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V Conferência Nacional de Mecânica dos Fluidos, Termodinâmica e Energia MEFTE 2014, 11–12 Setembro 2014, Porto, Portugal © APMTAC, 2014 Effects of viscous heating on the heat transfer between a rotating cylinder and the surrounding fluid medium AA Soares1,3, L Caramelo1,4, Abel Rouboa2,3 1 Department of Physics/ECT, Universidade de Trás-os-Montes e Alto Douro, Quinta de Prados, 5000-801 Vila Real, Portugal. 2 Department of Engineering/ECT, Universidade de Trás-os-Montes e Alto Douro, Quinta de Prados, 5000-801 Vila Real, Portugal. 3 INEGI/Faculty of Engineering, University of Porto, Porto, Portugal. 4 CITAB, University of Trás-os-Montes e Alto Douro, Quinta de Prados, 5000-801 Vila Real, Portugal. email: [email protected], [email protected], [email protected] ABSTRACT: This paper examines numerically the effects of viscous heating on the forced convection heat transfer between an unconfined rotating cylinder and the surrounding fluid medium. The forced convection heat transfer across an isothermal rotating cylinder is investigated for Reynolds numbers Re = 1 and 40, dimensionless rotational velocity 0 3 , Prandtl numbers Pr = 1 and 100, and Brinkman numbers 0 Br 1 .The isotherm patterns are presented for the varying values of the Brinkman number, Prandtl number and rotational velocity rate in the steady regime. The variation of the local and average Nusselt numbers with Reynolds number, Brinkman number, Prandtl number and rotational velocity rate are also presented for the range of conditions studied. KEY-WORDS: viscous heating, rotating cylinder, Prandtl number, Nusselt number, Brinkman number 1 INTRODUCTION In most practical situations, viscous heating (dissipation) always contribute, how so ever small, to the overall rate of heat transfer between rotating cylinders and the surrounding fluid medium. This contribution progressively increases with the increasing velocity of the imposed flow, i. e., with the increasing of the Reynolds number and/or the cylinder rotation rate. Hence, the temperature gradients induced by the strain rates are influenced by the Reynolds number and the cylinder rotation rate, which, in turn, influence the rate of heat transfer (or the Nusselt number). The viscous dissipation effects tend to be significant when either the viscosity is large or rate of shearing is large and/or when the fluid has a low thermal conductivity, which increases the temperature gradients. This phenomenon can also give rise to localhot (cold) spots which may be detrimental to the processing of temperature-sensitive materials. It has clearly been shown that in channel (pipe/duct) Brinkman numbers are of the order of 1 whereas in porous media flows they can be as large as 1000, e.g.[1-5]. Because the rotational velocity is not very high the problem of the flow past a rotating cylinder is closer to the channel flow situation, and choice of range 0 Br 1 is reasonable from a practical standpoint. The aims of the present study is to investigate numerically the heat transfer characteristics between the 2D laminar incompressible flow and a rotating circular cylinder taking into account the viscous heating, and then to further understand the corresponding underlying mechanism. For this study, we will concentrate on discussing the combined effects of viscous heating, rotational speed and Prandtl number on the heat transfer for a constant temperature imposed on the surface of the cylinder. Extensive results elucidating the effect of Brinkman, Reynolds and Prandtl numbers on the local and surface-averaged Nusselt numbers are presented and discussed herein. The results are found to be in a good agreement with numerical data for a rotating circular cylinder without viscous dissipation [6, 7]. 2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS Consider a uniform and incompressible flow of a Newtonian fluid past an unconfined rotating circular cylinder of diameter d. Cylinder is rotating in a counterclockwise direction with constant angular velocity . The approaching velocity of the fluid is U and the ambient temperature is assumed to be T . The surface temperature of the cylinder wall is Tw in the case of the isothermal boundary condition and the heat flux is q in the case of the isoflux boundary condition. The effect of temperature variation on thermophysical fluid properties (density , specific heat at constant temperature cp, thermal conductivity k, and viscosity ) is considered negligible. The flow is assumed to be steady, laminar and two dimensional. The two dimensional steady governing partial differential equations in their dimensionless form are written as follows: -Continuity equation u v 0 x y (1) -x-component of momentum equation uu x uv y p 1 2 u 2u x Re x 2 y 2 (2) p 1 2 v 2 v y Re x 2 y 2 (3) -y-component of momentum equation uv x vv y -energy equation u T T 1 2T 2T Br v x y RePr x 2 y 2 RePr (4a) where 2 v v u u 2 2 x y x y 2 2 (4b) is the viscous dissipation function for a incompressible flow. In physical terms, the relative measure of the heat produced due to viscous dissipation and that transferred by conduction is quantified by the Brinkman number (Br). The dimensionless groups appearing in equations (2) to (4) are defined as: -Reynolds number (Re) Re U d (5) -Prandtl number (Pr) Pr cp k (6) - Brinkman number (Br) for the constant temperature condition Br U 2 k (Tw T ) (7) Furthermore, the governing equations (1)-(4) have been rendered dimensionless using the following scaling variables: U for the velocities, U 2 pressure and d for the lengths. The temperature is made dimensionless by using Tw T . The physically realistic boundary conditions in dimensionless form for the flow across rotating circular cylinder are written as follows: at the inlet boundary u 1, v 0 and T 0 on the surface of the cylinder (8a) u sin( ), v cos( ) (8b) where the dimensionless rotational velocity , for the angular velocity of the cylinder with diameter d, is d 2 uw (8c) Thermal conditions