(1)
A.A.Atavin
Director
of Novosibirsk Branch of Institute for Water and Environmental Problems,
Morskoj Prospect, 2, Novosibirsk, 630090, Russia,
Tel. (383-2) 343484, Fax (383-2) 302005, E-mail: atavin@iwepsb.altai.su
V.V. Tarasevich
Associate
Prof. of Novosibirsk State University of Architecture and Civil Engineering
NGASU, Leningradskaya st., 113, Novosibirsk, 630008, Russia
Tel.: (383-2) 669411, Fax (383-2) 161107,
E-mail: tvv@iis.nsk.su
Abstract:
The large complex piping systems under unsteady operating mode are considered.
Such systems can exemplify the typical systems with distributed parameters.
General mathematical model of transients in piping systems is demonstrated. This
transient is described by the solution of the graph-defined mixed problem. The
dimensionless analysis is given for principal physics of the water hammer
equations' simplification. System with lumped parameters consisting from
inertial and elastic “blocks” is used for substitution the assumed piping
system with distributed parameters. The convergence of such a kind of
approximation is considered. The water hammer problem for single pipeline is
used for testing the obtained model. The comparison "thorough" system
with simplified one was realized for some models.
Keywords: pipe, system, distributed parameters, model making, lumped parameters
Many widespread piping systems with the distributed parameters (for example, network of water supply, thermal networks etc.) contain the thousands of elements and have very large sizes. The calculation of such systems requires significant computational resources and occupies a rather long time. But many problems arising in practice, for example, the problem of on-line control, require fast calculation and forecast of a system state.
Increase of
computer performance and the improvement of computational algorithms, results in
reduction of computational time of course. However the creation of
"fast" simplified models seems more perspective way. One of main
tendencies for creation such simplified models consists in replacement the
"original" systems with the distributed parameters by the systems with
the lumped parameters. The problem is to match the characteristics of this new
system so that parameters of process in it were "nearer" to parameters
of process in basic simulated system as possible.
The system of
pressure pipelines connected among themselves in some way, will be named the
piping system. The pumps, valves, throttles and other devices [
0
, 0
] can be located in the points of
connection, which are named nodes. The sizes of nodes are small in comparison
with the sizes of pipelines as a rule, therefore are simulated by systems with
the lumped parameters. The pipelines of the system have usually significant
length, so the parameters of flows in them depend from spatial variable;
therefore pipelines are simulated by systems with the distributed parameters.
Let us
assume that all the pipes and all the nodes of the system are numbered and index
i denotes pipe number and index j
denotes node number. The designation Rj will denote the set of numbers of such pipes,
which connected with node j.
We
consider a case of isothermal flow of weakly compressible fluid (for example,
water, oil) and assume the pressure p
and the volumetric discharge Q as the
parameters of flow. Then the well-known water hammer equations [
0
,
0
] can be
used for describing a liquid flow:
(1)
where r is the density of liquid, a is the velocity of water hammer wave [ 0 ], w is cross-section area of pipe, g is acceleration of gravity, z is pipe ordinate, l is hydraulic friction coefficient [0 ].
We shall designate through
the value of some quantity F
at that extremity of pipe i which
connected with node j (for example:
,
, and so on). Then
will designate a vector
compounded by components
(for example:
,
, etc), where iÎR.
Let
be a vector of intrinsic
parameters of node j. The functioning
of each node j is being described by the equation in a general view:
The initial data must be specified for each pipe and node at start time t=0:
Thus, the solution of mixed problem (1)–(4) describes the transient in considered piping system.
Let us rewrite the equations of a water hammer (1)–(2) in dimensionless form, having accepted dimensionless variables t¢=t/T, x¢=x/L, p¢=p/p*, Q¢=Q/V*w (index i is omitted here and further):
,
,
where
,
,
,
,
Here p* is typical pressure in pipe, V* is typical velocity, L is pipe length, T is typical process life.
