ɌȿKA. COMMISSION OF MOTORIZATION AND ENERGETICS IN AGRICULTURE – 2013, Vol. 13, No.4, 03-08
Choice of rational diameters of pipelines
for hydraulic system of city quarter water
Pavel Akimov
Volodymyr Dahl East-Ukrainian National University
Molodizhny bl., 20ɚ, Lugansk, 91034, Ukraine, e-mail: [email protected]
Received September 09.2013: accepted October 03.2013
S u m m a r y . The adequate mathematical model of
hydraulic characteristics is made for system of water
supply of city quarter. The method of integration of
model, method of optimization of parameters of system
is reasonable. It is shown that at the expense of a choice
of rational parameters of system of water supply
considerable economic effect can be gained.
Key
w o r d s : mathematical model, hydraulic
characteristics, water supply method of integration,
economic effect
INTRODUCTION
The system of water supply is the whole
complex of the constructions necessary for
supply by water of user, demanded quality and
in demanded quantity [8]. Therefore process
of its development is not only technically
difficult, but also responsible action to which it
is necessary to approach with the maximum
attentiveness and punctuality [21].
The analysis of methods of calculation of
systems of water supply showed that existing
methods not always meet the requirements of
high efficiency and profitability of work of
such schemes [9]. One of the perspective
directions of improvement of design and
modernization is use enough full mathematical
models of calculation of their characteristics,
both in a statics, and in dynamics [1].
OBJECTS AND PROBLEMS
Below possibility of modernization of
existing system of water supply of a quarter of
a city network with a choice of rational
parameters of its elements [23] is shown [11].
The scheme of system is given in Fig. 1.
Source of system of water supply is the
pump IE 50-160 NA [22] with a productivity
of 60 m3/hour, a pressure of 0.5 MPa on a
nominal mode [2]. In system the pump works
backwater 0.35 MPa discharge pressure
0.58 MPa.
Water supply in each house is defined,
proceeding from number of inhabitants and
norms of consumption [16].
Lengths of pipelines are determined by
the area plan, and their diameters by the
relevant standards [17].
Generally it is
diameter there are 100 mm, only a site from
pump station to the house No. 3 – 150 mm.
For a reasonable choice of the pump and
diameters of pipelines the maximum water
discharge in a network is defined [3]:
Qmax
Q0 K tp ,
where: Qmax , Q0
average discharge,
is maximum and
4
PAVEL AKIMOV
Fig. 1. Scheme of water supply system
Fig. 2. Water consumption characteristic
K tp
is the correction coefficient
considering unevenness of an expense in time.
For determination of coefficient a
number of experiments were carried out [12].
On an entrance to houses water gauge
(LLT-100 and LLT-150 depending on a
consumption of water and diameters of
pipelines) were established and measurements
of water discharge within days on weekdays
days, days off, and also a consumption of
water on months were carried out [15].
Characteristic dependences are given in Fig. 2 [21].
The error of measurement of an expense
didn't exceed 3% [10].
Statistical processing showed that the
average discharge differs from maximum by
=2.5 times [9]. Thus, for normal providing
with water of inhabitants of the area it is
necessary to increase an average discharge in
time and in its size to choose the pump and
diameters of pipelines [13].
Further the task is formulated thus: it is
necessary to determine rational parameters of
the considered hydraulic system which
CHOICE OF RATIONAL DIAMETERS OF PIPELINES FOR HYDRAULIC SYSTEM OF CITY
settlement scheme is given in fig. 1 [14].
Distances and sizes of selections in terminal
points are set. Pressure in points of selection
has to be not less than 0.425MPa. Parameters
of optimization are diameters of pipes.
Criterion function are the given expenses
which consist of the cost of pipes and electric
power cost for a year of operation of the
pipeline [5].
The cost of pipes pays off as [20]:
¦ dist
ij
˜ pricetube(dij ) ,
i, j
where: disti j is site length between knots
i and j ,
di j is diameter of a pipe on this site,
pricetube di j
is the cost of one meter
of a pipe with a diameter di j .
