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13.7.1.1: Friction Pressure Loss

The frictional pressure loss for a conduit can be calculated as
\[
    - \left. \dfrac{dP}{dx}\right|_f = \dfrac{S}{A} \tau_w
     \label{phase:eq:PLfriction}  \tag{28}
\]
Where \(S\) is the perimeter of the fluid. For calculating the frictional pressure loss in the pipe is  
\[
    - \left. \dfrac{dP}{dx} \right|_f = \dfrac{ 4\,\tau_w}{D}
     \label{phase:eq:PLfPipe}  \tag{29}
\]  
The wall shear stress can be estimated by
\[
    \tau_w = f \dfrac {\rho_m\, {U_m}^2 } {2}  
     \label{phase:eq:f}  \tag{30}
\]
The friction factor is measured for a single phase flow where the average velocity is directly related to the wall shear stress. There is not available experimental data for the relationship of the averaged velocity of the two (or more) phases and wall shear stress. In fact, this friction factor was not measured for the "averaged'' viscosity of the two phase flow. Yet, since there isn't anything better, the experimental data that was developed and measured for single flow is used. The friction factor is obtained by using the correlation
\[
    f = C \,\left( \dfrac{\rho_m\,U_m\,D}{\mu_m}\right)^ {-n}
     \label{phase:eq:fFurmula}  \tag{31}
\]
Where \(C\) and \(n\) are constants which depend on the flow regimes (turbulent or laminar flow). For laminar flow \(C=16\) and \(n=1\). For turbulent flow \(C=0.079\) and \(n=0.25\). There are several suggestions for the average viscosity.
\[
    \mu_m = \dfrac{\mu_G \, Q_G}{ Q_G + Q_L} + \dfrac{\mu_L \, Q_L}{ Q_G + Q_L}
     \label{phase:eq:dukler}  \tag{32}
\]  
suggest similar to equation (18) average viscosity as
\[
    \mu_{average} = \dfrac{1}{\dfrac{X}{\mu_G} + \dfrac{(1-X)}{\mu_L} }
     \label{phase:eq:muAvreF}  \tag{33}
\]
Or simply make the average viscosity depends on the mass fraction as  
\[
    \mu_m = X\, \mu_G + \left( 1 - X \right) \mu_L    
     \label{phase:eq:muMass}  \tag{34}
\]  
Using this formula, the friction loss can be estimated.

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.