Basics of Fluid Mechanics, 2014a

13.7. HOMOGENEOUS MODELS 549
There are several suggestions for the average viscosity. For example, Duckler suggest
the following
μ m =
μ G Q G
+
μ L Q L
(13.32)
Q G + Q L Q G + Q L
Duckler linear formula does not provide always good approximation and Cichilli suggest
similar to equation (13.18) average viscosity as
1
μ average =
(13.33)
X
μ G
+ (1−X)
μ L
Or simply make the average viscosity depends on the mass fraction as
Using this formula, the friction loss can be estimated.
μ m = Xμ G +(1− X) μ L (13.34)
13.7.1.2 Acceleration Pressure Loss
The acceleration pressure loss can be estimated by
− dP
dx ∣ = ṁ dU m
a
dx
(13.35)
The acceleration pressure loss (can be positive or negative) results from change of
density and the change of cross section. Equation (13.35) can be written as
− dP
dx ∣ = ṁ d
a
dx
(
ṁ
Aρ m
)
(13.36)
Or in an explicit way equation (13.36) becomes
⎡
pressure loss due to
− dP
density change
dx ∣ = ṁ 2 { }}
⎢ ( ){
a ⎣ 1 d 1
A dx ρ m
pressure loss due to
area change
{ }} {
1 dA
+
ρ m A 2 dx
⎤
⎥
⎦
(13.37)
There are several special cases. The first case where the cross section is constant,
dA/ dx =0. In second case is where the mass flow rates of gas and liquid is constant
in which the derivative of X is zero, dX/ dx =0. The third special case is for constant
density of one phase only, dρ L / dx =0. For the last point, the private case is where
densities are constant for both phases.