- Email: [email protected]
New optimum design for cooling system in thermoelectric thermal devices
Accepted Manuscript New optimum design for cooling system in thermoelectric thermal devices Mojtaba Babaelahi, Hamed Jafari
PII: DOI: Reference:
S2352-4316(18)30140-8 https://doi.org/10.1016/j.eml.2018.11.003 EML 415
To appear in:
Extreme Mechanics Letters
Received date : 26 June 2018 Revised date : 25 October 2018 Accepted date : 20 November 2018 Please cite this article as: M. Babaelahi and H. Jafari, New optimum design for cooling system in thermoelectric thermal devices, Extreme Mechanics Letters (2018), https://doi.org/10.1016/j.eml.2018.11.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
New Optimum Design for Cooling System in Thermoelectric Thermal Devices Mojtaba. Babaelahi*1, Hamed. Jafari Department of Mechanical Engineering, University of Qom, P.O. Box 6765-1719, Qom, Iran.
Abstract The problem of efficient cooling in the compact medium is a very critical problem in the design of thermoelectric thermal devices. In this paper, the thermal behavior of special heat sinks with exponential fin is considered for use in thermoelectric systems. To evaluation of cooling performance of the proposed cooling system, the energy balance of considered heat sinks with variable thermal conductivity including mass transfer is evaluated. The effective analytical method so-called differential transformation method is selected for the solution of the heat balance nonlinear differential equation. Comparisons of the analytical and numerical solution show the excellent accuracy of the selected analytical solution. Based on analytical results, the effects of operational and geometrical parameters on the performance of the heat sink in TEC are evaluated. To the selection of best optimum design with lowest volume, highest heat transfer and thermal efficiency, the Ant Colony Optimization procedure is selected for multi-objective optimization purposes. Keywords: Thermoelectric devices, thermal, heat sink, analytical, Ant Colony Optimization.
1. Introduction The thermoelectric cooling (TEC) devices are one of the thermal components that used for various applications. These applications can be simpli*
-Corresponding Author: Mojtaba Babaelahi (M. Babaelahi) [email protected]
fied in active cooling below ambient temperature or high-temperature precision. An important parameter in operation of thermoelectric heat devices is the temperature difference between the hot and the cold side of a Peltier element used in TEC. This temperature difference is affected by the performance of the cooling system used on the cold side of the Peltier element. In recent year, many types of research are performed for improving the performance of TEC. In this section, some of the bold research in this filed is presented. In general, the efficient methods used to optimization of TEC can be divided into three categories: 1) design and optimization of thermoelectric module's geometry, 2) Design and optimization of the heat sink and 3) change the working condition. In this section, some of these researches are reviewed. The theoretical investigation of thermoelectric cooling devices was performed by Chein and Huang [1]. In this research, the effect of gas and liquid cooling medium on the performance of the electronic cooling system was evaluated. Cheng and Lin [2], try to optimize the geometrical parameters of TES such as length, the number of modules, etc. They used the genetic algorithm optimization method for maximization of cooling performance. Lee [3] performed new dimensionless analysis for the thermoelectric cooling system and introduced some dimensionless group such as thermal conduction ratio, the convection conduction ratio, and load resistance ratio. Naphon and Wiriyasart [4] considered the effect of the rectangular fin on TEC for CPU, and the cooling liquid flow is analysis. This analysis was performed with and without thermoelectric cooling. The results show that the CPU performance was affected strongly by thermoelectric cooling performance. The irreversibility analysis of thermoelectric cooling devices performed by Wang et al. [5]. In this research, the first and second laws of thermodynamics are applied to TEC and the dimensionless entropy generation number introduced. Luo et al. [6] used a new novel TEC device for a truck cab. In this research, the performance of the TEC system is tested for ambient temperature between 46-30. The results show that the cooling per-
formance of the TEC varied between 0.4 to 0.8. Also, this paper presented that the cooling performance could be maximized by optimization of design parameters. Zhang [7], considered the optimization of the thermoelectric cooler with dimensionless analysis method. In this research, the iterative procedure method is a substitute with the straight-forward method. In the area of analytical solution, there are several types of researches that have been reviewed in the following paragraph. Dogonchi et al. [8] investigated the unsteady motion of a vertically falling non-spherical particle in the incompressible Newtonian media. The analytical results indicate that the velocity of the gold particle is higher than the copper and aluminum particles. Hosseinzadeh et al. [9] examined the effect of Brownian motion, thermophoresis phenomenon and Lewis number on MHD nanofluid flow heat transfer. In another article