V Conferência Nacional de Mecânica dos Fluidos, Termodinâmica e Energia

MEFTE 2014, 11–12 Setembro 2014, Porto, Portugal

APMTAC, 2014

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

1 40Re , dimensionless rotational velocities 0 3 , Prandtl numbers Pr = 1 and 100, and

Brinkman numbers 0 1Br . 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

local hot (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 1Br 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. 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

thermal boundary condition at surface of the cylinder is assumed to be isothermal with temperature wT .

The effect of temperature variation on thermophysical fluid properties (density , specific heat at constant

pressure 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:

Effects of viscous heating on the heat transfer between a rotating cylinder and the surrounding fluid medium

AA Soares1,3,4, L Caramelo1,4, Abel Rouboa2,3

1Department of Physics/ECT, University of Trás-os-Montes e Alto Douro, Quinta de Prados, 5000-801 Vila Real,

Portugal 2Department of Engineering/ECT, University of Trás-os-Montes e Alto Douro, Quinta de Prados, 5000-801 Vila

Real, Portugal 3INEGI, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal 4CITAB, University of Trás-os-Montes e Alto Douro, Quinta de Prados, 5000-801 Vila Real, Portugal

email: asoares@utad.pt, lcaramel@utad.pt, rouboa@utad.pt elsac@fe.up.pt

MEFTE 2014, 11–12 Set 2014, Porto, Portugal 131

– Continuity equation

0u v

x y

(1)

– x-component of momentum equation

2 2

2 2

1uu uv p u u

x y x Re x y

(2)

– y-component of momentum equation

2 2

2 2

1uv vv p v v

x y y Re x y

(3)

– energy equation

2 2

2 2

1T T T T Bru v

x y RePr RePrx y

(4a)

where

2 22

2 2u v v u

x y x y

(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)

U d

Re

(5)

– Prandtl number (Pr)

pc

Prk

(6)

– Brinkman number (Br) for the constant temperature boundary condition

2

( )w

UBr

k T T

. (7)

Furthermore, the governing equations (1)–(4) have been rendered dimensionless using the following

scaling variables: U for the velocities, 2U

for the pressures and d for the lengths. The temperature is

made dimensionless by using wT T .

The physically realistic boundary conditions in dimensionless form for the flow across a rotating

circular cylinder are written as follows:

– at the inlet boundary

1, 0 and 0u v T (8a)

– on the surface of the cylinder

sin( ), cos( )u v (8b)

where the dimensionless rotational velocity , for the angular velocity of the cylinder with diameter

d, is

132 MEFTE 2014, 11–12 Set 2014, Porto, Portugal

2

d

U

. (8c)

Thermal condition on the cylinder surface is given by

= 1T (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,

0, 0 and 0u v T

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

the drag coefficients, local and averaged Nusselt numbers for the thermal boundary condition

investigated.

In this present work, the Navier-Stokes equations have been solved using Fluent. The O-type mesh,

with structured 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, Br, and Pr. The semi-implicit method for the

pressure linked equations (SIMPLE) scheme was used for solving the pressure-velocity coupling. 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 RESULTS

3.1 Isothermal patterns

Representative plots showing the dependence of the isothermal contours, 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 of Br results in a

separation between the isothermal contours as a consequence of the increases of the temperature around

rotating cylinder. For fixed values of Re and Br numbers an opposite behaviour is observed with

increasing Prandtl number.

a) b)

Figure 1: Isothermal contours for α = 3, Re =1, Pr = 1.a) Br = 0, b) Br = 1.

MEFTE 2014, 11–12 Set 2014, Porto, Portugal 133

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 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.

134 MEFTE 2014, 11–12 Set 2014, Porto, Portugal

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 = 0, 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 for higher values of Re and lower values of Pr.

An increase in 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 5 and 6. 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.

MEFTE 2014, 11–12 Set 2014, Porto, Portugal 135

a) b)

Figure 7: 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 value of the Brinkman number lowers the values of local and average Nusselt numbers.

Brinkman and Prandtl numbers 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.

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136 MEFTE 2014, 11–12 Set 2014, Porto, Portugal