Theory and modeling of multiphase flows

1
Theory and modeling of multiphase flows
Payman Jalali Department of Energy and
Environmental Technology Lappeenranta University
of Technology Lappeenranta, Finland Fall 2006
2
Homogeneous flow theory
Problem 1 What is the total pressure drop in a
pipe carrying a homogeneous mixture of steam and
water with the following data Pipe length L1 m,
saturated water at entrance xi0, vertical tube
with diameter d1.2cm. Flow rate is Gf 270
kg/s.m2. The heat flux to the tube is ?6.5x105
W/m2. Friction factor is constant, Cf0.005. What
is the pressure drop through the evaporator for
the inlet pressures of a) 24 atm, and b) 68 atm.
3
Homogeneous flow theory
We integrate terms in pressure drop equation one
by one
Note that there are two terms constructing the
frictional pressure drop. The first term
corresponds to the saturated liquid phase (if the
phase change would not take place), and the
second term shows how heating would affect the
increase in frictional pressure drop (as it
increases the velocity of flow by generating
vapor). Revisit lecture notes of session 3.
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Homogeneous flow theory
The pressure drop due to heating (phase change)
becomes
This term comes from the acceleration of the
fluid. In general, if there are no phase and area
changes (no velocity change), fluid will not
accelerate. Here, we do not have the area change
and therefore its corresponding term is absent.
The only accelerating term is due to the phase
change which comes from the heating of the
mixture. Revisit the lecture notes of session 3.
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Homogeneous flow theory
The pressure drop due to variation of elevation
will be
6
Homogeneous flow theory
a) From steam table, we have
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Homogeneous flow theory
8
Homogeneous flow theory
Quality at exit is
This confirms that our assumption of a
liquid-vapor mixture is correct, otherwise if the
vapor was superheated in the exit we were not
allowed to use the above-mentioned
formulations. We can check from the steam tables
that the denominator in the general equation of
pressure drop is very close to 1 at this flow
rate and pressure.However, if the heating rate
and flow rate increase by a factor of 10 then we
may not easily neglect the denominator (effects
of compressibility and flashing will be
important).
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Homogeneous flow theory
b) From steam table, we have
Quality at exit is
10
Homogeneous flow theory
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Homogeneous flow theory
Problem 2 In the previous problem, how would be
the strategy of our solution for frictional
pressure drop if we could not consider friction
factor as a constant? For example, if we were
supposed to consider the average viscosity of the
mixture as
Solution If we look at the equation for pressure
drop
We had
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Homogeneous flow theory
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Homogeneous flow theory
This value is calculated for zL, which is
subtracted then from the value for z0 because we
have a definite integral. As you can see, a
variable friction factor makes the formulas
rather lengthy and complicated.