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The Unsteady Nonequilibrium Two-phase Flows
in Pipeline Systems
Vladimir V. Tarasevich
Novosibirsk University of Architecture and
Civil Engineering (NGASU)
NGASU, Leningradskaya, 113, Novosibirsk,
630008, Russia
tel.: (383-2) 66-47-87; fax: (383-2) 16-11-07;
e-mail: tvv@iis.nsk.su
Abstract
The two-phase unsteady flows of nonequilibrium heterogeneous medium are
considered. The new model of such flows based on hydraulic approach is
considered. This model can generate the family of various models for special
cases of streams.
The boundary conditions represent the system of the algebraic or/and obvious differential
equations. The stationary solution of a considered
problem acts usually as the initial data. The
relaxation method can be rather effective for solving of steady problem.
The effect of change of scale under
occurrence or disappearance of a phase produces significant computing
difficulties. The notion of "trigonometric concentration" is offered
for damping of this effect.
The method of calculation for the two-phase flows based on the method of
characteristics is considered. This method uses the two-scale joint grid: one
for liquid phase and other for vapour phase.
The examples of calculations are demonstrated. The results of
calculation for the case of outflow of hyperthermal liquid under sudden pipe
failure taking with appearance of vapour-liquid flow are demonstrated. The
results of calculations of flows in the system of industrial pipelines of
nuclear power plants are discussed.
Keywords: two-phase
flow, model, LIQUID, VAPOUR, pipe, system, calculation
Introduction
Two - phase flows may occur in the systems of industrial pipelines of
nuclear and heat power plants, pipelines of chemical industry, in the pipe
systems of oil deposit, etc.
The quite exact calculations are needed for efficient forecasting of
after-effects of control actions on these systems and possible emergencies.
The realisation of such calculations is possible on a basis of rather
adequate models of a wide range of application. On the other hand, it is
desirable, that, first, these models were rather simple for engineering
practice, secondly, were well joined to known models for special cases of
considered currents.
The family of mathematical models of vapour-liquid streams based on
unified foundations is described in the paper.
The mathematical
formulation of problem
The governing equations.
The following model is considered for general case of vapour-liquid
unsteady nonequilibrium heterogeneous flow with different velocities, pressures
and temperatures of phases.
The equation of continuity for each phase after some transformations can
be written in the form:
, (1)
where 
.
Here i is phase index (i=g for vapor and i=l for liquid); r is density; c is baroclinic velocity of
perturbation's propagation along i - th phase [1]; j is phase concentration; p is pressure; V is
velocity of phase medium.
The equation of motion transform to the form:
(2)
Neglecting the longitudinal heat transfer, after series of
transformations and simplifications the equation of energy conservation may be
written in form [1,2,3]:
, (3)
where T is absolute temperature, Cp,i is thermal capacity
under constant pressure,
s T is coefficient of volumetric thermal expansion.
In the formulas Fm, FJ, FH are
functions depending on solution, mass-, momentum- and heat-transfer and
friction between phases and friction between each phase and wall of pipe. These functions depend also
on two-phase flow structure which is described by the corresponding flow
pattern map [2].
The known hypothesis of the water hammer theory [4] were used under
obtaining of the equations (1) - (3).
The completing relationships
Thus there is the system of equations (1) - (3) under variables p , V ,
T, j
for each phase i. Besides them, the parameters describing the interphase
mass-and-heat transfer are the unknown for this system. The relationships on
the phase boundary are used for solving them and completing of system (1) - (3) finally.
The obvious geometrical relationships and
balances of mass, momentum and energy should
be fulfilled on the phase interface. First two
of them allow to choose the quantities j = jg , m = mg in the
capacity of supplementary parameters
characterising of «two-phaseness» of flow (here mg is mass flux into
vapour phase.
The energy balance gives the relationship for estimation of evaporation rate:
.
The momentum balance gives the relationship connecting the phase
pressures:
.
Specific realisations of functions fJ and fE
generate the various models of vapour-liquid flows. Thus one can choose the different models of flow by changing kind of
function fJ and fE without touching upon the rests parts
of model and/or program accordingly.
The various simplifications of equations (1) - (3) produce the family of
corresponding models for the various special cases (one-phase flow, isothermal
flow and so on ). In particular for the case of one-phase isothermal flows of
weakly-compressible liquid the equations (1) - (2) transform into well-known
equations of water hammer [4]. If in this case pressure under unsteady process will
fall up to cavitation pressure, this model will describe water hammer
accompanying vaporous cavitation of a flow.
The boundary conditions and initial data.
It is necessary to set still boundary conditions and
initial data for correct formulation of the problem [5]. The conditions,
describing the functioning of nodes of pipe system will act as boundary
conditions. These conditions represent the system of the algebraic or/and obvious differential
equations [1,6].
