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Correlations and optimization of a heat exchanger with offset fins by genetic algorithm combining orthogonal design
Accepted Manuscript Correlations and optimization of a heat exchanger with offset fins by genetic algorithm combining orthogonal design Du Juan, Yang Man-Ni, Yang Shi-Fang PII: DOI: Reference:
S1359-4311(16)30561-0 http://dx.doi.org/10.1016/j.applthermaleng.2016.04.074 ATE 8121
To appear in:
Applied Thermal Engineering
Received Date: Accepted Date:
5 March 2016 16 April 2016
Please cite this article as: D. Juan, Y. Man-Ni, Y. Shi-Fang, Correlations and optimization of a heat exchanger with offset fins by genetic algorithm combining orthogonal design, Applied Thermal Engineering (2016), doi: http:// dx.doi.org/10.1016/j.applthermaleng.2016.04.074
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Correlations and optimization of a heat exchanger with offset fins by genetic algorithm combining orthogonal design Du Juana , Yang Man-Nib, Yang Shi-Fang*,c a
School of Aeronautics and Astronautics Engineering, Guizhou Institute of Technology, Guiyang,Guizhou,550003,China
b
c
School of Materials Engineering, University of Manchester, Manchester,M139PL,UK
Chemical and Chemical Engineering Institute, Hubei University, Wuhan, Hubei,430062, China Corresponding author: Tel: +86 13627246526 E-mail addresses:[email protected]
Nomenclature
A
total heat transfer area, (m2)
Aff
free flow area, (m2)
C
heat capacity rate, (W/K)
Cp
specific heat at constant pressure, (J/kg K) hydraulic diameter,
f
friction factor, dimensionless
G
mass flow rate of fluid, ( kg/s) mass flux of the air based, fin height, (mm)
h
convective heat transfer coefficient of the medium, (W/m2 K)
j
Colburn factor, dimensionless 1
lf
offset value, (mm)
L
depth of the heat exchanger in flow direction, (mm)
n
fin frequency,fins per meter
N
number of fin layers for fluid
Nu
Nusselt number, dimensionless
NTU
number of transfer units
Pr
Prandtl number, dimensionless Total heat transfer rate, (W) the maximum possible heat transfer rate,(W)
Re
Reynolds number
s
fin width, (mm)
T
temperature, ( K )
t
fin thickness, (mm)
U
overall heat transfer coefficient, (W/m2 K)
V
volume flow, (m3/s)
Greek symbols
fin wrinkling angle, (º)
P
pressure drop, ( Pa )
T
temperature difference, ( K )
thermal conductivity, (W/m K) fin surface efficiency
o
surface effectiveness
2
fluid dynamic viscosity, (N s/m2)
density, (kg/m3)
porosity
heat exchanger effectiveness,
Subscripts 1
air side
2
oil side
c
cold side
h
hot side
in
inlet
m
mean value
min
minimum value
max
maximum value
out
outlet
Abstract：In this paper, the study is focused on a double flow plate-fin heat exchanger (PFHE) which heat transfer element is staggered offset fin, and heat transfer model and the energy equations for the structure have been established, seven geometric parameters such as the fin height, fin length and fin wrinkling angle are taken as the decision variables for optimization. A genetic algorithm (GA) combined with orthogonal design is used to search for the
optimal overall structure and the correlations about the fin heat transfer factor j and the friction factor f. The maximum total heat transfer rate and the minimum total pressure drop are taken as objective functions in the GA, respectively. Performance of the optimized structure is evaluated and correspondingly the heat transfer and hydrodynamic characteristics of the full-size PFHE are calculated by using a porous media approach. Numerical results 3
show that the total heat transfer rate of the optimized structure is improved about 6.2% comparing with the original design, the total pressure drop decreases by about 40% and the volume can reduce about 2.7%. Keywords: Plate-fin heat exchanger; Genetic algorithm; Orthogonal design; Optimization; Porous media.
fin geometries. Based on the regression of experimental data and deviation analysis, the experimental correlations of Colburn j-factor and Fanning friction f-factor about laminar flow and turbulent flow are obtained. Manson[5] put forward heat transfer and resistance experimental correlations of the offset fins in addition to
1.Introduction
Kays and London’s experiments. Dubrovsky and Vasiliev[6] established the
Offset fins are widely used for the
local Nusselt number and the thermal
plate-fin heat exchanger (PFHE) to improve
resistance loss experimental correlations by
heat transfer rate by using staggered offset
the experiment research on 11 different
fins as the extended surface to enlarge heat
staggered offset fins. Wieting[7] analysis 22
transfer area and regenerate thermal
different structural parameters of staggered
boundary layer in each column. However,
offset fins, gave the power law experimental
offset fins induce a large pressure drop
correlations on the heat transfer and flow
between the inlet and outlet of the PFHE.
resistance performance. Joshi and Webb[8]
Therefore, the heat transfer and pressure
built empirical correlations of Colburn
drop in the PFHE with offset fins need to be
j-factor and Fanning friction f-factor in
investigated. Furthermore, the overall
laminar and turbulent flow regimes
structure of the PFHE with offset fins must
according to the test data of 21 kinds of
be optimally designed so that it can
offset finned heat exchanger. Rational
reconcile these contradictory phenomena.
design equations are presented by Manglik[9] in the form of single continuous
Many studies have examined the PFHE
expressions covering the laminar, transition,
with offset fins. Since most of them used air
and turbulent flow regimes. In addition to
as the working fluid, few fin geometries are
regular finned structure parameters are
applicable to practical offset fins with
studied, Tension [10] researched for the
working fluids other than air. Kays and
"stagger position of incision" also and the
London[1] carried out early experimental
corresponding heat transfer correlations
investigations on the offset fin geometry and
were obtained. Guo Lihua[11] carried out
demonstrate a correlation based on the
experimental studies to three different types
experimental results. Later Manglik and
of the steel staggered offset fins, presented
Bergles[2] , London and Shah[3]，
empirical correlations about the j-factor and
Mochizuki and Yagi[4] et al. have made
f-factor of the offset fins for lubricant oil.
broad and profound researches on similar 4
Some special applications have higher
estimated the optimal geometry parameters
request for the volume of the PFHE. In
within an acceptable range while satisfying
order to make per unit volume in a larger
the above two objective functions. Xie et
quantity of heat, the PFHE overall structure
al.[18]minimized the total volume as well as
optimization become the focus of the
the total annual cost of a compact heat
present studies. Thus, various optimization
exchanger by considering three shape
methods have been proposed. Guo. et
parameters as decision variables. Wang et
al.[12] applied optimization design approach
al.[19]applied genetic algorithm to optimize
based on the field synergy theory and
primary energy saving annual total cost
genetic algorithm to improve the
saving, and carbon dioxide emission
shell-and-tube heat exchanger performance
reduction. Peng and Ling [20] successfully
and reduce the total cost. The optimal
used GA combined with back propagation
design leads to a significant cost cut and an
neural network for the optimal design of
improvement of the heat exchanger
heat exchanger by considering the minimum
performance. Sahin et al.[13] used Taguchi
total weight and total annual cost for a given
method to optimize the design parameters of
constrained condition as objective functions.
the heat exchanger with staggered offset
Recently, Xie et al.[21-24] designed
fins. In recent years, the multi-objective
micro-channel heat sinks and optimized
optimization has been successfully used to
different kinds of heat exchangers by using
many thermal systems. Gholap and
different optimization methods such as:
Khan[14] took energy consumption and
entropy generation minimization and the
material cost as the objective function of the
constructal theory[25,26].
two conflicts and carried out multi-objective optimization to get the optimal design parameters of a forced air heat exchanger. Hilbert et al.[15] also used a multi-objective optimization technique to maximize the heat transfer rate and to minimize the pressure drop in a tube bank heat exchanger. Najafi, et al.[16] applied genetic algorithm multi-objective optimization technique to provide the optimum geometric parameters of a PFHE. Sanaye and Hajabdollahi[17] applied multi-objective optimization to maximize the effectiveness and to minimize the pressure drop in a rotary regenerator and
This paper investigates the characteristics of heat transfer and pressure drop for a PFHE, the fin height, fin width, fin length, fin offset, and fin wrinkling angle are considered as optimization parameters within reasonable constraints, the total heat transfer rate and the total pressure drop are considered as two conflicting objective functions. Multi-objective optimization technique using genetic algorithm combined with orthogonal design is utilized in order to achieve a set of optimal solutions, called pareto multiple optimum solutions. The sensitivity analysis of the total heat transfer 5
rate and total pressure drop changed with
in table 1. Each layer is separated by
the design parameters was performed and
clapboard and two different flow channels
the results are reported and the correlations
are built. The oil-side fins are separated by
of the characteristics of heat transfer and
seals in the middle. (shown in Fig.3 and
pressure drop for the optimum structure
Fig.4)
were proposed. In addition, a series of Rectangular staggered offset fins are
comparative verification are also carried out in order to investigate the effect of some of the geometric variables on the objective functions by using CFD software Fluent.
used on air side and form a single-flow channel. In order to improve the thermal efficiency of the PFHE, the air-side fins are higher than the oil side, the fin appearance and front view are shown in Fig.5 and the
2. Modeling of the system
side view are shown in Fig.6. The main geometric parameters are provided by the
2.1 Physical Model and Assumptions
commission unit and shown in Table 2.
The PFHE with staggered offset fins studied in this paper exchanges heat under the working condition of an oil-to-gas heat transfer and the hot side is oil and the cold side is air. Hot oil flows across a trapezoidal offset fin while cold air flows across an rectangular offset fin in adjacent layers which are arranged staggered. The schematic diagrams of the PFHE are shown in Fig.1~Fig.6, the heat is transmitted to the partition through the oil-side fin surface, resulting in the temperature rise of the air side. The total heat transfer and pressure drop characteristics on the heat exchanger are to be found by numerical computations based on the suggested model. In order to
To simplify the calculation process, following assumptions are made for the analysis: (1) Mass conservation and energy equations and the Navier-Stokes can be used to analyze the physical processes. (2) Physical property variation of the fluids with temperature is neglected and the air is assumed to be ideal gas. (3) The fluid flow is single-phase, incompressible and the PFHE is operating under steady state condition in the computational domain. (4) The wall is considered as an ideal surface, which means there are no burrs, scarped edges, or adhesive substances and the thermal resistance and fouling effect are neglected.
avoid heat loss, the number of air-side fins layers are more than oil layer, and the oil
2.2 Data Reduction
side are 29 layers and the air side are 30 layers. In the optimization and analysis, 46# lubricating oil is used in hot side, which
In the analysis, to provide the heat transfer characteristics of the tested sample, ɛ-NTU method is used to determine the UA
main thermodynamic parameters are given 6
term of the heat exchanger[27]．The UA
Heat transfer area of both side are achieved
product was calculated using ɛ-NTU method
using the following relation:
for the unmixed cross-flow configuration.
