Optimizing the Design of Radiator using Genetic Algorithms
( Real –World Application )
Puneet Saxena
Graduate Student
Industrial Engineering Dept.
The University of Alabama
Tuscaloosa, AL 35487
Email: [email protected]
Phone: (205) 348 1659
Charles L. Karr
Associate Professor
Aerospace Engineering and
Mechanics Dept.
The University of Alabama
Tuscaloosa, AL
Email: [email protected]
Phone: (205) 348 0066
Abstract
This paper describes the application of genetic
algorithms to achieve the optimal design of a
radiator used in automobiles so as to achieve not
only the required performance but also to find a
cost effective solution. The performance of an
automobile radiator is a function of overall heat
transfer coefficient and total heat transfer area.
The basic thermodynamic equations are utilised
to enable the calculation of the overall heat
transfer coefficient of the vehicle radiator core
and thereafter the genetic algorithm is used for
manipulating the design parameters to achieve
the optimal solution.
1
INTRODUCTION
Radiators are heat exchangers responsible for controlling
engine-operating temperatures. The heat carried by the
cooling water jacket is generally 30% of the total energy
produced in the engine. This energy must be removed
constantly through the use of a heat exchanger, or a
radiator. A suitable radiator is used to achieve not only
the efficient performance of the engine but also the costeffective solution for the cooling system. In radiators, heat
carried by the coolant fluid is transported by convection
and conduction to the fin surface and from there by
thermal radiation into the atmosphere-free space. The hot
and cold fluids are separated by an impervious surface
and hence they are also referred to as surface heat
exchangers. In the case of a radiator, the hot fluid flows
inside the tubes and so the hot fluid is unmixed. However,
the cold fluid flows over the tubes and is free to be mixed.
The mixing tends to make the fluid temperature uniform
in transverse direction; therefore, the exit temperature of a
mixed stream exhibits negligible variation. The total heat
transfer rate between the fluids is dependent on the
Keith A. Woodbury
Associate Professor
Mechanical Engineering Dept.
The University of Alabama
Tuscaloosa, AL
Email: [email protected]
overall heat transfer coefficient and the total heat transfer
area.
The design optimization problem involves both explicit
constraints (such as fixed frontal area) and implicit
constraints (such as those specifying the heat transfer
coefficient). Once the geometry is selected, additional
constraints such as minimum and maximum values for the
fin pitch, minimum and maximum number of tubes and
the cross-section of the tubes are imposed, and thereafter
the problem reduces to that of solving the problem within
the ranges of variables specified to achieve the optimal
design. The overall heat transfer coefficient is dependent
on the number of tubes; in general, as the number of tubes
increases the heat transfer coefficient improves. However,
additional factors such as vibration damage (if the tubes
are very close together), the need to access the outer
surface of tubes for cleaning, and the limit on pressure
drop across the radiator affect the decision on the number
of tubes. Fins are attached to the tubes by brazing or
soldering, thereby imparting strength to the whole
assembly and enabling the exchanger to withstand high
pressure. Fins not only enhance the overall heat transfer
coefficient but also significantly increase the total heat
transfer area and thus help enhance the performance of a
radiator. However, if the fin pitch is high, the fluid in
between the fins will move at a lower velocity (for
constant pumping power) giving more time for fouling to
occur and it further becomes difficult to clean the
assembly. It is costly to have high fin pitch. Thus, fouling,
maintenance, manufacturing, and cost considerations limit
the fin pitch. The profile of the tube plays an important
role as it affects the contact area between the two fluids
without adding much cost, but the manufacturing process
again limits the kinds of profiles that can be adopted
economically.
The factor most often used to evaluate the performance of
the radiator is the product of overall heat transfer
coefficient, U, and the total heat transfer area, A. The
overall heat transfer coefficient is a function of the heat
transfer coefficient and a fouling factor. The fouling
factor is a constant for given environmental conditions
while the heat transfer coefficient can be calculated by
using the following set of equations:
St Pr G c 
Pr
2/3
Heat Transfer Coefficient, h =
Reynolds Number, Re
Mass Velocity, G
=
=
2/3
p
Dh G

A'  v
AC
where cp is specific heat at constant pressure, A' is frontal
area of radiator,  is density of air, v is velocity of air at
inlet, Ac is free-flow area of radiator, Pr is Prandtl
number, Dh is hydraulic diameter, St is Stanton number,
and  is the dynamic viscosity of the radiator fluid.
