisholm type models.

Moreover, if a laminar plug flow is assumed, a value of *C* = 0 can be easily derived that implies that the total pressure gradient is just the sum of the component pressure gradients based on plug length and component flow rate. This is a reasonable approximation

*dp dp dp dz dz dz*

0.5 0.5 *g l g l*

(192)

(193)

the geometric mean (*GM*) of the single-phase liquid and gas only pressure gradients.

*C*

The Lockhart-Martinelli (1949) correlation in its present form cannot be used to study a large set of data because it requires the use of charts and hence cannot be simulated numerically. As a result, Hemeida and Sumait (1988) developed a correlation between Lockhart and Martinelli parameters and *X* for a two-phase pressure drop in pipelines using the Statistical Analysis System (SAS). To calculate the parameter as a function of *X* using SAS software, their equation was

$$\phi = \exp\left[2.303a + bLn(X) + \frac{c}{2.30}(LnX)^2\right] \tag{194}$$

Where *a*, *b*, and *c* were constants. They selected the values of the constants *a*, *b*, and *c* according to the type of fluid and flow mechanisms (Table 8).

**Table 8.** Values of *a*, *b*, and *c* for Different Flow Mechanisms.

In Table 8, the first subscript refers to whether the liquid is laminar or turbulent while the second subscript refers to whether the gas is laminar or turbulent. Equation (194) enabled the development of a computer program for the analysis of data using the Lockhart-Martinelli (1949) correlation. Using this program, they analyzed field data from Saudi flow lines. The results showed that the improved Lockhart-Martinelli correlation predicted accurately the downstream pressure in flow lines with an average percent difference of 5.1 and standard deviation of 9.6%.

It should be noted that the Hemeida-Sumait (1988) model is not famous in the literature like other models such as the Chisholm (1967) model although it gave an accurate prediction of two-phase frictional pressure gradient.

#### *<IP_ADDRESS>. Modified Turner model*

Awad and Muzychka (2004b) arrived at the same simple form as the empirical Turner (1966) model, but with a different physical approach. Rather than model the fluid as two distinct fluid streams flowing in separate pipes, the researchers proposed that the two- phase frictional pressure gradient could be predicted using a nonlinear superposition of the component pressure gradient that would arise from every stream flowing alone in the same pipe, through application of the Churchill-Usagi (1972) asymptotic correlation method. This form was asymptotically correct for either phase as the mass quality varied from 0 *x* 1. Moreover, rather than approach the Lockhart-Martinelli parameter (*X*) from the point of view of the four flow regimes using simple friction models, they proposed using the Churchill (1977) model for the friction factor in smooth and rough pipes for all values of the Reynolds number. In this way, the proposed model was more general and contained only one empirical coefficient, the Churchill-Usagi blending parameter. The resulting model takes the form:

$$
\left(\frac{dp}{dz}\right)\_f = \left[\left(\frac{dp}{dz}\right)\_{f,l}^p + \left(\frac{dp}{dz}\right)\_{f,g}^p\right]^{1/p} \tag{195}
$$

Two-Phase Flow 315

*<IP_ADDRESS>. Modified Chisholm models*

correlation was proposed having the form:

flow, Saisorn and Wongwises (2008), and

above correlations should be:

interfacial pressure gradient.

parameter models.

*6.3.3. Interfacial pressure gradient*

or

Finally, in a recent series of studies by Saisorn and Wongwises (2008, 2009, and 2010),

for experimental data for slug flow, throat-annular flow, churn flow, and annular-rivulet

for experimental data for annular flow, liquid unstable annular alternating flow (LUAAF), and liquid/annular alternating flow (LAAF), Saisorn and Wongwises (2009). These correlations neglect the 1/*X*2 term that represents the limit of primarily gas flow in the Lockhart-Martinelli (1949) formulation. Neglecting this term ignores this important limiting case, which is an essential contribution in the Lockhart-Matrinelli modeling approach. As a result, at low values of *X*, the proposed correlations undershoot the trend of the data, limiting their use in the low *X* range. Thus, a more appropriate and generalized form of the

> 1 1 *l m A X X*

2 2 1 *m*

These formulations, Eqs. (200) and (201), can be considered extended Chisholm type models. They will be utilized in the next section as a means of modeling the two-phase flow

Gas-liquid two-phase flow will be examined from the point of view of interfacial pressure gradient. Recognizing that in a Lockhart-Martinelli reduction scheme, single-phase flow characteristics must be exhibited in a limiting sense, they will be subtracted from the experimental data being considered to illustrate some benefits of using the one and two

The two-phase frictional pressure gradient can be defined as a linear combination of three pressure gradients. These are the single-phase liquid, single-phase gas, and interfacial pressure gradient. The rationale for such a choice lies in the definition of the Lockhart-Martinelli approach, whereby, one obtains single-phase gas flow for small values of the

0.761 6.627 1 *l X*

1.666 2.844 1 *l X*

2

(198)

(199)

(200)

*AX X* (201)

2

2

2

*g*

or when written as a two-phase frictional liquid multiplier:

$$\left| \phi \right|^2 = \left[ 1 + \left( \frac{1}{X^2} \right)^p \right]^{1/p} \tag{196}$$

or when written as a two-phase frictional gas multiplier:

$$\boldsymbol{\phi}\_{\mathcal{S}}^{2} = [1 + (\boldsymbol{X}^{2})^{p}]^{1/p} \tag{197}$$

which are the same equations from the Turner approach, when *p* =1/ *n*. The main exception is that the values of *p* were developed for different flow regimes using the Churchill friction model to calculate *X*.

The principal advantages of the above approach over the Turner (1966) method are twofold. First, all four Lockhart-Martinelli flow regimes can be handled with ease because the Turner (1966) method leads to implicit relationships for the two mixed regimes. Second, since the friction model used is only a function of Reynolds number and roughness, broader applications involving rough pipes can be easily modeled. Using Eqs. (196) and (197), Awad (2007b) found that *p* 0.307 for large tubes and *p* 0.5 for microchannels, minichannels, and capillaries. The modified Turner model is also a one parameter correlating scheme. Recently, Awad and Butt (2009a, 2009b, and 2009c) have shown that the asymptotic method works well for petroleum industry applications for liquid-liquid flows, flows through fractured media, and flows through porous media. Moreover, Awad and Muzychka (2010a) have shown that the asymptotic method works well for two-phase gas-liquid flow at microgravity conditions.

Approximate equivalence between Eq. (188) and Eq. (196) (or Eq. (189) and Eq. (197)) can be found when *p* = 0.36, 0.3, 0.285, and 0.245 when *C* = 5, 10, 12, and 20, respectively. This yields differences of 3-9% rms. The special case of *p* = 1 leads to a linear superposition of the component pressure gradients that corresponds to *C* = 0. This limiting case is only valid for plug flows when plug length to diameter ratios exceed 15 (Walsh et al., 2009).

#### *<IP_ADDRESS>. Modified Chisholm models*

314 An Overview of Heat Transfer Phenomena

takes the form:

model to calculate *X*.

microgravity conditions.

pipe, through application of the Churchill-Usagi (1972) asymptotic correlation method. This form was asymptotically correct for either phase as the mass quality varied from 0 *x* 1. Moreover, rather than approach the Lockhart-Martinelli parameter (*X*) from the point of view of the four flow regimes using simple friction models, they proposed using the Churchill (1977) model for the friction factor in smooth and rough pipes for all values of the Reynolds number. In this way, the proposed model was more general and contained only one empirical coefficient, the Churchill-Usagi blending parameter. The resulting model

1/

(195)

(196)

*g X* (197)

, ,

*f fl fg*

2 1 1

2 2 1/ [1 ( ) ] *p p*

which are the same equations from the Turner approach, when *p* =1/ *n*. The main exception is that the values of *p* were developed for different flow regimes using the Churchill friction

The principal advantages of the above approach over the Turner (1966) method are twofold. First, all four Lockhart-Martinelli flow regimes can be handled with ease because the Turner (1966) method leads to implicit relationships for the two mixed regimes. Second, since the friction model used is only a function of Reynolds number and roughness, broader applications involving rough pipes can be easily modeled. Using Eqs. (196) and (197), Awad (2007b) found that *p* 0.307 for large tubes and *p* 0.5 for microchannels, minichannels, and capillaries. The modified Turner model is also a one parameter correlating scheme. Recently, Awad and Butt (2009a, 2009b, and 2009c) have shown that the asymptotic method works well for petroleum industry applications for liquid-liquid flows, flows through fractured media, and flows through porous media. Moreover, Awad and Muzychka (2010a) have shown that the asymptotic method works well for two-phase gas-liquid flow at

Approximate equivalence between Eq. (188) and Eq. (196) (or Eq. (189) and Eq. (197)) can be found when *p* = 0.36, 0.3, 0.285, and 0.245 when *C* = 5, 10, 12, and 20, respectively. This yields differences of 3-9% rms. The special case of *p* = 1 leads to a linear superposition of the component pressure gradients that corresponds to *C* = 0. This limiting case is only valid for

plug flows when plug length to diameter ratios exceed 15 (Walsh et al., 2009).

*dp dp dp dz dz dz*

*l X*

2

or when written as a two-phase frictional liquid multiplier:

or when written as a two-phase frictional gas multiplier:

*p p p*

1/

*p p*

Finally, in a recent series of studies by Saisorn and Wongwises (2008, 2009, and 2010), correlation was proposed having the form:

$$\left|\phi\right|^2 = 1 + \frac{6.627}{\mathcal{X}^{0.761}}\tag{198}$$

for experimental data for slug flow, throat-annular flow, churn flow, and annular-rivulet flow, Saisorn and Wongwises (2008), and

$$\left|\phi\right|\_{l}^{2} = 1 + \frac{2.844}{X^{1.666}}\tag{199}$$

for experimental data for annular flow, liquid unstable annular alternating flow (LUAAF), and liquid/annular alternating flow (LAAF), Saisorn and Wongwises (2009). These correlations neglect the 1/*X*2 term that represents the limit of primarily gas flow in the Lockhart-Martinelli (1949) formulation. Neglecting this term ignores this important limiting case, which is an essential contribution in the Lockhart-Matrinelli modeling approach. As a result, at low values of *X*, the proposed correlations undershoot the trend of the data, limiting their use in the low *X* range. Thus, a more appropriate and generalized form of the above correlations should be:

$$\left\|\phi\right\|^2\_l = 1 + \frac{A}{X^m} + \frac{1}{X^2} \tag{200}$$

or

$$\left\|\boldsymbol{\phi}\right\|\_{\mathcal{S}}^2 = 1 + A\boldsymbol{X}^m + \boldsymbol{X}^2 \tag{201}$$

These formulations, Eqs. (200) and (201), can be considered extended Chisholm type models. They will be utilized in the next section as a means of modeling the two-phase flow interfacial pressure gradient.

#### *6.3.3. Interfacial pressure gradient*

Gas-liquid two-phase flow will be examined from the point of view of interfacial pressure gradient. Recognizing that in a Lockhart-Martinelli reduction scheme, single-phase flow characteristics must be exhibited in a limiting sense, they will be subtracted from the experimental data being considered to illustrate some benefits of using the one and two parameter models.

The two-phase frictional pressure gradient can be defined as a linear combination of three pressure gradients. These are the single-phase liquid, single-phase gas, and interfacial pressure gradient. The rationale for such a choice lies in the definition of the Lockhart-Martinelli approach, whereby, one obtains single-phase gas flow for small values of the Lockhart-Martinelli parameter (*X*) and single-phase liquid flow for large values of the Lockhart-Martinelli parameter (*X*). While in the transitional region between 0.01 < *X* < 100, interfacial effects result in a large spread of data depending upon flow regime.

Beginning with

$$
\left(\frac{dp}{dz}\right)\_{f,tp} = \left(\frac{dp}{dz}\right)\_{f,l} + \left(\frac{dp}{dz}\right)\_{f,i} + \left(\frac{dp}{dz}\right)\_{f,g} \tag{202}
$$

Two-Phase Flow 317

(206)

*CX* (207)

(208)

*g i AX* (209)

*g,i*) for stratified flow

2 *l i*, *C X*

2 *g i*,

gas multiplier formulation, then a two parameter model is likely required.

