Arab J Sci Eng (2014) 39:2295–2306 DOI 10.1007/s13369-013-0741-8
RESEARCH ARTICLE - MECHANICAL ENGINEERING
Optimization of Laminar Free Convection in a Horizontal Cavity Consisting of Flow Diverters Using ICA Alimohammad Karami · Farzad Veysi · Saeed Mohebbi · Damoon Ghashghaei
Received: 21 May 2012 / Accepted: 11 July 2013 / Published online: 8 September 2013 © King Fahd University of Petroleum and Minerals 2013
Abstract The main focus of the present paper is to apply a novel optimization algorithm called as imperialist competitive algorithm (ICA). In this algorithm the cost function being a function of input parameters has to be optimized. In the present research work, our cost function is the laminar free convection heat transfer in a horizontal cavity with adiabatic vertical and isothermally horizontal walls and adiabatic diverters. The input parameters are the diverter angle with respect to horizon varying from 0◦ to 90◦ and Rayleigh number varying from 6 × 103 to 1.2 × 104 . After collecting data, the regression equation of averaged convection heat transfer is obtained as a function of the Rayleigh number and diverter angle. Subsequently, the cost function is optimized using the ICA. Results show that the proposed algorithm is powerful enough to be used for optimizing the cost function. According to the results, in order to obtain the maximum heat transfer, the diverter angle must be 27.9◦ whereas; the Rayleigh number must be 1.2 × 104 .
A. Karami Kermanshah Branch, Mechanical Engineering Department, Islamic Azad University, Kermanshah, Iran F. Veysi (B) Mechanical Engineering Department, Razi University, Kermanshah, Iran e-mail:
[email protected] S. Mohebbi Harsin Branch, Mechanical Engineering Department, Islamic Azad University, Kermanshah, Iran D. Ghashghaei Mechanical Engineering Department, Kermanshah University of Technology, Kermanshah, Iran
Keywords Laminar free convection · Horizontal cavity · Mach–Zehnder interferometer · Diverters · Optimization · Imperialist competitive algorithm (ICA)
List of Symbols A Surface area of the aluminum plate (m2 ) Surface area of each insulator placed right and left A of the aluminum plate (m2 ) C Gladstone–Dale coefficient e Thickness of each diverter (mm) H Length of the cavity (mm) Local heat transfer coefficient (W/m2 K) hy h ave Average heat transfer coefficient (W/m2 K) k Thermal conductivity of air L Length of each diverter (mm) N u y Local Nusselt number
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p P q qcond, hot wall, RL
qcond, hot wall, FB
qconv, hot wall
qcond, heater
qheater qL qnet qrad, hot wall R Ra T W x y
Pitch of diverters (mm) Pressure (Pa) Averaged convection heat flux (W/m2 ) Conduction heat transfer due to the contact between the hot wall and adiabatic walls (W) Conduction heat transfer due to the contacts between the hot wall and insulators placed front and back of the hot wall (W) Convection heat transfer obtained from the fringe patterns of the Mach–Zehnder interferometer (W) Conduction heat transfer due to the contact between the heater and insulator placed below the heater (W) Power of the heater (W) Overall thermal dissipation (W) Absolute value of the thermal dissipation (W) Radiation heat transfer from the hot wall (W) Gas constant (J/kg K) Rayleigh number based on the cavity side length Temperature (K) Cavity side length (mm) Direction normal to the hot surface Direction along the hot surface
Greek Symbols ε Fringe shift Emissivity of the aluminum plate ε σ Stefan–Boltzmann constant (W/(m2 K4 )) α Absorption coefficient of the aluminum plate (m2 /s) λ Laser wave length (m) θ Diverter angle (◦ ) x Thickness of each adiabatic wall (m) x Thickness of each insulator placed front and back of the hot wall (m) Thickness of the insulator placed below the x heater (m) Subscripts col Referrers to the colony f Film condition imp Referrers to the imperialist pop Referrers to the population ref Reference condition Sc Cold condition Sh Warm condition ∞ Ambient condition
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1 Introduction The free convection in a cavity consisting of diverters has been extensively studied using numerical simulations and experiments because of its importance in industrial applications. Some applications are solar collectors, fire research, electronic cooling, aeronautics, chemical apparatus, fenestration systems and construction engineering. Most of the papers in this field are substantially oriented toward the study of free convection in enclosed square or rectangular cavities. Horvat et al. [1] studied the turbulent free convection and fluid flow due to the internal heat generation in a square cavity. In the study, the turbulent fluid motion was modeled using large-Eddy simulation (LES) technique. The main focus of the obtained results was to investigate the effects of Prandtl number on free convection in the cavity. Hakan et al. [2] numerically investigated the free convection heat transfer in wavy enclosures with volumetric heat sources. The effects of volumetric heat sources on free convection heat transfer and flow structures in a wavy-walled enclosure were considered in the paper. Sultana and Hyder [3] conducted a numerical study on the