1 GEOMETRIC OPTIMISATION OF CONJUGATE COOLING CHANNELS WITH DIFFERENT CROSS-SECTIONAL SHAPES Department of Mechanical and Aeronautical Engineering, University of Pretoria, South Africa By Olabode Thomas OLAKOYEJO Prof. T. Bello – Ochende, Prof. J. P. Meyer 13 : 02 : 2013
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1
GEOMETRIC OPTIMISATION
OF CONJUGATE COOLING
CHANNELS WITH DIFFERENT
CROSS-SECTIONAL SHAPES
Department of Mechanical and Aeronautical Engineering,
University of Pretoria, South Africa
By
Olabode Thomas OLAKOYEJO
Prof. T. Bello – Ochende, Prof. J. P. Meyer
13 : 02 : 2013
Outline
• Introduction
• Background
• Motivation/Application
• Aims/Objective
• Objective functions
• Methodology
• Work done
• Results/Graphs
• Conclusions
• Future work
• Heat generating devices, such as high power electronic equipment and heat exchangers are
widely applicable in engineering fields e.g electronic chip cooling, power and energy sectors.
• Heat generation can cause overheating problems and thermal stresses and may leads to
system failure.
• Cooling of heat generating device critical challenge to thermal design engineers and
researchers.
• Heat generating devices are designed in such a way as to optimise the structural geometry by
packing and arranging array of cooling channels into given and available volume constraint
without exceeding the allowable temperature limit specified by the manufacturers.
• This translates into the maximisation of heat transfer density or the minimisation of overall
global thermal resistance, which is a measure of the thermal performance of the cooling
devices.
Heat generating devices and Thermal management
Introduction
H
L
W
sq
s
d
Global Volume V elElemental Volume v
Flow
Optimsation
parameters
Heat transfer
Performance
Characteristic
length scale
Conductive
heat transfer
Convective
heat transfer
H
L
W
sq
s
d
Global Volume V elElemental Volume v
Flow
hNu
k
Geometry
Shape
, , , hw h L d
Fig 1. The Nusselt Number (Nu) , a measure of heat transfer performance
Modern Heat Transfer: Geometry and Shape Optimisation
Introduction
Channel geometric design affects the thermal performance of Heat transfer
Geometry optimization of various shapes and sizes
Backgrounds: Constructal Theory and Design
• Bejan and Sciubba (1992),considered the optimization spacing of board to board of an array of parallel plate that can be fitted in a fixed volume in an electronic cooling system
• Muzychka (2005), analytical optimisation the geometry of circular and non-circular cooling channels.
• Ordonez (2004), Numerically, conducted a two-dimensional heat transfer analysis in a heat-generated volume with cylindrical cooling channels and air as the working fluid.
H
L
W
sq
s
d
Global Volume V elElemental Volume v
Flow
2d
HWn
4/1683.4 BeL
dopt
2/1
3
2'''* BeC
TTk
LQQ
is
Fig 2. Convectively conducting volume with cooling channels
2sd
HWn
Method of intersection of asymptotes
When the channel cross-sectional area is at optimum
inT
P
0h
d
When the channel cross-sectional area is large, D ∞
inT
P
When the channel characteristic dimension scale is small
and sufficiently slender, D 0, D << L
inT
P
Fig 3. Method of intersection of asymptotes
hd
0hd
h optd
H
L
W
sq
s
d
Global Volume V elElemental Volume v
Flow
hd
R
2 0h hR d d
2/3
h hR d d
hR d
opthd
Motivations/Applications
• The advent of high density components has required investigation of innovative techniques for removing heat from these devices
• Better and optimal performance
• Cost minimization
Applications
• Electronic cooling
• Compact heat exchanger,
• Automotive
• Nuclear power
Aims/Objectives
• Aim : To carry out theoretical and numerical optimization studies in conjugate
heat transfer in cooling channels with different cross-sections and under
varoius conditions
• Objectives : To minimise the dimensionless maximal excess of temperature or
global thermal resistance
The objectives will be conducted in two phases:
• Analytical (Theory) Analysis
• Numerical Analysis
• Optimisation process by suitable mathematical algorithm
Part 1 : Optimisation of Conjugate Heat Transfer In Cooling Channels with
Internal Heat Generation for Different Cross-sectional Shapes
Part 2 : Optimisation of Laminar-forced Convection Heat Transfer
Through a Vascularised Solid with Cooling Channels
Part 3 : Effect of flow orientation on forced convective heat transfer in
cooling channel with internal heat generation
Research activities/work done
PART 1
Optimisation of Conjugate Heat Transfer In Cooling Channels with Internal
Heat Generation for Different Cross-sectional Shapes
Numerical Modelling: Problem under consideration
H
L
W
sq
s
d
Global Volume V elElemental Volume v
Flow
L
H
W
sq
s Global Volume V
elElemental Volume v
Flow
cw
L
Global Volume V
elElemental Volume v
W
H
sq
flow
Fluid
PinT
L
Global Volume V
elElemental Volume v
flow
Fluid
PinT
q
W
H
Fig 4. Three-dimensional parallel channels with different cross section across a slab
with internal heat generation and forced flow.