One can see now, that equation of continuity (1) can be simplified up to a relation ¶Q/¶x=0 (in dimensional variables) when m<< 1, i.e. for enough slow oscillations (k<< 1, j is limited) or for small pressure drops (j << 1, k is limited). So, it is possible to neglect a fluid compressibility and to take into account only inertial forces on this length. Thus Bernoulli equation with an inertial term takes place here:
Here indexes “l” and “r” denote the quantities relating to the left (x=0) and right (x=L) extremities of pipe (see a,b,c,d Fig., a).
If m ~ 1, n << 1, (j×k ~ 1), i.e. for rather slow oscillations with a considerable pressure drop (k << 1, j >> 1), then the equation of impulse (2) can be simplified up to a relation ¶p/¶x=0 (in dimensional variables), that together with integrated the equation (1) along the length of the appropriate pipe reduces into the equation of oscillation of elastic liquid masses:
(6)
So, the inertial forces can be neglected on this length of a hydraulic-circuit system, since elastic forces dominate here.
If m ~ 1, and n ≥ 1, then it is necessary to use the complete equations of a water hammer for transient’s description.
If m >> 1, i.e. the high-frequency oscillations take place with great amplitudes of pressure, then that contradicts basis principles of the water hammer theory. Except for the case m >> 1, n << 1, when it is possible to take advantage of model with the elasticity prevailing (¶p/¶x=0 )
The try to take into account simultaneously both elastic and inertial properties of the pipe filled by a fluid can be implemented by replacement of a length of the actual pipe by three fragments: by two lengths of the inelastic pipe, filled by incompressible fluid (inertial blocks) and jointed by "a water tank" approximating elastic properties of all pipe (block of an elasticity), how it is shown in Fig., b. Using equation (3) for each part of inelastic pipe and equation (5) for “water tank”, one can obtain:
,
(8)
(9)
where
is the pressure in “water
tank”.
Let us divide all length of the pipeline into N
equal parts (see (a,b,c Fig.,d). Then length of each part will be equal D=L/N. Let average pressure equal
on that
site, Qk = Qk(t) is discharge at the beginning of this length, Qk+1 = Qk+1(t) is
discharge at the end of this length, where k=0,
,N–1.
It is obviously that Q0 º Ql
and QN º Qr.
One can obtain the next set of equations by describing the fluid current on each
k-th length by the model (8) - (10):
(10)
(11)
, when k=1,…,N–1
(12)
when
k=0,…,N–1
(13)
It is obvious, that the equations (10) – (12) tend to (2), and the equation (13) tends to (1) when N®¥ (D®0). Thus, the set of equations (10) – (13)is some discrete approximating of the equations (1) – (2), similarly finite-difference approximating, for example.
We shall consider the classical problem for a water
hammer in the simple pipe for the testing this model. Let's consider the case
without friction at first. Let us assume that the constant pressure pl=p0=const
is specified at left extremity of the pipe, and the law of valve closing is
specified for right extremity, in the following form
where V0 = const is initial velocity of a moving fluid in the pipe; f(t) is decreasing function describing the law of valve closing: f(t)=1 at t≤0, f(t)=0 at t>tcl and f(t) continuously decreases from 1 up to 0 under t varying from 0 up to tcl. Here tcl is the time of valve closing, 0 corresponds to the starting of the non-stationary process (valve closing).
The results of calculation for the case of instantaneous closing (tcl = 0) under different N are represented in Fig.2. One can see, that the solution of the problem (10)–(13) approximates more and more precisely the exact solution of (1)–(2) with N growth.
The results of calculations demonstrate, that the solution of the problem (10)–(13) takes into account high-frequency harmonics of the process not so well under small N, that is expressed in smoothing the plot of right angles as contrasted to by exact solution. However this tendency weakens with N growth. So, the shape of curve 4 is rather close to rectangular one for a solution with N=20.