non-Darcy free convection in a wavy enclosure consisting of two isothermal wavy walls using the finite-element method. Khanafer et al. [4] numerically investigated the free convection heat transfer in a cavity with a sinusoidal vertical wavy wall and filled with a porous medium. The finite-element formulation based on the Galerkin method was used to solve governing equations. Yousefi et al. [5] studied the effect of diverter angle on the free convection in a cavity with an adiabatic diverter by the laser interferometry method. The study was focused on the variations of diverter angle ranging from 0◦ to 90◦ and Rayleigh number from 1.5 × 105 to 4.5 × 105 . It was found that the average Nusselt number has an increasing trend with respect to Rayleigh number. Moreover, it was reported that, at each Rayleigh number, the maximum and minimum heat transfer occurs at the diverter angle of 45◦ and 90◦ , respectively. Mahmud et al. [6] investigated the heat transfer and laminar fluid flow characteristics inside a cavity made of two horizontal straight walls and two vertical wavy walls. In the paper, wavy walls were assumed to follow a profile of cosine curve. Corcione [7] numerically studied the free convection in an air-filled rectangular enclosure heated from below and cooled from above for a variety of thermal boundary conditions at the side walls. Obtained results were reported for several values of both width-to-height aspect ratio of the enclosure and Rayleigh number. Other investigations related to the present study can be found elsewhere [8–14]. The current study attempts to demonstrate the capability of the imperialist competitive algorithm (ICA), in order to obtain the optimum (maximum) laminar free convection in a horizontal cavity with adiabatic vertical and isothermally
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Fig. 1 Model geometry of the horizontal cavity consisting of diverters
horizontal walls and adiabatic diverters. The experiments have been carried out using the Mach–Zehnder interferometer. A schematic representation of the problem is shown in Fig. 1. The diverter length L, diverter width H , thickness of each diverter e, pitch of the diverter p, cavity side length W , diverter angles θ , and also the boundary conditions are represented in this figure. The ICA is a new evolutionary algorithm in the evolutionary computation field based on the human’s socio-political evolution. The proposed method for the optimization was developed using MATLAB functions. This method has some advantages, such as simplicity, accuracy, and time saving. ICA is presently one of the powerful tools widely used for the optimization of various heat transfer processes. Karami et al. [15] employed the imperialist competitive approach to optimize the average laminar free convection heat transfer from a horizontal isothermal cylinder located below an adiabatic ceiling, against the optimization variables including the ratio of the cylinder spacing from the adiabatic ceiling to its diameter (L/D) and Rayleigh number. Rezaei et al. [16] optimized the thermal performance of an air cooler equipped with butterfly inserts using ICA. The optimization parameters included the inclined angle of inserts and Reynolds number. Karami et al. [17] applied the ICA for the optimization of heat transfer in an air cooler equipped with classic twisted tape inserts. The main target of the paper was to optimize the heat transfer in an air cooler against the optimization variables including the twist ratio of the inserts and Reynolds number. Rezaei et al. [18] presented an ICA model to optimize heat transfer in an air cooler equipped with butterfly inserts. The optimization parameters included the inclined angle of inserts and Reynolds number. Karami et al. [19] developed ICA for the optimization of laminar free convection heat transfer in a vertical cavity with flow diverters. The study attempted to optimize the average heat transfer against the optimization variables including the angle of diverters (θ ) and Rayleigh number.
Fig. 2 Plain view of the Mach–Zehnder setup
2 Experimental Setup 2.1 Interferometer The experimental study is carried out using the Mach–Zehnder interferometry (MZI) technique. The interferometer consists of a light source, a micro lens, a pinhole, two doublets, three mirrors and two beam splitters. Figure 2 shows the interferometer setup. Beam splitters BS1 and BS2, along with plane mirrors M1 and M2 constituted the basic MZI. The laser beam gets expanded after passing through spatial filter and doublet1. The expanded beam is split into two equal beams by BS1. One beam passes through the test section and the other through the undisturbed field. These two beams, again, recombine at BS2. If the four optical plates, M1, M2, BS1, and BS2, are parallel, then infinite fringe interferograms will be formed. Further information about MZI can be found in [20–22]. The used light source was a 10mW Helium–Neon laser with a 632.8 nm wavelength. All the interferograms are digitized with a “ARTCAM-320P” 1/2” CCD camera with 3.2 M pixels. To acquire the interferograms a camera is connected to a PC. Figure 3 shows some of the interferograms which are recorded by the CCD camera.
