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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.
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