International Research Journal of Engineering and Technology (IRJET) Volume: 04 Issue: 03 | Mar -2017
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e-ISSN: 2395 -0056 p-ISSN: 2395-0072
Comparison of MOC and Lax FDE for simulating transients in Pipe Flows Er. Milanjit Bhattacharyya1, Dr. Mimi Das Saikia 2 1
PhD Research Scholar, Deptt. of Civil Engineering, Assam Down Town University, Guwahati, Assam, India. 2 Professor, Deptt. of Civil Engineering, Assam Down Town University, Guwahati, Assam, India
---------------------------------------------------------------------***-------------------------------------------------------------------2. Literature Review: Abstract -The method of characteristic (MOC) approach transforms the water hammer partial differential equations into ordinary differential equations along characteristic lines. The fixed-grid MOC is the most accepted procedure for solving the water hammer equations and has the attributes of being simple to code, efficient, accurate and provides the analysts with full control over the grid selection. Some authors are of the opinion that Lax Finite Difference Explicit method provides more convincing results for solving unsteady transient situations in pipe flow. Here an approach is made to compare the MOC and Lax FDE scheme of discretization for hydraulic transient governing equation, with the help of MATLAB as the programming tool and finally Lax FDE scheme is observed to be more effective. Key Words: Hydraulic pipe transients, water hammer, valve, numerical model, discharge, velocity, pressure etc.
1. INTRODUCTION: The variation in discharge and pressure head can be studied by solving the governing equations for hydraulic transients in a pipe using the Method of Characteristics for discretization of the partial differential equations and also by Lax Finite Difference Explicit method. Due to the nonlinearity of the governing equations, various numerical approaches have been developed for pipeline transient calculations, which include the Method of Characteristics (MOC), Finite Difference (FD) and Finite Volume (FV) etc. Among these methods, MOC proved to be the most popular among the water hammer analysts. In fact, out of the 14 commercially available water hammer software packages found on the world wide web, 11 are based on MOC, two are based on implicit FD method [11]. After the Finite Difference Equations (FDE) are obtained, the numerical models are developed using MATLAB. The models are then validated using lab data. Chudhury M.H. [13] advocated comparative effectiveness of Lax FDE method over MOC, which has been analyzed and observed here.
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The basic unsteady flow equations along pipe due to closing of the valve near the turbine are non-linear and hence its analytical solution is not possible. Watt C.S.et al (1980)[1] have solved for rise of pressure by MOC for only 1.2 seconds and the transient friction values have not been considered. Goldberg D.E. and Wylie B.(1983)[2] used the interpolations in time, rather than the more widely used spatial interpolations, demonstrates several benefits in the application of the method of characteristics (MOC) to wave problems in hydraulics. Chudhury M.H. and Hussaini M.Y.(1985)[3] solved the water hammer equations by MacCormack, Lambda, and Gabutti explicit FD schemes. Sibetheros I. A. et al. (1991) [4] investigated the method of characteristics (MOC) with spline polynomials for interpolations required in numerical water hammer analysis for a frictionless horizontal pipe. Silva-Arya W.F.and Choudhury M.H.(1997)[5] solved the hyperbolic part of the governing equation by MoC in one dimensional form and the parabolic part of the equation by FD in quasi-twodimensional form. Pezzinga G. (1999) [6] presented both quasi 2-D and 1-D unsteady flow analysis in pipe and pipe networks using finite difference implicit scheme. Pezzinga G. (2000) [7] also worked to evaluate the unsteady flow resistance by MoC. He used Darcy-Weisback formula for friction and solved for head oscillations up to 4 seconds only. Damping with constant friction factor is presented but not much pronounced, as the solution time was very small. Bergant A. et al (2001) [8] incorporated two unsteady friction models proposed by Zielke W. (1968) [9] and Brunone B. et al.(1991)[10] into MOC water hammer analysis. Zhao M. and Ghidaoui M.S. (2004)[11] formulated, applied and analyzed first and second –order explicit finite volume (FV) Godunov-type schemes for water hammer problems. They have compared both the FV schemes with MoC considering space line interpolation for three test cases with and without friction for Courant numbers 1, 0.5.0.1.They modeled the wall friction using the formula of Brunone B. et al (1991) [10]. It has been found that the First order FV Gadunov scheme produces identical results with MoC considering space line interpolation. They advocated that although different approaches such as FV, MOC, FD and finite element (FE) provide an entirely different framework for conceptualizing and representing the physics of the flow, the schemes that result from different approaches can be similar and even identical. Barr D.I.H. (1980)[12] formulated
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