Water Hammer Simulation For Practical Approach

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1 Water Hammer Simulation For Practical Approach C.D.Gălăţanu (1), Th.Mateescu (2) Technical University Gh.Asachi, Faculty of Civil Engineering, Department of Building Installations, Iaşi, Romania Abstract The theory for the water hammer is well know in the field, but for the practical approach, the authors consider that the graphical methods available in Romanian standards are difficult to be applied. Using MATLAB software with SIMULINK, like an open platform for dynamical simulations and numerical analysis, some original function and dynamical models was developed. The theory remains the same, but the solving method is very new. The first application is the valve closure, witch generates the pressures well over the steady state values. The boundary conditions consist of a large reservoir at the upstream end of the pipeline and a valve at the downstream end discharging to the atmosphere. An other important situation is the pump failure. Our calculation computes not only the maximum and minimum piezometric pressures (relative to atmospheric), but it reveals the evolution of the process. Mass conservation and momentum conservation are the fundamental equations used to analyse hydraulic transients. A fast model, with reasonable precision was developed for pipelines with pumps. This model represents an original contribution, the graphical programming improving the possibilities to analyse the phenomenon and giving more realistic results. The simulation remains open to the specific configuration, and in order to validate the software, some measurement (founded in literature) was used. Graphical interface was friendly designed, and details about the graphical programming was given, in order to be easy to be used and understanding. Keywords Water Hammer, Simulation, Graphical Programming. 1 Introduction There are many programs available to analyse water hammer problems [1,2]. A general condition for a specialized software to be accepted is to be simple to use, requiring short time to learn. In the authors opinion, this condition is restrictive and introduce an

2 important limitation for the modelling system. If we need a highly accurate simulation, the learning process need a long time if we want that such software to be used by the engineers. This aspect is an important disadvantage if we accept that this kind of problem is rare, and the learning must be repeated every time. Also, the solving method remains hidden. This paper has a different approach, using an open platform for dynamic simulation, developed for general purpose, but frequently used by the civil engineers. Using the classical theory, the calculus is accessible to be understand and improved with details. 2 The traditional graphical solution for water hammer A large number of authors [4, 5, 6] present the basic physics of conservation of momentum and conservation of mass. As partial differential equations they have a solution domain that is two dimensional, with one spatial dimension and the time dimension. Applying conventional solution techniques to these equations often results in divergence and large errors. b d Static head Dynamic head c a Figure 1 - The method of the calculation waves applied on the graphic form at a system with pump station: a the scheme of the system and the propagation of the calculus waves; b the graphical solution; c the variation in time of the over flow in section B; d the variation in time of the statical head in section B. Applying the method of characteristics allows these equation to be converted to ordinary differential equations, with a simple solution, obtained in general using graphical methods.

3 The figures 1, a, b, c, d, [6] illustrate the case of a pump equipped with a back flow preventer which closes it self instantaneously at the bock flow tendency. We can remark two types of boundary conditions: characteristic curve (CLA) in section A (at the tank), and the characteristic curve (CLB) in B, in which it is concentrated the pressure loss at the power failure of the pump, at different moments t = kπ. This method has important limitation, offering only some limited information about the phenomenon. 3 Theoretical consideration Further on we will use a direct demonstration [6] for the Jukovschi [4] relation between the waves velocity and pressure which doesn t use differential equations and then the relations used for the pipes. Jukovschi relation is a particular form deduced from Riemann invariants, adapted for faint compressible fluids (water and the others liquids). In this case, the compressibility can be treated like a linear phenomenon that leads to linear relations. For demonstration, in figure 2, we consider a semi-infinite pipe through the right, horizontal, with liquid at rest. At the left extremity, on apply constantly a over flow injection which generates a wave. Alternatively, we can imagine the perturbation being induced by the uniform motion through the right, with speed v, like a piston. v Figure 2 The diagram for the Jukovski equation Figure 2, like a reference for the demonstration which follows, represents the situation of the pipe at one time: T = T 0 + T, with T 0 the initial moment. The liquid which is initial in rest in section AB, with the length c x T (with T = T T 0 ), suffered the action of the waves: the speed and pressure will be modified from V 0 = 0 and p = P 0 to V = v and P = P0 + p. By the increasing of the pressure, the liquid compress itself and at the moment T will occupy the section A B. At the left extremity, in A, the pressure is P0 + p and in B is P0, the initial pressure un modified. We apply the pulse theorem ( (m x V) = F x T) for the liquid mass which is initially in the section AB and finally in A B: ρ A c T V 0) = ( P + p P ) A T (1) ( 0 0 where: ρ liquid density and A area of the cross section.

4 It results the simple relation: or with the notation p = ρ c v p = z q ρc z = A (2) (3) (4) The value z is called wave drag [6]. For the calculation of the transient motion on use, as usual, like descriptive parameters, the piezometric head (PN = P/(ρ x g), HN = ZN + PN) and the flows Q, respectively h and q for waves: In this case, for h and q we will obtain: c h = m q; m = (5) g A For m on use the name modulus of the wave drag or, more simple, wave drag. The relations from upper side are deduced using figure 2 for a direct wave. If we operate with a piston placed in the right of the figure and we move it through the left, the development of the phenomenon is physically the same, with the difference that the generated primary wave is an inverse wave and the value of the over flow wave is negative while the pressure wave remain positive. For concordance, for the inverses waves, the waves relations has to be written in the form: (6) p = z q ; h = m q i i To establish the formula of the propagation speed we apply to the same liquid mass a preserving relation. The liquid mass which is initially on the section AB of the pipe will be at the moment T = T 0 + T in the section included between A and B. In the section AA we have the mass ρ x q x T introduced, in the interval T, with the over flow injection. We write that the volume occupied by the injected liquid is equal with the deformation by compression of the liquid being initially between A and B, with B compressibility coefficient: i i A v T = B p ( A c T ) (7) Taking out p from the equation (7) we obtain: = = 1 c c A Bρ = 0 where ε is modulus of elasticity of the liquid. We obtained the well known formula of the propagation velocity of the waves in continuous mediums and, respectively in the undeformable pipe. For the motion in real pipes, the equation has to be completed with the influence of the pipe deformability. At the modification of the liquid pressure it takes place not only the ε ρ (8)