Problem under consideration
flow
Fluid0
y
T
0
x
T
0
y
T
P
inT
0
z
T
hd
2
s
L
h
w
0T
z
flow
Fluid0
y
T
0
x
T
0
y
T
P
inT
0
z
T
0T
z
hd
cw2
s
L
h
w
hd
sq
L
w
h
flow
Fluid
P inT
0T
z
0T
z
Periodic
Periodic
Symmetry
cw
ch
2/ 2s
1/ 2s
0
y
T
Lw
ch
cw
h
0
x
T''', q
Solid
0
y
T
0
z
T
0
y
T
flowFluid
PinT
21s
22s
x
yz
Fig 5 The three dimensional computational domain Elemental volume with cooling
channels
Objective functions and Assumptions
The objective is the minimisation of the global thermal resistance
max min
min 2
ink T T
fR
q L
min
min max, ,
opt opth elR f d v T
• Fluid flow and heat transfer :
• steady-steady state condition
• three dimensional.
• single phase
• Laminar
• Newtonian fluid with constant properties (Water)
• Micro-scale cooling channels
Assumptions
( P1.1)
Numerical Modelling /Analysis/Optimisation
y
x
z
Fig 6: The discretised 3-D computational domain
Numerical Modelling/Analysis
0u
2u u P u
2C u T k Tf Pf f
2 0k Ts
Govering Equations
Energy equation for a solid region is given as:
( P1.2)
( P1.3)
( P1.4)
( P1.5)
Numerical Modelling/Analysis
– Unit cell using symmetry
– Internal heat generation
T T
k ks fn n
The continuity of the heat flux at the interface between the solid and the liquid
is given as
0u
A no-slip boundary condition is specified at the wall of the channel,
• Boundary Conditions
20, ,
Beu u T T P P
x y in in outL
At the inlet ( x = 0 )
1 P atmout
At the outlet ( x = L ), zero normal stress
At the solid boundaries
0T
( P1.6)
( P1.7)
( P1.8)
( P1.9)
( P1.10)
Numerical Modelling/Analysis
Fig 7 The boundary conditions of the three dimensional Computational domain of the
cooling channel
Summary of Boundary Conditions
flow
Fluid0
y
T
0
x
T
0
y
T
P
inT
0
z
T
0T
z
hd
cw2
s
L
h
w
hd
sq
L
w
h
flow
Fluid
P inT
0T
z
0T
z
Periodic
Periodic
Symmetry
cw
ch
2/ 2s
1/ 2s
flow
Fluid0
y
T
0
x
T
0
y
T
P
inT
0
z
T
hd
2
s
L
h
w
0T
z
0
y
T
Lw
ch
cw
h
0
x
T''', q
Solid
0
y
T
0
z
T
0
y
T
flowFluid
PinT
21s
22s
x
yz
The constraint ranges are:
Optimisation Constraints
0.1 0.2, 50 500 , 0 , 0 m w m d w s wh
2 , ,el hv w L w d s
An elemental volume constraint is considered to compose
of elemental cooling channel of hydraulic diameter
HWN
hw
The number of channels in the structure arrangement can be defined as:
c
el
v
v
The void fraction or porosity of the unit structure can be defined as:
flow
Fluid
L
0
y
T
w
0
x
T
0
y
T
PinT
Tk qs z
cw2
s
h
0T
z
x
y
z
ch
( P1.11)
( P1.12)
( P1.13)
( P1.14)
Numerical analysis/ Grid independent tests
The numerical solution of the continuity, momentum and energy
Equations alongside with boundary conditions was obtained by
using a three dimensional commercial package FLUENT™
that employs a finite volume method.
The solution is said to be converged when
the normalized residual of the mass and
momentum equations falls below 10-6 and
that of the energy equation is less than 10-10.
Grid independent tests for several mesh
refinement were carried out to ensure
the accuracy of the numerical results.