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Fig. 3 Interferograms of the cavity consisting of diverters for Ra = 12,000 at a θ = 0◦ b θ = 30◦ c θ = 60◦ d θ = 90◦
3 Experiment Test Section The details of the horizontal cavity used in the experiments are shown schematically in Fig. 4. The length of each isothermal wall along the laser beam is chosen as 140 mm which causes the induced flow to be two-dimensional. Also, the wooden end caps with thermal conductivity of 0.05 W/m K [23] are installed on each aluminum plate bases to minimize the end effects. By passing electricity through the heater that is placed at the back of the aluminum plate and considering a relatively thick-walled (16 mm) aluminum plate, we achieve constant surface temperature. The uniformity of each plate’s surface temperature is experimentally validated by measuring it at three different locations. As shown in Fig. 4, the differences in temperature readings for each aluminum plate surface were about 0.1 ◦ C. The local surface temperatures of the heated aluminum plate are recorded via three type-K thermocouples, embedded vertically in the aluminum plate wall. Two other thermocouples of the same type are used to measure the ambient and the reference temperatures for data reduction. All the temperatures are monitored continuously in a PC by a selector switch and a “TESTO 177 T4” four-channel data logger. The laboratory pressure is recorded during all the experiments. The maximum uncertainties of temperature and pressure measurement for the present test condition are ± 0.1 ◦ C and ±100 Pa, respectively. In all of the experiments the heater voltage and
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current are recorded. In order to ascertain the accuracy of the measurements, the energy balance calculation for many cases is done by calculating laminar free convection heat transfer from the fringe patterns of the Mach–Zehnder interferometer and measuring electrical power input to the heaters. A comparison between the heat transfer coefficients obtained by two methods shows a complete agreement. Here, for the case of the Rayleigh number equal to 10,000 and diverter angle of 30◦ , the convective heat transfer obtained from the fringe patterns of the Mach–Zehnder interferometer is qconv, hot wall = 3.4132 W.
(1)
The thermal dissipation due to the radiation heat transfer from the hot wall is calculated as follow: 4 − α × Tsc4 ) qrad, hot wall = A × σ × (ε × TSh
(2)
where A = 0.14 m × 0.14 m, σ = 5.67 × 10−8 W/(m2 K4 ), ε = 0.1, α = 0.1, TSh = 347.35 K, Tsc = 311.15 K. Substituting the above values for the parameters in the Eq. (2), the following equation is obtained: qrad, hot wall = 0.5761 W.
(3)
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Fig. 4 Details of the horizontal cavity consisting of diverters used in the experiments
Moreover, the thermal dissipation due to the contacts between the hot wall and adiabatic walls is calculated as follows: qcond, hot wall, RL
T = A ×k× x
(4)
where A = 0.14 m × 0.14 m, k = 0.05 W/m, T = 0.1 K, x = 0.016.
where
Substituting the above values for the parameters into the Eq. (8), the following equation is obtained:
A = 0.14 m × 0.016 m × 2 m
qcond, heater = 2.9216 W
k = 0.05 W/m, T = 47.7 K, x = 0.035. Substituting the above values for the parameters into the Eq. (4), the following equation is obtained: qcond, hot wall, RL = 0.3053 W
(5)
Moreover, the heat dissipation due to the contacts between the hot wall and insulators placed front and left of the hot wall (along the laser beam) is calculated as follows: qcond, hot wall, FB
T = A×k× x
(6)
where k = 0.05 W/m, T = 47.7 K, x = 0.02.
qL = qrad, hot wall + qcond, hot wall, RL + qcond, hot wall, FB + qcond, heater .
qcond, hot wall, FB = 0.5342 W.
(7)
In addition, the thermal dissipation due to the contact between the heater and thermal insulator placed below the heater is calculated as follows: T x
(10)
Substituting the above values for the parameters into the Eq. (10), the following equation is obtained: qL = 4.3372 W
(11)
On the other hand, the power of the heater is calculated as follows: (12)
where
Substituting the above values for the parameters into the Eq. (6), the following equation is obtained:
qcond, heater = A × k ×
Using the following relation, the overall thermal dissipation is calculated as follows:
qheater = V × I
A = 0.14 m × 0.14 m,
(9)
(8)
V = 40.05 V, I = 0.197 A Substituting the above values for the parameters into the Eq. (12), the following equation is obtained: qheater = 7.8898 W.
(13)
Finally, the absolute value of the thermal dissipation is determined: qnet = qheater − qL .