Ah Lh Lc Nh 1 2nh ( H h th )
Correspondingly, the appropriate ɛ-NTU
(8)
Ac Lh Lc Nc 1 2nc ( H c tc )
relationship is:
1 exp
(9)
1 0.22 exp Cr NTU 0.78 1 Cr NTU
Total heat transfer area,
(1)
A Ah Ac
where, Cr Cmin / Cmax , assuming zero fouling factor and zero thermal resistance,
The rate of heat transfer (the first objective
NTU is number of transfer units and can be
function): Q Cmin (Th,1 Tc,1 )
obtained as: The pressure drop of the both side (the
1 1 1 UA ( hA )h ( hA )c
(2)
second objective function):
1 A f f h 1 A f cf C 1 m i n Cm i n NTU UA h h h A c h c A
Ph
(3)
2 f h LhGh2 h Dh
Pc
2 f c L cG 2 c c Dc
(10)
(11)
where ,1/UA is the thermal resistance of heat convection, and convective heat
3. Optimization
transfer coefficient is calculated using
hh jh Gh Cph Prh 2 / 3
(4)
3.1 Genetic Algorithm Genetic algorithm (GA) is a bionic
hc jc Gc Cpc Prc 2 / 3
(5)
Aff h ( H h th )( 1 nhth )Lh Nh
(6)
Affc ( H c tc )( 1 nctc )Lc Nc
(7)
global optimization algorithm inspired by natural selection and evolutionary processes, which was first conceived by Holland[28,29]. The idea is: imitating Darwin's theory of evolution and Mendel’s law of inheritance through operation mechanisms such as selection, crossover 7
and mutation, continuously improving the
PFHE system which lead to the maximum
fitness of the individuals in the population.
total rate of heat transfer and minimum total
Genetic algorithm begins from a population
pressure drop. The total heat transfer rate Q
which may be a potential solution set of
is considered as one of the objective
representative problems and the initial
functions and -Q is minimized through the
population is usually randomly generated.
optimization procedure which in turn leads
Population is made of a certain number of
to Q maximization. The operating
individuals and each individual is actually
conditions for the considered case are given
the entity with chromosome characteristics.
in Table 1. Both fluids are considered as hot
Chromosomes are the main carrier of
oil and cool air, respectively. Applicability
genetic material, and a collection of
of the geometry range is listed in Table 3.
multiple genes. Afterword, parents are
Considering the overall heat exchanger
selected according to their fitness values.
structure size cannot be larger than the
The higher fitness of an individual is, and
original model which the oil-side and
the greater possibility there is of being
air-side fin height is 2mm and 12.8mm
selected as the parent for the reproduction.
respectively, the optimized total height is
In the next step, the reproduced two
smaller than the original total height, so the
chromosomes combine together and form a
oil-side and air-side fin height can be
new chromosome, called the offspring.
desirable 2.6mm and 12mm respectively.
Since individuals with higher fitness have
Also because the oil-side fin is isosceles
more chance for being selected and produce
trapezoid and the biggest the angle is 90º, α=
offspring, the new population generated
45º ~ 90º and other parameters are based on
after reproduction possess more qualified
the principle of the little change of the
genes and consequently higher fitness. In
physical model size.
multi-objective genetic algorithm, the procedure is the same in addition that it has
4. Optimized results and
multiple fitness for each individual regarding different considered objectives.
discussions
(the workflow of the multi-objective genetic algorithm is shown in Fig.7.)
In each iteration, the values of optimization parameters are given new values within their specified constraints and
3.2 Multi-Objective Optimization
the considered objective functions are evaluated based on the new values. The
In the present work, a multi-objective genetic algorithm is utilized in order to
initial population size is considered to be 200, the maximum number of generations is
obtain optimal geometric parameters of the 8
500 and the generation gap is selected to be
4.1.1 Effect of the fin height
0.9. Select strategy uses a genetic algorithm
In order to detect geometric parameters
of crossover probability 0.7 and mutation
such as fin height, fin length, fin wrinkling
probability 0.5.
angle etc. on the influence of total heat
The optimization is terminated after 500 generations. Considering the given operating conditions in Table 2, The first and second objective function values after 500 times iteration are shown in Fig.8, some of the selected optimal design parameters obtained by utilizing the GA are given in
transfer and pressure drop, the oil-side inlet velocity of the PFHE is increased from 0.2m/s to 1m/s. Fig.9 shows the variation of the total rate of heat transfer and pressure drop versus the inlet velocity under constant flow rate and three different oil-side fin heights.
Table 4. Fig.8(a) illustrates the evolution process of the first objective function (the total rate of heat transfer).Fig.8(b) is the evolution process of the second objective function (the total pressure drop).Fig.8(c) shows the first objective function value and the second objective function values after 500 times iteration. From Fig.8(c), we can see that the optimal value fall between 5 and
It can be seen from the Fig. 9 that with the increase of flow velocity, the total heat transfer rate and pressure drop of the three kinds of fins always increasing, but when the fin height H2=2.6 mm, the total heat transfer rate is maximum and the total pressure drop is minimum. This results show that among the three different fins, the fin with H2 = 2.6 mm is the most optimal oil-side fin height.
6 sets of data.
4.1 Sensitivity Analysis
Fig.10. shows the variation of the total heat transfer rate and pressure drop versus the inlet velocity for the PFHE with the
In this section, considering the
three kinds of fin height. It can be seen that
operating condition and constant values
with the increase of flow velocity, the total
given in Table 1 and Table 2, the effects of
heat transfer rate of the three kinds of fins
some geometric variables on objective
will increase, and the total pressure drop
functions are investigated. In each section,
keeps stable. When the fin height H1=
the values of all parameters, except those
12.8mm, the total heat transfer rate is
which are selected for the investigation, are
maximum. The main reason is: when the
kept constant. By varying the values of the
other conditions are the same, the higher the
selected parameters, the sensitivity of each
fin, its heat transfer area is larger, also the
objective function with respect to the
more heat transfer. And the pressure drop is
parameters can be discussed.
mainly related to flow velocity and flow 9
direction of length, so when other
transfer rate of the three kinds of fins will
conditions are constant, in a small range of
increase, and the total pressure drop will
variation the fin height has negligible effect
increase too. When the fin wrinkling angle
on the pressure drop of the fluid. This can
85o , the total heat transfer rate is
be seen from the Fig.10(b).
maximum. The main reason is: when the other conditions are the same, the greater
4.1.2 Effect of the fin lenght
the fin angle, the more fin number and the
Fig.11. shows the variation of the total
more heat transfer. At the same time, the
heat transfer rate and the total pressure drop
thickness of the fin also makes the vortex
versus the inlet velocity for the heat
generation and shedding more obvious, so
exchanger with the three kinds of fin length.
the heat transfer effect will be increased
It can be seen that with the increase of flow
with the increase of fin wrinkling angle.
velocity, the total heat transfer rate of the three kinds of fins will increase, and the
4.2 Orthogonal design test
total pressure drop will increase too. But the
Due to involving seven parameters in
total pressure drop increases from the size
the process of optimization, it will be very
of l2 = 3.1 mm to l2 = 3.2 mm less than that
difficult to implement if the seven
of from l2 = 3.2 m to l2 = 4.2 mm and the
parameters are tested fully. For the reason
total heat transfer is almost the same. The
an orthogonal design method can be used to
main reason is: when the other conditions
find the optimal level combination. The
are the same, the total heat transfer rate and
basic idea is to arrange and analyze these
the total pressure drop are function of the
factors by using the orthogonal table and
velocity respectively. The offset fin periodic
some representative level combinations are
interrupt on flow direction, leading to the
selected from all level tests, that is to say,
periodic start and end at fin end boundary
through the part of the test results to analyze
layer, the shorter the fin length greatly
the overall situation and find the optimal
hinders the increase of the boundary layer
design parameters. In reality there are
and leads to the increase of heat transfer rate
various factors affecting heat transfer effect
and pressure drop simultaneously.
if the physical experimental conditions are
4.1.3 Effect of the fin wrinkling angle Fig.12 shows the variation of the total heat transfer rate and the total pressure drop versus the inlet velocity for the heat exchanger with the three kinds of fin wrinkling angle. It can be seen that with the increase of flow velocity, the total heat
taken into account. L15 orthogonal array is normally used in investigating effects of a 7
three-stage seven factor (3 ) combination. However, as is the case in many studies, some factors may be neglected thereby reducing the number of experiments and testing time [30]. As for seven parameter 10
and three-level array, L15(37) used in this
correlations about the fin heat transfer factor
study, consistency was maintained in
and the friction factor are obtained which
applying the Taguchi standard orthogonal
are suitable for this research of the geometry
design where a total of 15 sets of data were
parameters about the Reynolds number,
conducted. The design arrangement suitable
wrinkling angle, fin height, fin length.
4
for the study carried out L9(3 ) (combined
Consequently, the following equations are
with results of the section 4.1, the total is 10
obtained:
sets of data) together with its corresponding reactions are given in Table 5. The first
jcorr,1 28.5627 Re
1.178
parameter column of the table shows
H Dc
0.1125
(12)
air-side fin height, the second column gives oil-side fin height, the third column is for
f corr,1 0.9196 Re
oil-side fin length and the last column lists
0.3155
H Dc
-0.0127
(13)
the angle of oil-side fin. The farthermost right of the table are the values Q and
P .
H jcorr,2 0.1627 Re0.4821 0.4517 Dh
0.425
l Dh
(14)
From the Table 5, it can be seen that the Q value of the tenth group is larger and P is more moderate. Taking into
0.04547
f corr,2 147.08Re
account the actual requirements of the
1.052
0.7246
H Dh
0.3398
l Dh
volume of the heat exchanger and the
(15)
modification cost of the fin, the optimization results are taken as the tenth group parameters.
