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NEW METHODS TO COPE WITH TEMPERATURE ELEVATIONS IN HEATED
SEGMENTS OF FLAT PLATES COOLED BY BOUNDARY LAYER FLOW
Mohammad R. HAJMOHAMMADI a,*, Mohammad MOULOD b, Omid JONEYDI
SHARIATZADEH c, Seyed Salman NOURAZAR a
a: Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran.
b: Department of Mechanical engineering, Boukan Branch, Islamic Azad University, Boukan, Iran.
c: Department of Energy Technology, Faculty of Technology, Lappeenranta University of
Technology, P.O. Box 20, FI-53851 Lappeenranta, Finland
* Corresponding author, Email: mh.hajmohammadi@yahoo.com
This paper documents two reliable methods to cope with the rising
temperature in an array of heated segments with a known overall heat load
and exposed to forced convective boundary layer flow. Minimization of the
hot spots (peak temperatures) in the array of heated segments constitutes
the primary goal that sets the platform to develop the methods. The two
proposed methods consist of: 1) Designing an array of unequal heaters so
that each heater has a different size and generates heat at different rates,
and 2) Distancing the unequal heaters from each other using an insulated
spacing. Multi-scale design based on constructal theory is applied to
estimate the optimal insulated spacing, heaters size and heat generation
rates, such that the minimum hot spots temperature is achieved when subject
to space constraint and fixed overall heat load. It is demonstrated that the
two methods can considerably reduce the hot spot temperatures and
consequently, both can be utilized with confidence in industry to achieve
optimized heat transfer.
Keyword: Constructal design, Thermal performance, Peak temperature,
optimization
1. Introduction
The important role played by convection heat transfer in industry has attracted multiple activities
for maximizing this heat transfer mode subject to global constraints. In the last few decades,
researchers have addressed a large number of problems possessing multi-scale components [1] and
have indicated that the effort for maximizing the heat transfer density leads to the design of systems
with the best possible configuration. This is indeed the essential concept of constructal theory [2]. It
stipulates that for a finite-size system to persist in time, it must evolve in such a way that it provides
easier access to the imposed currents that flow through it. In such activities, the main objective is to
endow the flow configuration with a certain freedom to change, such that it provides easier and
greater access to its currents [3]. One of the factors that lower the access of the heat sources to the
convective heat transfer current revolves around developing the thermal boundary layer along the wall
when the flow sweeps the heat generating surfaces. This produces descend in the convective heat
1
transfer coefficient in the flow direction. It has been demonstrated that this problem is more sensitive
in laminar flows where the heat transfer coefficient, h, decreases rapidly along the surface when
compared to that in turbulent flows or even with natural convection flows. For example, for a laminar
flow with forced convection heat transfer along a flat plate, the convection heat transfer coefficient h
decreases as
-1/2
x
while for natural convection it decreases as
-1/4
x
where x is the distance along the
flow direction over a plate measured from the leading edge. Many methods have been proposed in the
past to control the development of the boundary layer, for example imposing suction.
It is clear that the consequence of the described descend in the heat transfer coefficient is the rising
temperature of the heated segments in the flow direction, which obviously reduces the thermal
performance. To cope with the forgoing, one must manipulate the location, distribution or the
geometry of the heaters. For example, some researchers have reported the effect of non-uniform
placement of discrete heat sources on the thermal performance, experimentally [5-9]. Bhowmik et al.
[6], for example, performed steady-state experiments to study general convective heat transfer
patterns from an in-line four simulated electronic chips in a vertical rectangular channel using water
as the working fluid. Other researchers have analyzed the same problems numerically [10-19]. Wang
et al. [10], for example, performed a numerical simulation to investigate the laminar natural
convection air cooling of a vertical plate with five wall-attached protruding, discretely heated
integrated circuit (IC) packages. Steady, natural convection from a discrete flush-mounted rectangular
heat source on the bottom of a horizontal enclosure was studied numerically by Sezai and mohamad
[13]. They found that the rate of heat transfer is not so sensitive to the vertical wall boundary
conditions. Sudhakar et al. [15] reported the results of a numerical investigation of the problem of
finding the optimum configuration for five discrete heat sources, mounted on a wall of a three-
dimensional vertical duct under mixed convection heat transfer, using artificial neural networks
(ANN). Recently, an analytical analysis was carried out by Hajmohammadi et al. [20, 21] to reveal
that by placing an insulated spacing placed between two heat sources cooled by in-tube [20] or
external [21] laminar forced convection, the rising temperature of the heat sources in the flow
direction can be attenuated significantly. Using constructal theory, these authors also quantified the
optimal insulated spacing that minimizes the peak temperature to cope with the rising temperature of
the heat sources in the flow direction. Although Hajmohammadi et al. [21] have managed to lower the
peak temperature of heat sources with unequal length but equal rate of heat generation cooled by
laminar forced convection, the present paper proposes an alternative opportunity by designing an
array of unequal heaters so that each heater has a different size and generates heat at different rates.
