# Thermal-Fluid Transport Phenomena between Twin Rotating Parallel Disks

Authors: Shuichi Torii; Wen-Jei Yang
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

This paper investigates thermal-fluid transport phenomena in laminar flow between twin rotating parallel disks from whose center a circular jet is impinged on the heated horizontal bottom disk surface. Emphasis is placed on the effects of the Reynolds number, rotation speed, and disk spacing on both the formations of velocity and thermal fields and the heat transfer rate along the heated wall surface. The governing equations are discretized by means of a finite-difference technique and are numerically solved to determine the distributions of velocity vector and fluid temperature under the appropriate boundary conditions. It is found from the study that (i) the recirculation zone which appears on the bottom disk moves along the outward direction with an increase in the Reynolds number, (ii) when the Reynolds number is increased, heat transfer performance is intensified over the whole disk surface and the minimum value of the heat transfer rate moves in the downstream direction, and (iii) the heat transfer rate is induced due to the disk rotation, whose effect becomes larger due to the upper disk rotation.


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## Body

## 1. Introduction

The impinging gas and liquid jets
are of general practical interest in many industrial fields, that is, in a wide
variety of industrial processes for cooling and heating.  Some industrial applications include the
thermal treatment of metals, cooling of internal combustion engines, and
thermal control of high-heat-dissipation electronic devices [1].  Both circular and planar liquid jets have
attracted research attention. Polat et al. [2] reviewed the numerical analysis
on laminar confined impinging jets of various configurations. In many practical
engineering applications, the working fluid is impinged by a coaxial round jet
for the purpose of controlling heat transfer
performance on two- and three-dimensional
channels (Ichimiya and Fukumoto [3];
Ichimiya and Yamada [4]).Meanwhile, laminar heat transfer in corotating and stationary
disk systems was studied theoretically and experimentally (Sim and Yang [5];
Mochizuki and Yang [6]; Mochizuki [7]; Prakash et al. [8]; Suryanarayana et
al., [9]). Both numerical and experimental results indicated a substantial
enhancement in heat transfer performance due to Coriolis, which is induced with
an increase in the Reynolds number and/or the rotational speed. Radial outward flow between two coaxial disks occurs in a number of situations
of engineering interest, including turbines, pumps, diffusions, rotating heat
exchangers, disk brakes, trust bearings, and so forth. Owen [10, 11] reviewed the literature pertinent to
fluid flow and heat transfer in rotating disk systems: the rotating disk
systems are classified into the free disk, rotor-stator system (which are used
to simulate turbine disks rotating near stationary casings) and rotating
cavities (which are used to simulate corotating turbine disks). Mochizuki and
Yang [6] investigated the momentum and heat transfer in radial flow through
parallel disks including the entrance effect. They obtained the heat transfer
and friction factors and at zero rotating speed and reported that the heat
transfer performance is comparable to those of high-performance compact
plate-fin surfaces of plain, louvered, and strip type, while the friction
factor is substantially higher than those of the compact surface. Heat transfer
and friction loss in multiple parallel disk assemblies were measured by Mochizuki
et al. [12] with the aid of a modified single-blow transient test method. They
obtained a schematic flow map to determine the variation of flow patterns along
the radial flow passage with the Reynolds number and disclosed three distinct
mechanisms of convective enhancement, that is, the laminar-flow, “second”
laminar-flow, and transition-turbulent flow enhancement. It was reported that
the heat transfer performance based on the overall behavior throughout the
entire flow channel exhibits characteristics of different convective mechanisms
dependent on the flow regimes.  Mochizuki
and Yang [13, 14] carried out theoretical and experimental study on radial flow
through two parallel disks with steady influx.
It was found that (i) for lower Reynolds number Re, the slug flow at the
inlet develops into a laminar profile toward the exit; (ii) as Re reaches a critical
value, Rec, vortexes separate alternately from both walls of the
radial channel and stretch a distance downstream.  Self-sustained flow oscillations are thus
induced which diminish downstream and a laminar profile is grossly amplified
and changed into a turbulent flow.
However, due to continuous enlargement of the flow cross-sectional area
in the radial direction, a reverse transition from turbulent to laminar flow
takes place.  However, to the
authors’ knowledge, there has been only a few information on thermal-fluid flow
transport phenomena through rotating disk with circular impinging jet.  To understand this transport phenomenon, it
is necessary to investigate the corresponding fluid flows because detailed
information on the heat and mass transfer is
of great importance to many engineering applications.The present study deals with
thermal-fluid transport phenomena in laminar flow between a rotating bottom
disk and a stationary or rotating upper disk from whose center a circular jet
is impinged on the axially rotating heated horizontal bottom disk surface.  Emphasis is placed on the effects of the
Reynolds number, rotation speed and disk spacing on both the formations of
velocity and thermal fields and the heat transfer rate along the heated wall
surface. A numerical method is employed to determine the velocity and
temperature profiles.


