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The frictional pressure gradient of two-phase flow can be related to that of either the vapor or liquid phase flowing alone in the channel [[#References|(Lockhart and Martinelli, 1949; Chisholm, 1967)]]. The frictional pressure gradients of the vapor or liquid phase flow in the channel, with their actual flow rate and properties, can be defined as
	
<center><math>-{{\left( \frac{d{{p}_{F}}}{dz} \right)}_{v}}=\frac{2{{f}_{v}}{{{\dot{{m}''}}}^{2}}{{x}^{2}}}{D{{\rho }_{v}}}\qquad\qquad(1)</math></center>
	
	<center><math>-{{\left( \frac{d{{p}_{F}}}{dz} \right)}_{\ell }}=\frac{2{{f}_{\ell }}{{{\dot{{m}''}}}^{2}}{{(1-x)}^{2}}}{D{{\rho }_{\ell }}}\qquad\qquad(2)</math></center>

where <math>{{f}_{v}}\text{ and }{{f}_{\ell }}</math> are, respectively, the friction factors for the vapor and liquid phases with their actual mass flux flowing in the channel alone.  

Similarly, the frictional pressure gradient in the channel – with the same total mass flow rate of the two-phase flow, but with the properties of the vapor or liquid phase – can be defined as 
	
<center><math>-{{\left( \frac{d{{p}_{F}}}{dz} \right)}_{vo}}=\frac{2{{f}_{v0}}{{{\dot{{m}''}}}^{2}}}{D{{\rho }_{v}}}\qquad\qquad(3)</math></center>
	
<center><math>-{{\left( \frac{d{{p}_{F}}}{dz} \right)}_{\ell o}}=\frac{2{{f}_{\ell 0}}{{{\dot{{m}''}}}^{2}}}{D{{\rho }_{\ell }}}\qquad\qquad(4)</math></center>

where <math>{f_{v0}}</math> is the vapor friction factor if the vapor phase with mass flux, <math>\dot{{m}''},</math> occupies the entire channel, whereas <math>{{f}_{\ell 0}}</math> is the liquid fraction factor if the channel is taken by liquid phase with mass flux <math>\dot{{m}''}</math> alone.  

Through the standard equations and charts for the single-phase flow, the friction factors defined in eqs. (1) – (4) can be related to the respective Reynolds numbers:
	
<center><math>{{\operatorname{Re}}_{v}}=\frac{\dot{{m}''}xD}{{{\mu }_{v}}}\qquad\qquad(5)</math></center>
	
<center><math>{{\operatorname{Re}}_{\ell }}=\frac{\dot{{m}''}(1-x)D}{{{\mu }_{\ell }}}\qquad\qquad(6)</math></center>
	
<center><math>{{\operatorname{Re}}_{vo}}=\frac{\dot{{m}''}D}{{{\mu }_{v}}}\qquad\qquad(7)</math></center>
	
<center><math>{{\operatorname{Re}}_{\ell o}}=\frac{\dot{{m}''}D}{{{\mu }_{\ell }}}\qquad\qquad(8)</math></center>

The relationships between the frictional factor and the Reynolds number are different for laminar and turbulent flow.
	
<center><math>f=\left\{ \begin{matrix}
   \frac{16}{\operatorname{Re}}\begin{matrix}
   {} & \begin{matrix}
   {} & {}  \\
\end{matrix} & \operatorname{Re}<2000  \\
\end{matrix}  \\
   \begin{matrix}
   0.079{{\operatorname{Re}}^{-0.25}} & \operatorname{Re}>2000  \\
\end{matrix}  \\
\end{matrix} \right.\qquad\qquad(9)</math></center>

The frictional pressure gradient of the two-phase flow can be related to those defined in eqs. (1) – (4) through pressure drop multipliers defined as
	
<center><math>\phi _{v}^{2}=\frac{d{{p}_{F}}/dz}{{{\left( d{{p}_{F}}/dz \right)}_{v}}}\qquad\qquad(10)</math></center>