the heat transfer and temperature distribution for semi-spherical convective-radiative porous fins are solved analytically by Atouei et al. [10]. Babaelahi and Eshraghi [11] used the analytical method for evaluation the effect of the convex parabolic fin on the thermal behavior of medical heat sink. Amiri et al. [12] examines both the numerical and the analytical approaches to the heat transfer in blood fluid flow containing nanoparticles in a porous media affected by the magnetic field. The coupled equations of the particle motion in fluid flow are evaluated by Hatami and Ganji [13] using the differential transformation method. The temperature distribution equation for a fully wet semi-spherical porous fin is presented with a new modified fin parameter by Hatami et al. [14]. Hatami and Ganji [15] presented the temperature distribution equation and refrigeration efficiency for fully wet circular porous fins. All of the researchers show that the TEC's performance is affected strongly by heat sink's temperature distribution and performance. Thus, in the present paper, a new design for the heat sink with an exponential fin is proposed for TEC's devices. To evaluation of cooling performance of the proposed system, the energy balance equation of considered heat sinks including mass transfer and variable thermal conductivity effect is considered.
The new novel analytical method named differential transformation method (DTM) used for an analytical solution of the energy balance equation. The Range-Kutta numerical solution is used for verification of the analytical solution. Finally, the optimum design point, with the lowest volume, highest heat transfer and thermal efficiency is selected with efficient multiobjective optimization procedure so-called Ant Colony Optimization.
2. Description of Problem and Solution Method The schematic view of the proposed heat sink for TEC devices with exponential fin is shown in Figure 1.
Figure 1: Thermoelectric cooling system sink and exponential fin
The cooling performance of heat sinks, strongly affected by temperature distribution in fins. For the evaluation of temperature distribution in considered heat sinks, the energy balance equation (included of conduction, convection, and mass transfer) is presented in equation (1). In this equation, the effect of variable thermal conductivity and variable heat transfer are considered [16]. (1) In Eq. (1), T is fin temperature, is fin's cross-section area, is fin's external surface area, is ambient temperature, is latent heat, ω is
humidity ratio. The variable thermal conductivity of the used material can be calculated based on thermal conductivity at the specific temperature and material parameter [17]: (2) The exponential fin's cross-section area and convective heat transfer area in equation (1), can be evaluated as below: (3) ⇒ (4) The simplification of equation (1) can be written as bellow: (5) In Eq. (5), the heat transfer coefficients due to convection (h) and mass transfer ( are linked through the Chilton-Clumberon relationship: (6) Where Le is the ratio of thermal diffusivity to mass diffusivity and named Lewis number. Sharqawy and Zubair [18] use the linear expression for calculation of as below: (7) (8) The simplified form of equation (5) can be evaluated with equation (9): (9) Where: (10)
The boundary conditions for the above nonlinear differential equation can be presented with equation (11):
(11) The governed thermal differential equation for exponentially fin with variable heat transfer coefficient (equation 9) is nonlinear. For the solution of this nonlinear differential equation a very high accuracy analytically solution so-called differential transformation method (DTM) is applied. The basic concept and principle of differential transformation and solution procedure can be found in [17]. Finally, the heat transfer rate heat sink can be calculated as below: (12) ∫ [
]
One of the important parameter for evaluation of fin's operation is thermal efficiency. This parameter is defined as the ratio of actual heat transfer from fin and heat transfer from ideal fin (with infinite thermal conductivity): (13) ] ∫ [ [
]
4. Optimization and Decision Making In heat sinks that used in in the thermoelectric cooling system, three parameters are very important in design procedure. These three parameters are heat sinks volume, thermal efficiency and heat transfer to ambient. In this paper, the multi-objective optimization process is applied for design with the lowest volume and highest thermal efficiency and heat dissipation. For multi-objective optimization, an efficient method based on the evolutionary algorithm known as Ant Colony Optimization (ACO) is used. The details and algorithm that used in ACO can be found in [19]. The multiobjective optimization results with ACO, present many optimization points (Pareto Front) that can be selected as the final optimum design point. The selection of optimum design from available ACO solutions required the decision-making process. In this paper, the LINMAP decision-making method was employed. In the LINMAP method, after non-
dimensionalization of all objectives, the distance of each solution on the solution frontier from the ideal unreachable point denoted by d is determined as follow: (14) √∑ In LINMAP method, the design point with a minimum distance from the ideal point is selected as a final optimal solution [20].