(14)
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Then, the mean relative error between the heat transfer obtained from the interferometer method and the energy balance is calculated by the following equation: Error % =
qnet − qconv, hot wall × 100 qconv, hot wall
(15)
After substituting the values obtained from Eqs. (1) and (14) into the Eq. (15), the Error % is found to be about 4.08 % indicating a good agreement between the values obtained by the two methods. Four sets of ten diverters of dimensions 167 mm ×14 mm ×1.5 mm, with thermal conductivity of 0.05 W/m K with angles of 0◦ , 30◦ , 60◦ and 90◦ with respect to the horizon, are built to use in the cavity for each experiment with its associated angle. In each set, the diverters are glued to a thin rod. Two windows are used on both sides of the cavity for preventing external air from entering the cavity. The entire experimental setup is located on an anti-vibration table under a condition of pneumatic isolation. In order to eliminate the effect of any other air disturbances on the experimental test section, the entire setup is housed within an open top Plexiglas box of dimensions 3 m ×1.5 m ×1.5 m.
number are determined as follows: dT 1 . h y = −kSh dx x=0 TSh − T∞
(18)
Then, the local Nusselt number is obtained from Nuy =
hyW . kf
(19)
∞ where, Tf is the film temperature and Tf = TSh +T , kSh and 2 kf are the thermal conductivity of air evaluated at the surface temperature TSh and the film temperature. The average Nusselt number is calculated from the following formula:
h ave
1 = H
H h y dy
(20)
0
where, y is the distance along the aluminum plate surface. Also, the averaged convection heat transfer is calculated from q = h ave (TSh − T∞ ).
(21)
5 Optimization Approach 4 Data Reduction The aim of the data reduction procedure is to determine the local and average Nusselt numbers. For the assurance of experiment repeatability, three interferograms with 10-s intervals are captured in each case. In order to determine the local and average Nusselt number on the hot wall, a code with MATLAB software is developed. The temperature of the interference fringes as well as their distance from the surface of the aluminum plate in a vertical direction is calculated by the method explained by Hauf and Grigull [20] and Eckert and Goldstein [21]. The local air temperature gradient at the aluminum plate surface is obtained from the following equation: dT dε dT = · (16) dx x=0 dε x=0 dx x=0 where ε is the fringe shift value. Using the ideal gas and Lorenz–Lorenz equations [20], dT dε x=0 is calculated as follow: 3C Lλ PR dT = (17) 2 L P dε x=0 3 C − 2ε 2 λ RTref x=0
where R = 287 J/kg K is the gas constant, P∞ is the ambient pressure, C is the Gladstone–Dale constant, and Tref is the temperature of reference fringe. It is worth mentioning that, due to the low temperature difference and also the atmospheric air pressure, the above equation is used for the ideal gas. The local heat transfer coefficient and the Nusselt
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An optimization problem can be easily described as finding an argument x whose relevant cost f (x) is optimum, and has been extensively used in many different situations such as industrial planning, resource allocation, scheduling, pattern recognition and so on. Different methods have been proposed to solve the optimization problem. Evolutionary algorithms, such as particle swarm optimization [24,25], taboo search [26,27], genetic algorithm [28,29], ant colony optimization [30,31], bees algorithm [32,33] and differential evolution [34,35], are a set of algorithms that have been introduced and suggested in the past decades for solving optimization problems in different science and engineering fields. 5.1 Imperialist Competitive Algorithm The ICA is an algorithm introduced for the first time in 2007 by Atashpaz-Gargari and Lucas [36] and used for optimizing inspired by imperialistic competition and which has considerable relevance to several engineering applications [15– 19]. Like other evolutionary algorithms, the proposed algorithm starts with an initial population. Population individuals called country are of two types: colonies and imperialists that all together form some empires. Imperialistic competition among these empires forms the basis of the proposed evolutionary algorithm. During this competition, weak empires collapse and powerful ones take possession of their colonies. Imperialistic competition hopefully converges to a state in which there exists only one empire and its colonies are in the same position and have the same cost as the imperialist. Using this algorithm, one can find the optimum condition of most
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functions. In this connection, the proposed model based on regression analysis is then embedded into the ICA to optimize the objective function [15]. The goal of optimization algorithms is to find an optimal solution in terms of the variables of the problem (optimization variables). Therefore, an array of variable values to be optimized is formed. In Genetic Algorithm terminology, this array is called a “chromosome”, but here the term “country” is used for this array. In an Nvar dimensional optimization problem, a country is an 1 × Nvar array. This array is defined by country = [ p1 , p2 , p3 , . . . , p Nvar ]
(22)
The variable values in the country are represented as floating point numbers. The cost of a country is found by evaluating the cost function f at the variables ( p1 , p2 , p3 , . . . , p Nvar ). Then cost = f (country) = f ( p1 , p2 , p3 , . . . , p Nvar ).
