Basics of Fluid Mechanics, 2014a

548 CHAPTER 13. MULTI–PHASE FLOW
13.7.1 Pressure Loss Components
In a tube flowing upward in incline angle θ, the pressure loss is affected by friction loss,
acceleration, and body force(gravitation). These losses are non-linear and depend on
each other. For example, the gravitation pressure loss reduce the pressure and thus the
density must change and hence, acceleration must occur. However, for small distances
(dx) and some situations, this dependency can be neglected. In that case, from equation
(13.25), the total pressure loss can be written as
friction acceleration gravity
{ }} { { }} { { }} {
dP
dx = dP
dP
dx ∣ +
dP
f
dx ∣ +
a
dx ∣ (13.27)
g
Every part of the total pressure loss will be discussed in the following section.
13.7.1.1 Friction Pressure Loss
The frictional pressure loss for a conduit can be calculated as
− dP
dx ∣ = S
f
A τ w (13.28)
Where S is the perimeter of the fluid. For calculating the frictional pressure loss in the
pipe is
− dP
dx ∣ = 4 τ w
(13.29)
f
D
The wall shear stress can be estimated by
τ w = f ρ 2
m U m
(13.30)
2
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
( ) −n ρm U m D
f = C
(13.31)
μ m
Where C and n are constants which depend on the flow regimes (turbulent or laminar
flow). For laminar flow C =16and n =1. For turbulent flow C =0.079 and n =0.25.