The previous method is subsequently coupled with the method proposed in [21] to examine the
opportunity of designing unequal heaters (unequal length and unequal heat generation) with insulated
spacing to each other.
2. Physical problem and mathematical formulation
In the majority of thermal engineering systems, the amount of heat transferred from a hot body to a
cold fluid is known a priori. In the context of electronics cooling, the primary objective is to control
the wall (heaters) temperature, such that the highest temperature in the package (the hot spot) does not
exceed a specified allowable value. This issue is more sensible in laminar boundary layer flows over
heated segments on plates, where the temperature of the heaters surface rises in the flow direction, as
the convection heat transfer coefficient h decreases in that direction. According to the constructal
2
theory, in order to minimize the peak temperature, the entire plate must operate at the same
temperature [22, 23]. Based on the so-called strategy, one method might be to manipulate the uniform
heat flux distribution over the plate, q''(x), by designing the heat sources (heaters) with different sizes
(li) and different rates of heat generation (
i
q
). Applying this method, the heater with the highest rate
of heat generation is placed at the leading edge of the plate, where h is highest, while the other heat
sources are located at the successive downstream regions of the plate, where h diminishes
as
-1/2
~)( xxh
.
Next, consider n unequal heater segments of lengths l1, l2 ,…and ln mounted on a plate to be heated by
a forced convection boundary layer flow as observable in Fig. 1. Again, the flow is assumed laminar
and two-dimensional with free stream velocity U∞ and free stream temperature T∞. The heaters
generate uniform heat rate per unit length at unequal rates of
i
q
which results in a stepwise
distribution of heat flux, q''(x) as shown in Fig. 1. The foregoing description can be expressed as,
nim
q
q
mxxlxmqxq i
iiiii ...,,2,1;1,,; )( 1
1
1===<<−=
′′
(1)
where mi represents the ratio of the heat generation rate of the i-th heater to that of the first heater and
xi is the location of the hot spots as marked with ‘H’ in Fig. 1. Besides, it is assumed that under the
circumstances of different heat flux distributions, the total heat removal rate from the plate is fixed,
and can be expressed as,
.
~
1
Constqql m
n
iii ==
∑
=
(2)
where
m
q
is the mean value of heat generation rates,
i
q
. In the case of non-uniform heat flux, the
temperature of the plate can be simply obtained by the classical relation [4],
∫
=
=
−
−
−
∞
′
−=
−
x
s
d
q
x
RePr
k
T
x
T
ξ
ξ
ξξ
ξ
0
3
2
4
3
2
1
x
3
1
)(
1
623
.0
)
(
(3)
Fig. 1. Multi-scale stepwise distribution of heat flux which corresponds to the heated segments with
variable rates of heat generation.
3
The temperature of the hot spots cab be obtained by using Eq. (1) and substituting ‘x’ by the location
of hot spots. In non-dimensional equation form, this corresponds to,
∑
∑=
−
=
−
−
−= i
ki
k
i
k
ki
n
i
i
i
ix
x
I
x
x
Imx
lm
T
1
4
3
3
4
,
3
1
4
3
1
3
4
,
3
1
2
1
1
~
~
1
~
ˆ
1.
~
1
~
(4)
where Im,n(x) is regularized incomplete beta function defined by
duuu
duuu
xI n
m
n
xm
nm 1
1
0
1
1
0
1
,)1(
)1(
)( −
−
−
−
∫
∫
−
−
=
(5)
and the dimensionless variables are given by
∞
∞
−
−
== TT
TT
T
L
lsx
lsx
L
s
~
;
),,,(
)
~
,
~
,
~
,
~
(
ξ
ξ
(6)
Here, L denotes the total length of the heaters, namely the length of the plate when the heaters are
flush mounted one after the other onto the plate and no insulated spacing is placed between the
heaters.
3. Optimization procedure
Consider two heaters with length ratios of
1
~
l
and
12
~
1
~ll −=
, and the heat generation rate ratio of
./
122
qqm
=
In this case, two hot spots are marked with H1 and H2 in Fig. 1 related to peak
temperatures
1
~
T
and
2
~
T
, respectively. Under these circumstances, the goal is to find the optimal value
of
1
~
l
and m2, in such a way that
)
~
,
~
(
~
21
TTmaxT
max
=
is minimized. Accordingly, the effect of the two
parameters on
1
~
T
and
2
~
T
is next studied. Fig. 2 depicts the effect of m2 on
1
~
T
and
2
~
T
for a constant
value of
1
~
l
in harmony with Eq. (4). As observed here, with increments in m2, the temperature
1
~
T
decreases and the temperature
2
~
T
elevates. This peculiar behavior is due to the fact that by increasing
m2, the amount of
1
q
is reduced, the temperature grows slowly along the first heater and therefore, the
level of
1
~
T
is reduced. In the same way, the direct consequence of increasing m2 is augmentation of
2
~
T
. Shown in Fig. 3 is the influence of
1
~
l
on
1
~
T
and
2
~
T
for a constant value of m2 linked to Eq. (4).