4.3 The corrections of the optimized model In section 4.1, the influences of
5. Optimization Results and Analysis 5.1 Result Analysis of the obtained correlations
different fin angles, fin length and fin height Fig.13 shows the comparison of the
on the performance of fin were qualitatively tested. In the section 4.2, the typical 10 sets of fin measurable data were listed by applying the orthogonal design method and the optimal fin geometric parameters with comprehensive performance were obtained. Here, the 10 sets of data were fitted and the
0.0527
correlations obtained in this paper against the correlations from Wieting[7] and Manglik[9]. The predictions from the correlations are reasonable good agreement with the open literature values at different Reynolds numbers, but some deviations are 11
observed. Fig.13(a) shows the comparison
results are significantly better than
results of the j-factor and f-factor before and
un-enhanced results. The above analysis
after optimization on air side. It is apparent
shows that the optimization results are
that the j-factor and f-factor values have
correct and feasible.
little change after optimized. But for the oil
From the above analysis and
side, the j-factor of the optimized model is significantly higher than that of before optimization at the same Reynolds number (see Fig.13(b)) and f-factor is higher than that of original model too(see Fig.13(c)). The main reasons for analysis are as follows: the first reason is that the used working fluids are different and the test medium in literature is air of Prandtl number 0.7 and the medium used in this paper is lubricating oil with Prandtl number greater than 100, and the literature[9] clearly pointed out whether the correlations based on the gas medium are suitable for the liquid medium has to be further verified. The
discussion, combined with the fin parameters in Table 2, the final optimization results are an optimal compromise of two conflicting objective functions of the total heat transfer and the total pressure drop. Considering limits of the manufacturing equipment, process conditions and the available investment, we can determine the optimal solution (12.8mm,2.6mm,3.45mm,3.2mm,1.24mm,1. 25mm,0.66mm,0.57mm,85º), as shown in Table 6. By contrast, we can see the change of the PFHE in the physical dimension is quite small.
second reason is that the form of fin is
Fig.14 shows comparing results of the
different and the fins are rectangular
heat transfer characteristics and flow
staggered array in the literature and the fins
characteristics for the original model and the
are trapezoidal staggered array in this paper;
optimized model of the PFHE. Fig. 14(a)
in addition, the contact thermal resistance
shows the comparison of the Nusselt
and the data error are also important factors.
number, Fig.14(b) is the comparison of the
Currently, the overall performance of the
total heat transfer rate, Fig.14(c) is the
enhanced heat exchanger is evaluated using
comparison of the total heat transfer
the criterion of goodness factor, j/f. The
coefficient. It can be seen from the graphs
baseline (un-enhanced) results are compared
that with the increase of flow velocity, the
to the enhanced performance in Fig.13(d). It
Nusselt number, the total rate of heat
can be seen from the figure that goodness
transfer and the total heat transfer
factors of optimized model are better than
coefficient are all increased. The Nusselt
that of the original model (before
number of the optimized model is increased
optimization) , in terms of its
by about 3.6%, the total heat transfer rate is
comprehensive performance, the optimized
relatively more 4%~11% and the total heat 12
transfer coefficient is higher about
same in the inlet, but surface heat transfer
4.5%~7.5% than the original model.
process of the optimized model cold air is much more sufficient than the original
Fig.14(d) shows comparing results of heat transfer resistance characteristics for the original model and the optimized model of the PFHE. It can be seen from the graph that with the increase of flow velocity the total pressure drop are all increased. Under the condition of the same flow, the total pressure drop of the optimized model is down by about 24% than the original model.
5.2 CFD numerical results and discussion
model. For the original model, keeping the air-side entrance velocity of 8 m/s, the oil-side entrance velocity are increased from 0.3 m/s to 1 m/s, and for the optimized model, the air-side entrance velocity is 8.678 m/s and the oil-side entrance velocity are increased from 0.231 m/s to 0.769 m/s, the corresponding the hydrodynamic characteristics are shown in Fig.16. The results obtained from Fluent in the form of the total heat transfer rate and the total the pressure drop are compared with the
In this section, the simulation model, meshing, boundary conditions and the simulation calculation method are all proposed in the literature [31].Here, these can be omitted. In the simulation calculation process, under the same volume flow, the entrance velocity corresponding relation of the original model and the optimized model is shown in Table 7 about oil side and air side. For the two kinds of model, when oil side entrance velocity of 0.9 m/s and 0.692 m/s respectively, air inlet velocity of 8 m/s and 8.678 m/s respectively, numerical calculation is implemented by using the CFD software Fluent, the plane y = 2.65 parallel to x-z is established after the convergence. The temperature distribution of the original model and the optimized
original model in Fig.16. From Fig.16(a), It is evident that the total heat transfer rate of the original model and optimized model are increased with the increase of flow velocity, and under the same volume flow, there is an increase of about 2% than before optimization. As can be seen from the Fig.16(b), the total pressure drop increases with the flow velocity and there is an decrease about 35% when compared with the original model. In addition, the range of pressure drop will be increased with the increase of flow velocity. The analyses above show that the improvement of the performance of the overall heat transfer for the PFHE is smaller than the decrease of the pressure drop when the fluid flow rate is constant.
model obtained from Fluent are compared as shown in Fig.15. It can be seen from the Fig.15 that the temperature of the two models air side and oil side are almost the 13
5.3 The simulation analysis with porous media Due to the growing size and complicacy of the PFHE, it is not feasible to simulate actual full-size PFHE, which
obtained as following (From local emulated data): oil side: P=640.92+695.9 porous coefficient: c1=1/α=1.33107
generally requires large computational resources and time. In this paper, two side
c2=190
porosity: =0.8914
fins can be simulated by porous media and
air side: P=0.47432+4.994
porous media technology is applied to solve
porous coefficient: c1=1/α=2.77107，
large-scale computing problems. The c2=104.16
original porous media model is given in literature[32] and the optimized porous media model can be obtained on base of modifying locally parameters of the original structure as shown in Table 6, and then input them into Workbench to be meshed, the porous medium local mesh model of original and optimized structure are shown
porosity:=0.9419
Under the boundary conditions listed in literature [32] and above porous parameters, the total heat transfer rate and the total pressure drop of the data are calculated respectively in different flow rate, the results are shown in Table 9 and Table 10.
in Fig.17. Table 9 is a simulation value contrast The above two porous media models are separately imported into the Fluent software and the boundary conditions are loaded as section 5.2, the viscous and inertia resistance coefficient of optimized porous media model can be obtained by local fin simulation results: the local simulation model as shown in literature [31], the oil-side fin length is 8mm and air side is 8.6mm, the fin local simulations are carried out and the results are shown in Table 8. Based on the analysis method of
of total heat transfer rate of the overall porous media models of the plate-fin heat exchanger before and after the optimization under the same volume flow. It can be seen that the heat exchange capacity of the heat exchanger is 6.2% larger than the before optimization. Table 10 is a simulation value contrast of total pressure drop of the overall porous media models of the plate-fin heat exchanger before and after the optimization under the same volume flow, the optimal design indicates an obvious effect with a
literature[32], for the optimized porous
40% decrease on total pressure drop and a
media model, the relationship between
2.7% decrease on total volume of the heat
entrance velocity and pressure drop can be
exchanger. The results are further evidence that the genetic algorithm combined with 14
orthogonal design to optimize the overall
the calculation results are in good
structure of the heat exchanger have great
agreements with the literature data. At the
engineering practical values.
same time, be it the heat transfer characteristics or resistance characteristics,
6. Conclusions
the optimized model is better than that of the original model.
Through using genetic algorithm
3D models of the overall heat
combined with the orthogonal design to
exchanger before optimization and after
optimize the overall structure of the PFHE
optimization are built up. Applying porous
with offset staggered fins, the total heat
medium technology and combining the local
transfer rate is maximized and the total
simulation results, the full-size numerical
pressure drop is minimized simultaneously,
simulations are carried out by FLUENT
so as to satisfy the engineering application.
software, and the numerical simulation
The following conclusions can be made.
results show that under the same volume
Mutli-objective genetic algorithm is
flow, the performance of the optimized
utilized in order to obtain optimal geometric
PFHE for the total rate of heat transfer
parameters of a PFHE system which leads
increased by about 6.2%, the total pressure
to the maximum total rate of heat transfer
drop decreased by about 40% and the
and minimum total pressure drop. The fin
volume reduced about 2.7%.