The total surface area through which heat exchange
occurs is dependent on the profile of both the tubes and
the fins, the number of tubes, the fin pitch and the number
of rows. The configuration of fins and tubes also affects
the performance, but the current study is confined to only
straight fins and inline tubes. Figure 1 shows the radiator
core having straight fins and tubes.
2
It is generally desired to find a solution for radiator design
that simultaneously meets both the performance
requirements and cost targets. Since a number of
parameters affect both the performance and the cost, it is
important to evaluate the search space thoroughly to
obtain the best possible solution. The radiator heat
transfer model is linearized about a known configuration
from a flattened tube / fin array (surface 9.68 –0.87)14.
This paper presents a solution approach in which a
genetic algorithm manipulates the parameters to find a
near-optimum solution. This study reveals the details of
the approach that solves the problem of searching the
cost-effective design of an automotive radiator for a predefined level of performance of a radiator. This is
accomplished by providing the details of the
configuration of the tubes and the fins along with the
details of the cross-section of the tube for the specific
design problem.
3
LITERATURE REVIEW
The automobile industry is a field in which an abundance
of research has been conducted. Since radiators play an
integral role in the operation of an automobile, these
devices have been explored extensively. The
concentration has always been on evaluating the radiator
together with the cooling system of the engine4. A work in
the early 1970’s focused on heat dissipation from a
radiator to cool vehicle engines1. Further, during that time
a computer program for selecting a radiator to provide a
desired level of engine cooling and for predicting the
engine cooling performance with a given radiator was
formulated2. Simulations were developed for evaluating
the performance of a radiator as a single part of an entire
cooling system. As time passed researchers started
analyzing the material of the radiator and now aluminum
is considered to produce the best performance based upon
the statistics available because of a better method of
manufacturing and new metallurgical combinations3.
Chiou9 conducted a study to understand the effect of the
tube length on the heat transfer capability of a heat
exchanger. Further, Emmaenthal and Hacho10 presented a
method to design the cooling system of an automobile
where the individual components were first described
using experimental data and then the study was carried
out to achieve the low cost design. But, genetic
algorithms have not been previously applied to the
problem of optimizing the design of a radiator.
4
Figure 1: Radiator core showing the straight fins and
tubes with air and water flowing at right angles to each
other.
PROBLEM STATEMENT
4.1
GENETIC ALGORITHM
PARTICULARS
CODING SCHEME
As described above the goal of the current effort is to find
a cost-effective design of the radiator having a desired
performance using a genetic algorithm. In this study the
parameters that define the performance and cost are the
number of tubes (nt), the fin pitch (pf), the length of the
cross-section of tube (lt) and the width of the cross-section
of tube (wt). The length of the binary string, which
represents these four parameters using standard binary
concatenated coding, is found by specifying the accuracy
of each parameter. The minimum and maximum values
for each parameter are problem specific and depend on
the constraints that exist on the design. Table 1 shows
pertinent information about the coding used in this study.
Table 1: Sub-string length for each parameter based on
chosen accuracy and maximum and minimum values.
SUBSTRING
PARAMETER Ap Umin Umax
LENGTH
nt
1
29
60
5
pf (per inch)
1
8
11
2
lt (mm)
1
8
15
3
wt (mm x 10-1) 1
15
30
4
The sub-string lengths of each of the four parameters are
concatenated, resulting in the total string length of fifteen.
Each string represents one possible solution to the
problem. The number of tubes and the fin pitch
significantly affect the performance and cost of the
radiator. Thus, these two parameters are placed adjacent
to one another to reduce the likelihood of destroying good
combinations of these two parameters by crossover.
Therefore the binary string obtained is described in Table
2.
Table 2: Position of each parameter in string.