This represents a simple one parameter model, whereby closure can be found with comparison with experimental data. Also, the simple asymptotic form of Eqs. (196) or (197) represents a one parameter model. If the interfacial effects can be modeled by Chisholm's proposed model or Eqs. (196) or (197), then all of the reduced data should show trends indicated by Eqs. (206) or (207). However, if data do not scale according to Eqs. (206) or (207), i.e. a slope of negative one for the liquid multiplier formulation or positive one for the

Muzychka and Awad (2010) extended Eqs. (206) and (207) to develop a simple two

2 *l i*, *m A X*

2 ,

These forms have the advantage that experimental data for a particular flow regime can be fit to the simple power law after removal of the single-phase pressure contributions (Muzychka and Awad (2010)). Also, the advantage of the *A* and *m* model over the Chisholm model (1967) is the Chisholm model (1967) is destined to fail as they do not scale with *X* properly when data deviate from the -1 and +1 slope. For example, this two parameter power law model can be use for the analysis of stratified flow data separated into different categories (t–t, l–t and l–l) in Cui and Chen (2010) for their study on a re-examination of the data of Lockhart-Martinelli. The researchers used the 619 data sets for horizontal flow. Their 619 data sets were classified based on the flow patterns as follows: 191 data sets for Annular flow, 277 data sets for Slug flow, 94 data sets for Wavy flow, 32 data sets for Bubbly flow,

The analysis is presented here for the stratified flow data because it has only 25 data points (the lowest number of data points for the different flow patterns: annular (191), slug (277),

wavy (94), bubbly (32), and stratified (25)). The interfacial component (

data of Lockhart-Martinelli is calculated as follows:

*m*

for the liquid multiplier formulation, or

for the gas multiplier formulation.

*<IP_ADDRESS>. Two parameter models*

leading to Eqs. (200) or (201).

and 25 data sets for Stratified flow.

or

parameter power law model such that:

Rearranging Eq. (202), we obtain

$$
\left(\frac{dp}{dz}\right)\_{f,i} = \left(\frac{dp}{dz}\right)\_{f,tp} - \left(\frac{dp}{dz}\right)\_{f,l} - \left(\frac{dp}{dz}\right)\_{f,g} \tag{203}
$$

Dividing both sides of Eq. (203) by the single-phase liquid frictional pressure gradient, we obtain

$$
\phi\_{1,i}^2 = \left(\frac{dp}{dz}\right)\_{f,i} / \left(\frac{dp}{dz}\right)\_{f,l} = \phi\_1^2 - 1 - \frac{1}{X^2} \tag{204}
$$

On the other hand, dividing both sides of Eq. (203) by the single-phase gas frictional pressure gradient, we obtain

$$\left(\rho\_{\mathcal{g},i}^2 = \left(\frac{dp}{dz}\right)\_{f,i} / \left(\frac{dp}{dz}\right)\_{f,\mathcal{g}} = \rho\_{\mathcal{g}}^2 - X^2 - 1 \tag{205}$$

Where *l,i2* and *g,i2* are two-phase frictional multiplier for the interfacial pressure gradient. This can be viewed as an extended form of the Chisholm model, where the interfacial contribution is what is to be modeled. The data defined using Eqs. (204) and (205) may then be modeled using one, two, or multi-parameter forms. We discuss these approaches below.

It should be noted that this analysis is useful to show that *g,i* does not exist at high values of *Xtt* for some correlations available in the literature such as the *g* correlation of Ding et al. (2009) to predict the pressure drop of R410A–oil mixtures in microfin tubes, the *g* correlation of Hu et al. (2008) to predict the pressure drop of R410A/POE oil mixture in micro-fin tubes, and the *g* correlation of Hu et al. (2009) to predict the pressure drop of R410A/oil mixture in smooth tubes because *2g,i* has negative values at high values of *Xtt* (Awad, 2010a, Awad, 2010b, and Awad, 2011). Also, it this analysis is useful to show that *l,i* does not exist at certain values of *Xtt* for some correlations available in the literature like the *l* correlation of Changhong et al. (2005) to predict the pressure drop in two vertical narrow annuli (Awad, 2012b).

#### *<IP_ADDRESS>. One parameter models*

Comparison with the Chisholm (1967) formulation gives:

Two-Phase Flow 317

$$\left|\phi\right|\_{l,i}^{2} = \frac{\text{C}}{\text{X}}\tag{206}$$

for the liquid multiplier formulation, or

$$
\boldsymbol{\phi}^2\_{\mathcal{g},i} = \mathbf{C} \mathbf{X} \tag{207}
$$

for the gas multiplier formulation.

This represents a simple one parameter model, whereby closure can be found with comparison with experimental data. Also, the simple asymptotic form of Eqs. (196) or (197) represents a one parameter model. If the interfacial effects can be modeled by Chisholm's proposed model or Eqs. (196) or (197), then all of the reduced data should show trends indicated by Eqs. (206) or (207). However, if data do not scale according to Eqs. (206) or (207), i.e. a slope of negative one for the liquid multiplier formulation or positive one for the gas multiplier formulation, then a two parameter model is likely required.

#### *<IP_ADDRESS>. Two parameter models*

Muzychka and Awad (2010) extended Eqs. (206) and (207) to develop a simple two parameter power law model such that:

$$\left\| \boldsymbol{\phi} \right\|\_{l,i}^{2} = \frac{A}{\mathbf{X}^{m}} \tag{208}$$

or

316 An Overview of Heat Transfer Phenomena

Rearranging Eq. (202), we obtain

pressure gradient, we obtain

micro-fin tubes, and the

annuli (Awad, 2012b).

*<IP_ADDRESS>. One parameter models*

Beginning with

obtain

Where *l,i2* and

Lockhart-Martinelli parameter (*X*) and single-phase liquid flow for large values of the Lockhart-Martinelli parameter (*X*).  The data defined using Eqs. (204) and (205) may then be modeled using one, two, or multi-parameter forms. We discuss these approaches below.

(2009) to predict the pressure drop of R410A–oil mixtures in microfin tubes, the

correlation of Hu et al. (2008) to predict the pressure drop of R410A/POE oil mixture in

does not exist at certain values of *Xtt* for some correlations available in the literature like the

*l* correlation of Changhong et al. (2005) to predict the pressure drop in two vertical narrow

(Awad, 2010a, Awad, 2010b, and Awad, 2011). Also, it this analysis is useful to show that

*dz dz*

>

*g,i2* are two-phase frictional multiplier for the interfacial pressure gradient.

*g* correlation of Hu et al. (2009) to predict the pressure drop of

*2g,i* has negative values at high values of *Xtt*

*g,i* does not exist at high values of

*g* correlation of Ding et al.

*g*

*l,i*

Dividing both sides of Eq. (203) by the single-phase liquid frictional pressure gradient, we

2 2 1, 1 2 , , 1 / 1 *i fi fl*

*dp dp*

,

It should be noted that this analysis is useful to show that

Comparison with the Chisholm (1967) formulation gives:

R410A/oil mixture in smooth tubes because

*Xtt* for some correlations available in the literature such as the

(202)

(203)

(204)

(205)

*dp dp dp dp dz dz dz dz*

*dp dp dp dp dz dz dz dz*

interfacial effects result in a large spread of data depending upon flow regime.

$$
\boldsymbol{\phi}^2\_{\mathcal{G},i} = \boldsymbol{A}\boldsymbol{X}^m \tag{209}
$$

leading to Eqs. (200) or (201).

These forms have the advantage that experimental data for a particular flow regime can be fit to the simple power law after removal of the single-phase pressure contributions (Muzychka and Awad (2010)). Also, the advantage of the *A* and *m* model over the Chisholm model (1967) is the Chisholm model (1967) is destined to fail as they do not scale with *X* properly when data deviate from the -1 and +1 slope. For example, this two parameter power law model can be use for the analysis of stratified flow data separated into different categories (t–t, l–t and l–l) in Cui and Chen (2010) for their study on a re-examination of the data of Lockhart-Martinelli. The researchers used the 619 data sets for horizontal flow. Their 619 data sets were classified based on the flow patterns as follows: 191 data sets for Annular flow, 277 data sets for Slug flow, 94 data sets for Wavy flow, 32 data sets for Bubbly flow, and 25 data sets for Stratified flow.

The analysis is presented here for the stratified flow data because it has only 25 data points (the lowest number of data points for the different flow patterns: annular (191), slug (277), wavy (94), bubbly (32), and stratified (25)). The interfacial component (*g,i*) for stratified flow data of Lockhart-Martinelli is calculated as follows:

$$
\phi\_{\mathcal{g},i} = \left(\phi\_{\mathcal{g}}^2 - \mathbf{X}^2 - \mathbf{1}\right)^{0.5} \tag{210}
$$

Two-Phase Flow 319

dimensionless parameters: the Laplace constant (*La*), and the liquid Reynolds number (*Rel*). Also, Venkatesan et al. (2011) adopted an approach that led to the development of *C* in circular tubes with *d* = 0.6, 1.2, 1.7, 2.6 and 3.4 mm using air and water as a function of the following dimensionless parameters: Weber number(*We*), superficial liquid Reynolds number (*Rel*), and superficial gas Reynolds number (*Reg*). In addition, Kawahara et al. (2011) used their two-phase frictional pressure drop data in a rectangular microchannel with a Tjunction type gas-liquid mixer to correlate the Chisholm constant (*C*) as a function of the following dimensionless parameters: Bond number(*Bo*), superficial liquid Reynolds number (*Rel*), and superficial gas Weber number (*We*g). But care must be taken because even with the introduction of additional variables, increased accuracy will not necessarily be obtained.

The drift flux model is a type of separated flow model. In the drift flux model, attention is focused on the relative motion rather than on the motion of the individual phases. The drift flux model was developed by Wallis (1969). The drift flux model has widespread application to bubble flow and plug flow. The drift flux model is not particularly suitable to a flow such as annular flow that has two characteristic velocities in one phase: the liquid film velocity and the liquid drop velocity. However, the drift flux model has been used for annular flows,

The drift flux model is the fifth example of the existing void fraction models. The Rouhani and Axelsson (1970) model is an instance for this type of model. In the drift-flux model, the

*(U)*, the phase distribution parameter *(Co)*, and the mean drift velocity *(ugj)* that includes the

*g o gj*

*U CU u*

The drift-flux correlations often present procedures to compute *Co* and *ugj*. Since the

*)* are calculated using method of solving of non-linear equation.

This model is known as the two-fluid model designating two phases or components. This model is an advanced predictive tool for liquid-gas two-phase flow in engineering applications. It is based on the mass, momentum and energy balance equations for every phase (Ishii, 1987). In this model, every phase or component is treated as a separate fluid with its own set of governing balance equations. In general, every phase has its own velocity, temperature and pressure. This approach enables the prediction of important nonequilibrium phenomena of two-phase flow like the velocity difference between liquid and gas phase. This prediction is important for two-phase flows in large shell sides of steam generators and kettle reboilers, where even different gas and liquid velocity directions exist.

effect of the relative velocity between the phases. The form of the drift-flux model is

expressions of *Co* and *ugj* are usually functions of the void fraction *(*

*)* is a function of the gas superficial velocity *(Ug)*, the total superficial velocity

(212)

*)*, the predictions of the

*6.3.4. The drift flux model*

but with no particular success.

*6.3.5. Two-fluid model*

void fraction *(*

void fraction *(*

Using this analysis, the interfacial component for the high pink triangle at the right hand side of stratified flow data separated into different categories (t–t, v–t and v–v) (Cui and Chen (2010)) does not exist. This is because (*g2*-*X2*-1) < 0 for this point so that the square root of a negative value does not exist. This means that there is an error in the measurement in one data point for the stratified flow at lt flow mechanisms of liquid-gas. In this analysis, the data points of tt, lt, and ll flow mechanisms of liquid-gas were fit with only one line instead of three different lines for each flow mechanism of liquid-gas (tt, lt, and ll) because tt has only one data point. As shown in Fig. 4, the fit equation was:

$$
\phi\_{\mathcal{g},i} = 2.1X^{0.678} \tag{211}
$$

However, drawing a different line of the interfacial component for the stratified flow for each flow mechanism of liquid-gas (tt, lt, and ll) will be more accurate. This analysis can be also done for other flow patterns: annular (191), slug (277), wavy (94), and bubbly (32).