height, fin width, fin length, fin offset, and fin corrugation angle are used as design
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Table 1 Operating conditions for the case study Parameters
Hot fluid(oil)
Cold fluid(air)
Volume flow, V (m3 / s)
0.015
1.573
Inlet temperature, Tin (K)
413
328
Density, (kg m3 )
844
1.06
Specific heat, Cp (kJ / (kg K))
2.013
1.0087
Viscosity, (N s m2 )
6566E-6
2.1E-6
Thermal conductivity of fluid，
0.1237
0.0305
113.4
0.6945
(W / (m K))
Prandtl number, Pr
Table 2
The detailed geometric parameters of the PFHE 18
Unit：mm
Parameter
H1 12.8
s1
l1
t1
1.24 3.45
0.1
lf1 0.66
H2 2
s2
l2
1.25 3.3 0.55
Table 3 The geometry ranges of the design parameters H1 1-12
H2 1-2.6
l1 1-3.8
lf2
l2
s1
s2
lf1
1-3.8
1-3
1-3
1-1.5
82.06°
Unit：mm
lf2 1-1.5
45-90°
Table 4 The optimization results after 500 iterations by GA algorithm lf2
Q
P
/W
/kPa
H1
H2
l1
l2
s1
s2
lf1
/mm
/mm
/mm
/mm
/mm
/mm
/mm
12.00
2.59
1.02 1.19
1.02 1.00 0.29 0.20 89.98 89384 31837
12.00
2.59
1.02 1.19
1.02 1.00 0.32 0.24 89.97 88527 28638
11.99
2.59
1.02 1.19
1.02 1.00 0.29 0.20 89.98 76906
/mm
/°
Pereto solvers
35743 11.99
2.59
1.02 1.19
1.02 1.00 0.29 0.20 89.98 89388
2.60
1.02 1.19
1.02 1.00 0.29 0.20 89.97 88654
11.76
2.60
3.77 3.80
2.99 2.99 1.50 1.50 45.06 29630
3221
11.76
2.60
3.48 3.80
2.99 2.51 1.50 1.50 45.06 29830
3308
11.76
2.60
3.48 3.76
2.94 2.99 1.50 1.50 45.06 29759
3225
31836 11.99 31869
19
11.76
2.60
3.48 3.80
3.00 2.99 1.50 1.50 45.06 29614
3221
11.76
2.60
3.77 3.80
2.99 2.94 1.50 1.50 45.20 29632
3222
Table 5 Number of tests
Four-factor and three-level orthogonal table
Parameter
Results
H1(mm) H2(mm) l2(mm) α(º)
Q (kW)
P (kPa)
P
1
10.8
2.4
2.2
65
29.75
47.63
2
10.8
2.5
3.2
75
34.45
53.84
3
10.8
2.6
4.2
85
40.73
67.26
4
11.8
2.4
3.2
85
41.84
71.79
5
11.8
2.5
4.2
65
30.36
44.20
6
11.8
2.6
2.2
75
35.81
52.41
7
12.8
2.4
4.2
75
36.57
55.82
8
12.8
2.5
2.2
85
43.83
70.70
9
12.8
2.6
3.2
65
31.06
42.38
10
11.8
2.6
3.2
85
42.49
67.26
0.625
0.640
0.613
0.583
0.687
0.683
0.655
0.620
0.733
20
Q/
0.632
Table 6. The geometry values of the design parameters for the optimized model H1
lf2
/mm /mm /mm /mm /mm /mm /mm
/°
H2
/mm The optimized model 11.8
2.6
l1
3.45
l2
s1
3.2 1.24
s2
1.25
lf1
0.66 0.57
85
Table 7 The inlet velocity corresponding relation of the original model and the optimized model under the same volume flow oil side inlet velocity original model
air side inlet velocity
optimized model
original model
optimized model
(m/s)
(m/s)
(m/s)
0.3
0.231
3
3.254
0.4
0.308
4
4.339
0.5
0.385
5
5.424
0.6
0.462
6
6.509
0.7
0.539
7
7.593
0.8
0.615
8
8.678
0.9
0.692
9
9.763
1.0
0.769
10
10.848
Table 8
(m/s)
The local simulation results in the two side fins
21
oil side
air side
inlet velocity
pressure drop
inlet velocity
pressure drop
(m/s)
(Pa)
(m/s)
(Pa)
0.6
647.38
6
52.73
0.7
801.8
7
64.48
0.8
967.81
8
78.58
0.9
1144.75
9
91.86
Table 9
Comparison of the total rate of heat transfer between before optimization and after optimization of the overall porous media model Oil side
编 original model flow No
Air side optimized
flow
original model 3
model
model m /h
optimized
kW
kW
m3/h
kW
kW
1
4.480
53.64
56.79
5106.3
53.52
56.65
2
4.570
54.76
57.97
5106.6
54.55
58.00
3
4.800
75.74
69.78
5265.0
72.23
69.94
4
5.317
76.63
81.71
5666.9
76.40
81.50
5
5.324
76.39
81.37
5660.6
76.56
81.59
6
5.330
80.65
85.85
5686.8
80.41
85.65
7
5.355
81.74
88.30
5675.0
81.41
88.33
8
5.360
81.54
95.67
5671.0
80.16
95.42
9
5.700
87.90
88.54
5623.1
87.66
88.75
10
6.420
89.16
95.84
5663.2
88.92
95.60
22
11
6.500
87.65
94.24
5664.4
87.42
94.01
12
6.508
87.98
93.21
5659.4
87.75
93.49
13
6.585
89.58
93.56
5696.6
89.33
93.82
Table 10
Comparison of the total rate of heat transfer between before optimization and after optimization of the overall porous media model Oil side
No
flow
original model
Air
optimized
flow
model
side
original model
optimized
model
m3/h
kPa
1
4.480
131.59
2
4.570
3
kPa
m3/h
kPa
76.35
5106.3
0.5419
0.7919
133.9
78.55
5106.6
0.5420
0.7917
4.800
145.1
82.47
5265.0
0.5668
0.8302
4
5.317
158.1
95.61
5666.9
0.6321
0.9315
5
5.324
167.0
95.75
5660.6
0.8527
0.9299
6
5.330
175.0
95.03
5686.8
0.6354
0.9367
7
5.355
159.3
95.65
5675.0
0.6344
0.9723
8
5.360
159.4
123.88
5671.0
0.6328
0.9326
9
5.700
186.3
104.13
5623.1
0.6328
0.9202
10
6.420
219.2
122.93
5663.2
0.6317
0.9308
11
6.500
201.2
125.24
5664.4
0.6321
0.9309
12
6.508
223.5
125.46
5659.4
0.6317
0.9296
13
6.585
227.3
127.52
5696.6
0.6370
0.9392
23
kPa
Oil outlet
Oil inlet
air outlet
air inlet
Oil-side fins Air-side fins
Fig.1.
The overall schematic of the PFHE
Air outlet Air inlet
Oil inlet
Fig.2.
Oil outlet
The adjacent two layers in the model of the whole
24
oil inlet
oil outlet
Fig.3.
The top view of the oil-side fins
(a) Fins profile
(b) The front view Fig.4.
The oil-side fins
25
(a) Fin profile
(b) The front view Fig.5.
The air-side fins
(a) Air side Fig.6.
Fig.7.
(b) Oil side
The oil-side view and the air-side view
The workflow of the multi-objective GA
26
4
-5.5
x 10
4400 4200
-6
Total pressure drop
Total rate of heat transfer
(Pa)
(W)
-6.5 -7 -7.5 -8
4000 3800 3600 3400
-8.5
3200
-9
3000
-9.5
0
50
100
150
200 250 300 Generation
350
400
450
500
2800
0
50
100
150
200 250 300 Generation
(a)
350
400
450
500
(b) 4
4
x 10
2
Objective function value
0 The first objective function value The second objective function value
-2
-4
-6
-8
-10
1
2
3
4
5
6
7
8
9
10
Population
(c) Fig.8. (a) Evolution process of the objective of the total rate of heat transfer. (b) Evolution process of the objective of the total pressure drop. (c) The first objective function value and the second objective function values after 500 times iteration.
27
55
65 50
H2=2.5mm
H2=2.5mm H2=2.6mm
Pressure drop (Kp)
Heat transfer rate (KW)
H2=2.4mm
H2=2.4mm
45
40
H2=2.6mm
60
55
50
35
45
30
0.2
0.4
0.6
0.8
0.2
1.0
0.4
0.6
0.8
1.0
Velocity (m/s)
Velocity (m/s)
(a) Velocity-total heat transfer rate
(b) Velocity-total pressure drop
Fig.9. Effects of the oil-side fin height H2 55
90 H1=10.8mm
50
H1=10.8mm 85
H1=11.8mm
H1=12.8mm
45
H1=12.8mm
Pressure drop (Kpa)
Heat transfer rate (KW)
H1=11.8mm
40
35
80
75
70
30
25 0.2
0.4
0.6
0.8
1.0
65 0.2
Velocity (m/s)
0.4
0.6
0.8
1.0
Velocity (m/s)
(a) Velocity-total heat transfer rate
(b) Velocity-total pressure drop
Fig.10. Effects of air-side fin height H1
28
52
l2=2.2mm
58
l2=3.2mm
l2=2.2mm
50
l2=4.3mm
Pressure drop (Kpa)
56
l2=4.2mm
48
46
44
54
52
50
42 0.4
0.5
0.6
0.7
0.8
0.4
0.9
0.5
0.6
0.7
0.8
0.9
Velocity (m/s)
Velocity (m/s)
(a) Velocity-total heat transfer
(b) Velocity-total pressure drop
Fig.11. Effects of the oil-side fin length l2
70 55
50 45
60
Pressure drop (Kpa)
Total heat transfer rate (KW)
Heat transfer (KW)
l2=3.2mm
40 35 30
50
40
30
25
20 20 15 0.2
0.4
0.6
0.8
1.0
10 0.2
Velocity (m/s)
(a)
0.4
0.6
0.8
1.0
Velocity (m/s)
Velocity-total heat transfer rate
(b) Velocity-total pressure drop
Fig.12. Effects of the oil-side offset fin wrinkling angle
29
0.20
800
1000
1200
1400
1600
1800 0.20
0.18
0.18
200
300
400
500
600
700 0.045
0.045
before optimization after optimization before optimization after optimization
0.16 0.14 0.12
j
0.040
0.14
0.035
0.12
0.030
0.030
0.025
0.025
0.020
0.020
0.015
0.015
0.02
0.010
0.010
0.00 1800
0.005
f
0.10
0.08
0.035
j
f
0.10
before optimization after optimization Manglik Wieting
0.040
0.16
0.08
0.06
0.06
0.04
0.04
j
0.02 0.00 800
1000
1200
1400
1600
200
300
400
Re
300
400
0.005 700
600
Re
(a) Comparison of air side 200
500
500
(b) Comparison of oil-side j-factor 600
700
0.5
200
before optimization after optimization Manglik Wieting
0.4 0.3
400
500
600
700 0.055
0.4
0.050
0.050
before optimization after optimization
0.3 0.045
0.045
0.040
0.040
0.035
0.035
0.030
0.030
0.025
0.025
0.2
f
j/f
0.2
300
0.055
0.5
0.1
0.1
200
300
400
500
600
200
700
300
400
(c) Comparison of oil-side f- factor Fig.13.
600
700
(d) Comparison of goodness factor
Comparison between the correlations presented in this paper and the open literature values
55
22
Q:Original model Q:Optimiz model Total heat transfer rate (KW)
Nusselt:Orignal model Nusselt:Optimized model
20
18 Nusselt
500
Re
Re
16
14
50
45
40
35
12
30 10 0.2
0.4
0.6
0.8
1.0
0.2
Velocity (m/s)
0.4
0.6
0.8
1.0
Velocity (m/s)
(a) Velocity-Nusselt number
(b) Velocity-total heat transfer rate 30
90 950
Original model Optimized model
80 850 Pressure (Kpa)
Total heat transfer rate (UA)
85
UA:Original model UA:Optimized model
900
800 750 700
75 70 65 60
650
55 600 0.2
0.4
0.6
0.8
50
1.0
0.2
Velocity (m/s)
(c)
0.4
0.6
0.8
1.0
Velocity (m/s)
Velocity-total heat transfer coefficient
(d) Velocity-total pressure drop
Fig.14. The oil-side thermal performance comparison between before optimization and after optimization.
(a) original model
(b) optimized model
Fig.15. Temperature distribution of the original model and optimized model under the same air velocity of 8m/s
31
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
18.0
0.2
17.4
17.4 17.2
17.0
17.0
16.8
16.8
16.6
16.6
16.4
16.4
16.2
16.2
16.0 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1 5000
original model optimized model
17.6
17.2
0.2
0.3
5000
17.8
original model optimized model
17.6
Total pressure drop (pa)
Total rate of heat transfer(W)
17.8
1.1 18.0
4000
4000
3000
3000
2000
2000
1000
1000
0
16.0 1.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1.1
Velocity (m/s)
Velocity (m/s)
(a) The total rate of heat transfer comparison (b) The total pressure drop comparison Fig.16.
The comparison of simulation value between before optimization and after optimization in CFD.
(a) Original model (b) Optimized model Fig.17. porous media mesh model before and after optimization
32
Research Highlights ►A double flow PFHE model whose heat transfer element is offset staggered fin have been established.►A GA combined with orthogonal design is used to search for the optimal overall structure. ►The correlations about the fin j-factor and f-factor are obtained. ►Numerical results show that the optimized heat transfer effect is better than before.