PARAMETER
nt
pf
lt
Positioning in string
4.2
1-5
k2
k3
k4
k5
= a weighting factor of 105
= a weighting factor of 103
= a weighting factor of 10
= a weighting factor of 1
The difference between the UA and UAdesired is weighted
by maximum factor as radiator’s under-performance and
over performance is highly undesired, while the higher
weightage is given to number of tubes represented in the
string, pf represents the fin pitch value in the string while
the higher weightage is given to the number of tubes as
compared to the fin pitch the number of tubes affect the
cost more severly. The profile of tube is given a low
penalty for poor design.
In the solution of this problem, a genetic algorithm first
generates a population of strings of given length using the
user-defined constraints. Each string is decoded to yield
actual parameters. The fitness of each string in the
population is found by evaluating the fitness function as
defined in Equation (1). Then, reproduction using
tournament selection, single-point crossover, and
mutation are used to generate subsequent generations and
search for acceptable values for nt, pf, lt and wt which
minimise the fitness function of Equation (1). Tournament
selection is executed by picking 15% of the strings from
the current population at random and comparing their
fitness values. The string with the lowest fitness value is
placed into the mating pool for the new population.
Single-point crossover is accomplished by randomly
picking two strings from the mating pool; then randomly
picking the crossover location in the string length based
on a probability of crossover of 0.9 and crossing the
strings at the location. A mutation operator with a
probability of 0.01 is used to introduce new genetic
material into the population.
wt
6 - 7 8 - 10 11 - 14
FITNESS FUNCTION
The parameters are sent to a mathematical model for
evaluation of the performance and the cost effectiveness
of the radiator, and in return a fitness value is assigned
defining the quality of solution represented by a given
binary string. The genetic algorithm then attempts to
achieve the desired performance with the minimum
number of tubes and fin pitch combination together with
the best profile of the cross-section of the tube. Here the
fitness function is defined as
Minimise {k1*[(UA - UAdesired)2 +1]1/2 + k2*nt + k3* pf +
k4*(17 -lt) + k5*(5-wt) }
…(1)
where
UAdesired = desired value of UA for radiator performance
k1
= a weighting factor of 107
5
RESULT
The accuracy of the genetic algorithm is tested by
comparing the solution obtained using a genetic algorithm
to the known practical result; one that is implemented
successfully in 1998 in India.
The goal set for the of the genetic algorithm was to
provide a cost-effective solution for the radiator, when the
performance desired from the radiator is 817 WC (product
of overall heat transfer coefficient and total heat transfer
area). The genetic algorithm was run for twenty-five
generations, using single-point crossover and a simple
mutation operator. An initial population of 100 strings
was randomly selected, where each string represented one
possible solution. The fitness value of the best string in a
generation is plotted against the function evaluations. The
results of this case are displayed in Figure 2.
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
Minimum
Fitness
0
10
0
20
0
30
0
40
0
50
0
60
0
70
0
80
0
90
0
10
00
11
00
12
00
13
00
14
00
15
00
16
00
17
00
18
00
19
00
20
00
21
00
22
00
23
00
24
00
Fitness ( x 105 )
When the genetic algorithm was run and compared to the
known practical solution the following results were
obtained (Table 3):
Function Evaluation
( Number of generation x Population size )
Figure 2: Best Fitness produced by the genetic algorithm
vs. function evaluation
Particulars of Simple Genetic Algorithm:
Population size=100
Number of generations=25
Probability of crossover=0.9
Probability of mutation=0.01
Chromosome length=15
Tournament size =15
Desired performance of radiator =817 WC
As seen in Figure 3, the offline performance shows better
convergence than the online performance. This is because
of the larger pool of diverse schemata are available in
larger population.
Table 3: Comparison of genetic algorithm to the known
practical solution for the 3-row radiator with straight fins
and inline tubes. (population size=100, number of
generations=25, probability of crossover=0.9, probability
of mutation=0.01)
GENETIC
PARAMETER
PRACTICAL
ALGORITHM
Number of Tubes
48
51
Fin pitch (per inch)
10
11
Length of cross12
11
section of tube
Width of cross2.5
1.8
section of tube
Fitness Value
1481052.5
1511163.2
The configuration resulting from the genetic algorithm is
slightly less economical than the practical known
solution. However, the result is near optimal and hence
the genetic algorithm is successful in providing a nearoptimal solution to the problem.