It should be noted that the nonexistence of the interfacial component for some data sets for any flow patterns: annular (191), slug (277), wavy (94), bubbly (32), and stratified (25)) means that there is an error in the measurement of some data sets of Lockhart-Martinelli although their paper has received nearly 1000 citations in journal papers.

**Figure 4.** Analysis of Stratified Flow Data Separated into Different Categories (t–t, v–t and v–v) Using Two Parameter Power Law Model (Muzychka and Awad (2010)).

#### *<IP_ADDRESS>. Multi-parameter models*

Multi-parameter models may be developed using both the Chisholm model and the modified Chisholm models, by correlating the constants *C*, *A*, and *m* with other dimensionless parameters. For example, Sun and Mishima (2009) adopted an approach that led to the development of *C* in the laminar flow region as a function of the following dimensionless parameters: the Laplace constant (*La*), and the liquid Reynolds number (*Rel*). Also, Venkatesan et al. (2011) adopted an approach that led to the development of *C* in circular tubes with *d* = 0.6, 1.2, 1.7, 2.6 and 3.4 mm using air and water as a function of the following dimensionless parameters: Weber number(*We*), superficial liquid Reynolds number (*Rel*), and superficial gas Reynolds number (*Reg*). In addition, Kawahara et al. (2011) used their two-phase frictional pressure drop data in a rectangular microchannel with a Tjunction type gas-liquid mixer to correlate the Chisholm constant (*C*) as a function of the following dimensionless parameters: Bond number(*Bo*), superficial liquid Reynolds number (*Rel*), and superficial gas Weber number (*We*g). But care must be taken because even with the introduction of additional variables, increased accuracy will not necessarily be obtained.

#### *6.3.4. The drift flux model*

318 An Overview of Heat Transfer Phenomena

Chen (2010)) does not exist. This is because (

0.01

*<IP_ADDRESS>. Multi-parameter models*

Two Parameter Power Law Model (Muzychka and Awad (2010)).

0.1

1

*g,i*

10

100

only one data point. As shown in Fig. 4, the fit equation was:

2 2 0.5 , ( 1)

0.678

*X* (210)

*g2*-*X2*-1) < 0 for this point so that the square root

*g i X* (211)

Stratified flow *C* = 20 *C* = 12 *C* = 10 *C* = 5 tt vt vv

*g,i* = 2.1*X*0.678

*g i g*

Using this analysis, the interfacial component for the high pink triangle at the right hand side of stratified flow data separated into different categories (t–t, v–t and v–v) (Cui and

of a negative value does not exist. This means that there is an error in the measurement in one data point for the stratified flow at lt flow mechanisms of liquid-gas. In this analysis, the data points of tt, lt, and ll flow mechanisms of liquid-gas were fit with only one line instead of three different lines for each flow mechanism of liquid-gas (tt, lt, and ll) because tt has

, 2.1

However, drawing a different line of the interfacial component for the stratified flow for each flow mechanism of liquid-gas (tt, lt, and ll) will be more accurate. This analysis can be also done for other flow patterns: annular (191), slug (277), wavy (94), and bubbly (32).

It should be noted that the nonexistence of the interfacial component for some data sets for any flow patterns: annular (191), slug (277), wavy (94), bubbly (32), and stratified (25)) means that there is an error in the measurement of some data sets of Lockhart-Martinelli

> 0.01 0.1 1 10 100 *X*

**Figure 4.** Analysis of Stratified Flow Data Separated into Different Categories (t–t, v–t and v–v) Using

Multi-parameter models may be developed using both the Chisholm model and the modified Chisholm models, by correlating the constants *C*, *A*, and *m* with other dimensionless parameters. For example, Sun and Mishima (2009) adopted an approach that led to the development of *C* in the laminar flow region as a function of the following

although their paper has received nearly 1000 citations in journal papers.

The drift flux model is a type of separated flow model. In the drift flux model, attention is focused on the relative motion rather than on the motion of the individual phases. The drift flux model was developed by Wallis (1969). The drift flux model has widespread application to bubble flow and plug flow. The drift flux model is not particularly suitable to a flow such as annular flow that has two characteristic velocities in one phase: the liquid film velocity and the liquid drop velocity. However, the drift flux model has been used for annular flows, but with no particular success.

The drift flux model is the fifth example of the existing void fraction models. The Rouhani and Axelsson (1970) model is an instance for this type of model. In the drift-flux model, the void fraction *()* is a function of the gas superficial velocity *(Ug)*, the total superficial velocity *(U)*, the phase distribution parameter *(Co)*, and the mean drift velocity *(ugj)* that includes the effect of the relative velocity between the phases. The form of the drift-flux model is

$$\alpha = \frac{\mathcal{U}\_{\text{g}}}{\mathcal{C}\_{o}\mathcal{U} + \boldsymbol{\mu}\_{\text{g}\dagger}} \tag{212}$$

The drift-flux correlations often present procedures to compute *Co* and *ugj*. Since the expressions of *Co* and *ugj* are usually functions of the void fraction *()*, the predictions of the void fraction *()* are calculated using method of solving of non-linear equation.

#### *6.3.5. Two-fluid model*

This model is known as the two-fluid model designating two phases or components. This model is an advanced predictive tool for liquid-gas two-phase flow in engineering applications. It is based on the mass, momentum and energy balance equations for every phase (Ishii, 1987). In this model, every phase or component is treated as a separate fluid with its own set of governing balance equations. In general, every phase has its own velocity, temperature and pressure. This approach enables the prediction of important nonequilibrium phenomena of two-phase flow like the velocity difference between liquid and gas phase. This prediction is important for two-phase flows in large shell sides of steam generators and kettle reboilers, where even different gas and liquid velocity directions exist.

## **6.4. Other methods**

There are other methods of analysis like integral analysis, differential analysis, computational fluid dynamics (CFD), and artificial neural network (ANN).

Two-Phase Flow 321

*6.4.4. Artificial Neural Network (ANN)*

**7. Summary and conclusions**

chapter represent the state of the art.

**Author details**

**Acknowledgement**

M.M. Awad

In recent years, artificial neural network (ANN) has been universally used in many applications related to engineering and science. ANN has the advantage of self-learning and self-organization. ANN can employ the prior acquired knowledge to respond to the new information rapidly and automatically. When the traditional methods are difficult to be carried out or sometimes the specific models of mathematical physics will not be thoroughly existing, the neural network will be considered as a very good tool to tackle these timeconsuming and complex nonlinear relations because neural network has the excellent characteristics of parallel processing, calculating for complex computation and self-learning. The development of any ANN model involves three basic steps. First, the generation of data required for training. Second, the training and testing of the ANN model using the information about the inputs to predict the values of the output. Third, the evaluation of the ANN configuration that leads to the selection of an optimal configuration that produces the best results based on some preset measures. The optimum ANN model is also validated using a larger dataset. In the area of two-phase flow, the applications of the ANN include the prediction of pressure drop (Osman and Aggour, 2002), identifying flow regimes (Selli and Seleghim, 2007), predicting liquid holdup (Osman, 2004) and (Shippen and Scott, 2004), and the determination of condensation heat transfer characteristics during downward

annular flow of R134a inside a vertical smooth tube (Balcilar et. al., 2011).

This chapter aims to introduce the reader to the modeling of two-phase flow in general, liquid-gas flow in particular, and the prediction of frictional pressure gradient specifically. Different modeling techniques were presented for two-phase flow. Recent developments in theory and practice are discussed. The reader of this chapter is encouraged to pursue the associated journal and text references for additional theory not covered, especially the state of the art and review articles because they contain much useful information pertaining to the topics of interest. Given the rapid growth in the research topic of two-phase flow, new models and further understanding in areas like nano fluids will likely be achieved in the near future. Although, for most design and research applications, the topics covered in this

*Mechanical Power Engineering Department, Faculty of Engineering, Mansoura University, Egypt*

The author acknowledges his Ph. D. supervisor, Prof. Yuri S. Muzychka, who introduced him to the possibilities of analytical modeling during his Ph. D. thesis. Also, the author gratefully acknowledges ASME International Petroleum Technology Institute (IPTI)

## *6.4.1. Integral analysis*

In a one-dimensional integral analysis, the form of certain functions which describe, for instance, the velocity or concentration distribution in a pipe is assumed first. Then, these functions are made to satisfy appropriate boundary conditions and the basic equations of fluid mechanics (continuity equation, momentum equation, and energy equation) in integral form. Single-phase boundary layers are analyzed using similar techniques.

## *6.4.2. Differential analysis*

The velocity and concentration fields are deduced from suitable differential equations. Usually, the equations are written for time-average quantities, like in single-phase theories of turbulence.

## *6.4.3. Computational Fluid Dynamics (CFD)*

Two-phase flows are encountered in a wide range of industrial and natural situations. Due to their complexity such flows have been investigated only analytically and experimentally. New computing facilities provide the flexibility to construct computational models that are easily adapted to a wide variety of physical conditions without constructing a large-scale prototype or expensive test rigs. But there is an inherent uncertainty in the numerical predictions due to stability, convergence and accuracy. The importance of a well-placed mesh is highlighted in the modeling of two-phase flows in horizontal pipelines (Lun et al., 1996).

Also, with the increasing interest in multiphase flow in microchannels and advancement in interface capturing techniques, there have recently been a number of attempts to apply computational fluid dynamics (CFD) to model Taylor flow such as van Baten and Krishna (2004), Taha and Cui (2006a, 2006b) and Gupta et al. (2009). The CFD package, Fluent was used in these numerical studies of CFD modeling of Taylor flow.

In addition, Liu et al. (2011) developed recently a new two-fluid two-component computational fluid dynamics (CFD) model to simulate vertical upward two-phase annular flow. The researchers utilized the two-phase VOF scheme to model the roll wave flow, and described the gas core by a two-component phase consisting of liquid droplets and gas phase. They took into account the entrainment and deposition processes by source terms of the governing equations. Unlike the previous models, their newly developed model included the influence of liquid roll waves directly determined from the CFD code that was able to provide more detailed and, the most important, more self-standing information for both the gas core flow and the film flow as well as their interactions. They compared predicted results with experimental data, and achieved a good agreement.

## *6.4.4. Artificial Neural Network (ANN)*

320 An Overview of Heat Transfer Phenomena

There are other methods of analysis like integral analysis, differential analysis,

In a one-dimensional integral analysis, the form of certain functions which describe, for instance, the velocity or concentration distribution in a pipe is assumed first. Then, these functions are made to satisfy appropriate boundary conditions and the basic equations of fluid mechanics (continuity equation, momentum equation, and energy equation) in integral

The velocity and concentration fields are deduced from suitable differential equations. Usually, the equations are written for time-average quantities, like in single-phase theories

Two-phase flows are encountered in a wide range of industrial and natural situations. Due to their complexity such flows have been investigated only analytically and experimentally. New computing facilities provide the flexibility to construct computational models that are easily adapted to a wide variety of physical conditions without constructing a large-scale prototype or expensive test rigs. But there is an inherent uncertainty in the numerical predictions due to stability, convergence and accuracy. The importance of a well-placed mesh is highlighted in the modeling of two-phase flows in horizontal pipelines (Lun et al.,

Also, with the increasing interest in multiphase flow in microchannels and advancement in interface capturing techniques, there have recently been a number of attempts to apply computational fluid dynamics (CFD) to model Taylor flow such as van Baten and Krishna (2004), Taha and Cui (2006a, 2006b) and Gupta et al. (2009). The CFD package, Fluent was

In addition, Liu et al. (2011) developed recently a new two-fluid two-component computational fluid dynamics (CFD) model to simulate vertical upward two-phase annular flow. The researchers utilized the two-phase VOF scheme to model the roll wave flow, and described the gas core by a two-component phase consisting of liquid droplets and gas phase. They took into account the entrainment and deposition processes by source terms of the governing equations. Unlike the previous models, their newly developed model included the influence of liquid roll waves directly determined from the CFD code that was able to provide more detailed and, the most important, more self-standing information for both the gas core flow and the film flow as well as their interactions. They compared

used in these numerical studies of CFD modeling of Taylor flow.

predicted results with experimental data, and achieved a good agreement.

computational fluid dynamics (CFD), and artificial neural network (ANN).

form. Single-phase boundary layers are analyzed using similar techniques.