5. The Solution with the Differential Transformation Method (DTM) For the analytical solution of governed differential equation (Eq. (9)), the differential transformation of all terms in Eq. (9) is performed as below: (15) ∑ ∑∑ ∑∑ ∑∑ ∑
From boundary conditions in Eq. (11), DTM of boundary condition can be evaluated as bellow: (16)
Where , is the auxiliary parameter and is calculated at the final step. With the solution procedure, the required coefficients are calculated as bellow: (17)
(
)
(
)
. . . . Thus, the temperature distribution can be presented in: ∑
(18)
6. Result and Discussion 6-1. Validation For the evaluation of analytical results accuracy, the 4-order Range-Kuta numerical solutions are used. The operational and geometrical parameters of the case study can be found in Table 1.
Table 1: geometrical and operation variables of case study
Fin Material aluminum Length Thickness (at base plate) Base plate temperature Ambient temperature ⁄ Convection coefficient Thus, the depended parameter can be calculated as below: (19) With applying boundary condition to the analytical solution, the auxiliary parameter v is calculated the equal to 0.9219. Comparison of the analytical solution and the numerical solution can be found in Table 2. The results show that the analytical solution implies very good agreement with the numerical solution. Table 2: Comparison of analytical and numerical results X/L
DTM
Numerical
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.92198 0.92205 0.92260 0.92408 0.92695 0.93170 0.93878 0.94867 0.96185 0.97880 1.00000
0.92198 0.92205 0.92260 0.92408 0.92695 0.93170 0.93878 0.94867 0.96185 0.97880 1.00000
Error Absolute Relative 4.48E-12 4.86E-12 4.48E-12 4.86E-12 4.48E-12 4.86E-12 4.48E-12 4.85E-12 4.48E-12 4.84E-12 4.48E-12 4.81E-12 4.49E-12 4.78E-12 4.49E-12 4.73E-12 4.38E-12 4.55E-12 3.64E-12 3.72E-12 0 0
6-2. Effect of Geometrical and Operational Parameter The effect of ambient temperature on temperature distribution at the different position is shown in Figure 2. Results show that, at lowest value for the distance from the base plate (x/L=1), the temperatures are equal to the base plate temperature (Tb=25) and the temperature is decreased with increase in
length (x/L<1) because of axial conduction resistance of fin. Also, with increasing in ambient temperature, the temperature difference between heat sink and ambient, and heat transfer rate are deceased; thus the local temperatures are increased. 25
Temperature ( C )
24.9 24.8 24.7 Ta = 10 Ta = 15 Ta = 20
24.6 24.5 24.4 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Non-dimensional Length
1
Figure 2: The effect of ambient temperature on temperature distribution (H = 60, Tb = 25, tb = 0.03)
The effect of heat sink's thickness on temperature distribution is shown in Figure 3. Results show that the local temperature is increased with an increase in thickness. This increase can be related to increases in heat transfer surface and heat transfer rate from the heat sink.