Dependence of cost of pipes is given in
Fig. 3 per one meter as function of their
diameter [23].
Fig. 3. Dependence of the price of pipes on diameter
5
Apparently, data are well described by
linear function that allows finding the cost of
pipes for any diameter.
The cost of the electric power paid off as
[19]:
pressure ˜ qtot
˜ 365 ˜ 24 ˜ pelect ,
eff ˜1000
where: pressure is pressure created by
the pump,
qtot is full expense at the pump exit,
eff is pump efficiency (value undertook),
pelect is the price of one kilowatt-hour
of the electric.
Then the full given expenses total can be
calculated as the sum of these expenses [18]:
total
¦dist ˜ pricetube(d ) ij
i, j
ij
pressure˜ qtot
˜365˜ 24˜ pelect .
eff ˜1000
The algorithm of
constructed as follows.
optimization
is
6
PAVEL AKIMOV
Input of basic data. Drawing up list of
unknown streams. The decision of system of
the equations for expenses. Creation of
criterion function. Creation of a matrix of
coefficients. Drawing up the equations for
pressure [3]. The received equations unite to
the massif. Drawing up list of unknown
pressure.
Pressure in all knots is considered as
unknown. As the number of unknown is one
more, than number of the equations, it is
necessary to set pressure in one of knots. It is
necessary equal to zero in a hub 10 (it is
possible to take any other knot). The received
system of the linear equations decides the
Gauss method which main idea consists in a
consecutive exception of unknown of this
system.
Then the size of the given expenses
which is output argument of criterion function
is determined by the formula given above.
Minimization of criterion function was
carried out by Hook and Jives which has been
a little changed according to specifics of the
solved task the method [5]. The method
includes two main stages – studied search and
search in a sample. Let's list the main stages of
a method [6].
• Initial values of coordinates (entrance
arguments of criterion function, in our case –
diameters of pipes) are set.
• In a cyclic order each variable with
increment addition changes.
• If after change value of criterion
function decreased, old value of a variable is
replaced with the new.
• If positive change wasn't successful,
negative change is carried out on 'xi
• If after change value of criterion
function decreased, old value of a variable is
replaced with the new.
• If positive change wasn't successful,
negative change is carried out on.
• If value of criterion function didn't
decrease and in this case, the size of
coordinate remains without changes.
• After carrying out such changes for all
variables, the stage of studied search comes to
an end.
• The changes of variables kept after
studied search, show the direction of
perspective minimization on all variables at
once. It is the direction of search in a sample.
• In this direction the following, bigger
step is taken.
• Procedure proceeds before achievement
of a minimum of criterion function.
In our case we can't set randomly the
step size as are limited to a discrete set of
existing diameters of pipes [4]. Therefore the
step for each variable, equal to distance
between this value of diameter and next, turns
out different. The size of a step turns out big
that does impossible carrying out search in a
sample. Therefore it is necessary to be limited
to studied search.
Search began with values of diameters of
the pipes which are really standing in system
[7]. In Fig. 4 dependence of values of criterion
function (the given expenses) on number of
steps of iterative process is shown
As a result of minimization, value of
criterion function decreased from 134284 to
110225, i.e. by 18%.
Optimum parameters of system turn out
the following:
Pump’s pressure - 0.455 MPa,
Pump’s power -10.1 kWt ,
Cost of tubes – 35682 grn,
Reduced cost – 110225 grn.
Diameters of pipes, distances between
knots and expenses between knots are
specified below
The researches executed by authors
showed that the offered technique of a choice
of rational parameters of difficult hydraulic
system allows reducing the given expenses
considerably.
CHOICE OF RATIONAL DIAMETERS OF PIPELINES FOR HYDRAULIC SYSTEM OF CITY
7
Fig. 4. Dependence of values of criterion function on number of iterations
CONCLUSIONS
1. The order of multiple parameter
optimization of system of water supply of a
site of the city area is developed, statistical
researches on water consumption within year,
month, week and days are executed.
2. It is shown that multiple parameter
optimization allows to lower the given
expenses only at the expense of a choice of
diameters of the pipeline by 18%.
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