**6.4. Other methods**

*6.4.1. Integral analysis*

*6.4.2. Differential analysis*

*6.4.3. Computational Fluid Dynamics (CFD)*

of turbulence.

1996).

In recent years, artificial neural network (ANN) has been universally used in many applications related to engineering and science. ANN has the advantage of self-learning and self-organization. ANN can employ the prior acquired knowledge to respond to the new information rapidly and automatically. When the traditional methods are difficult to be carried out or sometimes the specific models of mathematical physics will not be thoroughly existing, the neural network will be considered as a very good tool to tackle these timeconsuming and complex nonlinear relations because neural network has the excellent characteristics of parallel processing, calculating for complex computation and self-learning. The development of any ANN model involves three basic steps. First, the generation of data required for training. Second, the training and testing of the ANN model using the information about the inputs to predict the values of the output. Third, the evaluation of the ANN configuration that leads to the selection of an optimal configuration that produces the best results based on some preset measures. The optimum ANN model is also validated using a larger dataset. In the area of two-phase flow, the applications of the ANN include the prediction of pressure drop (Osman and Aggour, 2002), identifying flow regimes (Selli and Seleghim, 2007), predicting liquid holdup (Osman, 2004) and (Shippen and Scott, 2004), and the determination of condensation heat transfer characteristics during downward annular flow of R134a inside a vertical smooth tube (Balcilar et. al., 2011).

## **7. Summary and conclusions**

This chapter aims to introduce the reader to the modeling of two-phase flow in general, liquid-gas flow in particular, and the prediction of frictional pressure gradient specifically. Different modeling techniques were presented for two-phase flow. Recent developments in theory and practice are discussed. The reader of this chapter is encouraged to pursue the associated journal and text references for additional theory not covered, especially the state of the art and review articles because they contain much useful information pertaining to the topics of interest. Given the rapid growth in the research topic of two-phase flow, new models and further understanding in areas like nano fluids will likely be achieved in the near future. Although, for most design and research applications, the topics covered in this chapter represent the state of the art.

## **Author details**

M.M. Awad *Mechanical Power Engineering Department, Faculty of Engineering, Mansoura University, Egypt*

## **Acknowledgement**

The author acknowledges his Ph. D. supervisor, Prof. Yuri S. Muzychka, who introduced him to the possibilities of analytical modeling during his Ph. D. thesis. Also, the author gratefully acknowledges ASME International Petroleum Technology Institute (IPTI) scholarship awarded to him in 2005 and 2006. In addition, the author wants to thank the Editor, Prof. M. Salim Newaz Kazi, for inviting him to prepare this chapter.

Two-Phase Flow 323

*f ''e* electric force density, N/m2

*Fre* Electric Froude number

*g* gravitational acceleration, m/s2

modified Galileo number

*h* heat transfer coefficient, W/m2.K

*hlg* latent heat of voporization, J/kg *hsl* latent heat of melting, J/kg

*Jg* dimensionless vapor mass flux

*K* parameter in the Taitel and Dukler (1976) map

*K1*, *K2, K3* new non-dimensional constant of Kandlikar

dimensionless liquid plug length = *Ls/d*

*m* exponent in the modified Chisholm model

*Ma* Masuda number or dielectric Rayleigh number

*Ma* Homogeneous Equilibrium Model (HEM) Mach number

*K1*, *K2, K3* constants in the Cengel (1967) definition for viscosity

 modified Jacob number *K* mass transfer coefficient, m/s

*k* thermal conductivity, W/m.K

*G* mass flux, kg/m2.s

*Ga* Galileo number

*GM* Geometric mean *Gz* Graetz number

*HM* Harmonic mean *I* current, A

*Ja* Jacob number

*Kf* Boiling number *Ka* Kapitza number *Kn* Knudsen number *Kr* von Karman number *Ku* Kutateladze number *L* characteristic length, m

*L* length, m

*Lc* capillary length, m *Ls* liquid plug length, m

*La* Laplace number *Le* Lewis number

*Lo* dimensionless number

*m* mass flow rate, kg/s

*Mo* Morton number

*Hl* liquid holdup fraction

*Ga\**

*Ja\**

*L\**

*Fr\** ratio of Froude number to Atwood ratio

*Fo* Fourier number *FR* filling ratio *Fr* Froude number

## **Nomenclature**

*Mo* Morton number

322 An Overview of Heat Transfer Phenomena

*A* constant in the modified Chisholm model

*a* constant in Hemeida and Sumait (1988) correlation

*b* constant in Hemeida and Sumait (1988) correlation

*c* constant in Hemeida and Sumait (1988) correlation

**Nomenclature**

*A* area, m2

*A* Phillips parameter *a* Churchill parameter

*AM* Arithmetic mean *AR* Aspect ratio

*Bo* Bond number *Bod* Bodenstein number *C* Chisholm constant

*c* sound speed, m/s *CD* drag coefficient

*Ca* Capillary number *Cn* Cahn number *Co* Confinement number *Co* Convection number *Co*u Courant number *D* mass diffusivity, m2/s *d* pipe diameter, m

*Co* the phase distribution parameter *cp* constant-pressure specific heat, J/kg.K

E electric field strength, V/m

*EM* dimensionless number *Er* dimensionless number *EF* Enhanced factor *Eo* Eötvös number *Eu* Euler number

*f* Fanning friction factor *f* wave frequency, Hz

*E* two-phase heat transfer multiplier

*Ehd* EHD number or conductive Rayleigh number

*F* parameter in the Taitel and Dukler (1976) map

*C* constant

*Ar* Archimedes number *At* Atwood ratio *B* Phillips parameter *b* Churchill parameter

scholarship awarded to him in 2005 and 2006. In addition, the author wants to thank the

Editor, Prof. M. Salim Newaz Kazi, for inviting him to prepare this chapter.

Two-Phase Flow 325

*l*

*Subscripts*

*air* air *b* bubble

*D* Darcy

void fraction

volumetric quality

*x* mesh becomes finer, m

liquid film thickness, m

permittivity of free space (

dynamic viscosity, kg/m.s

kinematic viscosity, m2/s

surface tension, N/m

*0* vacuum or reference

*c* continuous phase

*d* dispersed phase *dh* hydraulic diameter

*go* gas only (all flow as gas)

*eq* equivalent *f* frictional *g* gas

*h* hydraulic *i* inner or inlet *i* interfacial

*a* acceleration

molecular mean free path length, m

*c* characteristic flow system time, s

physical property coefficient

*p* particle momentum response time, s *θ* inclination angle to the horizontal

*s* = /*0*)

*g2* two-phase frictional multiplier for gas alone flow

*2* two-phase frictional multiplier for liquid alone flow

*go2* two-phase frictional multiplier for total flow assumed gas

dimensionless parameter used in Baker flow pattern map

dimensionless parameter used in Baker flow pattern map

total liquid mass flow rate on both sides of the tube per unit length of tube

*lo2* two-phase frictional multiplier for total flow assumed liquid

*0* = 8.854 x 10-12 N/V2)

*s* dielectric constant (

permittivity, N/V2

pipe roughness, m

*c* ion mobility, m2/Vs

density, kg/m3

*t* time step size, s

*Nf* inverse viscosity number

*n* Blasius index *n* exponent

*Nu* Nusselt number *Oh* Ohnesorge number *p* fitting parameter *dp/dz* pressure gradient, Pa/m

*p* pressure drop, Pa

*Ph* phase change number *Po* Poiseuille constant *Pr* Prandtl number *Q* heat transfer rate, W *Q* volumetric flow rate, m3/s

*Pe* Peclet number

*q* heat flux, W/m2 *R* pipe radius, m *Re* Reynolds number *Ref* film Reynolds number *Rep* particle Reynolds number

*Ri* Richardson number

*Sc* Schmidt number *Sh* Sherwood number *St* Stanton number *Stk* Stokes number *Str* Strouhal number *Su* Suratman number

*T* temperature, K

*x* mass quality

concentration

thermal diffusivity, ms

*T* temperature difference, K *U* superficial velocity, m/s *ugj* mean drift velocity, m/s *v* specific volume, m3/kg

*X* Lockhart-Martinelli parameter *x* distance in x-direction, m

*X\** modified Lockhart-Martinelli parameter

/s

*Greek*

*S* slip ratio

dimensionless frictional pressure drop, Pa

*Re\** laminar equivalent Reynolds number

*T* parameter in the Taitel and Dukler (1976) map

*pf \**

## **8. References**

Agrawal, N., Bhattacharyya, S., and Nanda, P., 2011, Flow Characteristics of Capillary Tube with CO2 Transcritical Refrigerant Using New Viscosity Models for Homogeneous Two-Phase Flow, International Journal of Low-Carbon Technologies, 6 (4), pp. 243-248.

Two-Phase Flow 327

Aussillous, P., and Quere, D., 2000, Quick Deposition of a Fluid on the Wall of a Tube,

Awad, M. M., 2007a, Two-Phase Flow Modeling in Circular Pipes, Ph.D. Thesis, Memorial

Awad, M. M., 2007b, Comments on Condensation and evaporation heat transfer of R410A inside internally grooved horizontal tubes by M. Goto, N. Inoue and N. Ishiwatari,

Awad, M. M., 2010a, Comments on Experimental Investigation and Correlation of Two-Phase Frictional Pressure Drop of R410A-Oil Mixture Flow Boiling in a 5 mm Microfin Tube Int. J. Refrigeration 32/1 (2009) 150-161, by Ding, G., Hu, H., Huang, X., Deng, B.,

Awad, M. M., 2010b, Comments on Measurement and Correlation of Frictional Two-Phase Pressure Drop of R410A/POE Oil Mixture Flow Boiling in a 7 mm Straight Micro-Fin Tube by H.-t. Hu, G.-l. Ding, and K.-j. Wang, Applied Thermal Engineering, 30 (2-3),

Awad, M. M., 2011, Comments on "Pressure drop during horizontal flow boiling of R410A/oil mixture in 5 mm and 3 mm smooth tubes" by H-t Hu, G-l Ding, X-c Huang,

Awad, M. M., 2012a, Discussion: Heat Transfer Mechanisms During Flow Boiling in Microchannels (Kandlikar, S. G., 2004, ASME Journal of Heat Transfer, 126 (2), pp. 8-16),

Awad, M. M., 2012b, Comments on "Two-phase flow and boiling heat transfer in two

Awad, M. M., and Butt, S. D., 2009a, A Robust Asymptotically Based Modeling Approach for Two-Phase Liquid-Liquid Flow in Pipes, ASME 28th International Conference on Offshore Mechanics and Arctic Engineering (OMAE2009), Session: Petroleum

Awad, M. M., and Butt, S. D., 2009c, A Robust Asymptotically Based Modeling Approach for Two-Phase Flow in Porous Media, ASME Journal of Heat Transfer, 131 (10), Article (101014) (The Special Issue of JHT on Recent Advances in Porous Media Transport), Also presented at ASME 27th International Conference on Offshore Mechanics and Arctic Engineering (OMAE2008), Session: Offshore Technology, Petroleum Technology

Awad, M. M., and Muzychka, Y. S., 2004a, A Simple Two-Phase Frictional Multiplier Calculation Method, Proceedings of IPC2004, International Pipeline Conference, Track: 3. Design & Construction, Session: System Design/Hydraulics, IPC04-0721, Vol. 1, pp.

B. Deng, and Y-f Gao, Applied Thermal Engineering, 31 (16), pp. 3629-3630.

vertical narrow annuli", Nuclear Engineering and Design, 245, pp. 241-242.