Non-dimensional Temperature
1 0.99
tb = 0.02 tb = 0.025 tb = 0.03
0.98 0.97 0.96 0.95 0.94 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Non-dimensional Length
1
Figure 3: The effect of fin thickness on temperature distribution (H = 60, Tb = 25, Ta = 10)
The effect of base plate and ambient temperatures on the thermal efficiency at the specific thickness and ambient humidity ratio is presented in Figure 4. Results show that increases in ambient temperature caused decreasing in thermal efficiency and heat transfer rate. Because with increases in ambient temperature the temperature difference between the heat sink and ambient is decreased and thus the thermal efficiency and the heat transfer rate are increased.
96.4
Efficiency (%)
96.2 96 95.8 Tb = 35 Tb = 30 Tb = 25 Tb = 20 Tb = 15
95.6 95.4 95.2 95 3
6
9
12 15 18 21 24 27 30 33 36 Ambient temperature (C)
Figure 4: The variation of thermal efficiency at various ambient and base plate temperature (H = 60, tb = 0.025)
150
Heat transfer (W)
130
Tb = 35 Tb = 30 Tb = 25 Tb = 20 Tb = 15
110 90 70 50 3
6
9
12 15 18 21 24 27 30 33 36 Ambient temperature (C)
Figure 5: The variation of thermal efficiency at various ambient and base-plate temperature (tb = 0.025, H=60)
Figure 4 and Figure 5, show that increases in base plate temperature caused the rise in thermal efficiency and heat transfer. Because the heat transfer and thermal efficiency is the function of temperature difference and thermal resistance. Thus, with increases in base plate and ambient temperature, these values are improved. In general, the operation of the thermoelectric converters in high temperature conditions and lower ambient temperature will increase the thermoelectric cooling system's efficiency. The effect of ambient humidity ratio on thermal parameters is shown in Figure 6. With increasing in ambient air humidity ratio, the convective heat transfer coefficient is decreased, and thus thermal efficiency is decreased.
96.2 96.1
Efficiency (%)
96 95.9
H = 80 H = 70 H = 60
95.8 95.7 95.6 95.5 95.4 95.3 5
6
7
8 9 10 11 12 13 Ambient Temperature (C)
14
15
Figure 6: The variation of thermal efficiency at various ambient temperature and Humidity (tb = 0.025, Tb = 25)
The effect of thickness on the thermal efficiency of the considered heat sink is shown in Figure 7. Increasing in thickness causes improvement in heat transfer area and convective heat transfer. Thus the thermal efficiency is increased with an increase in this parameter. However, this increment is directly led to increasing the material costs, and a balance should be performed between increasing the efficiency of the thermoelectric generator and the associated costs.
Efficiency (%)
95.9
tb = 0.03 tb = 0.025 tb = 0.02
95.8 95.7 95.6 95.5 95.4 4
6
8
10 12 14 16 18 20 22 24 26 Ambient temperature (C)
Figure 7: The variation of thermal efficiency at various ambient temperature and fin thickness (H = 60, Tb = 25)
6-3. Multi-Objective Optimization For the multi-objective optimization with ACO, the selected objective function can be presented as below: f(1)=Volume (cm3 ), should be minimized f(2)=Heat transfer to Ambient (W), should be maximized f(3)=thermal efficiency (%), should be maximized The Pareto front from Ant Colony Optimization procedure is shown in Figure 8 and Figure 9.
Figure 8: The Pareto front with Ant Colony Multi-Objective Optimization
Figure 9: The normalized Pareto front with Ant Colony Multi-Objective Optimization
In Pareto front, the ideal point with the lowest volume (0.665 cm3), highest heat transfer to ambient (185.61 W) and highest thermal efficiency (97.37 %) is located out of the Pareto front. Thus with the principle of LINMAP method, the nearest point in Pareto front from this ideal point selected as the optimum point. The characteristics of this optimum point are presented in Table 2. Table 3: The decision variable and objective function at optimum point
Objective Function Decision Variables
Variable Volume (cm3) Heat transfer (W) Thermal efficiency (%) Thickness (m) Ambient Temperature Base Plate Temperature Humidity Ratio
Optimum Value 206.797 64.624 96.864 0.058 11.23 33.48 83.35
7. Conclusion In this paper, the problem of optimum design for special type of heat sinks that used in thermoelectric cooling devices is considered. For analysis, the proposed heat sink, the energy balance equation for the heat sink with the exponential profile is governed. The very excellent analytical method (DTM) is applied to the nonlinear differential equation. The comparison of analytical results and numerical solution show the good agreement between these results. Investigation the effect of geometrical and operation parameters show that improvements in base plate temperature, fin thickness led to increasing the thermal efficiency and heat transfer. Finally, the multiobjective optimization procedure with (ACO) algorithm is performed, and a suitable optimum point with the lowest volume and highest thermal efficiency and heat transfer are selected.
