A numerical investigation into the behaviour of cracks in uPVC pipes under pressure
- Authors: Cassa, Amanda Marilu
- Date: 2012-07-19
- Subjects: Pipe - Fluid dynamics , Numerical analysis , Piping , Finite element method
- Type: Thesis
- Identifier: uj:8843 , http://hdl.handle.net/10210/5255
- Description: D.Ing. , This study is a numerical investigation into the behaviour of cracks in uPVC pipes under pressure. This study is a continuation of a Masters dissertation which showed that leakage exponents vary significantly from the theoretical orifice exponent of 0.5 for cracks in pipes for different materials. This study looks at the behaviour of cracks in more detail and specifically with regard to the parameters of the pipe and crack. Using Finite Element Analysis the relationship between the pressure head and the leak area in pipes with longitudinal, spiral and circumferential cracks was investigated. It was found that the longitudinal, spiral and circumferential crack areas increase linearly with pressure. The slope of this linear relationship depends on various parameters, including loading state, pipe dimensions and pipe material properties. The effect that the individual pipe parameters had on the pressure-area slope was investigated. These parameters included the material properties of the pipe (Young’s modulus, Poisson’s ratio and longitudinal stress), the geometry of the pipe (internal diameter and wall thickness) as well as the geometry of the crack (length of the crack and the width of the crack). Once the effect of the pressure-area slope m is known, the link between the conventional leakage exponent N1 and the pressure-area slope m was further investigated and the effect of each parameter on the leakage exponent N1 was found. Using various data techniques the above data was combined and processed to find mathematical relationships that give reasonable descriptions of the pressure-area slopes of longitudinal, spiral and circumferential cracks. Once these equations for the pressure-area slopes were determined it was possible to obtain three new relationships for leakage from longitudinal, spiral and circumferential cracks.
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Numerical solutions for a class of nonlinear volterra integral equation
- Authors: Mamba, Hlukaphi Sithando
- Date: 2015-11-11
- Subjects: Volterra equations , Integral equations , Numerical analysis , Mathematical analysis
- Type: Thesis
- Identifier: uj:14546 , http://hdl.handle.net/10210/15077
- Description: M.Sc. (Applied Mathematics) , Numerous studies on linear and nonlinear Volterra integral equations (VIEs), have been performed. These studies mainly considered the existence and uniqueness of the solution, and numerical solutions of these equations. In this work, a class of nonlinear (nonstandard) Volterra integral equation that has received very little attention in the literature is considered. The existence and uniqueness of the solution for the nonlinear VIE is proved using the contraction mapping theorem in the space C[0; d]. Collocation methods, repeated trapezoidal rule and repeated Simpson's rule are used to solve the nonlinear (nonstandard) VIE. For the collocation solutions we considered two cases: implicit Euler method and implicit midpoint method. Examples are used to compare the performance of these methods and the results show that the repeated Simpson's rule performs better than the other methods. An analysis of the collocation solution and the solution by the repeated trapezoidal rule is performed. Su cient conditions for existence and uniqueness of the numerical solution are given. The collocation methods and repeated trapezoidal rule yield convergence of order one.
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