The convergence criterion for the
overall thermal resistance as the
quantity monitored
1max max
max
0.01ii
i
T T
T
27
27.5
28
28.5
0 1
7500 Cells
68388 Cells
108750 Cells
Sta
tic T
em
pe
ratu
re (
0C
)
ZL
Fig 8 : Grid independent tests
( P1.15)
0.001
0.0019
0 0.05 0.1
Be= 108
Pr = 1
Ordonez [15]
Present study
hd
L
300s
f
k
k
maxT
27
28
29
30
31
0 1 2 3 4 5
P = 50kPa
Cyl (
Sqr (
Cyl (
Sqr (
Cyl (
Sqr (T
max (
0C
)
vel ( mm
3 )
Fig 9a. Thermal resistance curves : present study
and ordoenez
Fig 9b Effect of optimised elemental
volume on the peak temperature
Numerical Results Findings /Graphs
max minT
Porosity Increasing
max minT
CASE STUDY 1: Cylindrical and square cooling channel
embedded in high-conducting solid
Fig 10 Effect of optimised hydraulic diameter and spacing on the peak temperature
27
28
29
30
31
0 50 100 150 200 250 300 350
P = 50kPa Cyl (
Sqr (
Cyl (
Sqr (
Cyl (
Sqr (
Tm
ax (
0C
)
dh ( m )
27
28
29
30
31
0 100 200 300 400 500
P = 50kPa
Cyl (
Sqr (
Cyl (
Sqr (
Cyl (
Sqr (
Tm
ax (
0C
)
s ( m )
Numerical Results Findings /Graphs
max minT
Porosity Increasing max minTPorosity Increasing
CASE STUDY 1: Cylindrical and square cooling channel
embedded in high-conducting solid
Numerical Results Findings /Graphs
CASE STUDY 2: Truagular cooling channel embedded in high-
conducting solid
28
30
0 3 6
I - R Triangle (
Equi Triangle (
I - R Triangle (
Equi Triangle (
Tm
ax (
0C
)
vel ( mm
3 )
28.5
30
0 0.015 0.03
I-R Triangle (
Equi Triangle (
I-R Triangle (
Equi Triangle (
Tm
ax (
0C
)
dh
/ L
Numerical Results Findings /Graphs
max minT
Porosity Increasing Porosity Increasing
max minT
Fig. 11 Effect of optimised hydraulic diameter and elemental volume on the peak temperature
27.6
27.8
28
28.2
0 10 20
Tm
ax (
0C
)
ARc
27.6
27.8
28
28.2
0.012 0.018
Tm
ax (
0C
)
dh/L
Numerical Results Findings /Graphs
max minT max min
T
CASE STUDY 3: Rectangular cooling channel embedded in high-
conducting solid
Fig. 12. Effect of optimised aspect ratio and hydraulic diameter on the peak temperature
Mathematical Optimisation:
• Standard optimization problem
2
0.1 0.2c h
el
v d
v w
optoptopth sdfR ,,min
DYNAMIC-Q Algorithm (by Prof. Snyman)
• Constraints
Porosity
min ; , ,.... ... , , 1 2
T nf x x x x X xi n i
x
0, 1,2,....j
g x j p
0, 1,2,....k
h x k q
Subject to
hoptd 0
hd
0 s
To search for the:
opts
( P1.16)
( P1.17)
( P1.18)
( P1.19)
Mathematical Optimisation
• DYNAMIC-Q Algorithm Very robust
CFD simulation
converged?
Setting design variables
Importing geometry and mesh to
FLUENT
Post-processing:
data and results processing
Yes
No
FLUENT Journal file
3-D CFD simulation ( solving model)
FLUENT Journal file
CFD simulation ( solving model)
Defining the boundary conditions
Geometry & mesh generation
GAMBIT Journal file
Mathematical optimisation
( Dynamic-Q Algorithm )
Optimisation
solution converged?
Start
No
Yes
Initialise the optimisation by
specifying the initial guess of the
design variables xo
Stop
Predicted new optimum design
variables and objective function f(x)
1
2
1, 1,....
2
1, 1,....
2
Tl l l l lT
Tl l l l l lT
i i i i
Tl l l l l lT
j j j j
f x f x f x x x x x A x x
g x g x g x x x x x B x x i p
h x h x h x x x x x C x x j q
Gradient based method
• Penalty function technique
• Approximation of numerical functions
by spherical quadratic function
• Forward differencing for gradient approximations.
• Automation of the process
Fig. 13. flow chart of numerical simulation
( P1.20)
Numerical Results Findings /Graphs
0, 1,2,....j
g x j p
10-5
10-4
10-3
109
1010
1011
Cyl (
Sqr (
Cyl (
Sqr (
Rm
in
Be
10-4
10-3
109
1010
1011
I-T Triangle (
E-T Triangle (
I-T Triangle (
E-T Triangle (
Rm
in
Be
Cylindrical, square , triangular and rectangular cooling channel
embedded in high-conducting solid
10-5
10-4
10-3
109
1010
1011
Rect ( Rect (
Rm
in
Be
Fig. 14 : Effect of dimensionless pressure difference on the minimised dimensionless global