Technology, OMAE2009-79072, Honolulu, Hawaii, USA, May 31-June 5, 2009. Awad, M. M., and Butt, S. D., 2009b, A Robust Asymptotically Based Modeling Approach for Two-Phase Gas-Liquid Flow in Fractures, 12th International Conference on Fracture (ICF12), Session: Oil and Gas Production and Distribution, ICF2009-646, Ottawa,

ASME Journal of Heat Transfer, 134 (1), Article No. (015501).

II, OMAE2008-57792, Estoril, Portugal, June 15-20, 2008.

475-483, Calgary, Alberta, October 4-8, 2004.

and Gao, Y., International Journal of Refrigeration, 33 (1), pp. 205-206.

Physics of Fluids, 12 (10), pp. 2367-2371.

pp. 260-261.

Canada, July 12-17, 2009.

University of Newfoundland, St. John's, NL, Canada.

International Journal of Refrigeration, 30 (8), pp. 1466.

Aussillous, P., and Quere, D., 2000, Quick Deposition of a Fluid on the Wall of a Tube, Physics of Fluids, 12 (10), pp. 2367-2371.

326 An Overview of Heat Transfer Phenomena

*le* laminar equivalent

*m* homogeneous mixture

*ll* laminar liquid-laminar gas flow type *lo* liquid only (all flow as liquid)

*lt* laminar liquid- turbulent gas flow type

*tl* turbulent liquid-laminar gas flow type

*tt* turbulent liquid-turbulent gas flow type

Agrawal, N., Bhattacharyya, S., and Nanda, P., 2011, Flow Characteristics of Capillary Tube with CO2 Transcritical Refrigerant Using New Viscosity Models for Homogeneous Two-Phase Flow, International Journal of Low-Carbon Technologies, 6 (4), pp. 243-248. Akbar, M. K., Plummer, D. A., and Ghiaasiaan, S. M., 2003, On Gas-Liquid Two-Phase Flow Regimes in Microchannels, International Journal of Multiphase Flow, 29 (5) pp. 855-865. Akers, W. W., Deans, H. A., and Crosser, O. K., 1959, Condensation Heat Transfer within Horizontal Tubes, Chemical Engineering Progress Symposium Series, 55 (29), pp. 171–

Al-Sarkhi, A., Sarica, C., and Magrini, K., 2012, Inclination Effects on Wave Characteristics in

Angeli, P., and Gavriilidis, A., 2008, Hydrodynamics of Taylor Flow in Small Channels: A Review, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of

Arrhenius, S., 1887, On the Internal Friction of Solutions in Water, Zeitschrift für

Annular Gas–Liquid Flows, AIChE Journal, 58 (4), pp. 1018-1029.

ASHRAE, 1993, Handbook of Fundamentals, ASHRAE, Atlanta, GA, Chap. 4.

Mechanical Engineering Science, 222 (5), pp. 737-751.

Physikalische Chemie (Leipzig), 1, pp. 285-298.

*l* liquid

*m* mean *max* maximum *min* minimum *o* outer or outlet

*o* oil *p* particle *p* plug

*Poise* Poiseuille flow *r* refrigerant *s* saturation *s* sound *slug* slug

*tp* two-phase

*w* wall *water* water

**8. References**

176.

Awad, M. M., and Muzychka, Y. S., 2004b, A Simple Asymptotic Compact Model for Two-Phase Frictional Pressure Gradient in Horizontal Pipes, Proceedings of IMECE 2004, Session: FE-8 A Gen. Pap.: Multiphase Flows - Experiments and Theory, IMECE2004- 61410, Anaheim, California, November 13-19, 2004.

Two-Phase Flow 329

Bandel, J., 1973, Druckverlust und Wärmeübergang bei der Verdampfung siedender Kältemittel im durchströmten waagerechten Rohr, Doctoral Dissertation, Universität

Bankoff, S. G., 1960, A Variable Density Single-Fluid Model for Two-Phase Flow with Particular Reference to Steam-Water Flow, Journal of Heat Transfer, 82 (4), pp. 265-272. Beattie, D. R. H., and Whalley, P. B., 1982, A Simple Two-Phase Frictional Pressure Drop Calculation Method, International Journal of Multiphase Flow, 8 (1), pp. 83-87. Becher, P., 2001, Emulsions: Theory and Practice, 3rd edition, Oxford University Press, New

Bico, J., and Quere, D., 2000, Liquid Trains in a Tube, Europhysics Letters, 51 (5), pp. 546-

Blasius, H., 1913, Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssikeiten, Forsch.

Bonfanti, F., Ceresa, I., and Lombardi, C., 1979, Two-Phase Pressure Drops in the Low

Borishansky, V. M., Paleev, I. I., Agafonova, F. A., Andreevsky, A. A., Fokin, B. S., Lavrentiev, M. E., Malyus-Malitsky, K. P., Fromzel V. N., and Danilova, G. P., 1973, Some Problems of Heat Transfer and Hydraulics in Two-Phase Flows, International

Brauner, N., and Moalem-Maron, D., 1992, Identification of the Range of 'Small Diameters' Conduits, Regarding Two-Phase Flow Pattern Transitions, International

Breber, G., Palen, J., and Taborek, J., 1980, Prediction of Horizontal Tube-Side Condensation of Pure Components Using Flow Regime Criteria, ASME Journal of Heat Transfer, 102

Bretherton, F. P., 1961, The Motion of Long Bubbles in Tubes, Journal of Fluid Mechanics, 10

Carson, J. K., Lovatt, S. J., Tanner, D. J., and Cleland, A. C., 2005, Thermal Conductivity Bounds for Isotropic, Porous Materials, International Journal of Heat and Mass

Catchpole, J. P., and Fulford, G. D., 1966, Dimensionless Groups, Industrial and

Cavallini, A., Censi, G., Del Col, D., Doretti, L., Longo, G. A., and Rossetto, L., 2002, Condensation of Halogenated Refrigerants inside Smooth Tubes, HVAC and R

Celata, G. P., 2004, Heat Transfer and Fluid Flow in Microchannels, Begell House, Redding,

Cengel, J., 1967, Viscosity of Liquid-Liquid Dispersions in Laminar and Turbulent Flow,

Chang, J. S., 1989, Stratified Gas–Liquid Two-Phase Electrohydrodynamics in Horizontal

Pipe Flow, IEEE Transactions on Industrial Applications, 25 (2), pp. 241–247.

Flowrate Region, Energia Nucleare, 26 (10), pp. 481-492.

Journal of Heat and Mass Transfer, 16 (6), pp. 1073-1085.

Communications in Heat and Mass Transfer, 19 (1), pp. 29-39.

Karlsruhe.

York, NY.

Gebiete Ingenieurw., 131.

(3), pp. 471-476.

(2), pp. 166-188.

CT.

Transfer, 48 (11), pp. 2150-2158.

Research, 8 (4), pp. 429-451.

Engineering Chemistry, 58 (3), pp. 46-60.

PhD Dissertation Thesis, Oregon State University.

550.

Bandel, J., 1973, Druckverlust und Wärmeübergang bei der Verdampfung siedender Kältemittel im durchströmten waagerechten Rohr, Doctoral Dissertation, Universität Karlsruhe.

328 An Overview of Heat Transfer Phenomena

Awad, M. M., and Muzychka, Y. S., 2004b, A Simple Asymptotic Compact Model for Two-Phase Frictional Pressure Gradient in Horizontal Pipes, Proceedings of IMECE 2004, Session: FE-8 A Gen. Pap.: Multiphase Flows - Experiments and Theory, IMECE2004-

Awad, M. M., and Muzychka, Y. S., 2005a, Bounds on Two-Phase Flow. Part I. Frictional Pressure Gradient in Circular Pipes, Proceedings of IMECE 2005, Session: FED-11 B Numerical Simulations and Theoretical Developments for Multiphase Flows-I,

Awad, M. M., and Muzychka, Y. S., 2005b, Bounds on Two-Phase Flow. Part II. Void Fraction in Circular Pipes, Proceedings of IMECE 2005, Session: FED-11 B Numerical Simulations and Theoretical Developments for Multiphase Flows-I, IMECE2005-81543,

Awad, M. M. and Muzychka, Y. S., 2007, Bounds on Two-Phase Frictional Pressure Gradient in Minichannels and Microchannels, Heat Transfer Engineering, 28 (8-9), pp. 720-729. Also presented at The 4th International Conference on Nanochannels, Microchannels and Minichannels (ICNMM 2006), Session: Two-Phase Flow, Numerical and Analytical Modeling, ICNMM2006-96174, Stokes Research Institute, University of Limerick,

Awad, M. M. and Muzychka, Y. S., 2008, Effective Property Models for Homogeneous Two

Awad, M. M., and Muzychka, Y. S., 2010a, Review and Modeling of Two-Phase Frictional Pressure Gradient at Microgravity Conditions, ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conference on Nanochannels, Microchannels, and Minichannels (FEDSM2010-ICNMM2010), Symposium 1-14 4th International Symposium on Flow Applications in Aerospace, FEDSM2010-

Awad, M. M., and Muzychka, Y. S., 2010b, Two-Phase Flow Modeling in Microchannels and Minichannels, Heat Transfer Engineering, 31 (13), pp. 1023-1033. Also presented at The 6th International Conference on Nanochannels, Microchannels and Minichannels (ICNMM2008), Session: Two-Phase Flow, Modeling and Analysis of Two-Phase Flow, ICNMM2008-62134, Technische Universitaet of Darmstadt, Darmstadt, Germany, June

Baker, O., 1954, Simultaneous Flow of Oil and Gas, Oil and Gas Journal, 53, pp.185-195. Balcilar, M., Dalkilic, A. S., and Wongwises, S., 2011, Artificial Neural Network Techniques for the Determination of Condensation Heat Transfer Characteristics during Downward Annular Flow of R134a inside a Vertical Smooth Tube, International Communications

Banasiak, K., and Hafner, A., 2011, 1D Computational Model of a Two-Phase R744 Ejector for Expansion Work Recovery, International Journal of Thermal Sciences, 50 (11), pp.

Phase Flows, Experimental and Thermal Fluid Science, 33 (1), pp. 106-113.

ICNMM2010-30876, Montreal, Canada, August 1-5, 2010.

in Heat and Mass Transfer 38 (1), pp. 75-84.

61410, Anaheim, California, November 13-19, 2004.

Orlando, Florida, November 5-11, 2005.

Ireland, June 19-21, 2006.

23-25, 2008.

2235-2247.

IMECE2005-81493, Orlando, Florida, November 5-11, 2005.

Chang, J. S., 1998, Two-Phase Flow in Electrohydrodynamics, in: Castellanos, A., (Ed.), Part V, Electrohdyrodynamics, International Centre for Mechanical Sciences Courses and Lectures No. 380, Springer, New York.