33
S1359-4311(16)30561-0 http://dx.doi.org/10.1016/j.applthermaleng.2016.04.074 ATE 8121
To appear in:
Applied Thermal Engineering
Received Date: Accepted Date:
5 March 2016 16 April 2016
Please cite this article as: D. Juan, Y. Man-Ni, Y. Shi-Fang, Correlations and optimization of a heat exchanger with offset fins by genetic algorithm combining orthogonal design, Applied Thermal Engineering (2016), doi: http:// dx.doi.org/10.1016/j.applthermaleng.2016.04.074
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Correlations and optimization of a heat exchanger with offset fins by genetic algorithm combining orthogonal design Du Juana , Yang Man-Nib, Yang Shi-Fang*,c a
School of Aeronautics and Astronautics Engineering, Guizhou Institute of Technology, Guiyang,Guizhou,550003,China
b
c
School of Materials Engineering, University of Manchester, Manchester,M139PL,UK
Chemical and Chemical Engineering Institute, Hubei University, Wuhan, Hubei,430062, China Corresponding author: Tel: +86 13627246526 E-mail addresses:[email protected]
Nomenclature
A
total heat transfer area, (m2)
Aff
free flow area, (m2)
C
heat capacity rate, (W/K)
Cp
specific heat at constant pressure, (J/kg K) hydraulic diameter,
f
friction factor, dimensionless
G
mass flow rate of fluid, ( kg/s) mass flux of the air based, fin height, (mm)
h
convective heat transfer coefficient of the medium, (W/m2 K)
j
Colburn factor, dimensionless 1
lf
offset value, (mm)
L
depth of the heat exchanger in flow direction, (mm)
n
fin frequency,fins per meter
N
number of fin layers for fluid
Nu
Nusselt number, dimensionless
NTU
number of transfer units
Pr
Prandtl number, dimensionless Total heat transfer rate, (W) the maximum possible heat transfer rate,(W)
Re
Reynolds number
s
fin width, (mm)
T
temperature, ( K )
t
fin thickness, (mm)
U
overall heat transfer coefficient, (W/m2 K)
V
volume flow, (m3/s)
Greek symbols
fin wrinkling angle, (º)
P
pressure drop, ( Pa )
T
temperature difference, ( K )
thermal conductivity, (W/m K) fin surface efficiency
o
surface effectiveness
2
fluid dynamic viscosity, (N s/m2)
density, (kg/m3)
porosity
heat exchanger effectiveness,
Subscripts 1
air side
2
oil side
c
cold side
h
hot side
in
inlet
m
mean value
min
minimum value
max
maximum value
out
outlet
Abstract：In this paper, the study is focused on a double flow plate-fin heat exchanger (PFHE) which heat transfer element is staggered offset fin, and heat transfer model and the energy equations for the structure have been established, seven geometric parameters such as the fin height, fin length and fin wrinkling angle are taken as the decision variables for optimization. A genetic algorithm (GA) combined with orthogonal design is used to search for the
optimal overall structure and the correlations about the fin heat transfer factor j and the friction factor f. The maximum total heat transfer rate and the minimum total pressure drop are taken as objective functions in the GA, respectively. Performance of the optimized structure is evaluated and correspondingly the heat transfer and hydrodynamic characteristics of the full-size PFHE are calculated by using a porous media approach. Numerical results 3
show that the total heat transfer rate of the optimized structure is improved about 6.2% comparing with the original design, the total pressure drop decreases by about 40% and the volume can reduce about 2.7%. Keywords: Plate-fin heat exchanger; Genetic algorithm; Orthogonal design; Optimization; Porous media.
fin geometries. Based on the regression of experimental data and deviation analysis, the experimental correlations of Colburn j-factor and Fanning friction f-factor about laminar flow and turbulent flow are obtained. Manson[5] put forward heat transfer and resistance experimental correlations of the offset fins in addition to
1.Introduction
Kays and London’s experiments. Dubrovsky and Vasiliev[6] established the
Offset fins are widely used for the
local Nusselt number and the thermal
plate-fin heat exchanger (PFHE) to improve
resistance loss experimental correlations by
heat transfer rate by using staggered offset
the experiment research on 11 different
fins as the extended surface to enlarge heat
staggered offset fins. Wieting[7] analysis 22
transfer area and regenerate thermal
different structural parameters of staggered
boundary layer in each column. However,
offset fins, gave the power law experimental
offset fins induce a large pressure drop
correlations on the heat transfer and flow
between the inlet and outlet of the PFHE.
resistance performance. Joshi and Webb[8]
Therefore, the heat transfer and pressure
built empirical correlations of Colburn
drop in the PFHE with offset fins need to be
j-factor and Fanning friction f-factor in
investigated. Furthermore, the overall
laminar and turbulent flow regimes
structure of the PFHE with offset fins must
according to the test data of 21 kinds of
be optimally designed so that it can
offset finned heat exchanger. Rational
reconcile these contradictory phenomena.
design equations are presented by Manglik[9] in the form of single continuous
Many studies have examined the PFHE
expressions covering the laminar, transition,
with offset fins. Since most of them used air
and turbulent flow regimes. In addition to
as the working fluid, few fin geometries are
regular finned structure parameters are
applicable to practical offset fins with
studied, Tension [10] researched for the
working fluids other than air. Kays and
"stagger position of incision" also and the
London[1] carried out early experimental
corresponding heat transfer correlations
investigations on the offset fin geometry and
were obtained. Guo Lihua[11] carried out
demonstrate a correlation based on the
experimental studies to three different types
experimental results. Later Manglik and
of the steel staggered offset fins, presented
Bergles[2] , London and Shah[3]，
empirical correlations about the j-factor and
Mochizuki and Yagi[4] et al. have made
f-factor of the offset fins for lubricant oil.
broad and profound researches on similar 4
Some special applications have higher
estimated the optimal geometry parameters
request for the volume of the PFHE. In
within an acceptable range while satisfying
order to make per unit volume in a larger
the above two objective functions. Xie et
quantity of heat, the PFHE overall structure
al.[18]minimized the total volume as well as
optimization become the focus of the
the total annual cost of a compact heat
present studies. Thus, various optimization
exchanger by considering three shape
methods have been proposed. Guo. et
parameters as decision variables. Wang et
al.[12] applied optimization design approach
al.[19]applied genetic algorithm to optimize
based on the field synergy theory and
primary energy saving annual total cost
genetic algorithm to improve the
saving, and carbon dioxide emission
shell-and-tube heat exchanger performance
reduction. Peng and Ling [20] successfully
and reduce the total cost. The optimal
used GA combined with back propagation
design leads to a significant cost cut and an
neural network for the optimal design of
improvement of the heat exchanger
heat exchanger by considering the minimum
performance. Sahin et al.[13] used Taguchi
total weight and total annual cost for a given
method to optimize the design parameters of
constrained condition as objective functions.
the heat exchanger with staggered offset
Recently, Xie et al.[21-24] designed
fins. In recent years, the multi-objective
micro-channel heat sinks and optimized
optimization has been successfully used to
different kinds of heat exchangers by using
many thermal systems. Gholap and
different optimization methods such as:
Khan[14] took energy consumption and
entropy generation minimization and the
material cost as the objective function of the
constructal theory[25,26].
two conflicts and carried out multi-objective optimization to get the optimal design parameters of a forced air heat exchanger. Hilbert et al.[15] also used a multi-objective optimization technique to maximize the heat transfer rate and to minimize the pressure drop in a tube bank heat exchanger. Najafi, et al.[16] applied genetic algorithm multi-objective optimization technique to provide the optimum geometric parameters of a PFHE. Sanaye and Hajabdollahi[17] applied multi-objective optimization to maximize the effectiveness and to minimize the pressure drop in a rotary regenerator and
This paper investigates the characteristics of heat transfer and pressure drop for a PFHE, the fin height, fin width, fin length, fin offset, and fin wrinkling angle are considered as optimization parameters within reasonable constraints, the total heat transfer rate and the total pressure drop are considered as two conflicting objective functions. Multi-objective optimization technique using genetic algorithm combined with orthogonal design is utilized in order to achieve a set of optimal solutions, called pareto multiple optimum solutions. The sensitivity analysis of the total heat transfer 5
rate and total pressure drop changed with
in table 1. Each layer is separated by
the design parameters was performed and
clapboard and two different flow channels
the results are reported and the correlations
are built. The oil-side fins are separated by
of the characteristics of heat transfer and
seals in the middle. (shown in Fig.3 and
pressure drop for the optimum structure
Fig.4)
were proposed. In addition, a series of Rectangular staggered offset fins are
comparative verification are also carried out in order to investigate the effect of some of the geometric variables on the objective functions by using CFD software Fluent.
used on air side and form a single-flow channel. In order to improve the thermal efficiency of the PFHE, the air-side fins are higher than the oil side, the fin appearance and front view are shown in Fig.5 and the
2. Modeling of the system
side view are shown in Fig.6. The main geometric parameters are provided by the
2.1 Physical Model and Assumptions
commission unit and shown in Table 2.
The PFHE with staggered offset fins studied in this paper exchanges heat under the working condition of an oil-to-gas heat transfer and the hot side is oil and the cold side is air. Hot oil flows across a trapezoidal offset fin while cold air flows across an rectangular offset fin in adjacent layers which are arranged staggered. The schematic diagrams of the PFHE are shown in Fig.1~Fig.6, the heat is transmitted to the partition through the oil-side fin surface, resulting in the temperature rise of the air side. The total heat transfer and pressure drop characteristics on the heat exchanger are to be found by numerical computations based on the suggested model. In order to
To simplify the calculation process, following assumptions are made for the analysis: (1) Mass conservation and energy equations and the Navier-Stokes can be used to analyze the physical processes. (2) Physical property variation of the fluids with temperature is neglected and the air is assumed to be ideal gas. (3) The fluid flow is single-phase, incompressible and the PFHE is operating under steady state condition in the computational domain. (4) The wall is considered as an ideal surface, which means there are no burrs, scarped edges, or adhesive substances and the thermal resistance and fouling effect are neglected.
avoid heat loss, the number of air-side fins layers are more than oil layer, and the oil
2.2 Data Reduction
side are 29 layers and the air side are 30 layers. In the optimization and analysis, 46# lubricating oil is used in hot side, which
In the analysis, to provide the heat transfer characteristics of the tested sample, ɛ-NTU method is used to determine the UA
main thermodynamic parameters are given 6
term of the heat exchanger[27]．The UA
Heat transfer area of both side are achieved
product was calculated using ɛ-NTU method
using the following relation:
for the unmixed cross-flow configuration.
