Investigation of flow through centrifugal pump impellers 2

CHAPTER INTRODUCTION 1.1 Background A centrifugal pump consists of a set of rotating vanes enclosed within a casing. The vanes impart energy to fluid through centrifugal force. The advantages of the centrifugal pumps over the reciprocating pump are based on the high and consistent flow rate, easy and effective operation control and low manufacturing and maintenance costs. The main disadvantage is its relatively low efficiency. Traditionally, centrifugal pump design depends on too many factors based on designer’s experience. The designer can not accurately predict the pump performance before it is tested. The pattern cost, manufacturing of prototype and testing are quite expensive and it may take many trials to obtain satisfactory or desirable results. To overcome this problem, Computational Fluid Dynamics (CFD) analyses are being increasingly used as an alternate tool in the design of centrifugal pumps. With increasing computer capability and rapid development of computational fluid dynamics, numerical simulation of the three-dimensional turbulent flow through impellers or even pump stage is becoming possible, and the complex internal flows in the impeller and pump stage are being well predicted to speed up the pump design procedure. Thus CFD is an important and useful tool for pump designers. Of a centrifugal pump stage, the impeller is the most important part. It rotates the liquid mass with the peripheral speed of its vane tips, thereby determining the head produced or the pump working pressure. However, the internal flow of the pump impellers is usually very complex and characterized by diffusion and strong swirl. Chapter Introduction Adverse pressure gradient and secondary flow always occur under off-design operating conditions, sometimes even under the design point. Therefore, it is necessary to investigate the internal flow of pump impellers by using some CFD approaches. It is also well known that the flows in centrifugal pump are three-dimensional turbulent flows, and a clear understanding of turbulent flow structures is essential for the optimization of the performance of centrifugal pumps. Recently as the rapid development of the technologies, more and more computational fluid dynamics (CFD) approach and turbulent models have been proposed and used frequently as a tool to simulate the turbulent flows numerically. However, it is found that most of researchers focused on the use of the standard k − ε model to simulate water flow through pump impellers; few of them took effort to apply different turbulence models on their simulation work and made comparison among these turbulence models. Therefore, studying the influences of different turbulence models to the pump performance is also an important aim of the present investigation. On the other hand, the pumped liquid always contains undissolved gas inside the pump impellers in a wide range of pump applications. Presently, the accurate prediction of the performance drop caused by two-phase flow is still a problem. Therefore, the knowledge of the performance of centrifugal pumps under gas/liquid two-phase flow conditions is of increasing interest in a wide range of industrial applications, especially in the chemical industries, the offshore oil production, and in relation to safety analyses in nuclear reactors. It is well known that the performance of standard centrifugal pumps decreases rapidly in the existence of entrained air. The deterioration begins as the gas fraction is Chapter Introduction only about 2-3% of the total volumetric flow and the total breakdown of the flow may occur when the gas fraction reaches about 8-10% of the total volumetric flow rate. For two-phase flow applications of centrifugal pumps, it is crucial to predict the influence of the gas fraction on the pump performance. However, this has not been fully done yet. Therefore, the accurate prediction of the flow characteristics in centrifugal pumps operating under gas-liquid two-phase flow conditions is imperative. 1.2 Literature Review Many researchers have been invo + ∫ ρu j nˆ jφd A − ∫ µ eff nˆ j d A = ∫ A A ∂t V ∂x j where nˆ j is surface outward normal vector and A and V are outer surface area and volume respectively. The first step in solving these equations numerically is to approximate them using discrete function. For example, the advection term can be approximated by, ∫ A ρu j nˆj φd A = ∑ mipφ ip ☎ ip 27 Chapter Theoretical Models of Single-Phase Flow where mip = ρu j nˆj ∆A is the discrete mass flow through a surface of the finite ✆ volume, ∆A is the surface area, φ ip is the discrete value of φ at the integration point, and the sum is over all the surfaces of the finite volume. To complete the discretisation of the advection term, the variable φ ip must be related to the dependent variables stored at the nodes of the element, φ n . As transported variables move with the flow, it is physically reasonable to approximate the variable at ip by the upstream nodal variable at n, to give φ ip = φ n . This is called the Upwind Difference Scheme (UDS) and it is first-order accurate. 