Two-Phase Flow 331

Cotton, J., Robinson, A. J., Shoukri, M., and Chang, J. S., 2005, A Two-Phase Flow Pattern Map for Annular Channels under a DC Applied Voltage and the Application to Electrohydrodynamic Convective Boiling Analysis, International Journal of Heat and

Cotton, J. S., Shoukri, M., Chang, J. S., and Smith-Pollard, T., 2000, Electrohydrodynamic (EHD) Flow and Convective Boiling Augmentation in Single-Component Horizontal Annular Channels, 2000 International Mechanical Engineering Congress and

Crowe, C. T., 2006, Multiphase Flow Handbook, CRC: Taylor & Francis, Boca Raton, FL. Cui, X., and Chen, J. J. J., 2010, A Re-Examination of the Data of Lockhart-Martinelli,

Davidson, W. F., Hardie, P. H., Humphreys, C. G. R., Markson, A. A., Mumford, A. R., and Ravese, T., 1943, Studies of Heat Transmission Through Boiler Tubing at Pressures from

Ding, G., Hu, H., Huang, X., Deng, B., and Gao, Y., 2009, Experimental Investigation and Correlation of Two-Phase Frictional Pressure Drop of R410A–Oil Mixture Flow Boiling in a 5 mm Microfin Tube, International Journal of Refrigeration, 32 (1), pp. 150-161. Drew, T. B., Koo, E. C., and McAdams, W. H., 1932, The Friction Factor for Clean Round

Duda, J. L., and Vrentas, J. S., 1971, Heat Transfer in a Cylindrical Cavity, Journal of Fluid

Dukler, A. E., Moye Wicks and Cleveland, R. G., 1964, Frictional Pressure Drop in Two-Phase Flow. Part A: A Comparison of Existing Correlations for Pressure Loss and Holdup, and Part B: An Approach through Similarity Analysis AIChE Journal, 10 (1),

Einstein, A., 1906, Eine neue Bestimmung der Moleküldimensionen (A New Determination

Einstein, A., 1911, Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Moleküldimensionen (Correction to My Paper: A New Determination of Molecular

English, N. J., and Kandlikar, S. G., 2006, An Experimental Investigation into the Effect of Surfactants on Air–Water Two-Phase Flow in Minichannels, Heat Transfer Engineering, 27 (4), pp. 99-109. Also presented at The 3rd International Conference on Microchannels and Minichannels (ICMM2005), ICMM2005-75110, Toronto, Ontario, Canada, June 13–

Fairbrother, F., and Stubbs, A. E., 1935, Studies in Electro-Endosmosis. Part VI. The Bubble-Tube Method of Measurement, Journal of the Chemical Society (Resumed), pp. 527-529. Fang, X., Xu, Y., and Zhou, Z., 2011, New Correlations of Single-Phase Friction Factor for Turbulent Pipe Flow and Evaluation of Existing Single-Phase Friction Factor

of Molecular Dimensions), Annalen der Physik (ser. 4), 19, pp. 289-306.

Correlations, Nuclear Engineering and Design, 241 (3), pp. 897-902.

Dimensions), Annalen der Physik (ser. 4), 34, pp. 591-592.

International Journal of Multiphase Flow, 36 (10), pp. 836-846.

Mass Transfer, 48 (25-26), pp. 5563-5579.

Exposition, HTD-366-4, pp. 177–184.

Pipe, Trans. AIChE, 28, pp. 56.

Mechanics, 45, pp. 261–279.

pp. 38-51.

15, 2005.

500-3300 Lbs, Trans. ASME, 65 (6), pp. 553-591.

Cotton, J., Robinson, A. J., Shoukri, M., and Chang, J. S., 2005, A Two-Phase Flow Pattern Map for Annular Channels under a DC Applied Voltage and the Application to Electrohydrodynamic Convective Boiling Analysis, International Journal of Heat and Mass Transfer, 48 (25-26), pp. 5563-5579.

330 An Overview of Heat Transfer Phenomena

235 (16), pp. 1737–1747.

18-20, 2007, Puebla, Mexico.

Transfer, 16 (2), pp. 347-358.

Engineering, 84 (24), pp. 91-92.

Energia Nucleare, 7 (6), pp. 407-425.

Multiphase Flow, 35 (12), pp. 1138-1148.

(6), pp. 1121-1128.

Claredon Press, Oxford.

Singapore.

Lectures No. 380, Springer, New York.

Dielectrics and Electrical Insulation, 1 (5), pp. 871-895.

Chang, J. S., 1998, Two-Phase Flow in Electrohydrodynamics, in: Castellanos, A., (Ed.), Part V, Electrohdyrodynamics, International Centre for Mechanical Sciences Courses and

Chang., J. S., and Watson, A., 1994, Electromagnetic Hydrodynamics, IEEE Transactions on

Changhong, P., Yun, G., Suizheng, Q., Dounan, J., and Changhua, N., 2005. Two-Phase Flow and Boiling Heat Transfer in Two Vertical Narrow Annuli, Nuclear Engineering and Design,

Charoensawan, P., and Terdtoon, P., 2007, Thermal Performance Correlation of Horizontal Closed-Loop Oscillating Heat Pipes, 9th Electronics Packaging Technology Conference (EPTC 2007), pp. 906-909, 10-12 December 2007, Grand Copthorne Waterfront Hotel,

Chen, Y., Kulenovic, R., and Mertz, R., 2009, Numerical Study on the Formation of Taylor Bubbles in Capillary Tubes, International Journal of Thermal Sciences, 48 (2), pp. 234- 242. Also presented at Proceedings of the 5th International Conference on Nanochannels, Microchannels and Minichannels (ICNMM2007), ICNMM2007-30182, pp. 939-946, June

Cherlo, S. K. R., Kariveti, S., and Pushpavanam, S., 2010, Experimental and Numerical Investigations of Two-Phase (Liquid–Liquid) Flow Behavior in Rectangular Microchannels, Industrial and Engineering Chemistry Research, 49 (2), pp. 893-899. Chisholm, D., 1967, A Theoretical Basis for the Lockhart-Martinelli Correlation for Two-Phase Flow, International Journal of Heat and Mass Transfer, 10 (12), pp. 1767-1778. Chisholm, D., 1973, Pressure Gradients due to Friction during the Flow of Evaporating Two-Phase Mixtures in Smooth Tubes and Channels, International Journal of Heat and Mass

Chisholm, D., 1983, Two-Phase Flow in Pipelines and Heat Exchangers, George Godwin in

Churchill, S. W., 1977, Friction Factor Equation Spans all Fluid Flow Regimes, Chemical

Churchill, S. W. and Usagi, R., 1972, A General Expression for the Correlation of Rates of Transfer and Other Phenomena, American Institute of Chemical Engineers Journal, 18

Cicchitti, A., Lombaradi, C., Silversti, M., Soldaini, G., and Zavattarlli, R., 1960, Two-Phase Cooling Experiments- Pressure Drop, Heat Transfer, and Burnout Measurements,

Cioncolini, A., Thome, J. R., and Lombardi, C., 2009, Unified Macro-to-Microscale Method to Predict Two-Phase Frictional Pressure Drops of Annular Flows, International Journal of

Colebrook, C. F., 1939, Turbulent Flow in Pipes, with Particular Reference to the Transition between the Smooth and Rough Pipe Laws, J. Inst. Civ. Eng. Lond., 11, pp. 133-156. Collier, J. G. and Thome, J. R., 1994, Convective Boiling and Condensation (3rd Edn),

Association with Institution of Chemical Engineers, London.

Fourar, M. and Bories, S., 1995, Experimental Study of Air-Water Two-Phase Flow Through a Fracture (Narrow Channel), International Journal of Multiphase Flow, 21 (4), pp. 621- 637.

Two-Phase Flow 333

Hayashi, K., Kurimoto, R., and Tomiyama, A., 2010, Dimensional Analysis of Terminal Velocity of Taylor Bubble in a Vertical Pipe, Multiphase Science and Technology, 22 (3),

Hayashi, K., Kurimoto, R., and Tomiyama, A., 2011, Terminal Velocity of a Taylor Drop in a

He, Q., Hasegawa, Y., and Kasagi, N., 2010, Heat Transfer Modelling of Gas–Liquid Slug Flow without Phase Change in a Micro Tube, International Journal of Heat and Fluid

Hemeida, A., and Sumait, F., 1988, Improving the Lockhart and Martinelli Two-Phase Flow Correlation by SAS, Journal of Engineering Sciences, King Saud University, 14 (2), pp.

Hewitt, G. F., and Roberts, D. N., 1969, Studies of Two-Phase Flow Patterns by Simultaneous

Hoogendoorn, C. J., 1959, Gas-Liquid Flow in Horizontal Pipes, Chemical Engineering

Howard, J. A., Walsh, P. A., and Walsh, E. J., 2011, Prandtl and Capillary Effects on Heat Transfer Performance within Laminar Liquid–Gas Slug Flows, International Journal of

Hu, H. -t., Ding, G. -l., and Wang, K. -j., 2008, Measurement and Correlation of Frictional Twophase Pressure Drop of R410A/POE Oil Mixture Flow Boiling in a 7 mm Straight

Hu, H. -t., Ding, G. -l., Huang, X. –c., Deng, B., and Gao, Y. –f., 2009, Pressure Drop During Horizontal Flow Boiling of R410A/Oil Mixture in 5 mm and 3 mm Smooth Tubes,

Hu, X., and Jacobi, A. M., 1996, The Intertube Falling Film: Part 1—Flow Characteristics, Mode Transitions, and Hysteresis, ASME Journal of Heat Transfer, 118 (3), pp. 616-625. Hulburt, E. T. and Newell, T. A., 1997, Modeling of the Evaporation and Condensation of Zeotropic Refrigerants Mixtures in Horizontal Annular Flow, ACRC TR-129, Air Conditioning and Refrigeration Center, University of Illinois at Urbana-Champaign. Irandoust, S., and Andersson, B., 1989, Liquid Film in Taylor Flow through a Capillary,

Ishii, M., 1987, Two-Fluid Model for Two Phase Flow, Multiphase Science and Technology, 5

Jayawardena, S. S., Balakotaiah, V., and Witte, L., 1997, Pattern Transition Maps for

Kandlikar, S. G., 1990, A General Correlation for Saturated Two-Phase Flow Boiling Heat Transfer inside Horizontal and Vertical Tubes, ASME Journal of Heat Transfer 112, (1)

Kandlikar, S. G., 2001, A Theoretical Model to Predict Pool Boiling CHF Incorporating Effects of Contact Angle and Orientation, ASME Journal of Heat Transfer, 123 (12), pp.

Micro-Fin Tube, Applied Thermal Engineering, 28 (11-12), pp. 1272-1283.

Industrial & Engineering Chemistry Research, 28 (11), pp. 1684-1688.

Microgravity Two-Phase Flow, AIChE Journal, 43 (6), pp. 1637-1640.

Vertical Pipe, International Journal of Multiphase Flow, 37 (3), pp. 241-251.

pp. 197-210.

423-435.

(1), pp. 1-63.

pp. 219–228.

1071–1079.

Flow, 31 (1), pp. 126–136.

Science, 9, pp. 205-217.

Flash and X-Ray Photography, AERE-M2159.

Heat and Mass Transfer, 54 (21-22), pp. 4752-4761.

Applied Thermal Engineering, 29 (16), pp. 3353-3365.

Hayashi, K., Kurimoto, R., and Tomiyama, A., 2010, Dimensional Analysis of Terminal Velocity of Taylor Bubble in a Vertical Pipe, Multiphase Science and Technology, 22 (3), pp. 197-210.

332 An Overview of Heat Transfer Phenomena

Chemistry, 60 (3), pp. 71-78.

New York, NY, Chapter 5.

60 (574), pp. 2117-2124.

Refrigeration, 24 (7), pp. 628-638.

University of Illinois at Urbana-Champaign.

(10), pp. 1605-1624.

637.

Fourar, M. and Bories, S., 1995, Experimental Study of Air-Water Two-Phase Flow Through a Fracture (Narrow Channel), International Journal of Multiphase Flow, 21 (4), pp. 621-

Friedel, L., 1979, Dimensionless Relationship for The Friction Pressure Drop in Pipes during Two-Phase Flow of Water and of R 12, Verfahrenstechnik, 13 (4), pp. 241-246. Fulford, G. D., and Catchpole, J. P., 1968, Dimensionless Groups, Industrial and Engineering

García, F., García, R., Padrino, J. C., Mata, C., Trallero J. L., and Joseph, D. D., 2003, Power Law and Composite Power Law Friction Factor Correlations for Laminar and Turbulent Gas-Liquid Flow in Horizontal Pipelines, International Journal of Multiphase Flow, 29

García, F., García, J. M., García, R., and Joseph, D. D., 2007, Friction Factor Improved Correlations for Laminar and Turbulent Gas-Liquid Flow in Horizontal Pipelines,

Ghiaasiaan, S. M., 2008, Two-Phase Flow, Boiling and Condensation in Conventional and

Glielinski, V., 1976, New Equations for Heat and Mass Transfer in Turbulent Pipe and

Gnielinski, V., 1999, Single-Phase Convective Heat Transfer: Forced Convection in Ducts, Heat Exchanger Design Updates, Heat Exchanger Design Handbook, Begell House,

Goto, M., Inoue, N., and Ishiwatari, N., 2001, Condensation and Evaporation Heat Transfer of R410A inside Internally Grooved Horizontal Tubes, International Journal of

Graham, D. M., Kopke, H. P., Wilson, M. J., Yashar, D. A., Chato, J. C. and Newell, T. A., 1999, An Investigation of Void Fraction in the Stratified/Annular Flow Regions in Smooth Horizontal Tubes, ACRC TR-144, Air Conditioning and Refrigeration Center,

Grimes, R., King, C., and Walsh, E., 2007, Film Thickness for Two Phase Flow in a Microchannel, Advances and Applications in Fluid Mechanics, 2 (1), pp. 59-70. Gunther, A., and Jensen, K. F., 2006, Multiphase Microfluidics: From Flow Characteristics to

Gupta, R., Fletcher, D. F., and Haynes, B. S., 2009, On the CFD Modelling of Taylor Flow in

Han, Y., and Shikazono, N., 2009a, Measurement of the Liquid Film Thickness in Micro Tube Slug Flow, International Journal of Heat and Fluid Flow, 30 (5), pp. 842–853. Han, Y., and Shikazono, N., 2009b, Measurement of the Liquid Film Thickness in Micro Square Channel, International Journal of Multiphase Flow, 35 (10), pp. 896-903. Haraguchi, H., Koyama, S., and Fujii, T., 1994, Condensation of Refrigerants HCF C 22, HFC 134a and HCFC 123 in a Horizontal Smooth Tube (2nd Report, Proposals of Empirical Expressions for the Local Heat Transfer Coefficient), Transactions of the JSME, Part B,

Chemical and Materials Synthesis, Lab on a Chip, 6 (12), pp. 1487-1503.

