Trans. Tianjin Univ. 2010, 16: 056-060 DOI 10.1007/s12209-010-0011-8
Application of Rectangular Integral to Numerical Simulation of Hydraulic Transients in Complex Pipeline Systems* WAN Wuyi (万五一)1,2，MAO Xinwei (毛欣炜)1，CUI Xiuhong (崔秀红)3 (1. College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China; 2. College of Engineering, Florida State University, FL 32310, USA; 3. Zhejiang University of Technology, Hangzhou 310032, China) © Tianjin University and Springer-Verlag Berlin Heidelberg 2010
Abstract：The hydraulic oscillation of surge tank was analyzed through numerical simulation. A rectangular integral scheme was established in order to improve the numerical model. According to the boundary control equation of surge tank, the rectangular integral scheme omits the second-order infinitesimal and simplifies the solving process. An example was provided to illustrate the rectangular integral scheme, which is compared with the traditional trapezoid integral scheme. Appropriate numerical solutions were gained through the new scheme. Results show that the rectangular integral scheme is more convenient than the trapezoid integral one, and it can be applied to the numerical simulation of various surge tanks in complex pipeline systems. Keywords：rectangular integral; trapezoid integral; surge tank; hydraulic oscillation
The numerical simulation of transient flow is important to control hydraulic pressure and velocity of water supply pipeline systems. Arithmetic method, graphical method, finite element method, and characteristics method were used to simulate pressure pipe unsteady flow. In fact, characteristics method was the primary one applied to numerical simulation of hydraulic transients[1]. The method was greatly improved by application to practical simulation of unsteady flow[2,3]. In order to reduce the oscillation of pressure and flow, surge tank is widely adopted in pressure pipeline systems[4]. Numerical methods were adopted to simulate water surface oscillation in surge tanks[5,6]. The critical steady section of surge tank is usually the key to numerical simulation[7]. Moreover, the hydraulic pressures and surge waves were investigated through model test and numerical method[8]. The free water surface oscillations in pipe network systems were simulated with analogous numerical method[9,10]. The previous literature focused on the hydraulic unsteady simulation of simple surge tank with only one intake and one outlet. The boundary control equation is simple and can be solved by ordinary trapezoid integral scheme. In fact, there are many kinds of complicated surge tanks, such as manholes, pump pool, and diversion well, which may be connected with more than two pipe-
lines. This kind of surge tank has complex boundary control equations. It is difficult to solve these equations through ordinary method. Therefore, the rectangular integral scheme was established by omitting the second-order infinitesimal of continuity equation of surge tank. The new scheme optimizes the equations and solving process, and it is convenient to be applied to some complicated surge tanks.
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Basic principles and solution
1.1 Basic partial differential equations For water supply pipeline systems, the continuity equation and motion equation are the basis of unsteady flow numerical simulation, both of which are partial differential equations. Characteristics method was widely employed to solve the partial differential equations. Continuity equation of pressure flow: ∂h ∂h a 2 ∂v =0 v + + v sin α + (1) ∂x ∂t g ∂x Motion equation of pressure flow: ∂h ∂v ∂v fv | v | (2) g +v + + =0 2D ∂x ∂x ∂t where h is the piezometric head; v the flow velocity; α the pipe slope; a the wavespeed of water-hammer;
Accepted date: 2009-01-07. *Supported by National Natural Science Foundation of China (No.50709029). WAN Wuyi, born in 1976, male, Dr, associate Prof. Correspondence to WAN Wuyi, E-mail:
[email protected].