5 modification of the volume of water, but also of the pipe, in the section subdued to the waves actions. If in the right side of the equation (8) is introduced the second term which had to express the volume modification because of the pipe deformation on the wave action p, it obtains the formula which take account only of the cross deformation. c0 c = ε D 1+ (9) E e where: D is the pipe diameter; E modulus of elasticity of the pipe material; e the width of the walls of the pipe; ε - the wave velocity in the non-deflecting pipe, in concordance with the equation (8). In order to validate our results, the value the wave velocity is c 0 = 1439 m/s in continuous medium, respectively the non-deflecting pipe and c = 1000 m/s, in average, in real pipes. This equation will be use directly in the SIMULINK models. 4 The slowly closure of the downstream valve This situation is frequently presented in the manuals. For water distribution systems, the situation is not very typical (in opposition with hydraulical power plant). For this reason, we use this simple water hammer problem in order to verify the possibilities of the simulation tools. Figure 3 SIMULINK model for the valve closure The main aspect revealed is that in every step (with autosize option, but smaller than 0.01s) suppose that the flow is calculated. The block Flow use the valve closure characteristic. A very powerful block is Wave reflection, witch introduce a transport delay of the input values. The results are available in figure 4. The particular aspect of the output is generated by

6 the valve characteristic and the speed of the closure. Presure (bar) The flow decreasing Time (s) Figure 4 Water hammer simulation for the valve closure 5 The pump failure The SIMULINK model is presented in figure 5. Apparently is very complex, but the connections are realised in a logical way, close to the phenomenon. The principal calculus stages which can be observed in the SIMULINK model (figure 5) are :! the pump work coupled at the equalizing tank. The pump characteristic is a polynomial type, modelled by the coefficients : k1, k2, k3, k4. (Matlab function Pump with tank );! pump head depends by the rotative speed of the pump ( lowering, after the second 1);! kinetic energy of the pump is consumed during the water flow is evacuated (block Kinetical Energy ) ;! rotative speed of the pump is corresponding with the remaining kinetic energy;! at the lowering of the flow through zero, a back flow preventing blocks the back flow ( block Saturation );! variation of the flow (of the pulls) generates an overpressure (blocks Derivative, Gain, Gain 2 );! failure condition is generated in the moment 1 ( block Step ).

7 Figure 5 SIMULINK model for the pump failure Presure (m H 2 O) Time (s) Flow (m 3 /h) Time (s) Figure 6 The pump failure simulation

8 6 Conclusions The utilization of SIMULINK for the modeling of transient process is apparently difficult. The graphical programming, with specialized blocks available for calculations and visualization allows the modeling of the elementary phenomenons which cooperate to the development of the phenomenon on the whole. Using the theories of the energy and the pulls conservation, the obtained results are realistic. The reflection of the waves, for instance, is realized by specific SIMULINK blocks, which introduce a delay. The advantage of this approach method is the fact that it can be applied at different systems without need of specific knowledge of programming. The simulation remains open to the specific configuration, and in order to validate the software, some measurement will be realised in water distribution system. 7 References Mateescu Th.& Profire M. (2000). Alimentari cu apă, Iaşi, Editura Cermi 5. Iamandi C. and all (2002), Hidraulica Instalaţiilor, Bucureşti, Editura Tehnică 6. Cioc D. & Anton A. (2001), Reţele Hidraulice Calcul, Optimizare şi Siguranţă, Timişoara, Editura Orizonturi Universitare 7. Mateescu Th. (1996) Calculul Instalaţiilor Sanitare, Iaşi, Editura Gh.Asachi 8. Mateescu Th. (1989) Instalaţii sanitare şi de gaze, Editura Institutului Politehnic 8 Presentation of Authors Gălăţanu Cătălin-Daniel is Professor since 2003, Assoc.Prof between 1999 and 2003 at Technical University Gh.Asachi of Iasi, Deparment of Building Installations, Faculty of Civil Engineering, teaching Automation and Measurement for Buiding Installations, Computer Aided Design, author of four books and manuals, three patents. Theodor Mateescu is tenured professor at Technical University Gh.Asachi of Iasi, Department of Building Installations, teaching Water Supply and Sewerage for Buildings, Natural Gases Network, PhD Coordinator, Expert of Educational and Cultural Ministry, Expert of Public Ministry, Member in the Distance Learning Commision of National Council of Academical Evaluation, Member in Technical Commision CIB W062 - Water Supply and Drainage, Member of Central European Academy of Science and Art, Vice President of Romanian Association of Buiding Installation Engineering, president of Modova Branch.

For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then:

For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then: Hydraulic Coefficient & Flow Measurements ELEMENTARY HYDRAULICS National Certificate in Technology (Civil Engineering) Chapter 3 1. Mass flow rate If we want to measure the rate at which water is flowing