Nomenclature Constants defined in Eq.(10) kg
w
.kg a1
Fin area m 2
fin's external surface area
B
Parameter defined in Eq.(10) K Parameter defined in Eq.(10) kg
w
.kg a1
specific heat of incoming moist air stream
w
h
The heat transfer coefficient
k
The thermal conductivity of the fin w / m .k
L
Fin length m
/ m 2 .k
j .kg 1.k 1
Lewis number
m
Fin parameter defined in Eq.(10) Fin parameter defined in Eq.(10)
P
Fin perimeter m Heat transfer through fin W Efficiency (%)
H T
b V
Relative humidity of ambient air Temperature K Base temperature (K) Fin thickness at base Width of the fin Auxiliary parameter which calculates at final step Fin volume m 3
Dimensional distance parameter m X Non-dimensional distance parameter Greek letters x
Slope of the thermal conductivity-temperature curve 1/ k
Non-dimensional parameter defined in Eq.(10) Non-dimensional Temperature defined in Eq.(10) humidity ratio of air
kg
w
.kg a1
humidity ratio of base temperature
subscripts a Ambient air c Cross section Dew point s Fin surface
References [1] R. Chein, G. Huang, Thermoelectric cooler application in electronic cooling, Applied Thermal Engineering 24 (2004) 2207-2217. [2] Y. Cheng, W. Lin, Geometric optimization of thermoelectric coolers in a confined volume using genetic algorithms, Applied Thermal Engineering 25 (2005) 2983-2997. [3] H. Lee, Optimal design of thermoelectric devices with dimensional analysis, Applied Energy 106 (2013) 79-88. [4] P. Naphon, S. Wiriyasart, Liquid cooling in the mini-rectangular fin heat sink with and without thermoelectric for CPU, International Comunication in Heat and mass transfer 36 (2009) 166-171. [5] X. Wang, J. Yu, M. Ma, Optimization of heat sink configuration for thermoelectric cooling system based on entropy generation analysis, International Journal of Heat and Mass Transfer 63 (2013) 361-365. [6] L. Qinghai, W. Yanjin, Z. Pengfei, A novel thermoelectric airconditioner for a truck cab, International Conference on Advances in Energy Engineering, 2010. [7] H.Y. Zhang, A general approach in evaluating and optimizing thermoelectric coolers, International Journal of Refrigeration 33 (2010) 1187-1196. [8] A.S. Dogonchi , M. Hatami , Kh. Hosseinzadeh, G. Domairry, Nonspherical particles sedimentation in an incompressible Newtonian medium by Padé approximation, Powder Technology 278 (2015) 248–256. [9] Kh. Hosseinzadeh, A.Jafarian Amiri, S.Saedi Ardahaie, D.D. Ganji, Effect of variable lorentz forces on nanofluid flow in movable parallel plates utilizing analytical method, Case Studies in Thermal Engineering 10 (2017) 595-610. [10] S.A. Atouei, Kh. Hosseinzadeh, M. Hatami, Seiyed E. Ghasemi, S.A.R. Sahebi, D.D. Ganji, Heat Transfer Study on Convective–Radiative SemiSpherical Fins with Temperature-Dependent Properties and Heat Generation using Efficient Computational Methods, Applied Thermal Engineering 89 (2015) 299-305
[11] M. Babaelahi, E. Eshraghi, Optimum analytical design of medical heat sink with convex parabolic fin including variable thermal conductivity and mass transfer, Extreme Mechanics Letter, Extreme mechanics Letter 15 (2017) 83-90 [12] A.J. Amiri, A. Saedi, A. Amooie, K. Hosseinzadeh, D.D. Ganji, Investigating the effect of adding nanoparticles to the blood flow in presence of magnetic field in a porous blood arterial, Informatics in Medicine Unlocked 10 (2017) 71-81. [13] M. Hatami, D.D. Ganji, Motion of a spherical particle in a fluid forced vortex by DQM and DTM, Particuology 16(2014) 206-212. [14] M. Hatami, GH.R. Mehdizadeh Ahangar, D.D. Ganji, K. Boubaker, Refrigeration efficiency analysis for fully wet semi-spherical porous fins, Energy Conversion and Management 84 (2014) 533–540. [15] M. Hatami, D.D. Ganji, Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis, International