Document 6527942
Transcription
Document 6527942
V Conferência Nacional de Mecânica dos Fluidos, Termodinâmica e Energia MEFTE 2014, 11–12 Setembro 2014, Porto, Portugal © APMTAC, 2014 Development length in channel flows of inelastic fluids described by the Sisko viscosity model LL Ferrás1, C Fernandes1, O Pozo1, AM Afonso2, MA Alves2, JM Nóbrega1, FT Pinho3 1 Instituto de Polímeros e Compósitos/I3N, Universidade do Minho, Campus de Azurém 4800-058 Guimarães, Portugal [email protected], [email protected], [email protected], [email protected] 2 Departamento de Engenharia Química, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465, Porto, Portugal [email protected], [email protected] 3 Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465, Porto, Portugal [email protected] ABSTRACT: This work presents a numerical study regarding the dimensionless development length, =L/H required for fully-developed channel flow of inelastic non-Newtonian fluids described by the generalized Newtonian model with the Sisko viscosity equation. The simulations were carried out for generalized Reynolds numbers in the range 0 Re gen 100 , for a flow behaviour index, n , in the range 0.25 n 2 and for an infinite dimensionless shear rate viscosity, , varying in the range 0 5 . A new non-linear relationship between and Re was derived and for specific values of n, new exact solutions are also presented for the velocity profile under fully-developed flow conditions. KEY-WORDS: Development length, Sisko model, channel flow. 1 INTRODUCTION The length required to achieve a fully developed flow in pipes and channels has been for a long time a subject of interest (see [1] and [2] and the references therein). Durst et al. [1] derived a correlation for the development length of Newtonian fluids in pipe and channel flows of Newtonian fluids, and Poole and Ridley [2], extended the work of Durst et al. [1] to the pipe flow of inelastic fluids described by the power-law model. Common to these works is that the development length is determined on the basis of the velocity profile, specifically when the velocity at the center of the channel reaches 99% of the fullydeveloped value. In this work we devise a correlation for the development length of fluids described by the Sisko model [3] in planar channels. 2 GOVERNING EQUATIONS The governing equations for steady, incompressible, laminar and isothermal flows are the conservation of mass equation u 0 (1) and the momentum equation, u uu p τ , t (2) where u is the velocity vector, is the fluid density, p is the pressure and τ is the extra stress tensor, that is given by, τ u u T 2 D (3) where D is the symmetric rate of strain tensor and is the shear viscosity model that follows the Sisko model, K n 1 (4) where K is the flow consistency index, is the infinite shear rate viscosity, n is the flow behaviour index and 2 tr D2 ( tr is the trace of a tensor) is the shear rate. 3 NUMERICAL PROCEDURE The exact velocity profiles for the fully developed flow, required to identify the fully developed conditions, can only be determined analytically for specific values of the exponent, n= 1/3, 1/2, 1 and 2, but, the exact shear rate profile can be obtained for a wider range of flow index, namely n=1/4, 1/3, 1/2, 2/3, 3/4, 1, 3/2. When an analytical solution is not possible for fully-developed flow, the shear rate value was used to compute numerically the pressure gradient and the velocity at the centre of the channel, through a numerical method. For the numerical solution of the system of governing equations ((1)-(4)) we used the Finite Volume Method together with the SIMPLE procedure of Patankar [4] to couple the velocity and pressure fields [5]. The numerical computations were performed with the opensource OpenFOAM software [6]. 4 RESULTS AND DISCUSSION The development length was defined as the axial distance required for the velocity to reach 99% of the calculated maximum value. The Reynolds number adopted is the generalized version usually employed for power-law fluids, cf. [7], n 6 U 2 n H n 4n 2 (4) Re gen , K n and the dimensionless infinite shear rate viscosity, , is defined as /( UH ) , where U is the flow average velocity and H is the channel width. 1 semi-analytical solution numerical results 0.5 0 2 2.5 1.5 u/U 1 0.1 0.2 0.3 0.4 0.5 0 0 0.1 0.2 y/H 0.3 0.4 2.5 u/U 1.3 y=0.99u analytical numerical results 6 x/H 8 10 0.3 0.4 0.5 5 1.4 1.3 1.2 1.2 1.2 4 0.2 1.5 1.4 u/U 1.4 2 0.1 y/H 1.5 1 0 0 0.5 y/H 1.5 1.3 1 0.5 0.5 0 0 5 1.5 u/U u/U 2 1 u/U 2 1.5 0 2 4 6 x/H 8 10 0 2 4 6 8 10 x/H Figure 1: (Top) Comparison between the semi-analytical and the numerical solutions for three different values of . (Bottom) Centreline velocity profile along the channel length. In Fig. 1 (Top) we show a good agreement between the numerical and the analytical results, for the fully-developed velocity profile. Fig. 1 (Bottom) presents the Centreline velocity profile along the channel length for three different values of , with Re gen 1 and n 0.5 . For 1 , 2.5 and 5 we obtained a development length, =L/H, of 0.835,0.746 and 0.747 , respectively. This shows that the development length decreases with increase of , which was expected due to the growth of momentum diffusion along the channel thickness promoted by the increase of . REFERENCES [1] F Durst, S Ray, B Unsal, and OA Bayoumi, (2005). The Development Lengths of Laminar Pipe and Channel Flows, ASME J. Fluids Eng., 127, 1154-1160. [2] RJ Poole, and BS Ridley, (2007). Development Length Requirements for Fully-Developed Laminar Pipe Flow of Inelastic Non-Newtonian Liquids, ASME J. Fluids Eng., 129, 1281–1287. [3] A.W. Sisko, (1958). The flow of lubricating greases. Ind. Engng Chem. 50, 1789–1792. [4] S.V. Patankar, (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington D.C.. [5] PJ Oliveira, FT Pinho, and GA Pinto, (1998). Numerical simulation of non-linear elastic flows with a general collocated finite-volume method, J. Non-Newtonian Fluid Mech., 79, 1–43 [6] The OpenFOAM Foundation, www.openfoam.org [7] P. Ternik, (2009) . Planar sudden symmetric expansion flows and bifurcation phenomena of purely viscous shear thinning fluids, J. Non-Newtonian Fluid Mech. 157, 15–25.