Analysis of wave damping in pipeline having different pipe materials configuration under water hammer conditions

Hydraulic transient occurs whenever there is a sudden change in the flow velocity resulting in variation of pressure and flow in a water conductor system. Experiments have been conducted in a straight pipeline having material of Mild Steel (MS) and Glass Fibre Reinforced Plastic (GRP) pipelines and their combined configurations. From experiments, it has been observed that there is a smooth and strong damping of pressure waves in the pipeline system. Experimental results were compared with results obtained for classical water hammer equations solved in MATLAB and analyzed that there are several dissipative factors, other than fluid viscosity, responsible for strong damping of pressure wave amplitude. Further, an improvement in the governing equation of water hammer in a closed conduit was proposed by incorporating a different wave damping coefficient ( α ). The modified governing equations have been solved for each water hammer cycle using MATLAB. The numerical simulation results show that proposed approach gives better agreements between the experimental and computational results for all investigated cases.


Introduction
Hydraulic transient or water hammer occur whenever there is a sudden alteration in the flow velocity resulting in variation of pressure and discharge in a water conductor system for hydropower and water supply plants [1,2]. It is very important to keep hydraulic transient with in safe limits in a water conductor system for a viewpoint of hydropower system safety, effective operation and to increase its life span. Uncontrolled transient events may cause serious impact on both civil and mechanical infrastructure of the plants [3][4][5]. It is difficult for power and water utilities to reduce water hammer formation in water conductor system due to the variation in demand. The analysis of hydraulic transient is essential for the selection of penstock material, its pressure class and the specifications of surge protection devices. Formation of hydraulic transient in penstock is greatly influenced by its material [6,7]. More rigid material means higher transient formation and elastic materials forms lower transient waves. The viscoelastic pipes suppress water hammer pressure effectively because of its low characteristics impedance and fast damping compared with metallic pipes [8,9].
The general simplified continuity and momentum equations used for transient flow analysis in a closed conduits consist of two hyperbolic partial differential equations shown as Eq. (1) and (2) respectively [10,11] which does not provide direct solution by any numerical method for given boundary conditions. The water hammer wave velocity (celerity) for an elastic pipeline with circular cross section can be estimated with Eq. (3) [12,13].
where, = A cross section flow area, = g gravitational constant, = R pipeline resistance coefficient, a = wave velocity (celerity), ( ) Researchers across the globe have studied the water hammer impact on penstock material experimentally or analytically. Mitosek et al. [14] conducted experiments for water hammer analysis in two different material pipelines and proposed some modification in governing equation of the water hammer for smooth damping of pressure waves. Adamkowski et al. [15] conducted the experiments to study the damping effect of the pressure waves in the pipeline and concluded that the classical water hammer theory cannot predict the damping effect properly.
Gong et al. [8] proposed a surge suppression technique by using metallic-viscoelastic-metallic configuration in water distribution system. Larson et al. [9] studied transient effect in water and sewage pipes by measuring pressure and strain in pipes made by different materials and concluded that the response of pipes during transient events can be analyzed by using linear elastic theory with related modulus of the elasticity. Duan et al. [16] highlighted the effect of unsteady friction in viscoelastic pipes by implementing quasi 2D numerical method and concluded that the water hammer equations were not capable to exhibits the damping behavior during water hammer events. Several researchers had studied the effect of the shear stress in wave damping and smoothing of the unsteady flow [14,17,18]. It was further identified that apart from shear stress, there are some other factors which influences the damping and smoothing of pressure waves in the water hammer phenomenon. The friction force only influences the amplitude of celerity, whereas, there are some dissipative effect which affect the wave damping. There are various sources of energy dissipation in water hammer events. In this paper, the effect of material elasticity, one of dissipative effect is discussed and its effect is incorporated in the water hammer equations. This source is related with both fluid and pipe wall material elasticity which can be taken into account by introducing variable wave pressure velocity in the continuity equation of water hammer. The water hammer experiments were carried out for metallic (MS) and viscoelastic (GRP) pipeline and results obtained using improved MOC codes were validated.

Experimental Set up
The experimental set up for hydraulic transient test was designed and fabricated in the Department of Hydro and Renewable Energy at Indian Institute of Technology Roorkee. The experimental set up consist of two straight pipelines with different materials, a butterfly valve (7) mounted on the downside of the pipeline, an air vessel (5) and pressure sensors (10). The water hammer experiments were carried out for metallic (MS) and viscoelastic (GRP) pipeline. The detail characteristics of the pipe materials are given in Table 1.
The pipes were fixed to the floor with supports. The water supply was given to the experimental setup from the reservoir (8) by a centrifugal pump (1). The butterfly valve (7)

MODIFIED WATER HAMMER EQUATION
The water hammer events can be considered as a 1D phenomenon in which both the bulk modulus of elasticity and fluid density change less within a large range of pressure values [19,20]. The celerity or pressure wave velocity is usually considered to be constant throughout in the numerical simulation. As per the result which is obtained from Eq.  It can be analyzed and observed that the time of pressure increase is shorter than the time of pressure decrease. This difference is related with the pressure wave amplitude, because as the amplitude decreases, it disappears with time. As these time intervals have different lengths, their sum, being the total wave period is constant with time. The asymmetry observed in ( ) t H must be related with the celerity. Therefore, instead of = a constant, should be assumed in simulation process [14,21]. By taking the accounts of varying gradients of function ( ) because of dissipation. The principal of energy conservation can also be written as Eq. (4) [22].
During compression wave damping coefficient (α1) may be defined as per Eq. In the phase of decompression, where an increment in the flow velocity occurs ( ) the store energy due to the elasticity and pipe material is returned as the kinetic energy of the following water. However, due to dissipation, only small fraction of the stored energy is return back. Thus, the energy balance and change in pressure due to water hammer events can be expressed as Eqs. (7) and (8) respectively.  Step Step Step Step  Table 2.
Numerical Results obtained for all material pipelines at flow velocity equal to 0.5 m/s are plotted along with experimental results as shown in Fig. 4. It may be seen that, high damping in amplitude of pressure wave was observed for higher value of wave damping coefficient.
Similarly, for each pipe configuration and flow velocity condition the results obtained from modified MOC method using four different wave damping coefficient were compared with experimental results. The smaller value of α is observed in metallic pipeline and higher in viscoelastic material.

EXPERIMENTAL VALIDATION
The wave damping coefficient is evaluated for all experimental cases. It can be concluded that the proposed approach by considering variable wave velocity in the analysis of water hammer gives better agreement between experimental and numerical results.

Funding
The first author expresses sincere thanks to Department of Hydro and Renewable Energy and Quality Improvement Program Centre, IIT, Roorkee, India for providing research facilities and to the All India Council for Technical Education (AICTE) Government of India for providing financial assistance in the form of research scholarship.