The solution under N=1, that
corresponds to model (7) – (9), gives a noticeable deviation both on
amplitude, and on frequency from an exact solution (see curve 2 in Fig.2). This deviation is diminishing with N
growth. So, already the curve 3 for a
case N=2 (see Figure 1, c)
gives the good concordance by maximum amplitude and by frequency (the frequency
discrepancy becomes more appreciable with growth of time). These discrepancies
will be diminishing under N
increasing.
In real-world situations, in the presence of the
viscosity forces and enough smoothly varying closing of gate valves, the
parameters of transient will be smoother. The solution of simplified problem
will become more adequate in this case. For example, the solution of considered
problem for the more realistic case (smooth valve closing and taking into
account the viscosity forces) is represented in Fig.. One can see that the case N=2
gives already the sufficiently satisfactory result. Therefore it will suffice to
not go beyond the case N=2
or even N=1 for practical
calculations.
The universality of presentation for
arbitrary complex hydraulic-circuit system takes place under such approximating
since the main equations (1)–(2) are substituted
for (10)–(13) by described above method. Then transient in pipe became
simulated by the system with lumped parameters. In this case one can consider
the pipe as a kind of node with a vector of intrinsic parameters
. The boundary conditions (3),
representing the operation of controlling devices, as well as the exposures on a
hydraulic-circuit system remain the same. Thus, the "pure" system with
lumped parameters (3),
(10) – (13) with initial data (4) is used for piping
system simulation
Besides, the improved theory of hydraulic circuits [ 0 ] (taking into account inertial component of pressure drop and presence in nodes of water and air tanks, surge chambers etc.) will be applicable for description of dynamic processes in an arbitrary hydraulic-circuit system within such a schematizing.
Many of real-world hydraulic-circuit systems have
large extension and complex structure. The direct simulation of dynamic
processes in such systems on the basis of a solution of the set of a partial
derivative differential equations (1) – (2) requires rather significant
computational burden. The decomposition and agglomeration techniques can help
really to calculate transients in such systems, which reduce
"dimension" of simulated systems [0
] quite much.
The simplification of the main differential equations (1) – (2) with the aid of replacement of differential partial equations by the ordinary differential equations (10) – (13) (model of system with lumped parameters) is other approach which allows to calculate dynamic processes in complex hydraulic-circuit systems rather effectively.
References
[1] Atavin, A.A. and Tarasevich, V.V. The Numerical Modeling of Flows in the Complex Pipe Networks. In: Proc. of IAHR XXVIII Biennial Congress, 22-27 August 1999 in Graz, Austria. Abstract Volume. Papers on CD-Rom, Graz, 1999, p. 110.
[2] Tarasevich, V.V. The Simulation and Mathematical Modelling of the Complex Pipe Systems. In: Proc. 5th (IMACS) World Congress on Scientific Computation, Modelling and Applied Mathematics, Berlin, August 1997 (Ed.: Sydow, A.), vol. 3. (Computational Physics, Chemistry and Biology), 115-120.
[3] Zhukowsky, N.E. About a Water Hammer in Water Pipes. - Journal of Polytechnic Society, 1899, No.5. (in Russian).
[4] Streeter, V.L. and Wylie, E.B. Hydraulic Transients. Mc Graw-Hill, 1968.
[5] Merenkov, A.P. and Hasiliev, V.Ja. Theory of Hydraulic Circuits. – Moscow, Nauka, 1985. (in Russian).
(c) (d)
Fig.
1 A scheme of substitution for lengthy pipe.
Fig. 2 Comparison exact solution with the approximate solution by formulas (1)–(13) 1 is exact solution; 2 – when N=1; 3 – when N=2 (see Fig, c); 4 – when N=20.
Fig. 3 Comparison “precise” numerical solution of (1) – (2) with the approximate solution by formulas (10) – (13): 1 is “precise” solution; 2 is solution N=1; 3 is solution with N=2; 4 is solution with N=4.