(23)
The flowchart of the ICA algorithm is shown in Fig. 5. To start the optimization algorithm the initial population of size Npop is generated. The Nimp of the most powerful countries to form the empires is selected. The remaining Ncol of the population will be the colonies each of which belongs to an empire. Now, there are two types of countries: imperialist and colony. To form the initial empires, the colonies are then divided among imperialists, based on their powers. That is the initial number of colonies of an empire should be directly proportionate to its power. To divide the colonies among imperialists proportionally, the normalized cost of an imperialist are defined by Cn = cn − max{ci }, where cn is the cost of nth imperialist and Cn is its normalized cost. Having the normalized cost of all imperialists, the normalized power of each imperialist [16] is defined by N imp C n (24) pn = Ci . i=1 From another point of view, the normalized power of an imperialist is the portion of colonies that should be possessed by that imperialist. Then, the initial number of colonies of an empire will be N .C.n = round { pn .Ncol }
(25)
where N .C.n , is the initial number of colonies of the nth empire and Ncol is the number of all colonies. To divide the colonies for each imperialist, N .C.n of the colonies is chosen randomly and is given them to it. These colonies along with the imperialist will form nth empire. A schematic representation of the initial population of each empire can be observed in Fig. 6. As shown in this figure, bigger (powerful) empires have more colonies while smaller (weaker) ones have less. As
Fig. 5 Procedure of the proposed algorithm
mentioned, imperialist countries start to improve their colonies. This fact has been modeled by moving up all the colonies toward the imperialist. This movement is shown in Fig. 7, where the colony moves toward the imperialist by x units. The new position of the colony is shown in a darker color. The direction of the movement is the vector from colony toward imperialist. In this figure, x is a random variable with uniform or any proper profile [36]. Then, for x x ∼ U (0, β × d)
(26)
where β is a number greater than 1 and d is the distance between colony and imperialist. When β > 1, it causes the colonies to get closer to the imperialist state from both sides. To search different points around the imperialist a random amount of deviation was added to the direction of movement. Figure 8 shows the new direction. In this figure, θ is a random number with uniform or any proper profile. Then θ ∼ U (−γ , γ )
(27)
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Fig. 6 Generating the initial empires: the more colonies an imperialist possess, the bigger its relevant mark [36]
Table 1 Optimization variables in heat transfer and their levels Optimization variables
Fig. 7 Movement of colonies toward their relevant imperialists
Fig. 8 Movement of colonies toward their relevant imperialist in a randomly deviated direction
where γ is a parameter that adjusts the deviation from the original direction. Nevertheless, the values of β and γ are arbitrary; in most of our implementation a value of about 2 for β and about π/4 (Rad) for γ have resulted in good convergence of countries to the global minimum.
Notation
Levels
Rayleigh number
Ra
6,000, 8,000, 10,000, 12,000
Diverter angle (◦ )
θ
0, 30, 60, 90
Due to some limitations in regard with the time and manufacturing expenses, it was decided to select only four angles for the diverters. Moreover, same limitations forced us to select the Rayleigh number. The limitations imposed upon us are due to the current and voltage to be used for supplying the heat flux. Therefore, we use four levels of Rayleigh number (Ra) ranging from 6 × 103 to 1.2 × 104 , with the four levels of the diverter angle (θ ) from 0◦ to 90◦ as optimization variables, and the averaged convection heat transfer (q ) as output variable. It is important to be mentioned that, since the temperature difference between the diverters is negligible, the radiation heat transfer between the diverters can be ignored. Moreover, since the shape factor between the two walls (cold and hot walls) is assumed to be unity (in fact the shape factor is less than unity due to the presence of the diverters), according to Table 1, the heat transfer between the diverters and walls is not taken into consideration. In order to develop the ICA model, 16 values for the averaged convection heat transfer are collected experimentally. The values of the averaged heat transfer are shown in Table 2. The correlation of the averaged heat transfer in terms of the Rayleigh number (Ra) and diverter angle (θ ), taken as the sensitivity ratio, is developed as follows: q = 127 − 0.0195Ra + 3.04θ − 0.0855θ 2
6 ICA Optimization Results and Discussion In order to use the ICA, the optimization (input) and output variables with their corresponding levels must be determined.
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+ 0.000521θ 3 + 0.000002Ra 2 W/m2
(28)
The statistical information of the above equation is shown in Table 3.
