This work presents a novel computational model for the 3D flow in a rigid stator Progressing Cavity Pump (PCP), using an element based finite volume method, which includes the relative motion between rotor and stator. Usual flow models in PCPs consider a Poiseuille flow along the seal lines, i.e., along the positive clearance between cavities in order to predict the internal slip and then, the volumetric efficiency for different pressures, rotations and fluid viscosities. Furthermore, some attempts for more detailed models including computational solutions for the flow in simplified geometries can be encountered in literature. These approaches include, treating cavities as parallel plates or computing the flow between two static cavities, in all cases considering steady state flow, which is a strong hypothesis in this case. Nevertheless no models considering the solution for the full transient 3D Navier-Stokes equations and the relative motion between rotor and stator were encountered. The main challenge at this point was the imposition of the mesh motion and mesh generation process, mainly, because of the mesh quality control (element distortion) in regions near the seal lines, or in the clearance regions between rotor and stator. The model developed is capable to predict accurately the volumetric efficiency and the viscous looses as well as provide detailed information of pressure and velocity fields inside this device. Furthermore, the present model could be used to predict the hydraulic performance of an elastomeric progressing cavity pump after stator wear or deformation and allow for the development of a computational model for the fluid-structure interaction which permits the analysis of the non-rigid stator case. Introduction Progressing Cavity Pumping is being more and more used in oil production, mainly in heavy oil fields, due to its numerous technical advantages. Simplest models for PCP design, firstly presented by Moineau (1930), are based on calculating the slippage across the pump, considering a Hagen-Poiseuille flow in the sealing region, which is subtracted from the volume displaced, giving the volumetric flow pumped. As differential pressure increases, so does the slippage, and the relation between differential pressure across the pump and net volumetric flow pumped, can be calculated. After Moineau's models, several attempts for more precise fluid dynamic and fluid-structure interaction models have been presented. For oil production applications works due to Robello Samuel & Saveth (1998), Olivet et al. (2002), Gamboa et al. (2002) and Gamboa et al. (2003) constitute the main references in this field of research. Robello Samuel & Saveth (1998) developed optimal relationships between the pitch and the diameter of the stator to achieve a maximum flowrate for multilobe pumps. Olivet et al. (2002) performed an experimental study and obtained characteristic curves and instantaneous pressure profiles along metal to metal pumps for single- and two-phase flow conditions. Gamboa et al. (2002) presented some attempts of flow modeling within a PCP using Computational Fluid Dynamics with the aim of getting a better comprehension of the flow inside the pump. Nevertheless, attempts for developing a three-dimensional model including rotor motion were failed even for rigid stator (this means constant clearance) due to the complexity of the geometry, mesh motion and (may be) the inadequateness or limitations of the numerical approach used to solve the governing equations. In virtue of this, Gamboa et al. (2003) presented simplified models for single phase flow considering the possibility of variable gap due to elastomeric stator deformation. The basic approach does not differ too much from previous works based on metallic stator, but the slippage is calculated cavity by cavity and the possibility of a variation of the clearance as function of differential pressure is considered. In this way they were able to reproduce the characteristic non-linear behavior of volumetric flow versus differential pressure in a PCP with elastomeric stator.
Progressing Cavity Pumps (PCP) are positive-displacement pumps customarily employed in the petroleum industry for artificial lift of high viscosity oils, or fluids with moderate degree of gas and/or sediments. Basically, an elastomeric PCP consists of a metallic helical rotor and a twin helix rubber stator. For this kind of PCP, flow is strongly modified by structural deformations undergone by the stator. For automation, design and operation optimization purposes, any mathematical model able to accurately describe and predict flow and structural phenomena inside these pumps is welcome. However, due to its intrinsically coupled non-linear multiphysics -flow and structural fields, friction, wear, heat generation -analysis of the real dynamic interaction is very complex and experimental measuring is extremely hampered by the elastomeric nature of the stator. Therefore, as an attempt to describe and numerically evaluate this dynamic behavior, the present work features a simplified three-dimensinoal computational model for analysis of the fluid-structure interaction (FSI) in single-lobe elastomeric progressing cavity pumps. The governing unsteady and three-dimensional fluid dynamic equations, for laminar or turbulent flow, are solved using an Element-based Finite Volume Method in a moving and deformable mesh, to account for the relative motion between rotor and stator, and the stator deformation as well. Once a pressure distribution on the stator inner surface is obtained, for an initial non-deformable fluid domain/geometry, it is then used to evaluate the corresponding structural deformations of the elastomeric stator. In the present model, only radial deformations are allowed, being explicitly evaluated following an elastic linear model. This procedure is continuously repeated until "converged" geometrical configuration and flow field are reached before a new time step is advanced. In addition to dynamically provide detailed information on flow and structural fields (pressure, velocity, deformation and tension distributions), operational parameters such as volumetric efficiency, viscous loses and torque, for example, are precisely predicted by the model developed as function of several parameters (rotor and stator dimensions, pump rotation and differential pressure, fluid viscosity, and elastomer bulk modulus). Numerical results are presented and compared against experimental findings, illustrating the behavior of the main fields, for the different governing parameters.
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