As observed here, with increments in
1
~
l
, the temperature
1
~
T
augments, while
2
~
T
exhibits a descendant-
ascendant trend. The rise in
1
~
T
is clear since the temperature climbs over a longer heater. The reason
behind the decreasing-increasing behavior of
2
~
T
is that for low values of
1
~
l
, as a result of increasing
1
~
l
, a higher proportion of the total heat load is transferred to the fluid near the leading edge of the
plate where h is more vigorous. On the contrary, when
1
~
l
exceeds a specified value, elevations in
1
~
T
,
which also have a bearing on
2
~
T
, compensates the so-called reducing trend, and eventually leads to the
augmentation of
2
~
T
.
Switching the attention to the diagrams in the tandem of Figs. 2 and 3, one can now conclude that the
optimal point is reached by intersecting the
1
~
T
curve with the
2
~
T
curve. Thereby, the governing
4
Fig. 2. The effect of m2 on
1
~
T
and
2
~
T
for a constant value of
5.0
~
1
=l
when two heat sources are designed
with the ratio of heat generation rate, m2.
Fig. 3. The effect of
1
~
l
on
1
~
T
and
2
~
T
for a constant value of m2 = 0.5 when two heat sources are designed
with the length ratio of
1
~
l
.
equation for optimization in this section is
12 ~~ TT =
, which in conformity with Eq. (4) delivers the
following relation,
)
~
1()1(1
~4
3
1
3
4
,
3
12
2
1
1lIml −−+=
(7)
The optimization process delineated here can be generalized to a larger number of heaters attached to
a plate because the extension requires solving a Lagrange system of equations. The resulting system
5
could be solved using a numerical approach of Newton’s. It is also clear that; in the case of distancing
the heaters with insulated spacing, mi must be substituted by zero.
4. Results and discussion
In this section, the numerical results of the proposed optimization procedure that led to the
minimum level of the peak temperature are presented and the direct impact of the controlling
parameters is investigated. In this regard, the outcome of utilizing the proposed method utilized alone
and utilized in combination with the method of distancing the heaters by insulated spacing. It must be
mentioned that the insulating segment is in fact an approximation for the existence in such
applications of a very low thermal conductivity substrate
4.1 Method I: Unequal heaters without insulated spacing
Figure 4 contains the optimized values for the ratio of the heat generation rate of the two heaters,
opt
m,2
, which minimizes the peak temperature, for various values of the length ratio of the two heated
segments,
.
~
1
l
The results mapped in Fig. 4, predict larger
opt
m,2
for higher values of
.
~
1
l
An alternative
facet of the optimization is also evident from the decreasing-increasing variation of
max
~
T
with respect
to
.
~
1
l
i.e., there is an optimal value of
1
~
l
that maximizes the efficacy of Method I for minimizing the
peak temperature,
max
~
T
. When the number of unequal heaters is increased to three, the optimal values
which minimize the peak temperature for various values of
,
~
1
l
are found in Fig. 5. The optimization
results of Figs. 4 and 5 are summarized in tab. 1 when two, three and infinite number of unequal
heaters are utilized under the platform of Method I. Upon comparing the results listed in this table, the
advantage of increasing the number of unequal heaters is recognizable. It is also observed that the
upper bound for the efficacy of Method I for minimizing the peak temperature in the limiting case of
n = ∞ where the heat flux distribution is nearly continuous, (q''
-1/2
~x
), produces a 32 % reduction in
the peak temperature.
Fig. 4. Optimal values for ratio of the heat generation rate of the two heaters with respect to the variation
of
1
~
l
.
6
Fig. 5. Optimal values which minimize the peak temperature of three heat sources with respect to the
variation of
1
~
l
.
4.2 Method II: Unequal heaters with insulated spacing
In this section, it is assumed that two unequal heaters with the ratios of
1
~
l
and
12 ~
1
~ll −=
are spaced
with an insulated spacing
1
~
s
to each other. In addition, the ratio of the heat generation rate at the two
heaters corresponds to m2.. In fact, the main goal in this section is to examine whether the
combination of the present method (designing the unequal heaters) and the method proposed in [21]
(distancing the heaters of unequal length but equal rate of heat generation) is superior to each of the
independent methods. To do this, a multi-parametric optimization is performed by relaxing the flux
ratio and the insulated spacing. Fig. 6a shows the variation of the optimal values of
1
~
s
and m2 which
minimize
max
~
T
for a wide range of
.