## 2. Governing Equations and Numerical Method

Consider thermal-fluid transport
phenomena in a pair of coaxial parallel disks ofDo in diameter in
which the bottom wall rotating at an angular frequency Ω is heated at constant wall temperature Tw and is impinged from the circular pipe of Di in diameter and the
stationary upper wall is insulated. Here, the spacing between the disks is
represented by h. The physical configuration of the coaxial disks with the
impinging jet and the coordinate system of the flow are shown in
Figure 1. The
following assumptions are imposed in the formulation of the problem based on
the characteristics of the flow: it is an incompressible laminar flow; fluid
properties are constant because Tw−Tinlet<5K; there is
uniform velocity and uniform inlet fluid temperature Tinlet at the
injection nozzle tip and negligible axial conduction (due to the high Peclet
number).  Then, the governing
differential equations for mass, momentum, and energy can be expressed in the
tensor formation as continuity equation:
(1)∂Ui∂xi=0, momentum equation:
(2)Uj∂Ui∂xj=−1ρ∂P∂xi+ν∂∂xj(∂Ui∂xj), energy equation:
(3)Ui∂T∂xi=a∂∂xi(∂T∂xi). An isothermal
laminar flow is assumed as the inlet condition. The boundary conditions in the
stationary and rotating disks are specified as
(4)U=V=0,W=Ww,T=Tw,atbottomwall,U=V=W=0,orW=−Ww,∂T∂z=0,atupperwall,U=Uin,V=0,W=0,T=Tinlet,attheimpingingnozzle,∂U∂r=0,V=W=0,∂T∂r=0,atcenter(r=0),∂U∂r=0,∂V∂r=0,∂W∂r=0,∂T∂r=0,atoutlet,thatis,r=Do/2.Figure 1
Schematic diagrams of
two-dimensional channel wall impinged by planar dual jets and their coordinates.A set of the governing equations
is solved using the control-volume-based formulation of Patankar [15]. In this
procedure, the domain is discretized by a series of control volumes, with each
control volume containing a grid point. Each differential equation is expressed
in an integral manner over the control volume, and profile approximations are
made in each coordinate direction, leading to a system of algebraic equations
that can be solved in an iterative manner. The semi-implicit method for pressure-linked
equations (SIMPLEs) algorithm is employed to couple the pressure and velocity
fields [15]. A staggered grid is considered such that the velocity components
are located at the control volume faces, whereas pressure and temperature are
located at the centers of control volumes to avoid the velocity-pressure
decoupling. A power law interpolation scheme is used to evaluate the values of
variables at the control volume interfaces. The discretized equations are
solved with a line-by-line and the TDMA (tridiagonal-matrix algorithm). The
convergence criteria of the residuals of all equations are asssumed to be less
than 10−5 of total inflow rates. Computation reveals only a slight
difference when the grid system is properly changed from 50×50×20(r,θ,z) to
100×100×40, resulting in a grid-independent solution.
Hence, a grid system of 50×50×20
with uniformly distributed nodal points is employed here to also save
computation time. Based on the dataset obtained here, visualization of the flow
and thermal fields is carried out using a commercially available 2D graphics
software tool.The numerical computation was performed on a personal
computer using air as the working fluid(Pr=0.71). The parameters used in the
present study are radial ratio Do/Di=4, dimensionless
disk height h/Di=0.05~0.1, Reynolds number Re=300~2000, and rotational Reynolds number Ret=0~3000. Note
that calculation is focused in the vicinity of the impinging jet region in
which the forced convection flow is dominant.Simulations with grids of
various degrees of coarseness, as mentioned earlier, were conducted to
determine the required resolution for grid-independent solutions.  Throughout the Reynolds number range
considered here, the maximum error was estimated to be about±2% by comparing the solutions on regular and
fine grids with twice of the grid points.