Ah Lh Lc Nh 1 2nh ( H h th )
Correspondingly, the appropriate ɛ-NTU
(8)
Ac Lh Lc Nc 1 2nc ( H c tc )
relationship is:
1 exp
(9)
1 0.22 exp Cr NTU 0.78 1 Cr NTU
Total heat transfer area,
(1)
A Ah Ac
where, Cr Cmin / Cmax , assuming zero fouling factor and zero thermal resistance,
The rate of heat transfer (the first objective
NTU is number of transfer units and can be
function): Q Cmin (Th,1 Tc,1 )
obtained as: The pressure drop of the both side (the
1 1 1 UA ( hA )h ( hA )c
(2)
second objective function):
1 A f f h 1 A f cf C 1 m i n Cm i n NTU UA h h h A c h c A
Ph
(3)
2 f h LhGh2 h Dh
Pc
2 f c L cG 2 c c Dc
(10)
(11)
where ,1/UA is the thermal resistance of heat convection, and convective heat
3. Optimization
transfer coefficient is calculated using
hh jh Gh Cph Prh 2 / 3
(4)
3.1 Genetic Algorithm Genetic algorithm (GA) is a bionic
hc jc Gc Cpc Prc 2 / 3
(5)
Aff h ( H h th )( 1 nhth )Lh Nh
(6)
Affc ( H c tc )( 1 nctc )Lc Nc
(7)
global optimization algorithm inspired by natural selection and evolutionary processes, which was first conceived by Holland[28,29]. The idea is: imitating Darwin's theory of evolution and Mendel’s law of inheritance through operation mechanisms such as selection, crossover 7
and mutation, continuously improving the
PFHE system which lead to the maximum
fitness of the individuals in the population.
total rate of heat transfer and minimum total
Genetic algorithm begins from a population
pressure drop. The total heat transfer rate Q
which may be a potential solution set of
is considered as one of the objective
representative problems and the initial
functions and -Q is minimized through the
population is usually randomly generated.
optimization procedure which in turn leads
Population is made of a certain number of
to Q maximization. The operating
individuals and each individual is actually
conditions for the considered case are given
the entity with chromosome characteristics.
in Table 1. Both fluids are considered as hot
Chromosomes are the main carrier of
oil and cool air, respectively. Applicability
genetic material, and a collection of
of the geometry range is listed in Table 3.
multiple genes. Afterword, parents are
Considering the overall heat exchanger
selected according to their fitness values.
structure size cannot be larger than the
The higher fitness of an individual is, and
original model which the oil-side and
the greater possibility there is of being
air-side fin height is 2mm and 12.8mm
selected as the parent for the reproduction.
respectively, the optimized total height is
In the next step, the reproduced two
smaller than the original total height, so the
chromosomes combine together and form a
oil-side and air-side fin height can be
new chromosome, called the offspring.
desirable 2.6mm and 12mm respectively.
Since individuals with higher fitness have
Also because the oil-side fin is isosceles
more chance for being selected and produce
trapezoid and the biggest the angle is 90º, α=
offspring, the new population generated
45º ~ 90º and other parameters are based on
after reproduction possess more qualified
the principle of the little change of the
genes and consequently higher fitness. In
physical model size.
multi-objective genetic algorithm, the procedure is the same in addition that it has
4. Optimized results and
multiple fitness for each individual regarding different considered objectives.
discussions
(the workflow of the multi-objective genetic algorithm is shown in Fig.7.)
In each iteration, the values of optimization parameters are given new values within their specified constraints and
3.2 Multi-Objective Optimization
the considered objective functions are evaluated based on the new values. The
In the present work, a multi-objective genetic algorithm is utilized in order to
initial population size is considered to be 200, the maximum number of generations is
obtain optimal geometric parameters of the 8
500 and the generation gap is selected to be
4.1.1 Effect of the fin height
0.9. Select strategy uses a genetic algorithm
In order to detect geometric parameters
of crossover probability 0.7 and mutation
such as fin height, fin length, fin wrinkling
probability 0.5.
angle etc. on the influence of total heat
The optimization is terminated after 500 generations. Considering the given operating conditions in Table 2, The first and second objective function values after 500 times iteration are shown in Fig.8, some of the selected optimal design parameters obtained by utilizing the GA are given in
transfer and pressure drop, the oil-side inlet velocity of the PFHE is increased from 0.2m/s to 1m/s. Fig.9 shows the variation of the total rate of heat transfer and pressure drop versus the inlet velocity under constant flow rate and three different oil-side fin heights.
Table 4. Fig.8(a) illustrates the evolution process of the first objective function (the total rate of heat transfer).Fig.8(b) is the evolution process of the second objective function (the total pressure drop).Fig.8(c) shows the first objective function value and the second objective function values after 500 times iteration. From Fig.8(c), we can see that the optimal value fall between 5 and
It can be seen from the Fig. 9 that with the increase of flow velocity, the total heat transfer rate and pressure drop of the three kinds of fins always increasing, but when the fin height H2=2.6 mm, the total heat transfer rate is maximum and the total pressure drop is minimum. This results show that among the three different fins, the fin with H2 = 2.6 mm is the most optimal oil-side fin height.
6 sets of data.
4.1 Sensitivity Analysis
Fig.10. shows the variation of the total heat transfer rate and pressure drop versus the inlet velocity for the PFHE with the
In this section, considering the
three kinds of fin height. It can be seen that
operating condition and constant values
with the increase of flow velocity, the total
given in Table 1 and Table 2, the effects of
heat transfer rate of the three kinds of fins
some geometric variables on objective
will increase, and the total pressure drop
functions are investigated. In each section,
keeps stable. When the fin height H1=
the values of all parameters, except those
12.8mm, the total heat transfer rate is
which are selected for the investigation, are
maximum. The main reason is: when the
kept constant. By varying the values of the
other conditions are the same, the higher the
selected parameters, the sensitivity of each
fin, its heat transfer area is larger, also the
objective function with respect to the
more heat transfer. And the pressure drop is
parameters can be discussed.
mainly related to flow velocity and flow 9
direction of length, so when other
transfer rate of the three kinds of fins will
conditions are constant, in a small range of
increase, and the total pressure drop will
variation the fin height has negligible effect
increase too. When the fin wrinkling angle
on the pressure drop of the fluid. This can
85o , the total heat transfer rate is
be seen from the Fig.10(b).
maximum. The main reason is: when the other conditions are the same, the greater
4.1.2 Effect of the fin lenght
the fin angle, the more fin number and the
Fig.11. shows the variation of the total
more heat transfer. At the same time, the
heat transfer rate and the total pressure drop
thickness of the fin also makes the vortex
versus the inlet velocity for the heat
generation and shedding more obvious, so
exchanger with the three kinds of fin length.
the heat transfer effect will be increased
It can be seen that with the increase of flow
with the increase of fin wrinkling angle.
velocity, the total heat transfer rate of the three kinds of fins will increase, and the
4.2 Orthogonal design test
total pressure drop will increase too. But the
Due to involving seven parameters in
total pressure drop increases from the size
the process of optimization, it will be very
of l2 = 3.1 mm to l2 = 3.2 mm less than that
difficult to implement if the seven
of from l2 = 3.2 m to l2 = 4.2 mm and the
parameters are tested fully. For the reason
total heat transfer is almost the same. The
an orthogonal design method can be used to
main reason is: when the other conditions
find the optimal level combination. The
are the same, the total heat transfer rate and
basic idea is to arrange and analyze these
the total pressure drop are function of the
factors by using the orthogonal table and
velocity respectively. The offset fin periodic
some representative level combinations are
interrupt on flow direction, leading to the
selected from all level tests, that is to say,
periodic start and end at fin end boundary
through the part of the test results to analyze
layer, the shorter the fin length greatly
the overall situation and find the optimal
hinders the increase of the boundary layer
design parameters. In reality there are
and leads to the increase of heat transfer rate
various factors affecting heat transfer effect
and pressure drop simultaneously.
if the physical experimental conditions are
4.1.3 Effect of the fin wrinkling angle Fig.12 shows the variation of the total heat transfer rate and the total pressure drop versus the inlet velocity for the heat exchanger with the three kinds of fin wrinkling angle. It can be seen that with the increase of flow velocity, the total heat
taken into account. L15 orthogonal array is normally used in investigating effects of a 7
three-stage seven factor (3 ) combination. However, as is the case in many studies, some factors may be neglected thereby reducing the number of experiments and testing time [30]. As for seven parameter 10
and three-level array, L15(37) used in this
correlations about the fin heat transfer factor
study, consistency was maintained in
and the friction factor are obtained which
applying the Taguchi standard orthogonal
are suitable for this research of the geometry
design where a total of 15 sets of data were
parameters about the Reynolds number,
conducted. The design arrangement suitable
wrinkling angle, fin height, fin length.
4
for the study carried out L9(3 ) (combined
Consequently, the following equations are
with results of the section 4.1, the total is 10
obtained:
sets of data) together with its corresponding reactions are given in Table 5. The first
jcorr,1 28.5627 Re
1.178
parameter column of the table shows
H Dc
0.1125
(12)
air-side fin height, the second column gives oil-side fin height, the third column is for
f corr,1 0.9196 Re
oil-side fin length and the last column lists
0.3155
H Dc
-0.0127
(13)
the angle of oil-side fin. The farthermost right of the table are the values Q and
P .
H jcorr,2 0.1627 Re0.4821 0.4517 Dh
0.425
l Dh
(14)
From the Table 5, it can be seen that the Q value of the tenth group is larger and P is more moderate. Taking into
0.04547
f corr,2 147.08Re
account the actual requirements of the
1.052
0.7246
H Dh
0.3398
l Dh
volume of the heat exchanger and the
(15)
modification cost of the fin, the optimization results are taken as the tenth group parameters.