2.5 Boundary Conditions The boundary conditions of single-phase flow are specified as follows: Inlet Boundary. A constant mass flow rate is specified at the inlet of calculation domain. Different mass flow rates are specified to study pump design and off-design conditions. The turbulent kinetic energy k, diffusion rate ε and the specific dissipation rate ω in the inlet of calculation domain are defined as: k in = 0.005u in2 µ ε in = C (k / Lm ) ω in = (1 → 10) u in Lm 28 Chapter Theoretical Models of Single-Phase Flow where u in is inlet velocity, C µ is an experimental constant and is 0.09 and Lm is the Prandtl’s mixing length scale and is assumed to be 0.5 inlet hydraulic diameters. (Wang et al., 2002). Outlet Boundary. In the outlet of calculation domain, the gradients of the velocity components and k, ε , ω are assumed to be zero respectively by following the suggestion of Keimasi and Taeibi (2001). ∂u j ∂n = (j=1,2,3) ∂k ∂ε ∂ω = = =0 ∂n ∂n ∂n Solid Walls. For surface of blade, hub and casing, relative velocity components are set as zero. Also the standard wall function is used near the walls when the standard k − ε model is implemented. For the k − ω and SST models, the turbulence kinetic energy k was set to zero at the walls and ω is obtained by the following relation suggested by Menter (1994): ω = 60υ / β y 2p at y=0 (2.17) where y p is the distance to the next point away from the wall, υ is the molecular kinematic viscosity, and β1 is a constant with the value of 0.075. Periodic Condition. In the pairs of the boundaries of the computational domain before and after impeller, the periodic conditions are applied: ϕ |left = ϕ | right where ϕ = u , v, w, k , ε , ω , p 29 [...]... specified at the inlet of calculation domain Different mass flow rates are specified to study pump design and off-design conditions The turbulent kinetic energy k, diffusion rate ε and the specific dissipation rate ω in the inlet of calculation domain are defined as: 2 k in = 0.005u in 3 4 µ 3 2 ε in = C (k / Lm ) ω in = (1 → 10) u in Lm 28 Chapter 2 Theoretical Models of Single-Phase Flow where u in is...Chapter 2 Theoretical Models of Single-Phase Flow ˆ where mip = ρu j n j ∆A is the discrete mass flow through a surface of the finite ¥ volume, ∆A is the surface area, φ ip is the discrete value of φ at the integration point, and the sum is over all the surfaces of the finite volume To complete the discretisation of the advection term, the variable φ ip must be... and is assumed to be 0.5 inlet hydraulic diameters (Wang et al., 20 02) Outlet Boundary In the outlet of calculation domain, the gradients of the velocity components and k, ε , ω are assumed to be zero respectively by following the suggestion of Keimasi and Taeibi (20 01) ∂u j ∂n = 0 (j=1 ,2, 3) ∂k ∂ε ∂ω = = =0 ∂n ∂n ∂n Solid Walls For surface of blade, hub and casing, relative velocity components are set... nodes of the element, φ n As transported variables move with the flow, it is physically reasonable to approximate the variable at ip by the upstream nodal variable at n, to give φ ip = φ n This is called the Upwind Difference Scheme (UDS) and it is first-order accurate 2. 5 Boundary Conditions The boundary conditions of single-phase flow are specified as follows: Inlet Boundary A constant mass flow. .. (1994): ω = 60υ / β 1 y 2 p at y=0 (2. 17) where y p is the distance to the next point away from the wall, υ is the molecular kinematic viscosity, and β1 is a constant with the value of 0.075 Periodic Condition In the pairs of the boundaries of the computational domain before and after impeller, the periodic conditions are applied: ϕ |left = ϕ | right where ϕ = u , v, w, k , ε , ω , p 29 . portion of the cross-diffusion term of Eq. (2. 16): ) 10 , 1 2 max( 20 2 − ∇ ∇ = ω ω ρσ ω ω k CD k The values of the constants k σ , ω σ , 2 ω σ , * β , β , γ in Eqs. (2. 15) and (2. 16). 41 .0 = κ , * 2 1 * 1 1 / / β κ σ β β γ ω − = The constants of set 2 ( 2 φ ) are (standard ε − k ): 0 . 1 2 = k σ , 856 . 0 2 = ω σ , 0 828 . 0 2 = β Chapter 2 Theoretical Models of Single-Phase. ] ) [( ) ( * k k P Uk t k k ∇ + • ∇ + − = • ∇ µ σ µ ρω β ρ (2. 15) and ω ω σ ρ ω µ σ µ βρω υ γ ω ρ ω ω ∇ ∇ − + ∇ + • ∇ + − = • ∇ k F P U t k t 1 ) 1 ( 2 ] ) [( ) ( 2 1 2 (2. 16) Chapter 2 Theoretical Models of Single-Phase Flow 24 where