67MATHEMATICAL MODELS AND CRITERIA FOR WATER DISTRIBUTION QUALITY IN LARGE MAIN IRRIGATION CANALS
In the article it's mentioned that mathematical models of water distribution in channels of irrigation systems under discrete water supply conditions have been developed. These include the models of direct and kinematical waves, diffusive & convectional model, and complete model of unsteady water flow in channel sections. Also, the criteria of quality of water distribution in the channel sections, for lateral water intakes and pumping stations have been defined. For them, discrete conditions, initial and border conditions, scopes of variables and limitations on the water distribution process in channels have been determined.
INTRODUCTION
Due to the change in water supply over the years when there is a lot of water or low water, there is a need for such a allocation of water resources, so that they are enough to irrigate crops on the ground. The parameters of the flow of water in the channels are distributed in time and space, i.e. the parameters of the water flow depend on the distance and time of movement. Hydrotechnical (water intake, blocking, etc.) structures are located in separate folds along the length of channels of irrigation systems. Modern water distribution theory (characteristics, qualitative indicators, models and methods of water distribution) in irrigation systems is based mainly on the continuous supply of water to their consumers and is based on equations of inseparability and the amount of movement of water flow in the channels and continuity of processes in space and time. In most irrigation systems, in the presence or shortage of water, the water supply regime to consumers is carried out discretely in time (at a certain time to a particular consumer), so the parameters of water flow in such systems depend on the discretion of the facilities. At present, with the development of the theory of optimal management of complex systems with different characteristics (concentrated, distributional, discrete, etc.) it is possible to create a special theory of optimal distribution of water in irrigation systems in the conditions of discrete water supply to consumers. Modern computer technologies and numerical methods (splinths, generalized functions, numerical algorithms, databases and graphic representations of data) allow to create special systems of mathematical
A. Zh. Seitov
Chirchik State Pedagogical Institute, Chirchik, Uzbekistan
B. R. Khanimkulov
Chirchik State Pedagogical Institute, Chirchik, Uzbekistan
ABSTRACT
modeling and optimal distribution of water in irrigation systems based on the development of the theory of management of complex systems. Consider a section of the canal (Figure 1 a,b), which has five water intakes.
METHODOLOGY
The task of distributing water taking into account the discreteness of the water supply will consider how to provide at a time T for each water intake the flow of water qi, i.e. a step-by-step change in the flow of water, with minimal variation of the water level on the channel.
q^t-T) q^if-T)
a)
J il <h H1 • t
J L Q2 t
J k t
J 1 Qi h1 J t
À q5 t
b)
Figure 1. Discrete water supply in the canal sections RESULTS AND DISCUSSION
Model of a direct wave. Let us consider the formulation of the problem of water distribution in the channel section, which takes into account the discreteness of
water supply with a delay in the distribution of water flow along the channel length. The discreteness of water flow is formulated using a discrete unit function q, l(t-T).
As a mathematical model of the channel section, we consider a one-dimensional differential equation in the form dQ(xJ) + vdQ{xJ) dt
ox
■ = q{x,t),
(1)
where Q (x, t)- is the change in water flow in the channel section, v- is the flow velocity.
Initial condition
<9(x,0) = O0 (X), (2)
where Oo(x) - is the initial distribution of water flow in the channel section Boundary condition
0(0,0 = 0,(0, (3)
where Ot (0~ is the change in water flow at the beginning of the channel section
Variable Definition Area
x>0, t> 0, v>0. (4)
The water discharge at the points of water intake of the channel section q (x, t) under conditions of discreteness of water distribution has the form
q(x,t) = -Zql 8(x-ai )1 (t-T).
i=i
(5)
Equation (1) takes into account the main property of the irrigation canal, such as the delay in water flow along the length of the canal, the farther the target is considered from the initial one, the greater the delay in water flow.
The fundamental solution of equation (1), using the Green's function, has the form [1]
G(x, £ 0 = 1(* - mvt - (X- £)), (6)
where 5 (x)- is the Dirac delta function.
The analytical solution of equation (1), in the presence of its fundamental solution, is determined as follows
11
0(x,t)= \\G(x,£,t-T)w(£,i)d$dTt
(V)
t00
where w (x, t)- is the standardizing function for the boundary value problem (1) - (4), which has the form
w(x,t)= q(x,t) +O0(x,t)S(t)-v d(x)Oi(t) (8)
Consider the properties of the solution of equation (1). The change in water flow at the beginning of the channel section under zero initial conditions propagates
with a speed v along the channel length. The wave front in this case does not change in length, Fig. 2 (a, b) and 3. in isometry shows the propagation of the wave of water flow along the length and time.
In this case, the main task of water distribution is the timely interception of water flow by taps as the water flows along the length of the channel.
Kinematic wave model. The equation of the kinematic wave, in the case of lateral taps, is written as follows
dco(xj) dQ(x,t)_
dt + dx "qWh (9)
where Q (x, t)- is the change in water flow in the channel section, v is the flow velocity.