## 3. Results and Discussion

The effect of Reynolds number on the local Nusselt number along the
bottom disk in the absence of rotation is depicted in Figures2(a) and
2(b) for
h/Di=0.05
and 0.10, respectively. One observes
that (i) the local Nusselt number is increased over the whole heated surface
with an increase in the Reynolds number, as seen in
Figure 2(a), and (ii) the
same trend appears in the case of the wider disk, but heat transfer performance
is substantially lower than that in the narrow disk, as seen in
Figure 2(b). These characteristics are in accordance with that
reported by Ichimiya and  Fukumoto [3]. The streamwise location of
the minimum heat transfer coefficient moves in the downstream direction, as the
Reynolds number is increased. This is caused due to the presence of the
recirculation zone, as shown in the following figures. In other words, this
zone moves along the outward direction with an
increase in the Reynolds number. This
behavior becomes clearer for the velocity and temperature fields. Figure
3, for
h/Di=0.05, illustrates the velocity vector and isotherms in the
r−z cross-section of the dick. The corresponding results for h/Di=0.10
are depicted in Figure 4. Here, Θ=1 and 0 in
Figures 3 and
4 correspond to the
heated wall temperature and the fluid temperature at the injection nozzle tip,
respectively. Notice that the velocity components are normalized by dividing it
by the fluid velocity at the injection nozzle tip. It is observed that a
recirculation zone appears near the outlet of the injection nozzle and is
extended with the increase of the channel width. Thermal boundary layer is
developed along the heated wall and the temperature gradient at each axial
location is intensified with a decrease in the channel width. This implies
enhancement of heat transfer performance, as seen in Figure
2(a).Local Nusselt Number on the Disk Surface, (a)h/Di=0.05 and (b) h/Di=0.10.
(a)(b)Velocity and
temperature distributions in stationary disk ath/Di=0.05, (a)
Re=360, (b) Re=2000.
(a)(b)Velocity and
temperature distributions in stationary disk ath/Di=0.10, (a)
Re=360, (b) Re=2000.
(a)(b)Figure5, for Re=2000, illustrates
local Nusselt number Nu on the rotating disk surface with the rotating Reynolds
number Ret, as the parameter. (a) and (b) of Figure 5 correspond to
numerical results at h/Di=0.05 and 0.10. Heat transfer
performance for h/Di=0.05 becomes larger than that for h/Di=0.10.
One observes that the Nusselt number is gradually diminished along the flow. For the narrow channel, there is no effect of disk
rotation on heat transfer rate on its plate. On the contrary, the heat transfer
rate is intensified by the disk rotation for the channel with the wide
distance, as seen in Figure 5(b). The corresponding numerical results
in the
r−z cross-section of the disk are illustrated in Figures
6 and
7 at h/Di=0.05
and 0.10, respectively. (a) and (b) in
each figure correspond to the results of the velocity and thermal distributions,
respectively.  It is observed that the
velocity and thermal fields in rotating disk, for h/Di=0.05, are the
same as that in the stationary disk, while the corresponding fields for h/Di=0.10
are affected by the disk rotation, and the recirculation zone appears in the
outlet of the rotating disk, resulting in a substantial deformation of the
velocity and thermal fields.  The
presence of the recirculation zone induces the temperature gradient near the
heated wall at the outlet of the disk, resulting in an increase of heat transfer
performance, as seen in Figure 5(b). Thus, the effect of disk rotation on heat
transfer performance is found to occur in the wider disk spacing. This trend
became larger in the lower Reynolds number region (not shown).Local Nusselt
Number on the Rotating Disk Surface forRe=2000, (a) h/Di=0.05 and (b) h/Di=0.10.
(a)(b)Velocity and
temperature distributions in rotating disk ath/Di=0.05, Re=2000 and
Ret=2000.
(a)
velocity distribution(b)
temperature distributionVelocity and
temperature distributions in rotating disk ath/Di=0.10, Re=2000 and
Ret=2000.
(a)
velocity distribution(b)
temperature distributionNext task is to study the
effect of upper wall rotation on heat transfer rate and thermal and velocity
fields. The local Nusselt number forh/Di=0.1 and Re=360 is depicted
in Figure 8 in the same form as Figure
5. One observes that when the upper and
lower walls are simultaneously rotated in the opposite direction, the Nusselt
number is intensified along the wall, particularly near the impinging region.
The drastic enhancement of heat transfer performance becomes clearer in
velocity and thermal fields.  The corresponding
numerical results in the r−z cross-section of the disk are illustrated in Figures
9(a) and 9(b) in each
figure correspond to the results of the velocity and
thermal distributions, respectively. It is observed that the velocity in the vicinity
of in rotating disk walls is increased along the exit due to the opposite flow
in the center region of the parallel disk, resulting in an amplification of
temperature gradient near the wall.Figure 8
Local Nusselt Number on the rotating disk
surface forRe=360 and h/Di=0.1.Velocity and temperature distributions in
rotating disks forh/Di=0.10 and Re=360.
(a)(b)


## 4. Summary

Numerical simulation has been
employed to investigate thermal-fluid transport phenomena in laminar flow
between a rotating bottom disk and a stationary or rotating upper disk, from
whose center a circular jet is impinged on the axially rotating heated
horizontal bottom disk surface. Consideration is given to the effects of the
Reynolds number, rotation speed, and disk spacing on both the formations of
velocity and thermal fields and the heat transfer rate along the heated bottom
surface. The results are summarized as follows.(1) The
recirculation zone appears on the stationary disk. This zone moves along the outward direction with an increase in the Reynolds
number.(2) The heat transfer rate
approaches the minimum rate due to the presence of the recirculation zone. When
the Reynolds number is increased, heat transfer performance is intensified over
the whole disk surface and the streamwise location of the minimum heat transfer
rate moves in the downstream direction. This trend yields for the channels with
different width.(3) The heat transfer rate is
induced due to the disk rotation, whose effect becomes larger in the wider disk
spacing.(4) Further enhancement heat
transfer causes when the upper and lower walls are simultaneously rotated in
the opposite direction.



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*Source: 406809-2008-07-14.xml*