~
1
l
For further clarification, a one-to-one comparison is made in
Fig. 6b between the optimization results associated with the case of the two isolated methods and the
case of combined methods. The comparison over the optimal values of m2 reveals that; in the case of
method II, a larger optimal ratio of the heat generation rate at the heaters is predicted in comparison
with the case of method I is utilized alone. The comparison over the minimized peak temperature in
each cases reveals that the method II is superior than the method I and the method used in [21] for
minimizing the peak temperature when
.78.0
~
45.0 1<< l
In addition, it can be also realized that when
,
75.0
~
1
<l
usage of method I is superior than the method proposed in [21] in lowering the level of peak
temperature. In contrast, when
,75
.0
~
1
>l
the method used in [21] is more efficient in reducing the peak
temperature in comparison with method I.
1. Possible future extensions
Two possible extensions can be envisioned for the future. One deals with considering the effect of
axial conduction and conjugate heat transfer in the heaters. The other pertains to applying the
proposed methods to other types of fluid flows crossing over the heated segments, such as internal
flows or cross flows over the heated walls.
7
.Fig. 6. a) Variations of optimum values of
1
~
s
, m2 and their corresponding maximum temperature
according to variation of
1
~
l
b) Comparison of optimum values of m2 and maximum temperature in
different cases. The results of Ref. [21] correspond to the case of distancing the heaters with unequal
length, but having the equal rate of heat generation.
2. Conclusions
The main conclusions that may be drawn from this work are the implementation of two reliable
methods to cope with the temperature elevations of heated segments (heaters) under a laminar
boundary layer flow. In the first method, it is proposed to design the unequal heaters with different
heat generation rates, such that the heater with the highest rate of heat generation is placed at the
leading edge of the plate, while the others are placed at downstream regions of the plated. The optimal
length ratio of the heaters and the optimal ratio of the heat generation at the heaters which minimize
the peak temperature are obtained. According to the optimized results, it is shown that utilizing this
method reduces the peak temperature up to 15% when two heaters are utilized, while the peak
temperature reduction is reported up to 21% when the method is applied to three heaters. The second
scheme consists in distancing the previous unequal heaters with an insulated spacing. It is shown that
when the unequal heaters are distanced b insulated spacing, the scheme provides higher efficiency on
reducing the peak temperature for certain range of length ratio of the two heaters.
8
Nomenclature
A
-Total area of plate, [m2]
s
-Insulated spacing between heaters,
[m]
h
-Convection heat transfer coefficient, [W m-
2 K-1]
T
i
-Terminal point temperature of
heater number i, [K]
I
m,n
-Incomplete beta function
max
T
-Plate maximum temperature, [K]
k
-Thermal conductivity, [W m-1 K-1]
T
s
-Plate temperature, [K]
l
-Heater size, [m]
T
∞
-Free stream temperature, [K]
l
i,cr
-Critical value of l
i
, [m]
U
∞
-Free stream velocity [ms-1]
L
-Sum of heater lengths, [m]
x
-Cartesian coordinates, [m]
n
-Number of heaters
Greek symbols
m
-Ratio of the heat generation rate
ν
-Fluid kinematic viscosity, [m2s-1]
Pr
-Prandtl number
Subscripts
qꞌ ,qꞌꞌ
-Heat flux, [W m-2]
i=0,1,2,
…,n
-Heaters index
q
-Heat transfer per unit of length, [W m-2]
(~)
-Dimensionless variables
Re
x
-Reynolds number (= U
∞
x /
ν)
References
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Applied Thermal Engineering, 27 (2007), pp. 1708–1714.
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Nature, Cambridge University Press,
Cambridge, England, UK, 2000.
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sources in a liquid cooled rectangular channel, Applied Thermal Engineering, 25 (2005), pp. 2532–
2542.
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47 (2006), pp. 3447–3455
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Heat Mass Transfer, 32 (2005), pp. 1244–1252
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251–257.
9
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Table. 1. Optimization results demonstrating the optimal size and optimal rate of heat
generation associated with two, three and countless number of heat sources.
Peak temperature
reduction (%)
max
~
T
opt
l,3
~
opt
m
,3
opt
l,2
~
opt
m
,2
opt
l
,1
~
n
15.43
0.8457
-
-
0.731
0.471
0.269
2
21.02
0.7898
0.5619
0.2856
0.3281
0.4556
0.11
3
31.54
0.6846
-
-
-
-
-
∞
10