4.3 The corrections of the optimized model In section 4.1, the influences of
5. Optimization Results and Analysis 5.1 Result Analysis of the obtained correlations
different fin angles, fin length and fin height Fig.13 shows the comparison of the
on the performance of fin were qualitatively tested. In the section 4.2, the typical 10 sets of fin measurable data were listed by applying the orthogonal design method and the optimal fin geometric parameters with comprehensive performance were obtained. Here, the 10 sets of data were fitted and the
0.0527
correlations obtained in this paper against the correlations from Wieting[7] and Manglik[9]. The predictions from the correlations are reasonable good agreement with the open literature values at different Reynolds numbers, but some deviations are 11
observed. Fig.13(a) shows the comparison
results are significantly better than
results of the j-factor and f-factor before and
un-enhanced results. The above analysis
after optimization on air side. It is apparent
shows that the optimization results are
that the j-factor and f-factor values have
correct and feasible.
little change after optimized. But for the oil
From the above analysis and
side, the j-factor of the optimized model is significantly higher than that of before optimization at the same Reynolds number (see Fig.13(b)) and f-factor is higher than that of original model too(see Fig.13(c)). The main reasons for analysis are as follows: the first reason is that the used working fluids are different and the test medium in literature is air of Prandtl number 0.7 and the medium used in this paper is lubricating oil with Prandtl number greater than 100, and the literature[9] clearly pointed out whether the correlations based on the gas medium are suitable for the liquid medium has to be further verified. The
discussion, combined with the fin parameters in Table 2, the final optimization results are an optimal compromise of two conflicting objective functions of the total heat transfer and the total pressure drop. Considering limits of the manufacturing equipment, process conditions and the available investment, we can determine the optimal solution (12.8mm,2.6mm,3.45mm,3.2mm,1.24mm,1. 25mm,0.66mm,0.57mm,85º), as shown in Table 6. By contrast, we can see the change of the PFHE in the physical dimension is quite small.
second reason is that the form of fin is
Fig.14 shows comparing results of the
different and the fins are rectangular
heat transfer characteristics and flow
staggered array in the literature and the fins
characteristics for the original model and the
are trapezoidal staggered array in this paper;
optimized model of the PFHE. Fig. 14(a)
in addition, the contact thermal resistance
shows the comparison of the Nusselt
and the data error are also important factors.
number, Fig.14(b) is the comparison of the
Currently, the overall performance of the
total heat transfer rate, Fig.14(c) is the
enhanced heat exchanger is evaluated using
comparison of the total heat transfer
the criterion of goodness factor, j/f. The
coefficient. It can be seen from the graphs
baseline (un-enhanced) results are compared
that with the increase of flow velocity, the
to the enhanced performance in Fig.13(d). It
Nusselt number, the total rate of heat
can be seen from the figure that goodness
transfer and the total heat transfer
factors of optimized model are better than
coefficient are all increased. The Nusselt
that of the original model (before
number of the optimized model is increased
optimization) , in terms of its
by about 3.6%, the total heat transfer rate is
comprehensive performance, the optimized
relatively more 4%~11% and the total heat 12
transfer coefficient is higher about
same in the inlet, but surface heat transfer
4.5%~7.5% than the original model.
process of the optimized model cold air is much more sufficient than the original
Fig.14(d) shows comparing results of heat transfer resistance characteristics for the original model and the optimized model of the PFHE. It can be seen from the graph that with the increase of flow velocity the total pressure drop are all increased. Under the condition of the same flow, the total pressure drop of the optimized model is down by about 24% than the original model.
5.2 CFD numerical results and discussion
model. For the original model, keeping the air-side entrance velocity of 8 m/s, the oil-side entrance velocity are increased from 0.3 m/s to 1 m/s, and for the optimized model, the air-side entrance velocity is 8.678 m/s and the oil-side entrance velocity are increased from 0.231 m/s to 0.769 m/s, the corresponding the hydrodynamic characteristics are shown in Fig.16. The results obtained from Fluent in the form of the total heat transfer rate and the total the pressure drop are compared with the
In this section, the simulation model, meshing, boundary conditions and the simulation calculation method are all proposed in the literature [31].Here, these can be omitted. In the simulation calculation process, under the same volume flow, the entrance velocity corresponding relation of the original model and the optimized model is shown in Table 7 about oil side and air side. For the two kinds of model, when oil side entrance velocity of 0.9 m/s and 0.692 m/s respectively, air inlet velocity of 8 m/s and 8.678 m/s respectively, numerical calculation is implemented by using the CFD software Fluent, the plane y = 2.65 parallel to x-z is established after the convergence. The temperature distribution of the original model and the optimized
original model in Fig.16. From Fig.16(a), It is evident that the total heat transfer rate of the original model and optimized model are increased with the increase of flow velocity, and under the same volume flow, there is an increase of about 2% than before optimization. As can be seen from the Fig.16(b), the total pressure drop increases with the flow velocity and there is an decrease about 35% when compared with the original model. In addition, the range of pressure drop will be increased with the increase of flow velocity. The analyses above show that the improvement of the performance of the overall heat transfer for the PFHE is smaller than the decrease of the pressure drop when the fluid flow rate is constant.
model obtained from Fluent are compared as shown in Fig.15. It can be seen from the Fig.15 that the temperature of the two models air side and oil side are almost the 13
5.3 The simulation analysis with porous media Due to the growing size and complicacy of the PFHE, it is not feasible to simulate actual full-size PFHE, which
obtained as following (From local emulated data): oil side: P=640.92+695.9 porous coefficient: c1=1/α=1.33107
generally requires large computational resources and time. In this paper, two side
c2=190
porosity: =0.8914
fins can be simulated by porous media and
air side: P=0.47432+4.994
porous media technology is applied to solve
porous coefficient: c1=1/α=2.77107，
large-scale computing problems. The c2=104.16
original porous media model is given in literature[32] and the optimized porous media model can be obtained on base of modifying locally parameters of the original structure as shown in Table 6, and then input them into Workbench to be meshed, the porous medium local mesh model of original and optimized structure are shown
porosity:=0.9419
Under the boundary conditions listed in literature [32] and above porous parameters, the total heat transfer rate and the total pressure drop of the data are calculated respectively in different flow rate, the results are shown in Table 9 and Table 10.
in Fig.17. Table 9 is a simulation value contrast The above two porous media models are separately imported into the Fluent software and the boundary conditions are loaded as section 5.2, the viscous and inertia resistance coefficient of optimized porous media model can be obtained by local fin simulation results: the local simulation model as shown in literature [31], the oil-side fin length is 8mm and air side is 8.6mm, the fin local simulations are carried out and the results are shown in Table 8. Based on the analysis method of
of total heat transfer rate of the overall porous media models of the plate-fin heat exchanger before and after the optimization under the same volume flow. It can be seen that the heat exchange capacity of the heat exchanger is 6.2% larger than the before optimization. Table 10 is a simulation value contrast of total pressure drop of the overall porous media models of the plate-fin heat exchanger before and after the optimization under the same volume flow, the optimal design indicates an obvious effect with a
literature[32], for the optimized porous
40% decrease on total pressure drop and a
media model, the relationship between
2.7% decrease on total volume of the heat
entrance velocity and pressure drop can be
exchanger. The results are further evidence that the genetic algorithm combined with 14
orthogonal design to optimize the overall
the calculation results are in good
structure of the heat exchanger have great
agreements with the literature data. At the
engineering practical values.
same time, be it the heat transfer characteristics or resistance characteristics,
6. Conclusions
the optimized model is better than that of the original model.
Through using genetic algorithm
3D models of the overall heat
combined with the orthogonal design to
exchanger before optimization and after
optimize the overall structure of the PFHE
optimization are built up. Applying porous
with offset staggered fins, the total heat
medium technology and combining the local
transfer rate is maximized and the total
simulation results, the full-size numerical
pressure drop is minimized simultaneously,
simulations are carried out by FLUENT
so as to satisfy the engineering application.
software, and the numerical simulation
The following conclusions can be made.
results show that under the same volume
Mutli-objective genetic algorithm is
flow, the performance of the optimized
utilized in order to obtain optimal geometric
PFHE for the total rate of heat transfer
parameters of a PFHE system which leads
increased by about 6.2%, the total pressure
to the maximum total rate of heat transfer
drop decreased by about 40% and the
and minimum total pressure drop. The fin
volume reduced about 2.7%.