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Table 2 Applied experimental data No
Rayleigh number, Ra
Diverter angle, θ (◦ )
Averaged heat transfer, q (W/m2 )
1
6,000
0
2
6,000
30
3
6,000
60
64.2740
4
6,000
90
52.3251
5
8,000
0
96.1760
6
8,000
30
7
8,000
60
8
8,000
90
9
10,000
0
124.546
10
10,000
30
174.144
11
10,000
60
148.625
12
10,000
90
104.026
13
12,000
0
195.424
14
12,000
30
210.453
15
12,000
60
172.362
16
12,000
90
131.954
90.3070 112.117
122.725 68.7960 59.4789
Fig. 9 The physical interpretation of the ICA in terms of heat transfer parameters Table 4 The selected optimal parameters of proposed ICA model Number of total countries
Table 3 Statistical information of the regression equation Statistical information
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Number of initial imperialist countries
8
Number of epochs (decades)
8
Value
Revolution rate
0.3
R2
94.8 %
Assimilation coefficient
2
Adjusted R 2
92.1 %
Assimilation angle
0.5
13.7985
Cost function
±q a
STD
a The minus sign refers to maximum heat transfer and the plus one refers to minimum heat transfer
According to Eq. (28), increasing the Rayleigh number (Ra) increases the heat transfer. With respect to the effect of diverter angle, the maximum heat transfer occurs at a diverter angle of 30◦ and the minimum heat transfer at an angle of 90◦ . These behaviors can be described as follows. It can be observed that because of the insulated wall, the circulating force between cold and hot walls weakens and the diverters prevent hot air from reaching the cold wall. With increasing the diverter angle to 30◦ , because of the creation of a chimney effect between the cold and hot wall, heat transfer increases enormously. Again, with increasing the angle from 30◦ to 90◦ , the distance between the tips of the diverters and the heated wall will be decreased. The ability of the air circulating flow to pass though this gap will be decreased due to a decrease in its velocity and, consequently, the heat transfer decreases. Furthermore, the regression equation is embedded into the ICA to be optimized. In this paper among all the numerical optimization algorithms, the ICA is preferred due to its high convergence speed in attaining the optimum solution. The convergence time in this algorithm is so little that it cannot be compared with other optimization techniques. Therefore, using this algorithm helps in saving of time. Moreover, appropriate setting of all the main para-
meters in the algorithm results in a more accurate optimum solution as compared to other optimization techniques. However, if the adjusted R 2 of the regression equation is too low, then the results obtained using the ICA cannot be compared with the experimental results. In the ICA terminology, the country is basically a vector of input parameters:
Xi =
Rai θi
(29)
Population is defined as the collection of countries competing internally in order to minimize the costs involved in becoming an imperialist. It also indicates the optimum level of the input parameters that give maximum heat transfer. After the competition between the countries is complete, the second stage starts. The second stage is external competition between imperialists. In this stage, the imperialist having the least cost is taken as the winner, as shown in Fig. 9; this is equivalent to the maximum performance obtained in this study. The main parameters used in ICA model are listed in Table 4 and the optimization results are shown in Table 5.
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q (W/m2 )
Moreover, the optimum value of diverter angle is obtained by differentiating the Eq. (28) with respect to θ parameter as follows: ∂q =0 ∂θ resulting in the following equation:
Table 5 Optimization results Optimized parameters
Ra
θ
(◦ )
Optimized values For maximum heat transfer 12,000
21.79
212.00
For minimum heat transfer
87.71
42.45
6,000
3.04 − 0.171 θ + 0.001563 θ 2 = 0 Figure 10 shows the minimum and mean cost of all imperialists. As it can be observed from these results, the diverter angle must be 21.79◦ whereas; the Rayleigh number must be 1.2 × 104 . In order to validate the obtained results, the optimum value of the Rayleigh number (Ra) is obtained from the Eq. (28) by differentiating it with respect to Ra parameter as follows: ∂q =0 ∂ Ra
(30)
resulting in the following equation: − 0.0195 + 0.000004Ra = 0
(31)
which results in the following value for the Rayleigh number: Ra = 4,875
(32)
Since the above value for the Rayleigh number is out of the considered range of Rayleigh number (6,000 ≤ Ra ≤ 12,000), so the optimum value for the Rayleigh number is considered as 6,000. It is worth mentioning that, the minimum value of the averaged heat flux corresponds to the above value of the Rayleigh number (Ra = 6,000). Moreover, since the Eq. (31) shows absolutely increasing trend, maximum heat transfer corresponds to the maximum value of the Rayleigh number (Ra = 12,000).