Microchannels, Chemical Engineering Science, 64 (12), pp. 2941-2950.

International Journal of Multiphase Flow, 33 (12), pp. 1320-1336.

Channel Flow, International Chemical Engineering, 16 (2), pp. 359–367.

Miniature Systems, Cambridge University Press, New York.

Kandlikar, S. G., 2004, Heat Transfer Mechanisms During Flow Boiling in Microchannels, ASME Journal of Heat Transfer, 126 (2), pp. 8-16.

Two-Phase Flow 335

Li, W., and Wu, Z., 2010, A General Correlation for Adiabatic Two Phase Flow Pressure Drop in Micro/Mini-Channels, International Journal of Heat and Mass Transfer, 53 (13-

Li, W., and Wu, Z., 2011, Generalized Adiabatic Pressure Drop Correlations in Evaporative Micro/Mini-Channels, Experimental Thermal and Fluid Science, 35 (6,) pp. 866–872. Lin, S., Kwok, C. C. K., Li, R. Y., Chen, Z. H., and Chen, Z. Y., 1991, Local Frictional Pressure Drop during Vaporization for R-12 through Capillary Tubes, International Journal of

Liu, Y., Cui, J., and Li, W. Z., 2011, A Two-Phase, Two-Component Model for Vertical Upward Gas–Liquid Annular Flow, International Journal of Heat and Fluid Flow, 32

Lockhart, R. W., and Martinelli, R. C., 1949, Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes, Chemical Engineering Progress

Lombardi, C., and Ceresa, I., 1978, A Generalized Pressure Drop Correlation in Two-Phase

Lombardi, C., and Carsana, C. G., 1992, Dimensionless Pressure Drop Correlation for Two-Phase Mixtures Flowing Upflow in Vertical Ducts Covering Wide Parameter Ranges,

Lun, I., Calay, R. K., and Holdo, A. E., 1996, Modelling Two-Phase Flows Using CFD,

Ma, X., Briggs, A., and Rose, J. W., 2004, Heat Transfer and Pressure Drop Characteristics for Condensation of R113 in a Vertical Micro-Finned Tube with Wire Insert, International

Mandhane, J. M., Gregory, G. A., and Aziz, K., 1974, A Flow Pattern Map of Gas-Liquid Flow in Horizontal Pipes, International Journal of Multiphase Flow, 1 (4),

Marchessault, R. N., and Mason, S. G., 1960, Flow of Entrapped Bubbles through a Capillary,

Martinelli, R. C., and Nelson, D. B., 1948, Prediction of Pressure Drop during Forced-

McAdams, W. H., Woods, W. K. and Heroman, L. C., 1942, Vaporization inside Horizontal

Mishima, K., and Hibiki, T., 1996, Some Characteristics of Air-Water Two-Phase Flow in Small Diameter Vertical Tubes, International Journal of Multiphase Flow, 22 (4), pp.

Mudawwar, I. A., and El-Masri, M. A., 1986, Momentum and Heat Transfer across Freely-Falling Turbulent Liquid Films, International Journal of Multiphase Flow 12 (5), pp.

Moody, L. F., 1944, Friction Factors for Pipe Flow, Trans. ASME, 66 (8), pp. 671-677.

14), pp. 2732-2739.

(4), pp. 796–804.

pp. 537-553.

703-712.

771-790.

Multiphase Flow, 17 (1), pp. 95-102.

Symposium Series, 45 (1), pp. 39-48.

Applied Energy, 53 (3), pp. 299-314.

Flow, Energia Nucleare, 25 (4), pp. 181-198.

Heat and Technology, 10 (1-2), pp. 125-141.

Communications in Heat and Mass Transfer, 31, pp. 619-627.

Circulation Boiling of Water, Trans. ASME, 70 (6), pp. 695-702.

Tubes. II -Benzene-Oil Mixtures, Trans. ASME, 64 (3), pp. 193-200.

Industrial & Engineering Chemistry, 52 (1), pp. 79-84.

Li, W., and Wu, Z., 2010, A General Correlation for Adiabatic Two Phase Flow Pressure Drop in Micro/Mini-Channels, International Journal of Heat and Mass Transfer, 53 (13- 14), pp. 2732-2739.

334 An Overview of Heat Transfer Phenomena

1085.

York, NY.

Netherlands.

51 (9), pp. 2428-2440.

and Washington, D. C.

Kotloturbostroenie, 3, pp. 10–12.

Mass Transfer, 25 (7), pp. 945–960.

Mechanics – Soviet Research, 1, pp. 29-50.

ASME Journal of Heat Transfer, 126 (2), pp. 8-16.

of Heat Transfer, 134 (1), Article No. (015502).

Heat Transfer Engineering, 32 (7-8), pp. 585-594.

Chemical Engineering Science, 60 (22), pp. 5895–5916.

ASME Journal of Heat Transfer, 132 (8), Article No. (081501).

Kandlikar, S. G., 2004, Heat Transfer Mechanisms During Flow Boiling in Microchannels,

Kandlikar, S. G., 2010a, Scale Effects on Flow Boiling Heat Transfer in Microchannels: A Fundamental Perspective, International Journal of Thermal Sciences, 49 (7), pp. 1073-

Kandlikar, S. G., 2010b, A Scale Analysis Based Theoretical Force Balance Model for Critical Heat Flux (CHF) During Saturated Flow Boiling in Microchannels and Minichannels,

Kandlikar, S. G., 2012, Closure to Discussion of `Heat Transfer Mechanisms During Flow Boiling in Microchannels (2012, ASME J. Heat Transfer, 134, p. 015501), ASME Journal

Kandlikar, S. G., Garimella, S., Li, D., Colin, S., and King, M. R., 2006, Heat Transfer and

Kawahara, A., Sadatomi, M., Nei, K., and Matsuo, H., 2011, Characteristics of Two-Phase Flows in a Rectangular Microchannel with a T-Junction Type Gas-Liquid Mixer,

Keilin, V. E., Klimenko, E. Yu., and Kovalev, I. A., 1969, Device for Measuring Pressure Drop

Kew, P., and Cornwell, K., 1997, Correlations for the Prediction of Boiling Heat Transfer in Small-Diameter Channels, Applied Thermal Engineering, 17 (8-10), pp. 705–715. Kleinstreuer, C., 2003, Two-Phase Flow: Theory and Applications, Taylor & Francis, New

Kreutzer, M. T., 2003. Hydrodynamics of Taylor Flow in Capillaries and Monoliths Channels, Doctoral dissertation. Delft University of Technology, Delft, The

Kreutzer, M. T., Kapteijn, F., Moulijin, J. A., and Heiszwolf, J. J., 2005a, Multiphase Monolith Reactors: Chemical Reaction Engineering of Segmented Flow in Microchannels,

Kreutzer, M. T., Kapteijn, F., Moulijin, J. A., Kleijn, C. R., and Heiszwolf, J. J., 2005b, Inertial and Interfacial Effects on Pressure Drop of Taylor Flow in Capillaries, AIChE Journal,

Kutateladze, 1948, On the Transition to Film Boiling under Natural Convection,

Kutateladze, S. S., 1972, Elements of Hydrodynamics of Gas-Liquid Systems, Fluid

Lazarek, G. M., and Black, S. H., 1982, Evaporative Heat Transfer, Pressure Drop and Critical Heat Flux in a Small Vertical Tube with R-113, International Journal of Heat and

Lefebvre, A. H., 1989,. Atomization and Sprays. Hemisphere Publishing Corp., New York

and Heat Transfer in Two-Phase Helium Flow, Cryogenics, 9 (2), pp. 36-38.

Fluid Flow in Minichannels and Microchannels, Elsevier, Oxford, UK.

Muradoglu, M., Gunther, A., and Stone, H. A., 2007, A Computational Study of Axial Dispersion in Segmented Gas–Liquid Flow, Physics of Fluids, 19 (7), Article No. (072109).

Two-Phase Flow 337

Renardy, M., Renardy, Y., and Li, J., 2001, Numerical Simulation of Moving Contact Line Problems Using a Volume-of-Fluid Method, Journal of Computational Physics, 171 (1),

Revellin, R., and Haberschill, P., 2009, Prediction of Frictional Pressure Drop During Flow Boiling of Refrigerants in Horizontal Tubes: Comparison to an Experimental Database,

Rezkallah, K. S., 1995, Recent Progress in the Studies of Two-Phase Flow at Microgravity

Rezkallah, K. S., 1996, Weber Number Based Flow-Pattern Maps for Liquid-Gas Flows at Microgravity, International Journal of Multiphase Flow, 22 (6), pp. 1265–1270. Rouhani S. Z., and Axelsson, E., 1970, Calculation of Volume Void Fraction in the Subcooled and Quality Region, International Journal of Heat and Mass Transfer, 13 (2), pp. 383-

Sabharwall, P., Utgikar, V., and Gunnerson, F., 2009, Dimensionless Numbers in Phase-Change Thermosyphon and Heat-Pipe Heat Exchangers, Nuclear Technology, 167 (2),

Saisorn, S., and Wongwises, S., 2008, Flow Pattern, Void Fraction and Pressure Drop of Two-Phase Air-Water Flow in a Horizontal Circular Micro-Channel, Experimental Thermal

Saisorn, S., and Wongwises, S., 2009, An Experimental Investigation of Two-Phase Air-Water Flow Through a Horizontal Circular Micro-Channel, Experimental Thermal and

Saisorn, S., and Wongwises, S., 2010, The Effects of Channel Diameter on Flow Pattern, Void Fraction and Pressure Drop of Two-Phase Air-Water Flow in Circular Micro-Channels,

Salman, W., Gavriilidis, A., and Angeli, P., 2004, A Model for Predicting Axial Mixing During Gas–Liquid Taylor Flow in Microchannels at Low Bodenstein Numbers,

Sardesai, R. G., Owen, R. G., and Pulling, D. J., 1981, Flow Regimes for Condensation of a Vapour Inside a Horizontal Tube, Chemical Engineering Science, 36 (7), pp. 1173-1180. Scott, D. S., 1964, Properties of Co-Current Gas- Liquid Flow, Advances in Chemical

Selli, M. F., and Seleghim, P., Jr., 2007, Online Identification of Horizontal Two-Phase Flow Regimes Through Gabor Transform and Neural Network Processing, Heat Transfer

Shah, M. M., 1982, Chart Correlation for Saturated Boiling Heat Transfer: Equations and

Shannak, B., 2009, Dimensionless Numbers for Two-Phase and multiphase flow. In: International Conference on Applications and Design in Mechanical Engineering

Sherwood, T. K., Pigford, R. L., and Wilke, C. R., 1975, Mass Transfer, Mc-Graw Hill, New

Experimental Thermal and Fluid Science, 34 (4), pp. 454-462.