height, fin width, fin length, fin offset, and fin corrugation angle are used as design
References
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Table 1 Operating conditions for the case study Parameters
Hot fluid(oil)
Cold fluid(air)
Volume flow, V (m3 / s)
0.015
1.573
Inlet temperature, Tin (K)
413
328
Density, (kg m3 )
844
1.06
Specific heat, Cp (kJ / (kg K))
2.013
1.0087
Viscosity, (N s m2 )
6566E-6
2.1E-6
Thermal conductivity of fluid，
0.1237
0.0305
113.4
0.6945
(W / (m K))
Prandtl number, Pr
Table 2
The detailed geometric parameters of the PFHE 18
Unit：mm
Parameter
H1 12.8
s1
l1
t1
1.24 3.45
0.1
lf1 0.66
H2 2
s2
l2
1.25 3.3 0.55
Table 3 The geometry ranges of the design parameters H1 1-12
H2 1-2.6
l1 1-3.8
lf2
l2
s1
s2
lf1
1-3.8
1-3
1-3
1-1.5
82.06°
Unit：mm
lf2 1-1.5
45-90°
Table 4 The optimization results after 500 iterations by GA algorithm lf2
Q
P
/W
/kPa
H1
H2
l1
l2
s1
s2
lf1
/mm
/mm
/mm
/mm
/mm
/mm
/mm
12.00
2.59
1.02 1.19
1.02 1.00 0.29 0.20 89.98 89384 31837
12.00
2.59
1.02 1.19
1.02 1.00 0.32 0.24 89.97 88527 28638
11.99
2.59
1.02 1.19
1.02 1.00 0.29 0.20 89.98 76906
/mm
/°
Pereto solvers
35743 11.99
2.59
1.02 1.19
1.02 1.00 0.29 0.20 89.98 89388
2.60
1.02 1.19
1.02 1.00 0.29 0.20 89.97 88654
11.76
2.60
3.77 3.80
2.99 2.99 1.50 1.50 45.06 29630
3221
11.76
2.60
3.48 3.80
2.99 2.51 1.50 1.50 45.06 29830
3308
11.76
2.60
3.48 3.76
2.94 2.99 1.50 1.50 45.06 29759
3225
31836 11.99 31869
19
11.76
2.60
3.48 3.80
3.00 2.99 1.50 1.50 45.06 29614
3221
11.76
2.60
3.77 3.80
2.99 2.94 1.50 1.50 45.20 29632
3222
Table 5 Number of tests
Four-factor and three-level orthogonal table
Parameter
Results
H1(mm) H2(mm) l2(mm) α(º)
Q (kW)
P (kPa)
P
1
10.8
2.4
2.2
65
29.75
47.63
2
10.8
2.5
3.2
75
34.45
53.84
3
10.8
2.6
4.2
85
40.73
67.26
4
11.8
2.4
3.2
85
41.84
71.79
5
11.8
2.5
4.2
65
30.36
44.20
6
11.8
2.6
2.2
75
35.81
52.41
7
12.8
2.4
4.2
75
36.57
55.82
8
12.8
2.5
2.2
85
43.83
70.70
9
12.8
2.6
3.2
65
31.06
42.38
10
11.8
2.6
3.2
85
42.49
67.26
0.625
0.640
0.613
0.583
0.687
0.683
0.655
0.620
0.733
20
Q/
0.632
Table 6. The geometry values of the design parameters for the optimized model H1
lf2
/mm /mm /mm /mm /mm /mm /mm
/°
H2
/mm The optimized model 11.8
2.6
l1
3.45
l2
s1
3.2 1.24
s2
1.25
lf1
0.66 0.57
85
Table 7 The inlet velocity corresponding relation of the original model and the optimized model under the same volume flow oil side inlet velocity original model
air side inlet velocity
optimized model
original model
optimized model
(m/s)
(m/s)
(m/s)
0.3
0.231
3
3.254
0.4
0.308
4
4.339
0.5
0.385
5
5.424
0.6
0.462
6
6.509
0.7
0.539
7
7.593
0.8
0.615
8
8.678
0.9
0.692
9
9.763
1.0
0.769
10
10.848
Table 8
(m/s)
The local simulation results in the two side fins
21
oil side
air side
inlet velocity
pressure drop
inlet velocity
pressure drop
(m/s)
(Pa)
(m/s)
(Pa)
0.6
647.38
6
52.73
0.7
801.8
7
64.48
0.8
967.81
8
78.58
0.9
1144.75
9
91.86
Table 9
Comparison of the total rate of heat transfer between before optimization and after optimization of the overall porous media model Oil side
编 original model flow No
Air side optimized
flow
original model 3
model
model m /h
optimized
kW
kW
m3/h
kW
kW
1
4.480
53.64
56.79
5106.3
53.52
56.65
2
4.570
54.76
57.97
5106.6
54.55
58.00
3
4.800
75.74
69.78
5265.0
72.23
69.94
4
5.317
76.63
81.71
5666.9
76.40
81.50
5
5.324
76.39
81.37
5660.6
76.56
81.59
6
5.330
80.65
85.85
5686.8
80.41
85.65
7
5.355
81.74
88.30
5675.0
81.41
88.33
8
5.360
81.54
95.67
5671.0
80.16
95.42
9
5.700
87.90
88.54
5623.1
87.66
88.75
10
6.420
89.16
95.84
5663.2
88.92
95.60
22
11
6.500
87.65
94.24
5664.4
87.42
94.01
12
6.508
87.98
93.21
5659.4
87.75
93.49
13
6.585
89.58
93.56
5696.6
89.33
93.82
Table 10
Comparison of the total rate of heat transfer between before optimization and after optimization of the overall porous media model Oil side
No
flow
original model
Air
optimized
flow
model
side
original model
optimized
model
m3/h
kPa
1
4.480
131.59
2
4.570
3
kPa
m3/h
kPa
76.35
5106.3
0.5419
0.7919
133.9
78.55
5106.6
0.5420
0.7917
4.800
145.1
82.47
5265.0
0.5668
0.8302
4
5.317
158.1
95.61
5666.9
0.6321
0.9315
5
5.324
167.0
95.75
5660.6
0.8527
0.9299
6
5.330
175.0
95.03
5686.8
0.6354
0.9367
7
5.355
159.3
95.65
5675.0
0.6344
0.9723
8
5.360
159.4
123.88
5671.0
0.6328
0.9326
9
5.700
186.3
104.13
5623.1
0.6328
0.9202
10
6.420
219.2
122.93
5663.2
0.6317
0.9308
11
6.500
201.2
125.24
5664.4
0.6321
0.9309
12
6.508
223.5
125.46
5659.4
0.6317
0.9296
13
6.585
227.3
127.52
5696.6
0.6370
0.9392
23
kPa
Oil outlet
Oil inlet
air outlet
air inlet
Oil-side fins Air-side fins
Fig.1.
The overall schematic of the PFHE
Air outlet Air inlet
Oil inlet
Fig.2.
Oil outlet
The adjacent two layers in the model of the whole
24
oil inlet
oil outlet
Fig.3.
The top view of the oil-side fins
(a) Fins profile
(b) The front view Fig.4.
The oil-side fins
25
(a) Fin profile
(b) The front view Fig.5.
The air-side fins
(a) Air side Fig.6.
Fig.7.
(b) Oil side
The oil-side view and the air-side view
The workflow of the multi-objective GA
26
4
-5.5
x 10
4400 4200
-6
Total pressure drop
Total rate of heat transfer
(Pa)
(W)
-6.5 -7 -7.5 -8
4000 3800 3600 3400
-8.5
3200
-9
3000
-9.5
0
50
100
150
200 250 300 Generation
350
400
450
500
2800
0
50
100
150
200 250 300 Generation
(a)
350
400
450
500
(b) 4
4
x 10
2
Objective function value
0 The first objective function value The second objective function value
-2
-4
-6
-8
-10
1
2
3
4
5
6
7
8
9
10
Population
(c) Fig.8. (a) Evolution process of the objective of the total rate of heat transfer. (b) Evolution process of the objective of the total pressure drop. (c) The first objective function value and the second objective function values after 500 times iteration.
27
55
65 50
H2=2.5mm
H2=2.5mm H2=2.6mm
Pressure drop (Kp)
Heat transfer rate (KW)
H2=2.4mm
H2=2.4mm
45
40
H2=2.6mm
60
55
50
35
45
30
0.2
0.4
0.6
0.8
0.2
1.0
0.4
0.6
0.8
1.0
Velocity (m/s)
Velocity (m/s)
(a) Velocity-total heat transfer rate
(b) Velocity-total pressure drop
Fig.9. Effects of the oil-side fin height H2 55
90 H1=10.8mm
50
H1=10.8mm 85
H1=11.8mm
H1=12.8mm
45
H1=12.8mm
Pressure drop (Kpa)
Heat transfer rate (KW)
H1=11.8mm
40
35
80
75
70
30
25 0.2
0.4
0.6
0.8
1.0
65 0.2
Velocity (m/s)
0.4
0.6
0.8
1.0
Velocity (m/s)
(a) Velocity-total heat transfer rate
(b) Velocity-total pressure drop
Fig.10. Effects of air-side fin height H1
28
52
l2=2.2mm
58
l2=3.2mm
l2=2.2mm
50
l2=4.3mm
Pressure drop (Kpa)
56
l2=4.2mm
48
46
44
54
52
50
42 0.4
0.5
0.6
0.7
0.8
0.4
0.9
0.5
0.6
0.7
0.8
0.9
Velocity (m/s)
Velocity (m/s)
(a) Velocity-total heat transfer
(b) Velocity-total pressure drop
Fig.11. Effects of the oil-side fin length l2
70 55
50 45
60
Pressure drop (Kpa)
Total heat transfer rate (KW)
Heat transfer (KW)
l2=3.2mm
40 35 30
50
40
30
25
20 20 15 0.2
0.4
0.6
0.8
1.0
10 0.2
Velocity (m/s)
(a)
0.4
0.6
0.8
1.0
Velocity (m/s)
Velocity-total heat transfer rate
(b) Velocity-total pressure drop
Fig.12. Effects of the oil-side offset fin wrinkling angle
29
0.20
800
1000
1200
1400
1600
1800 0.20
0.18
0.18
200
300
400
500
600
700 0.045
0.045
before optimization after optimization before optimization after optimization
0.16 0.14 0.12
j
0.040
0.14
0.035
0.12
0.030
0.030
0.025
0.025
0.020
0.020
0.015
0.015
0.02
0.010
0.010
0.00 1800
0.005
f
0.10
0.08
0.035
j
f
0.10
before optimization after optimization Manglik Wieting
0.040
0.16
0.08
0.06
0.06
0.04
0.04
j
0.02 0.00 800
1000
1200
1400
1600
200
300
400
Re
300
400
0.005 700
600
Re
(a) Comparison of air side 200
500
500
(b) Comparison of oil-side j-factor 600
700
0.5
200
before optimization after optimization Manglik Wieting
0.4 0.3
400
500
600
700 0.055
0.4
0.050
0.050
before optimization after optimization
0.3 0.045
0.045
0.040
0.040
0.035
0.035
0.030
0.030
0.025
0.025
0.2
f
j/f
0.2
300
0.055
0.5
0.1
0.1
200
300
400
500
600
200
700
300
400
(c) Comparison of oil-side f- factor Fig.13.
600
700
(d) Comparison of goodness factor
Comparison between the correlations presented in this paper and the open literature values
55
22
Q:Original model Q:Optimiz model Total heat transfer rate (KW)
Nusselt:Orignal model Nusselt:Optimized model
20
18 Nusselt
500
Re
Re
16
14
50
45
40
35
12
30 10 0.2
0.4
0.6
0.8
1.0
0.2
Velocity (m/s)
0.4
0.6
0.8
1.0
Velocity (m/s)
(a) Velocity-Nusselt number
(b) Velocity-total heat transfer rate 30
90 950
Original model Optimized model
80 850 Pressure (Kpa)
Total heat transfer rate (UA)
85
UA:Original model UA:Optimized model
900
800 750 700
75 70 65 60
650
55 600 0.2
0.4
0.6
0.8
50
1.0
0.2
Velocity (m/s)
(c)
0.4
0.6
0.8
1.0
Velocity (m/s)
Velocity-total heat transfer coefficient
(d) Velocity-total pressure drop
Fig.14. The oil-side thermal performance comparison between before optimization and after optimization.
(a) original model
(b) optimized model
Fig.15. Temperature distribution of the original model and optimized model under the same air velocity of 8m/s
31
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
18.0
0.2
17.4
17.4 17.2
17.0
17.0
16.8
16.8
16.6
16.6
16.4
16.4
16.2
16.2
16.0 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1 5000
original model optimized model
17.6
17.2
0.2
0.3
5000
17.8
original model optimized model
17.6
Total pressure drop (pa)
Total rate of heat transfer(W)
17.8
1.1 18.0
4000
4000
3000
3000
2000
2000
1000
1000
0
16.0 1.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1.1
Velocity (m/s)
Velocity (m/s)
(a) The total rate of heat transfer comparison (b) The total pressure drop comparison Fig.16.
The comparison of simulation value between before optimization and after optimization in CFD.
(a) Original model (b) Optimized model Fig.17. porous media mesh model before and after optimization
32
Research Highlights ►A double flow PFHE model whose heat transfer element is offset staggered fin have been established.►A GA combined with orthogonal design is used to search for the optimal overall structure. ►The correlations about the fin j-factor and f-factor are obtained. ►Numerical results show that the optimized heat transfer effect is better than before.
33