Based on these results, the genetic algorithm can be used
to determine the configuration of a radiator, for which we
have no solution, when the performance desired from the
radiator is know, say 1500 WC (product of overall heat
transfer coefficient and total heat transfer area). Here a
genetic algorithm is run for forty generations with a
population size of 50, initially selected at random. The
results of this case are displayed in Figure 4.
900
800
70
65
700
60
55
( x 10 5 )
50
500
Online Performance
Fitness
400
300
45
40
35
30
25
20
200
15
10
Offline Performance
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240
5
0
10
0
20
0
30
0
40
0
50
0
60
0
70
0
80
0
90
0
10
00
11
00
12
00
13
00
14
00
15
00
16
00
17
00
18
00
19
00
20
00
21
00
22
00
23
00
24
00
100
0
Fitness (x 105)
600
Function Evaluation
Function Evaluation
(Number of generation x Population size)
( Number of generation x Population size )
Figure 3: Comparison of online and offline performance
of genetic algorithm vs. function evaluation
Figure 4: Best Fitness produced by the genetic algorithm
vs. function evaluation
The configuration of radiator obtained after running the
genetic algorithm is given in Table 4.
10. K. D. Emmenthal and W. Hucho. A Rational
Approach to Automotive Radiator Systems Design,
Society of Automobile Engineers – 740088
Table 4: Solution provided by genetic algorithms for the
5-row radiator with straight fins and inline tubes.
(population size=100, number of generations=100,
probability of crossover=0.9, probability of mutation=0.1)
SOLUTION
PROVIDED BY
PARAMETER
GENETIC
ALGORITHM
Number of Tubes
52
Fin pitch (per inch)
9
Length of cross-section of tube
10
Width of cross-section of tube
3
Fitness Value
1520972
11. D. E. Goldberg (1989). Genetic Algorithms in Search,
Optimization, and Machine Learning.
6
CONCLUSIONS
A genetic algorithm was developed to search for the
optimal design of a radiator with pre-defined performance
characteristics and cost constraints. The validity of the
approach was tested against a problem with a known
solution. The genetic algorithm produced near-optimum
result for the problem; a solution which matched the bestknown practical solution. Thereafter, the genetic
algorithm is used for finding the optimal design
parameters for a radiator with desired performance
criteria and cost constraints. Therefore, it is concluded
that a genetic algorithm can be used successfully to find
near-optimum solutions in the realm of radiator design.
References
1. R. A. Beard and G. J. Smith, A Method of Calculating
the Heat Dissipation from Radiators to Cool Vehicle
Engines, Society of Automobile Engineers – 710208
2. Charles N. Kurland, Computer Program for Engine
Cooling Radiator Selection, Society of Automobile
Engineers – 710209
3. Performance of Aluminium Automotive Radiators,
Society of Automobile Engineers – 790400
4. Engine Cooling System Design for Heavy Duty Trucks,
Society of Automobile Engineers – 770023.
5. J. P. Holman (1986). Heat Transfer.
6. P. L. Balaney. Thermal Engineering.
7. G. F. Hewitt, G. L. Shires and T. R. Bott. Process Heat
Transfer.
8. M. Necati Ozisik (1985). Heat Transfer - A Basic
Approach, New York, McGraw-Hill Inc.
9. Jiunn P. Chiou. Correction Factor to Unit Core Heat
Transfer Capability of Heat Exchanger Core Due to
Variation of Tube Length, Society of Automobile
Engineers – 750884
12. C. L. Karr, J. C. Phillips. Scheduling and Resource
Allocation with Genetic Algorithms, Presentation at the
Society of Mechanical Engineers Annual Meeting.
13. B. K. Hodge, Robert P. Taylor, (3 rd edition) Analysis
& Design of Energy Systems.
14. W. M. Kays and A. L. London, 2 nd ed. (1964),
Compact Heat Exchangers.