Initial condition
0(X,0) = O0 (X), ©(x,0) = ©„(*), (10)
where Qo(x)- is the initial distribution of water flow in the channel section
Boundary condition
0(0,0 = G(0, (ii)
where Qi(t)- is the change in water flow at the beginning of the channel section
Variable Definition Area
x>0, t> 0, v>0. (12)
Water flows at the points of water intake of the channel section q (x, t) under the conditions of discreteness of water distribution have the form
q(x,t) = -¿<7, S(x - at )l(i - T).
i=1
Equation (9) takes into account such basic properties of the irrigation canal as the delay and transformation of water flow along the length of the canal. In this case, the water flow rate changed in the initial section of the channel section leads to a change in the water flow in other sections of the channel section after a certain time with delay, which gradually changes in time (Fig. 2-3).
Qfx,t)
Q(x,t)
0 Oj
a)
b)
Fig. 2. The propagation of a square wave along the length of the channel
Q(x,t)
to+T5+T4
Fig. 3. The propagation of complex waves along the length of the channel
The convection-diffuse model is based on neglecting the inertial terms of the equations and has the form
dO , (dOdK\dO K2 d20
dt
dK dh
T- = q(x,t),
(13)
dx 2b\0\ dx2
where K- is the flow module.
The flow modulus K (x, z) characterizes the magnitude of the friction forces and is determined by the following formula k = a ■ c.Jr, where R = © / is the
hydraulic radius of the channel; %- wetted perimeter of the channel; C- is the Shezy coefficient.
To determine the Shezy coefficient, there is a whole series of empirical formulas [2].
As one of them can beat accepted we use the famous Pavlovsky formula
c = -Ry, y = 2,5 J n -0,13-0,75-J rUu -0,ll (14)
n v '
where n- is the channel roughness coefficient.
Initial condition
0(x,0) = O0 (x), cd(x,0) = cd0(x), (15)
where Qo(x)- is the initial distribution of water flow in the channel section.
Boundary condition
0(0,0=a (o, (i6)
where Qi(t) is the change in water flow at the beginning of the channel section.
Variable Definition Area
x>0, t> 0, v>0. (17)
Water flows at the points of water intake of the channel section, under conditions of discreteness of water distribution, have the form
q(x, t) = -X qt S(x - at )l(i - T).
(18)
A complete model of the transient motion of a water stream in a canal section. The condition of the main canal section is characterized by an unsteady flow of water and is described by the system of differential equations of Saint-Venant
(19)
dO dt dx
1 (dq + 7vdq
g(D
dt
dx
+
1-
c
& dx
1 (da^ i + —
B
dx
h = const
K:
q
v = — ,c -
gw
CD V B '
where: 0=0(x ,t) - is the flow rate of water; z=z (x, t) -is the coordinate of the free surface; g- gravitational constant; i- is the slope of the bottom; B = B (z) is the width of the flow along the surface of a living section; a o (z) -is the living cross-sectional area of the stream;
c = c (z) is the propagation velocity of small waves; K = K (z) is the flow modulus.
Partial differential equations of hyperbolic type in system (19) are the equations of conservation of mass and momentum of the flow, which are a mathematical model of the unsteady movement of water in the open channel section .
The flow rate Q (x, t) and the ordinate of the free surface z (x, t) are selected here as functions determining the flow. The independent variables are the longitudinal coordinate x and time t.
The channel is defined by the ordinate of the bottom z0(x) and the width of the cross section B (x, t) at a distance z (vertical) from the bottom of the channel.
Then:
- depth of flow: h (x, t) =z (x, t)-z0(x)
- cross-sectional area of the stream: co (x, h) = f0h B (x, z) dz;
- average flow rate: v=0 co;
- propagation velocity of small waves: c= -Jgco IB ;
- bottom slope i=-dz0\dx.
The characteristic form of equations (19) has the form [1]
K
,2
di y } dx K
dz , , .dz — + (v±c)— dt dx
= ((p--^)gco-(v + c)q.
(20)
where cp =
\(dco^
i +
B
v
dx )u -
h = const
Initial conditions are specified as:
z(x,0) = z0(x), Q(x,0) = Q0(x) (21)
where: Oo (x), z0(x)~are known functions.
The boundary conditions at the points xi= 0 and x2= 1 are written as
Q(0,t)=uM Q(U)=u2(t) (22)
Water flows at the points of water intake of the channel section under conditions of discreteness of water distribution have the form
q(x,t) = -iqiS(x-ai)\{t-T). (23)
i=1
In the presented models, an analytical solution of equations (9), (13) and (20) under the indicated boundary conditions is absent, since the hydraulic parameters of the water flow is a nonlinear function depending on the shape of the cross section of the channel section.
From the expression of the lateral water intakes (23) it is seen that consumers are provided with a discrete water supply in time in the form of a step function. With step functions, to solve the problem of optimal water distribution, it is necessary to formulate criteria for the quality of water distribution in the channels of irrigation systems in the conditions of discrete water supply to consumers and a system of restrictions.