WAN Wuyi et al: Application of Rectangular Integral to Numerical Simulation of Hydraulic Transients in Complex Pipeline Systems
g the acceleration of gravity; D the pipe diameter; f the Darcy-Weisbach friction factor; t time; x the distance along pipe. 1.2 Equation conversion based on characteristic line The continuity equation and motion equation constitute the basic partial differential hyperbolic equations, which include two dependent variables(flow velocity v and piezometric head h )and two independent variables (distance along pipe x and time t ). It is difficult to establish the analytical solution of these equations at present. The partial differential equations can be transformed into ordinary differential equations by the integral along the subsidiary characteristic line. The ordinary differential equations can be written according to the characteristic line as g dh dv fv | v | + + =0 (3) 2D a dt dt g dh dv fv | v | − + + =0 (4) 2D a dt dt Eq.(3) is based on the forward characteristic line dx / dt = a , and Eq.(4) is based on the reverse characteristic line dx / dt = −a . 1.3 Solution of hydraulic transients Eq.(3) and Eq.(4) are transformed into finite difference equation by the integral along the characteristic line, in which the first-order approximation is adopted to simplify the equation. The finite difference equation is a h(i ,t +Δt ) − h(i −1,t ) + (Q(i ,t +Δt ) − Q(i −1,t ) ) + gA f Δx (5) Q(i −1,t ) | Q(i −1,t ) |= 0 2 gDA2 a (Q(i ,t +Δt ) − Q( i +1,t ) ) − h(i ,t +Δt ) − h(i +1,t ) − gA f Δx Q(i +1,t ) | Q(i +1,t ) |= 0 (6) 2 gDA2
Fig.1 shows the x-t digital plane grid[1]. The iteration starts generally from the initial time, in which every node’s Q and h values are known. The variables after the time step Δt can be calculated according to the known parameters. Similarly, the variables at time t + Δt can be calculated according to the previous time step parameters at time t . Hydraulic variables Q and h can be calculated at any time as long as demanded. The finite time increment Δt should be controlled small enough to meet the numerical precision. It is usually subject to the formula Δt ≤ 4 DA f Q [1]. The reliability of characteristics method is proved by application to the simulation of pressure pipelines. As shown in Fig.2, the comparison between calculation and experiment testified the numerical method[1].
Fig.1
Fig.2
x-t digital plane grid
Comparison between calculation and experiment[1]
where A is the pipe section area; Q the flow of pipe, Q = Av ; Δx the step distance; Δt the time step; sub- 2 Improvement of boundary equations scripts i and t are the series number of node and time. 2.1 Boundary control equations In order to apply the equation to computer calculation, The continuity equation is the primary boundary the equations can be abbreviated as control equation of surge tank, which is usually ordinary h(i ,t +Δt ) = CP − BQ(i ,t +Δt ) (7) differential equation. The trapezoid integral scheme is (8) h(i ,t +Δt ) = CM + BQ(i ,t +Δt ) widely adopted in order to transform ordinary differential where B = a / gA and R = f Δx (2 gDA2 ) , CP and CM are equations into finite difference equations. The trapezoid constants, which can be computed according to the previintegral scheme is a standard format and can provide satous known variables, isfactory solution for the equation, but it is difficult to be CP = h(i −1,t ) + BQ(i −1,t ) − RQ(i −1,t ) |Q(i-1,t)| applied to some complex systems. For complex branch CM = h(i +1,t ) − BQ(i +1,t ) + RQ(i +1,t ) |Q(i-1,t)| pipeline systems, the trapezoid integral scheme consists —57—
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of many unknown variables, which make the boundary control equation complicated. In fact, the rectangular integral scheme is also available for transforming ordinary differential equations into finite difference equations. And it is shown that the rectangular integral scheme is simpler than the trapezoid integral one. An example is provided to illustrate the rectangular integral scheme. Fig.3 shows a simple surge tank connected with pipelines. The free water surface in the surge tank will fluctuate if downstream valve closes suddenly. The basic boundary control equation of surge tank can be written as