(34)
Solving the above equation, two values are obtained for θ parameter as given below: θ = 22.339, 87.066
(35)
Then, the following situations can be considered as follows: θ θ θ θ
= 22.34, Ra = 22.34, Ra = 87.06, Ra = 87.06, Ra
= 6,000 = 12,000 = 6,000 = 12,000
(36)
Substituting each set of above values in Eq. (28), the maximum and minimum heat transfers are obtained. A comparison between the ICA and theoretical results is presented in Table 6. According to these results, the calculated values of the Ra and θ parameters by the ICA are very close to the ones obtained by the Eq. (36).
7 Conclusions In this paper, a novel optimization technique called ICA was utilized in order to optimize the laminar free convection heat transfer in a horizontal cavity consisting of flow diverters, based on experimental data. A correlation was developed to obtain a relationship between the averaged heat transfer as
Fig. 10 Mean and minimum cost of all imperialists versus epochs for a minimum heat transfer b maximum heat transfer
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(33)
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Table 6 Comparison between the ICA optimization results and the theoretical ones Optimized parameters
RaICA
MRE (%)
θICA
θTheory (◦ )
MRE (%)
qICA
qTheory
MRE (%)
12,000
0
21.79
22.34
2.46
212.00
212.21
0.11
6,000
0
87.71
87.06
0.75
42.45
42.35
0.24
RaTheory
Optimized values Maximum heat transfer 12,000 Minimum heat transfer
6,000
output and two input variables namely the Rayleigh number and diverter angle. The correlation was then taken into the ICA to be optimized. According to the optimization results, in order to obtain the maximum heat transfer, the diverter angle must be 27.9◦ whereas; the Rayleigh number must be 1.2 × 104 .
References 1. Horvat, A.; Kljenak, I.; Marn, J.: Two-dimensional large-eddy simulation of turbulent natural convection due to internal heat generation. Int. J. Heat Mass Transf. 44, 3985–3995 (2001) 2. Hakan, F.; Abu-Nada, E.; Varol, Y.; Chamkha, A.: Natural convection in wavy enclosures with volumetric heat sources. Int. J. Therm. Sci. 50, 502–514 (2011) 3. Sultana, Z.; Hyder Md, N.: Non-darcy free convection inside a wavy enclosure. Int. Commun. Heat Mass 34, 136–146 (2007) 4. Khanafer, K.; Al-Azmi, B.; Marafie, A.; Pop, I.: Non-Darcian effects on natural convection heat transfer in a wavy porous enclosure. Int. J. Heat Mass Transf. 52, 1887–1896 (2009) 5. Yousefi, T.; Nezhad, S.M.; Bigharaz, M.; Ebrahimi, S.: An experimental study on the effect of partition angle on free-convection heat transfer in a partitioned cavity by laser interferometry method. In: 10th Biennial Conference on Engineering Systems Design and Analysis ESDA, July 12–14, Istanbul, Turkey (2010) 6. Mahmud, S.; Fraser, R.A.: Magnetohydrodynamic free convection and entropy generation in a square porous cavity. Int. J. Heat Mass Tranf. 47, 3245–3256 (2004) 7. Corcione, M.: Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above. Int. J. Therm. Sci. 42, 199–208 (2003) 8. Oztop, H.F.; Dagtekin, I.; Bahloul, A.: Comparison of position of a heated thin plate located in a cavity for natural convection. Int. Commun. Heat Mass 31, 121–132 (2004) 9. Frederick, R.L.: Natural convection in an inclined square enclosure with a partition attached to its cold wall. Int. J. Heat Mass Transf. 32, 87–94 (1989) 10. Shahid, H.; Naylor, D.: Energy performance assessment of a window with a horizontal Venetian blind. Energy Build. 37, 836–843 (2005) 11. Karayiannis, T.G.; Ciofalo, M.; Barbaro, G.: On natural convection in a single and two zone rectangular enclosure. Int. J. Heat Mass Transf. 35, 1645–57 (1992) 12. Safer, N.; Woloszyn, M.; Roux, J.: Three-dimensional simulation with a CFD tool of airflow phenomena in a single floor double-skin façade equipped with a venetian blind. Sol. Energy 79, 193–203 (2005) 13. Ganzarolli, M.M.; Milanez, L.F.: Natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides. Int. J. Heat Mass Transf. 38, 1063–1073 (1995)