	
<center><math>\phi _{\ell }^{2}=\frac{d{{p}_{F}}/dz}{{{\left( d{{p}_{F}}/dz \right)}_{\ell }}}\qquad\qquad(11)</math></center>
	
	
<center><math>\phi _{vo}^{2}=\frac{d{{p}_{F}}/dz}{{{\left( d{{p}_{F}}/dz \right)}_{vo}}}\qquad\qquad(12)</math></center>
	
	
<center><math>\phi _{\ell o}^{2}=\frac{d{{p}_{F}}/dz}{{{\left( d{{p}_{F}}/dz \right)}_{\ell o}}}\qquad\qquad(13)</math></center>
	
Two commonly used parameters in two-phase flow investigations are the [[Two-Phase Flow Patterns in Horizontal Tubes|Martinelli parameter]], <math>X</math>, which was defined as
<center><math>X={{\left[ \frac{{{(d{{p}_{F}}/dz)}_{\ell }}}{{{(d{{p}_{F}}/dz)}_{v}}} \right]}^{1/2}}</math></center> 
and the Chisholm parameter, <math>Y</math>,
	
<center><math>Y={{\left[ \frac{{{(d{{p}_{F}}/dz)}_{\ell o}}}{{{(d{{p}_{F}}/dz)}_{vo}}} \right]}^{1/2}}\qquad\qquad(14)</math></center>
[[Image:Chapter11j_(5).jpg|thumb|400 px|alt=Lockhart-Martinelli correlations for pressure drop|Figure 1: Lockhart-Martinelli correlations for pressure drop.]]

Parameter <math>X</math>, the Martinelli parameter, is a ratio of pressure drops of single-phase flow terms. As can be seen from eqs. (10) – (13), the pressure drop in two-phase flow can be determined if any one of the four multipliers is known. A generalized method to determine the frictional pressure gradient multiplier was proposed by [[#References|Lockhart and Martinelli (1949)]], who related the frictional multipliers <math>{{\phi }_{v}}</math> and <math>{{\phi }_{\ell }}</math> to the Martinelli parameter <math>X</math> as shown in Fig. 1. It can be seen that the trends for <math>{{\phi }_{v}}</math> and <math>{{\phi }_{\ell }}</math> are different because <math>{{\phi }_{v}}</math> increases with increasing <math>X</math>, but <math>{{\phi }_{\ell }}</math> decreases with increasing <math>X</math>. The multiplier curves also depend on whether the liquid-phase alone flow and the vapor-phase alone flow are laminar or turbulent. There are four curves for <math>{{\phi }_{v}}</math> and <math>{{\phi }_{\ell }}</math> and each corresponds to the combination of laminar (viscous) and turbulent flow for the vapor- or liquid-phases-alone flows in the channel. For example, <math>{{\phi }_{\ell ,vt}}</math> represents the multiplier in the liquid alone pressure drop for cases where the liquid-phase flowing alone in the channel is laminar (viscous) but the vapor phase flowing alone in the channel is turbulent. [[#References|Chisholm (1967)]] correlated the curves of [[#References|Lockhart and Martinelli (1949)]] and recommended the following relationships:
	
<center><math>\phi _{\ell }^{2}=1+\frac{C}{X}+\frac{1}{{{X}^{2}}}\qquad\qquad(15)</math></center>
	
	
<center><math>\phi _{v}^{2}=1+CX+{{X}^{2}}\qquad\qquad(16)</math></center>
	
where <math>C</math> is a dimensionless constant that depends on the combination of the natural and the phase-alone flows. The value of the constant <math>C</math> recommended by [[#References|Chisholm (1967)]] can be found in Table 1. The correlation by [[#References|Lockhart and Martinelli (1949)]] can provide a good prediction when <math>{{\mu }_{\ell }}/{{\mu }_{v}}>1000</math> and <math>\dot{{m}''}<100\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s}</math>. Alternative correlations should be used when the two-phase flow falls outside these ranges. 