The processes occurring in the channels relate to systems with distributed parameters, and the optimal distribution of water between consumers in the channels relate to the problems of optimal control of systems with distributed parameters. For them, we formulate criteria for the quality of water distribution as criteria for the quality of control of systems with distributed parameters.
The quality control criteria for systems with distributed parameters are generally written in the form of the sum of integral functionals [3]
T L
/ = J Ji7 (x,t,Q(x,t),u(x,t))dxdt +
0 0
T T
+ \F2 {t,Q{0,0,«! {t)]dt + \F, (t,Q{L,t),ul (t))dt+ (24)
0 0
L
+
0
where Fh i=l,...,4 - are given continuous functions of their arguments, the first component being quality criteria for distributed control actions, the second and third for boundary controls, and the fourth for final states of the controlled process.
Channel section. The quality criterion for hydraulic processes of water distribution in the channel section can be written as follows
■\F4(x,Q(x,T))dx
T i r
/j =JJ [z(x,t)-z*fdxdt, (25)
0 0
where z (x, t) is the actual change in water level in the channel section; z* - is the set value of the water level.
Functional (25) shows the quality of the water level change in the channel section during the entire process of water distribution. The solution to the problem of minimizing the functional (25) is to reduce the excessive fluctuation of water levels in the channel section.
The next quality criterion for changing the water level in the channel section is
the integral deviation of the water level at the end of the process T from the given
*
distribution of the water level z (x,)
10
j [z(x,T)-z\x)}
dx,
Similarly for the flow rate of water in the channel section
T 1
h
If |C?M-C?*
dxdt,
0 0 /
j [Q(x,T)-Q\X)]
dx,
(26)
(27)
(28)
where 0(x,t) - is the actual change in water flow in the channel section; O* -set value of water flow.
Functional (27) and (28) show the quality of the change in water flow in the channel section at the beginning, along the length and end of the water distribution process. The solution to the problem of minimizing functionals (27) and (28) is to reduce excessive fluctuations in water flow in the channel section.
In addition to integral quality criteria, the following analogues of criteria (25) -(28) can be used
I
h =
z(x,T)-z0(x)*
h
0{x,t)-Q* h=\Q{x,T)-Q*
(29)
Side water intakes. For lateral water intakes, the quality standard of the water distribution process can be used to choose the root-mean-square integral deviation of the actual water flow from the planned (limited) values for the period [0, T] of the water distribution
N T . . 7
(30)
7=1 0
Cl,
where q, (t) - is the actual value of the water flow rate of the i-th side intake; is the planned value of the water flow rate of the i-th water intake. An analogue of criterion (30) is the expression
h=b,w-<7;
Pumping station. To determine the operating modes of the pumping unit, at different angles of rotation of the blades of the axial pump, it is necessary to have their load characteristics N = f (Q). They can be built using the universal
0
characteristic, which shows the dependences H = f (Q) and rj = f (Q) according to the well-known power formula on the shaft of the pump unit
OH
11 , (31)
where Q- is the flow rate of the pump; H - pump head; r|- is the efficiency of the pump.
The criteria for optimizing the water supply process at pumping stations are h = I>, (32)
isNr
i2=\qhc-q'\ (33)
where N, - is the power consumption of the i-th working pump unit; 0"c, (f -the actual and required water flow rate of the pumping station. These criteria evaluate the operation of pumping stations in channel sections bounded by pumping stations.
Technological restrictions on the operation modes of the channel sections are of the form
_min ^ /„„ A ^ max
2, <z\Xl,t)<Zl ,
(34)
0Гш<0,(х„ t)<0- (35)
where 0,mm, 6>,max minimum and maximum allowable water flow rates on the i-th section of the channel; z""" , z"'ax - minimum and maximum permissible ordinates of the free surface of the water on the i-th section of the channel. The main restrictions on hydraulic structures are
a,MllH <Qi(t) <аГх, (36)
where at"1", a,max -are the minimum and maximum permissible openings of the gates of hydraulic structures.
The main restrictions at pumping stations are
(л j min jrPmin Pmin 1 , i'w i'\ -ктР/А P Ли - l'jrmax .rPimx Ртах \
N, ,N, ,w )<\N\t\N, {tlwj (Oj^l^, ,N, Ji (37)
where n""" и n"'ax - the minimum and maximum number of running pumping
units;
n-г"" и nfmax - minimum and maximum set of numbers of working pumping
units;
(///"""' и i//'"""x - the minimum and maximum set of angles of rotation of the blades of the axial pumps of the i-th NS.
CONCLUSION
Thus, mathematical models of the irrigation canal section, criteria for the quality of water distribution in the canal sections are developed, discreteness
conditions and restrictions on the water distribution process in the canals are determined, which are the basis for the development of mathematical models of the optimal distribution of water in the channels of irrigation systems under conditions of discreteness of water supply to consumers.
REFERENCES
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2. Arkhangelsk V.A. Calculations of unsteady motion in open channels. M.-L.: Izd.AN SSSR, 1947, 134 p.
3. Serazetdinov TK Optimization of systems with distributed parameters. - M .: Nauka, 1977, 480 p.
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