The stationary solution of a considered problem acts
usually as the initial data [6,7].
The finding of this solution is far from being trivial problem especially for
two-phase flows [1,2,7]. The relaxation method can be
rather effective in this case.
effect of scale change
Only second and third terms in (1)
play a role for one-phase flow, but first term becomes sharply included at the
moment of occurrence of a second phase and begins to prevail in the equations
(1) in view of rather large factor r c2
. This effect is caused by physical phenomenon, since the velocity of
propagation of perturbations in medium sharply falls under occurrence of the
second phase, i.e. temporary scale changes. This phenomenon creates significant
computing difficulties for stages of origin or disappearance of a phase.
The notion of "trigonometric
concentration" q is offered:
,
that allows to have instead of (1)
- (3) more "smooth" system of the equations, that provides
"smooth connection " of the second phase.
The method
of calculation.
The method of calculation (on the basis of a method of the
characteristics), using concurrent grid for liquid and gas phases was developed
by the author. Thus the mesh width is determined by minimum speed of
perturbation's propagation, which corresponds to speed of perturbation's
propagation in vapour (gas) phase. So the grid for a liquid phase uses not
neighboring but distant points of a grid. At the approach to borders regularity
of the calculation is infringed; therefore in the vicinity of boundaries the
calculation is made under the special schemes, taking into account reflection
from borders.
Results of
computations
The outflow of superheated water from the pipe under sudden
depressurization is considered. This situation is simulated by the next
problem. Let's consider the single pressurized horizontal pipe of length L,
filed by rest superheated liquid (V0=0) with temperature T0
(T0 >100°C) under the pressure P0 (P0 >Patm).
In the initial moment of time the left end of the
pipeline suddenly became destroyed and connected with atmosphere. So the left
boundary condition became under t > t0 : 
. The right boundary condition remains 
and 
(heat-insulated end).
The result of calculations of initial stage of process
is represented on figure 1 in dimensionless relative parameters. The further
development of process is submitted on figure 2. One can see that the
boiling-up of a liquid ensues from the fall of pressure, and the flow turns
into two-phase stream behind a wave of downturn of pressure.
Fig.1. The profiles of flow parameters at ct/L=0.15.
Fig.2. The profiles of flow parameters at
ct/L=0.7.
This method was applied also for calculations of unsteady flows the
system of industrial pipelines of power nuclear plant. This system consists from more than
2000 pipes and similar number of nodes. Source of stochastic disturbances was
the operating of throttle valve located on outlet pipe of reactor coolant pump. Some
results are represented on figure 3, where curve 1 is the pressure in the
source of disturbances, curve 2 is the pressure in dead end of outlet pressure
manifold and curve 3 is the pressure in tee of this manifold.
Fig.3. The pressure in different points
of pipe system.
Conclusion
Thus, the family of mathematical models is obtained
describing unsteady flows in pipelines with a various degree of specification
and unified by common approach. This set of models covers cases from water
hammer and isothermal flows of one-phase liquid up to the two-phase
nonequilibrum flow. It provides compatibility of models with transition from
one kind currents to other also is a basis for design of uniform families of
algorithms and programs, suitable for accounts of wide class of streams. These models are well
joined to known engineering models, are rather simple for practical computing,
and at the same time with a sufficient degree of accuracy simulate a wide
spectrum of two-phase flows.
References
[1]
V.V.Tarasevich,
The calculation of the flows of two-phase
mixtures in pipe systems // Proc. 1st All-Russia seminar about dynamics of
space and nonequilibrium flows of liquid and gas. Miass, Russia, 5-7 October, 1993,. p.111-113 (in Russian).
[2]
Yu. N. Kuznetsov.
Heat Exchange in the Problem of Safety of Nuclear Reactors. Energoatomizdat,
Moscow, 1990. (in Russian).
[3]
R.I.Nigmatullin.
The Dynamics of Multi-phase Mediums, in two parts. Nauka, Moscow, 1987. (In
Russian).
[4]
N.E.Zukowsky, About the water hammer in water-supply
pipelines // Proc. 4th Russian water-supply congress. Moscow, Russia, 1899 (in Russian).
[5]
B.L.Rozhdestvensky,
N.N.Yanenko, The systems of quasilinear equations and their applications to gas
dynamics. Nauka, Moscow, 1978. (in Russian).
[6]
A.F.Voevodin,
S.M.Shugrin, The numerical methods of calculating of one-dimensional
systems. Nauka (Siberian Division),
Novosibirsk, 1981. (in Russian).
[7]
A.P.Merenkov,
V.Y.Hasilev, The theory of the hydraulic circuits. Nauka, Moscow, 1985. (in
Russian)