<div style="display:inline; float:right;">     '''Table 1: Value of <Math>C</math> in eqs. (15) and (16)'''.
{| class="wikitable" border="1"   
| align="center" style="background:#f0f0f0;"|'''Liquid'''
| align="center" style="background:#f0f0f0;"|'''Vapor'''
| align="center" style="background:#f0f0f0;"|'''Subscripts'''
| align="center" style="background:#f0f0f0;"|'''<math>C</math>'''
|-
| Turbulent||Turbulent||tt||20
|-
| Viscous||Turbulent||vt||12
|-
| Turbulent||Viscous||tv||10
|-
| Viscous||Viscous||vv||5
|-
|}</div>
For cases where <math>{{\mu }_{\ell }}/{{\mu }_{v}}>1000</math> and <math>\dot{{m}''}>100\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s},</math> the following correlation proposed by [[#References|Chisholm (1973a)]] should be used:
	
<center><math>\phi _{\ell 0}^{2}=1+({{Y}^{2}}-1)[B{{x}^{(2-n)/2}}{{(1-x)}^{(2-n)/2}}+{{x}^{2-n}}]\qquad\qquad(17)</math></center>
	
where <math>n</math> is the exponent in the friction factor-Reynolds number relationship (<math>f{{\operatorname{Re}}^{n}}</math> = constant). According to eq. (9), <math>n</math> equals 1 for laminar flow and 0.25 for turbulent flow. The parameter <math>B</math> is given by
	
<center><math>B=\left\{ \begin{matrix}
   \begin{matrix}
   \frac{55}{\sqrt{{\dot{{m}''}}}} & 0<Y<9.5  \\
\end{matrix}  \\
   \begin{matrix}
   \frac{520}{Y\sqrt{{\dot{{m}''}}}} & 9.5<Y<28  \\
\end{matrix}  \\
   \begin{matrix}
   \frac{15000}{{{Y}^{2}}\sqrt{{\dot{{m}''}}}} & Y>28\begin{matrix}
   {} & {}  \\
\end{matrix}  \\
\end{matrix}  \\
\end{matrix} \right.\qquad\qquad(18)</math></center>

For cases where <math>{{\mu }_{\ell }}/{{\mu }_{v}}<1000,</math> the following correlation developed by [[#References|Friedel (1979)]] using a database of 25,000 points can provide a better prediction: 
	
<center><math>\phi _{\ell 0}^{2}={{C}_{1}}+\frac{3.24{{C}_{2}}}{\text{F}{{\text{r}}^{0.045}}\text{W}{{\text{e}}^{0.035}}}\qquad\qquad(19)</math></center>

where 
	
<center><math>{{C}_{1}}={{(1-x)}^{2}}+{{X}^{2}}\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)\left( \frac{{{f}_{v0}}}{{{f}_{\ell 0}}} \right)\qquad\qquad(20)</math></center>
	
<center><math>{{C}_{2}}={{x}^{0.78}}{{(1-x)}^{0.24}}{{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{0.91}}{{\left( \frac{{{\mu }_{v}}}{{{\mu }_{\ell }}} \right)}^{0.19}}{{\left( 1-\frac{{{\mu }_{v}}}{{{\mu }_{\ell }}} \right)}^{0.7}}\qquad\qquad(21)</math></center>
	
<center><math>\text{Fr}=\frac{{{{\dot{{m}''}}}^{2}}}{gD{{\rho }^{2}}}\qquad\qquad(22)</math></center>
	
<center><math>\text{We}=\frac{{{{\dot{{m}''}}}^{2}}D}{\rho \sigma }\qquad\qquad(23)</math></center>