Chemical Engineering Journal, 101 (1-3), pp. 391-396.

Further Study, ASHRAE Trans., 88, Part I, pp. 185-196.

(ICADME), Penang, Malaysia, 11–13 October, 2009.

International Journal of Refrigeration, 32 (3) pp. 487–497.

Conditions, Journal of Advances in Space Research, 16, pp. 123-132.

pp. 243-263.

393.

pp. 325-332.

and Fluid Science, 32 (3), pp. 748-760.

Fluid Science, 33 (2), pp. 306-315.

Engineering, 4, pp. 199-277.

York, NY, USA.

Engineering, 28 (6), pp. 541–548.

Renardy, M., Renardy, Y., and Li, J., 2001, Numerical Simulation of Moving Contact Line Problems Using a Volume-of-Fluid Method, Journal of Computational Physics, 171 (1), pp. 243-263.

336 An Overview of Heat Transfer Phenomena

Francis Group, Boca Raton, FL.

and Mechanics) 16, pp. 355–358.

Facilities, 19 (1), pp. 33-40.

Stanford University, CA.

Cambridge, MA.

Heat Transfer, Part II, pp. 363-368.

University of Illinois/Urbana, IL.

of Multiphase Flow, 37 (8), pp. 888–897.

44010, March 13-17, 2011, Honolulu, Hawaii, USA.

Petroleum Science and Technology, 20 (1-2), pp. 1-15.

Physical Properties, Advances in Heat Transfer, 6, pp. 503-564.

Applied Thermal Engineering, 31 (11-12), pp. 1894-1905.

Journal of Heat and Fluid Flow, 28 (5), pp. 1060-1072.

(072109).

Muradoglu, M., Gunther, A., and Stone, H. A., 2007, A Computational Study of Axial Dispersion in Segmented Gas–Liquid Flow, Physics of Fluids, 19 (7), Article No.

Muzychka, Y. S., Walsh, E., Walsh, P., and Egan, V., 2011, Non-boiling Two Phase Flow in Microchannels, in Microfluidics and Nanofluidics Handbook: Chemistry, Physics, and Life Science Principles, Editors: Mitra, S. K., and Chakraborty, S., CRC Press Taylor &

Ohnesorge, W., 1936, Formation of Drops by Nozzles and the Breakup of Liquid Jets, Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) (Applied Mathematics

Oliemans, R., 1976, Two Phase Flow in Gas-Transmission Pipelines, ASME paper 76-Pet-25, presented at Petroleum Division ASME meeting, Mexico, September 19-24, 1976. Ong, C. L., and Thome, J. R., 2011, Experimental Adiabatic Two-Phase Pressure Drops of R134a, R236fa and R245fa in Small Horizontal Circular Channels, Proceedings of the ASME/JSME 2011 8th Thermal Engineering Joint Conference (AJTEC2011), AJTEC2011-

Osman, E. A., 2004, Artificial Neural Network Models for Identifying Flow Regimes and Predicting Liquid Holdup in Horizontal Multiphase Flow, SPE Production and

Osman, E. A., and Aggour, M. A., 2002, Artificial Neural Network Model for Accurate Prediction of Pressure Drop in Horizontal and Near-Horizontal-Multiphase Flow,

Ouyang, L., 1998, Single Phase and Multiphase Fluid Flow in Horizontal Wells, PhD Dissertation Thesis, Department of Petroleum Engineering, School of Earth Sciences,

Owens, W. L., 1961, Two-Phase Pressure Gradient, ASME International Developments in

Petukhov, B. S., 1970, Heat Transfer and Friction in Turbulent Pipe Flow with Variable

Phan, H. T., Caney, N., Marty, P., Colasson, S., and Gavillet, J., 2011, Flow Boiling of Water in a Minichannel: The Effects of Surface Wettability on Two-Phase Pressure Drop,

Phillips, R. J., 1987, Forced Convection, Liquid Cooled, Microchannel Heat Sinks, Master's Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology,

Pigford, R. L., 1941, Counter-Diffusion in a Wetted Wall Column, Ph. D. Dissertation, The

Quan, S. P., 2011, Co-Current Flow Effects on a Rising Taylor Bubble, International Journal

Quiben, J. M., and Thome, J. R., 2007, Flow Pattern Based Two-Phase Frictional Pressure Drop Model for Horizontal Tubes. Part II: New Phenomenological Model, International

Shippen, M. E., and Scott, S. L., 2004, A Neural Network Model for Prediction of Liquid Holdup in Two-Phase Horizontal Flow, SPE Production and Facilities, 19 (2), pp 67-76.

Two-Phase Flow 339

in Horizontal Pipelines, International Journal of Multiphase Flow, 26 (6), pp. 1019-

Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., and Sadowski, D. L., 1999, Gas-Liquid Two-Phase Flow in Micro-Channels, Part 1: Two-Phase Flow Pattern, International

Turner, J. M., 1966, Annular Two-Phase Flow, Ph.D. Thesis, Dartmouth College, Hanover,

Ullmann, A., and Brauner, N., 2007, The Prediction of Flow Pattern Maps in Mini Channels,

van Baten, J. M., and Krishna, R., 2004, CFD Simulations of Mass Transfer from Taylor Bubbles Rising in Circular Capillaries, Chemical Engineering Science, 59 (12), pp. 2535–

Vandervort, C. L., Bergles, A. E., and Jensen, M. K., 1994, An Experimental Study of Critical Heat Flux in very High Heat Flux Subcooled Boiling, International Journal of Heat and

Venkatesan, M., Das, Sarit K., and Balakrishnan, A. R., 2011, Effect of Diameter on Two-Phase Pressure Drop in Narrow Tubes, Experimental Thermal and Fluid Science, 35 (3),

Wallis, G. B., 1969, One-Dimensional Two-Phase Flow, McGraw-Hill Book Company, New

Walsh, E. J., Muzychka, Y. S., Walsh, P. A., Egan, V., and Punch, J., 2009, Pressure Drop in Two Phase Slug/Bubble Flows in Mini Scale Capillaries, International Journal of

Wang X. -Q., and Mujumdar A. S., 2008a, A Review on Nanofluids - Part I: Theoretical and Numerical Investigations, Brazilian Journal of Chemical Engineering, 25 (4), pp. 613-

Wang X. -Q., and Mujumdar A. S., 2008b, A Review on Nanofluids - Part II: Experiments and Applications, Brazilian Journal of Chemical Engineering, 25 (4), pp. 631-648. Wei, W., Ding, G., Hu, H., and Wang, K., 2007, Influence of Lubricant Oil on Heat Transfer Performance of Refrigerant Flow Boiling inside Small Diameter Tubes. Part I:

Experimental Study, Experimental Thermal and Fluid Science, 32 (1), pp. 67–76. Weisman, J., Duncan, D., Gibson, J., and Crawford, T., 1979, Effects of Fluid Properties and Pipe Diameter on Two-Phase Flow Patterns in Horizontal Lines, International Journal of

Whalley, P. B., 1987, Boiling, Condensation, and Gas-Liquid Flow, Clarendon Press,

Wilson, M. J., Newell, T. A., Chato, J. C., and Infante Ferreira, C. A., 2003, Refrigerant Charge, Pressure Drop and Condensation Heat Transfer in Flattened Tubes,

Whalley, P. B., 1996, Two-Phase Flow and Heat Transfer, Oxford University Press, UK. White, F. M., 2005, Viscous Fluid Flow, 3rd edition, McGraw-Hill Book Co, USA.

International Journal of Refrigeration, 26 (4), pp. 442–451.

Journal of Multiphase Flow 25 (3), pp. 377-394.

Mass Transfer, 37 (Supplement 1), pp. 161-173.

Multiphase Flows, 35 (10), pp. 879-884.

Multiphase Flow 5 (6), pp. 437-462.

Multiphase Science and Technology, 19 (1), pp. 49-73.

1036.

NH, USA.

2545.

pp. 531-541.

York.

630.

Oxford.

in Horizontal Pipelines, International Journal of Multiphase Flow, 26 (6), pp. 1019- 1036.

Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., and Sadowski, D. L., 1999, Gas-Liquid Two-Phase Flow in Micro-Channels, Part 1: Two-Phase Flow Pattern, International Journal of Multiphase Flow 25 (3), pp. 377-394.

338 An Overview of Heat Transfer Phenomena

Engineering, 86 (3), pp. 576–582.

the Hydraulics Divsion - ASCE, 102 (5), pp. 657-664.

Chemical Engineering Science 61 (2), pp. 665-675.

Engineering Science, 61 (2), pp. 676-687.

pp 67-76.

263-269.

768.

107 (2), pp. 424-430.

(3), pp. 485-498.

Mechanics, 10 (2), pp. 161-165.

Heat Transfer Engineering, 24 (6), pp. 46–59.

Shippen, M. E., and Scott, S. L., 2004, A Neural Network Model for Prediction of Liquid Holdup in Two-Phase Horizontal Flow, SPE Production and Facilities, 19 (2),

Sobieszuk, P., Cygański, P., and Pohorecki, R., 2010, Bubble Lengths in the Gas–Liquid Taylor Flow in Microchannels, Chemical Engineering Research and Design, 88 (3), pp.

Spelt, P. D. M., 2005, A Level-Set Approach for Simulations of Flows with Multiple Moving Contact Lines with Hysteresis, Journal of Computational Physics, 207 (2), pp. 389-404. Stephan, K., and Abdelsalam, M., 1980, Heat Transfer Correlation for Natural Convection

Suo, M., and Griffith, P., 1964, Two Phase Flow in Capillary Tubes, ASME Journal of Basic

Swamee, P. K., and Jain, A. K., 1976, Explicit Equations for Pipe Flow Problems, Journal of

Taha, T., and Cui, Z. F., 2006a, CFD Modelling of Slug Flow inside Square Capillaries,

Taha, T., and Cui, Z. F., 2006b, CFD Modelling of Slug Flow in Vertical Tubes, Chemical

Taitel, Y., 1990, Flow Pattern Transition in Two-Phase Flow, Proceedings of 9th International

Taitel, Y., and Dukler, A. E., 1976, A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas–Liquid Flow, AIChE Journal, 22 (1), pp. 47–55. Talimi, V., Muzychka, Y. S., and Kocabiyik, S., 2012, A Review on Numerical Studies of Slug Flow Hydrodynamics and Heat Transfer in Microtubes and Microchannels,

Tandon, T. N., Varma, H. K., and Gupta. C. P., 1982, A New Flow Regime Map for Condensation Inside Horizontal Tubes, ASME Journal of Heat Transfer, 104 (4), pp. 763-

Tandon, T. N., Varma, H. K., and Gupta. C. P., 1985, Prediction of Flow Patterns During Condensation of Binary Mixtures in a Horizontal Tube, ASME Journal of Heat Transfer,

Taylor, G. I., 1932, The Viscosity of a Fluid Containing Small Drops of Another Fluid,

Taylor, G. I., 1961, Deposition of a Viscous Fluid on the Wall of a Tube, Journal of Fluid

Thome, J. R., 2003, On Recent Advances in Modeling of Two-Phase Flow and Heat Transfer,

Tran, T. N., Wambsganss, M. W., and France, D. M., 1996, Small Circular- and Rectangular-Channel Boiling with Two Refrigerants, International Journal of Multiphase Flow, 22

Tribbe, C., and Müller-Steinhagen, H. M., 2000, An Evaluation of the Performance of Phenomenological Models for Predicting Pressure Gradient during Gas-Liquid Flow

Proceedings of the Royal Society of London, Series A, 138 (834), pp. 41-48.

Heat Transfer Conference (IHTC9), Jerusalem, Vol. 1, pp. 237-254.

International Journal of Multiphase Flow, 39, pp. 88-104.

Boiling, International Journal of Heat and Mass Transfer, 23 (1), pp. 73–87.

Yan, Y.-Y., and Lin, T.-F., 1998, Evaporation Heat Transfer and Pressure Drop of Refrigerant R134a in a Small Pipe, International Journal of Heat and Mass Transfer, 41 (24). pp. 4183–4193.

**Chapter 12**