14. Aydin, O.; Unal, A.; Ayhan, T.: Natural convection in rectangular enclosures heated from one side and cooled from the ceiling. Int. J. Heat Mass Transf. 5, 2345–2355 (1999) 15. Karami, A.; Rezaei, E.; Mahmoudinezhad, S.; Yousefi, T.: Optimization of free convection heat transfer in a horizontal cylinder beneath an adiabatic ceiling, using an Imperialist Competitive Algorithm. J. Chem. Eng. Jpn. 45, 401–407 (2012) 16. Rezaei, E.; Karami, A.; Shahhosseni, M.; Aghakhani, M.: The optimization of thermal performance of an air cooler equipped with butterfly inserts by the use of Imperialist Competitive Algorithm. Heat Transf. Asian Res. 41, 214–226 (2012) 17. Karami, A.; Rezaei, E.; Shahhosseni, M.; Aghakhani, M.: Optimization of heat transfer in an air cooler equipped with classic twisted tape inserts using Imperialist Competitive Algorithm. Exp. Therm. Fluid Sci. 38, 195–200 (2012) 18. Rezaei, E.; Karami, A.; Shahhosseni, M.; The use of Imperialist Competitive Algorithm for the optimization of heat transfer in an air cooler equipped with butterfly inserts. Aust. J. Basic Appl. Sci. 6, 293–301 (2012) 19. Karami, A.; Yousefi, T.; Ghashghaei, D.; Rezaei, E.: The application of Imperialist Competitive Algorithm in optimizing the free convection heat transfer in a vertical cavity with flow diverters. Int. J. Model. Optim. 1, 289–295 (2011) 20. Hauf, W.; Grigull, U.: Optical methods in heat transfer. Adv. Heat Transf. 6, 133–366 (1970) 21. Eckert, E.R.; Goldstein, R.J.: Measurements in Heat Transfer, 2nd edn. McGraw-Hill, New York (1972) 22. Flack, R.D.: Mach–Zehnder interferometer errors resulting from test section misalignment. Appl. Opt. 17, 985–987 (1978) 23. Bever, M.B.: Encyclopedia of Materials Science And Engineering, vol. 7. Pergamon, Oxford (1986) 24. Gudisz, V.G.; Venayagamoorthy, G.K.: Comparison of particle swarm optimization and back propagation as training algorithms for neural networks. In: IEEE, SIS, pp. 110–117. Indianapolis, Indiana, USA. (2003) 25. Parsopoulos, K.E.; Vrahatis, M.N.: Particle swarm optimization method in multiobjective problems. In: ACM, SAC, pp. 603–607. (2002) 26. Pezzini, P.; Gomis-Bellmunt, O.; Sudrià-Andreu, A.: Optimization techniques to improve energy efficiency in power systems. Renew. Sustain. Energy Rev. 15, 2028–2041 (2011) 27. Chelouah, R.; Siarry, P.: Search applied to global optimization. Eur. J. Oper. Res. 123, 256–270 (2000) 28. Muttil, N.; Chau, K.W.: Neural network and genetic programming for modelling coastal algal blooms. Int. J. Environ. Pollut. 28, 223– 238 (2006) 29. Wu, C.L.; Chau, K.W.: A flood forecasting neural network model with genetic algorithm. Int. J. Environ. Pollut. 28, 261–273 (2006) 30. Zhang, B.; Chen, D.; Zhao, W.: Iterative ant-colony algorithm and its application to dynamic optimization of chemical process. Comput. Chem. Eng. 29, 2078–2086 (2005) 31. Dorigo, M.; Blum, C.: Ant colony optimization theory: a survey. Theor. Comput. Sci. 344, 243–278 (2005) 32. Pham, D.T.; Soroka, A.J.; Ghanbarzadeh. A.; Koç, E.; Otri, S.; Packianather, M.: Optimising neural networks for identification of
123
2306
Arab J Sci Eng (2014) 39:2295–2306
wood defects using the Bees Algorithm. In: IEEE, International Conference on Industrial Informatics, Singapore (2006) 33. Pham, D.T.; Ghanbarzadeh, A.; Koç, E.; Otri, S.: Application of the Bees Algorithm to the training of radial basis function networks for control chart pattern recognition. In: 5th CIRP International Seminar on Intelligent Computation in Manufacturing Engineering (CIRP ICME). Ischia, Italy (2006) 34. Rui, X.; Ganesh, K.; Venayagamoorthy G.K.; Donald, C.: Modeling of gene regulatory networks with hybrid differential evolution and particle swarm optimization. Neural Netw. 20, 917–927 (2007)
123
35. Yanling,W.U.; Jiangang, L.U.; Youxian, S.: An improved differential evolution for optimization of chemical process. Chin. J. Chem. Eng 16, 228–234 (2008) 36. Atashpaz-Gargari, E.; Lucas, C.: Imperialist Competitive Algorithm: an algorithm for optimization inspired by imperialistic competition. In: IEEE Congress on Evolutionary Computation. Singapore (2007)