∫
t +Δt t
Qi (t )dt = sΔz + ∫
t +Δt t
Qi +1 (t )dt
(9)
ing to the trapezoid integral scheme, the boundary control equation can be written as Eq.(10). Coupling the basic characteristic equation and energy equation, the numerical model of surge tank can be written as Q(i ,t ) + Q(i ,t +Δt ) + Q( i +1,t +Δt ) Δz Q (10) = s + (i +1,t ) 2 2 Δt h(i ,t +Δt ) = h(i ,t ) + Δz (11) h(i +1,t +Δt ) = h(i +1,t ) + Δz
(12)
h(i ,t +Δt ) = CP − BQ(i ,t +Δt )
(13)
h(i +1,t +Δt ) = CM + BQ( i +1,t +Δt )
(14)
The above model includes five correlative equations. Eq.(10) consists of three unknown variables, Q(i ,t +Δt ) , Q(i +1,t +Δt ) and Δz . The variable Δz can be gained only by combining all the other equations. It is difficult to solve the equations if the surge tank is connected with multibranch pipelines and subject to more correlative equations. 2.3 Rectangular integral scheme The rectangular integral scheme is another available method, though Eq.(10) is a standard scheme with the following form:
where Qi (t ) is the discharge process in upstream section i ; Qi +1 (t ) the discharge process in downstream section i+1; s the section area of surge tank; z the water level of surge tank; Δz the increment of water level during time Δt . Fig.4 shows the basic principle of conversion from ordinary differential equation to finite difference equation. The water volume increment can be represented by 1 rectangle ABD1C1 or trapezoid ABDC . In this figure, Q(i ,t ) sΔz = [(Q( i ,t ) − Q( i +1,t ) ) + (Q( i ,t +Δt ) − Q( i +1,t +Δt ) )]Δt (15) 2 is the discharge of upstream section i at time t , and In Eq.(15), the term sΔz is the water volume inQ(i +1,t +Δt ) is the discharge of downstream section i+1 at crement of surge tank, which is the difference between time t + Δt . influent volume and effluent volume in time step Δt . Considering that discharge is a continuous process in both upstream and downstream section, Q(i ,t +Δt ) = Q(i ,t ) + ΔQ(i ,t ) , Q(i +1,t +Δt ) = Q( i +1,t ) + ΔQ( i +1,t ) , Eq.(15) can be written as 1 sΔz = Δt (Q( i ,t ) − Q( i +1,t ) ) + Δt (ΔQ( i ,t ) − ΔQ( i +1,t ) ) ( 16) 2 Eq.(16) can be simplified by omitting second-order Fig.3 Simple surge tank connected with pipelines
infinitesimal as sΔz = (Q(i ,t ) − Q( i +1,t ) )Δt
(17)
The water volume increment in Eq.(10) is represented by the area of trapezoid ABDC shown in Fig.4. Analogously the water volume increment in Eq.(17) is represented by the area of rectangle ABD1C1 shown in Fig.4. In fact, the above equations are equivalent, since Δt is a finite time increment. The equations of surge tank are simplified as Δz = (Q(i ,t ) − Q( i +1,t ) ) Δt s (18) Fig.4 Finite differential of water volume
2.2
Trapezoid integral scheme (traditional method) Trapezoid integral is generally applied to transform integral equation into finite differential equation. Accord—58—
h( i ,t +Δt ) = h( i ,t ) + Δz
(19)
h( i +1,t +Δt ) = h( i +1,t ) + Δz
(20)
Q( i ,t +Δt ) = (CP − h( i ,t +Δt ) ) B
(21)
Q( i +1,t +Δt ) = (h( i +1,t +Δt ) − CM ) B
(22)
WAN Wuyi et al: Application of Rectangular Integral to Numerical Simulation of Hydraulic Transients in Complex Pipeline Systems
Eq.(18) contains only one unknown variable, so it is easy to individually work out Δt with the equation, and the other equations are simplified accordingly based on the known Δt . For the surge tank with multi-inlet and multi-outlet, the equations can be extended as J
K
j =1
k =1
Δz = (∑ Q( j ,i ,t ) − ∑ Q( k ,i +1,t ) ) Δt s
both the trapezoid integral scheme and rectangular integral scheme. The hydraulic oscillation shown in Fig.6 and Fig.7 is solved through the two methods. Fig.6 shows the water surface oscillation in the surge tank, and Fig.7 shows the flow oscillation in the inlet. This example illustrates that the rectangular integral scheme is available (23) for simulating the hydraulic oscillation in surge tank.