==Bounds on Two-Phase Flow==

The advantage of the pressure drop correlations based on the separated-flow model is that it is applicable for all flow patterns. This flexibility is accompanied by low accuracy. [[#References|Awad and Muzychka (2005)]] developed rational bounds for two-phase pressure gradients.  The lower bound of the friction pressure drop is
	
<center><math>{{\left( \frac{dp}{dz} \right)}_{F,lower}}=\frac{0.158{{{\dot{{m}''}}}^{1.75}}{{(1-x)}^{1.75}}\mu _{\ell }^{0.25}}{{{D}^{1.25}}{{\rho }_{\ell }}}{{\left[ 1+{{\left( \frac{x}{1-x} \right)}^{0.7368}}{{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{0.4211}}{{\left( \frac{{{\mu }_{v}}}{{{\mu }_{\ell }}} \right)}^{0.1053}} \right]}^{2.375}}\qquad\qquad(24)</math></center>
		
where <math>D</math> is the diameter of the tube. 
 
The upper bound of the friction pressure drop is
 	
<center><math>{{\left( \frac{dp}{dz} \right)}_{F,upper}}=\frac{0.158{{{\dot{{m}''}}}^{1.75}}{{(1-x)}^{1.75}}\mu _{\ell }^{0.25}}{{{D}^{1.25}}{{\rho }_{\ell }}}{{\left[ 1+{{\left( \frac{x}{1-x} \right)}^{0.4375}}{{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{0.25}}{{\left( \frac{{{\mu }_{v}}}{{{\mu }_{\ell }}} \right)}^{0.0625}} \right]}^{4}}\qquad\qquad(25)</math></center>

An acceptable prediction of pressure drop can be obtained by averaging the maximum and minimum values, i.e., 

<center><math>{{\left( \frac{dp}{dz} \right)}_{F,ave}}=\frac{0.79{{{\dot{{m}''}}}^{1.75}}{{(1-x)}^{1.75}}\mu _{\ell }^{0.25}}{{{D}^{1.25}}{{\rho }_{\ell }}}</math></center>

<center><math>\left. \cdot \left\{ {{\left[ 1+{{\left( \frac{x}{1-x} \right)}^{0.7368}}{{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{0.4211}}{{\left( \frac{{{\mu }_{v}}}{{{\mu }_{\ell }}} \right)}^{0.1053}} \right]}^{2.375}} \right.+{{\left[ 1+{{\left( \frac{x}{1-x} \right)}^{0.4375}}{{\left( \frac{{{\rho }_{\ell }}}{{{\rho }_{v}}} \right)}^{0.25}}{{\left( \frac{{{\mu }_{v}}}{{{\mu }_{\ell }}} \right)}^{0.0625}} \right]}^{4}} \right\}\qquad\qquad(26)</math></center>
		
Equation (26) could predict the pressure drop with the root mean square error of 26.4%.	

==References==

Awad, M.M., and Muzychka, Y.S., 2005, “Bounds on Two-Phase Flow: Part I Fractional Pressure Gradient in Circular Tubes,” ''Proceedings of International Mechanical Engineering Congress and Exposition'', Orlando, FL (DVD).

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

Chisholm, D., 1967, “A Theoretical Basis for the Lockhart-Martinelli Correlation for Two-Phase Flow,” ''International Journal of Heat and Mass Transfer'', Vol. 10, pp. 1767- 1778. 

Chisholm, D., 1973a, “Pressure Gradients Due to Friction During the Flow of Evaporating Two-Phase Mixtures in Smooth Tubes and Channels,” ''International Journal of Heat and Mass Transfer'', Vol. 16, pp. 347-348. 

Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA

Friedel, L., 1979, “Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two-Phase Pipe Flow,” ''European Two-Phase Flow Group Meeting'', Paper E2, Ispra, Italy. 

Lockhart, R.W., and Martinelli, R.C., 1949, “Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes,” ''Chemical Engineering Progress Symposium Series'', Vol. 45, pp. 39-48. 

==Further Reading==

==External Links==
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