NUMERICAL SOLUTION OF SOME FLOW PROBLEMS AND SIMULATION OF TWO-PHASE FLOW SYSTEMS

Abstract

The thesis running into Eleven Chapters is an attempt to study simulation of single and two phase flow systems in a pipe network and numerical solution of some flow problems of incompressible second-order fluids, using computer oriented numerical methods. Chapter I is introductory and, deals with the fundamental concepts of single phase flow and two-phase systems and of non-Newtonian second-order fluids. It also provides basic physical laws governing the pipe network problem and the tensor technique of the transformations of the governing equations (consisting of the constitutive equation, continuity equation and equation of conservation of mass and momentum) for the incompressible second-order fluids. The subsequent content of the thesis which forms the main contribution, is classified under parts A and B which deal with the steady single and two-phase flow systems and flows of second-order fluids, respectively. Part A consists of four Chapters and Part B consists of six more Chapters. The detailed coverage of Chapters in parts A and B is as follows: Chapter II is devoted to the simulation of hydraulic flow in pipe network. A typical network problem has been taken from the open literature. The problem has been translated into model equations. using . physical laws governing pipe network flows. Hazen-Williams empirical correlation is used to describe pressure drop correlation for flow of water in pipe lines. The resulting model equations. come Out to be a set of simultaneous mixed linear and non-linear (MLNL) equations which has been solved using four methods namely, Hardy-Cross method, Newton-Raphson method, Newton type linearisation method and Globally convergent method. Algorithm of these solvers have been generated alongwith detailed computer codes in FORTRAN - 77 language. The results obtained from these methods are compared with the results available in literature. The strength and weaknesses of each of these methods have also been identified and discussed. The Globally convergent method is found to be the most efficient as it converges to the solution from widely different sets of initial values. To further check the efficiency and stability of Globally convergent method, a more complex hydraulic pipe network problem having 46 nodes 15 loops has also been solved. Hence, for all subsequent studies of two-phase flow networks, the Globally convergent method has been employed. The procedure developed and tested for hydraulic network is extended to cover two-phase flow in pipe network by suitably modifying the pressure drop correlation. Three representative two-phase flow systems of gas-liquid, liquid-liquid and solid-liquid have been studied. The pressure drop correlations for these two-phase flow systems have been critically reviewed and selected for present work. Chapter III covers the simulation of steam-water flows in pipe network. A two-phase multiplier alongwith Hazen-William equation is used to compute pressure drop in the pipes of the network. In Chapter IV an oil-water flow is simulated in pipe network. At low oil concentration where the flow is found to be Newtonian, pressure drop in the pipes is obtained using Darcy-Weisbach equation alongwith friction factor for Newtonian fluid. In Chapter V solid-water (slurry) flow is simulated in pipe network. Slurry exhibit non-Newtonian fluid flow behaviour described by power law. A properly fitted rheological equation is used to obtain pressure drop flow rate relationship. Chapters VI-XI are devoted to the numerical solution of steady flows of the second-order fluid between two enclosed rotating discs for the cases (i) when the upper disc is stationary and lower disc rotates, (ii) when there is suction on the upper stationary disc and an equal and uniform injection on the lower rotating disc, ii (iii) when upper disc rotates and lower disc is stationary, • (i ) when there is suction on the upper rotating disc and equal and uniform injection on the lower stationary disc, (v) when both discs rotate either uniformly in opposite sense or with different magnitude in either sense, (vi) when there is suction at the upper disc and an equal injection at the lower disc and both the discs rotate with different magnitude in either sense. The equations involved in this part have been non-dimensionlised and the interpretation of the result in case of second-order fluids is based, wherever possible, on the experimental values of the material constants provided by the experiments conducted by Markowitz(see Truesdell) for the solution of poly-iso-butylene in cetane at 30° C for different concentrations. The governing differential equations are converted into a set of difference equations by using finite difference approkimations for the derivatives, Starting from the known values of flow functions for small values of the Reynolds number, the solution is extended for larger values of Reynolds numbers by making use of Newton-Raphson iterative method and Gauss-elimination method. Effects of second-order forces and those due to suction and injection on the velocity field have been investigated in detail in the regions of recirculation and no recirculation for the cases of radial outflow and inflow and are shown graphically: Comparison between the numerical solution and the approximate solution has also been made..............