on the cylinder surface are given by T=1 (8d) at the exit boundary The standard default option in Fluent known as “outflow” has been used at the exit boundary. Qualitatively, this option is similar to the homogeneous Neumann conditions, that is, u v T 0, 0 and 0 x x x (8e) The numerical solutions of equations (1)-(4) together with the above-noted boundary conditions yields the pressure, velocity and temperature fields which in turn can be processed further to obtain the values of individual and total drag coefficients, surface vorticity, local and averaged Nusselt number for the two thermal boundary conditions. In this present work, the Navier-Stokes equations have been solved using Fluent. The O-type mesh, with unstructured quadrilateral cells of non-uniform mesh spacing, has been used. The two-dimensional, steady, laminar, segregated solver module of Fluent was used to solve the incompressible viscous flow on the full computational domain for varying conditions of Re member, Br member, and Pr number. The semi-implicit method for the pressure linked equations (SIMPLE) scheme was used for solving the pressure-velocity decoupling. The second order upwind scheme has been used to discretize the convective terms in the governing equations. The Gauss-Siedel point-by-point iterative method in conjunction with the algebraic multi-grid (AMG) method solver was been used to solve the system of algebraic equations. Relative convergence criteria of 10–9 for the continuity and x – and y -components of the velocity were prescribed. In addition, the values of the drag coefficients were also monitored and only when these values had stabilized to four significant digits were these values finally accepted. 3 3.1 RESULTS Isothermal patterns Representative plots showing the dependence of the isothermal contours and in the vicinity of the rotating cylinder (α = 3) on the Reynolds number (Re), Brinkman number (Br), and Prandtl number (Pr) are presented in Figures 1-4.For fixed values of the Re and Pr numbers, an increasing value of the Br results in a remoteness from the cylinder of the isothermal contours as a consequence of the increases of the temperature around rotating cylinder. For fixed values of the Re and Br numbers an opposite behaviour is observed with increasing Prandtl number (Pr). a) b) Figure 1: Isothermal contours for α = 3, Re =1, Pr = 1.a) Br = 0, b) Br = 1. a) b) Figure 2: Isothermal contours for α = 3, Re =1, Pr = 100.a) Br = 0, b) Br = 1. a) b) Figure 3: Isothermal contours for α = 3, Re =40, Pr = 1.a) Br = 0, b) Br = 1. a) b) Figure 4: Isothermal contours for α = 3, Re =40, Pr = 100.a) Br = 0, b) Br = 1. 3.2 Local Nusselt number Figure 5 shows that the effect of Brinkman number Br on the local Nusselt number profiles over the cylinder surface for dimensionless rotational velocity α = 3. It was shown that the higher values of Br decreases the local Nusselt number for both conditions Re=1 and Pr = 1, and Re = 40 and Pr = 100. Despite the higher Prandtl number (Pr =100) the aforementioned behaviour is more pronounced for Re=1 and Pr = 1 than for Re = 40 and Pr = 100. The negative values of Nu can be attributed to the role of viscous dissipation, which is equivalent to that of an energy source, due to the internal heating effect of viscous dissipation on fluid temperature. Consequently, an increase in viscous dissipation changes the overall heat balance. As a result, for fixed values of α, Re and Pr, when Br exceeds a critical value, the heat generated internally by the viscous dissipation process will overcome the effect of the cylinder surface heating under favourable conditions, resulting in negative Nu. a) b) Figure 5: Local Nusselt number Nu for α = 3. a) Re = 1 and Pr =1, b) Re = 40 and Pr =100. 3.3 Average Nusselt number For a fixed value of Br and α, the magnitude of the average Nu number increases with increasing Re number for both Pr =1 and 100. This can be explained as when Re number increases the inertia of flow increases thereby increasing the heat transfer. The decrease in the average Nu number with Br is more pronounced at higher values of α, Re and Pr numbers. On increasing the value of the Br number, the average Nu number decreases for the fixed values of the Re α and Pr numbers. The negative values of Nu indicate that heat transfer occurs from fluid to the cylinder, see Figure 1. The local and average Nusselt numbers increase with increasing values of Reynolds, Prandtl, and with the decreasing values of the Brinkman number and rotational velocity. a) b) Figure 6: Average Nusselt number Nu for Pr =1. a) α = 0 and b) α = 3. a) b) Figure 6: Average Nusselt number Nu for Pr = 100. a) α = 0 and b) α = 3. 4 CONCLUSIONS This paper presents the effect of viscous dissipation Br on heating on the forced convection heat transfer between an unconfined rotating cylinder and the surrounding fluid medium. Increasing values of the Brinkman number lower the value of local and average Nusselt numbers. Brinkman number and Prandtl number have opposite effects on the distribution of the isothermal around rotating cylinder and consequently on the local and average Nusselt numbers. Broadly, the rate of heat transfer increases with the increasing Reynolds and Prandtl numbers, and with the decreasing values of the Brinkman number and rotational velocity. The dependence is strongest on the Reynolds number; however it is possible to control the rate of heat transfer by imposing to the flow the appropriate combinations of the rotational velocity, Reynolds number, Prandtl number , and Brinkman number in a given application. REFERENCES [1] PM Coelho, FT Pinho (2009). A generalized Brinkman number for non-Newtonian duct flows. J. NonNewtonian Fluid Mech. 156, 202–206. [2] RP Chhabra, AA Soares, FM Ferreira, L Caramelo (2007). Effects of viscous dissipation on heat transfer between an array of long circular cylinders and power law fluids. Canadian Journal of Chemical Engineering 85, 808-816. 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