h( j ,i ,t +Δt ) = h( j ,i ,t ) + Δz ( j = 1, 2," , J )
(24)
h( k ,i +1,t +Δt ) = h( k ,i +1,t ) + Δz (k = 1, 2,", K )
(25)
Q( j ,i ,t +Δt ) = (C jP − h( j ,i ,t +Δt ) ) B j
( j = 1, 2," , J )
(26)
Q( k ,i +1,t +Δt ) = (h( k ,i +1,t +Δt ) − CkM ) Bk
(k = 1, 2,", K )
(27)
where subscripts j and k are series number of inlet and outlet. Every equation can be solved separately, since Δz can be gained individually with Eq.(23). 2.4 Analysis and discussion Eq.(10) and Eq.(18) are two finite difference schemes about boundary control equations of surge tank. Besides unknown variable Δz , Eq.(10) consists of two unknown variables and two known variables, and Eq. (18) consists of only two known variables. The difference between the two equations is the omitted secondorder infinitesimal. According to Eq.(10) and Eq.(18), the trapezoid integral scheme presents the second-order numerical precision and the rectangular integral scheme presents the first-order numerical precision. The trapezoid method is based on both the known time t and unknown time t + Δt , but the rectangular method is based only on the known time t . The term Δz can be gained individually with Eq.(18), then the other equations can be solved separately through substitution. Obviously, the rectangular integral scheme is more convenient than the trapezoid integral scheme, because it simplifies the equations and solving process.
3
Case study
In order to illustrate that the rectangular integral scheme is more convenient than the trapezoid integral scheme, the hydraulic oscillation in a complex surge tank is simulated based on two different methods. As shown in Fig.5, the surge tank is connected with one inlet and three outlets. In this case, the surge tank is 2.0 m2 in section area, and every branch pipe is 300 m in length and 0.5 m in diameter. The time step Δt = 0.125 s is adopted in
Fig.5
Complex surge tank with multi-branch pipelines
Fig.6
Water surface oscillation in surge tank
Fig.7 Flow oscillation in inlet
The trapezoid integral scheme is generally adopted as a standard method in solving the continuity equations of surge tanks. The method is proved available by application to many practical problems. As shown in Fig.6, the rectangular integral scheme can also provide an approximate numerical solution. In fact, the water surface fluctuations are the result of alternate change in water volume, which is due to the volume difference between inlets and outlets. The volume increment can be represented by both rectangular integral scheme and trapezoid —59—
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integral scheme; however, the rectangular method is sim- ［2］ Bergant A, Simpson A R, Vitkovsky J. Developments in pler than the trapezoid method. As shown in Fig.4, the unsteady pipe flow friction modeling [J]. Journal of Hyrectangle or trapezoid is an approximate expression of the draulic Research, 2001, 39(3): 249-257. volume increment during the time step Δt . According to ［3］ Zhang Yongliang, Vairavamoorthy K. Transient flow in the integral principle, the smaller the time step Δt , the rapidly filling air-entrapped pipelines with moving boundamore precise the result. For the computer numerical ries [J]. Tsinghua Science and Technology, 2006, 11(3): simulation, the precision is usually satisfactory, since the 313-323. time step is subject to the formula Δt ≤ 4 DA / f Q . Many ［4］ Moghaddam M A. Analysis and design of a simple surge practical examples show that Δt = 0.2 s is small enough tank [J]. International Journal of Engineering, Transacto achieve appropriate numerical solutions. tions A, 2004, 17(4): 339-346.
4
Conclusions
The rectangular integral scheme is established by omitting the second-order infinitesimal of boundary control equation, which is applied to the numerical model of hydraulic oscillation in complex surge tanks. It can greatly simplify the equations and solving process. An example is presented to illustrate the rectangular integral scheme. Numerical results show that the water surface fluctuation is due to the volume difference between inlets and outlets, which can be represented by both the rectangular integral scheme and trapezoid integral scheme. The rectangular method is proved simpler than the trapezoid method. Therefore, it is convenient to extend the numerical model for multi-branch surge tanks with the rectangular integral scheme. References ［1］ Wylie E B, Streeter V L. Fluid Transients [M]. McGrawHill International Book Company, New York, 1978.
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