Journal of Refrigeration 40 (2014) 140-151. [16] F.P. Incropera, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, Inc. 1981. [17] A. A. Joneidi, D. D. Ganji, M. Babaelahi, Differential Transformation Method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity, International Communications in Heat and Mass Transfer, 36 (2009) 757-762. [18] M. H. Sharqawy, S. M. Zubair, Efficiency and optimization of an annular fin with combined heat and mass transfer–An analytical solution, International Journal of Refrigeration, 30 (2007) 751-757. [19] M. Schlueter, Nonlinear Mixed Integer based Optimization Technique for Space Applications, Ph.D. Thesis, School of Mathematics, University of Birmingham (UK) (2012) [20] M. Babaelahi, H. Sayyaadi, Analytical closed-form model for predicting the power and efficiency of Stirling engines based on a comprehensive
numerical model and the genetic programming, Energy, 98 (2016) 324-339.
PII: DOI: Reference:
S2352-4316(18)30140-8 https://doi.org/10.1016/j.eml.2018.11.003 EML 415
To appear in:
Extreme Mechanics Letters
Received date : 26 June 2018 Revised date : 25 October 2018 Accepted date : 20 November 2018 Please cite this article as: M. Babaelahi and H. Jafari, New optimum design for cooling system in thermoelectric thermal devices, Extreme Mechanics Letters (2018), https://doi.org/10.1016/j.eml.2018.11.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
New Optimum Design for Cooling System in Thermoelectric Thermal Devices Mojtaba. Babaelahi*1, Hamed. Jafari Department of Mechanical Engineering, University of Qom, P.O. Box 6765-1719, Qom, Iran.
Abstract The problem of efficient cooling in the compact medium is a very critical problem in the design of thermoelectric thermal devices. In this paper, the thermal behavior of special heat sinks with exponential fin is considered for use in thermoelectric systems. To evaluation of cooling performance of the proposed cooling system, the energy balance of considered heat sinks with variable thermal conductivity including mass transfer is evaluated. The effective analytical method so-called differential transformation method is selected for the solution of the heat balance nonlinear differential equation. Comparisons of the analytical and numerical solution show the excellent accuracy of the selected analytical solution. Based on analytical results, the effects of operational and geometrical parameters on the performance of the heat sink in TEC are evaluated. To the selection of best optimum design with lowest volume, highest heat transfer and thermal efficiency, the Ant Colony Optimization procedure is selected for multi-objective optimization purposes. Keywords: Thermoelectric devices, thermal, heat sink, analytical, Ant Colony Optimization.
1. Introduction The thermoelectric cooling (TEC) devices are one of the thermal components that used for various applications. These applications can be simpli*
-Corresponding Author: Mojtaba Babaelahi (M. Babaelahi) [email protected]
fied in active cooling below ambient temperature or high-temperature precision. An important parameter in operation of thermoelectric heat devices is the temperature difference between the hot and the cold side of a Peltier element used in TEC. This temperature difference is affected by the performance of the cooling system used on the cold side of the Peltier element. In recent year, many types of research are performed for improving the performance of TEC. In this section, some of the bold research in this filed is presented. In general, the efficient methods used to optimization of TEC can be divided into three categories: 1) design and optimization of thermoelectric module's geometry, 2) Design and optimization of the heat sink and 3) change the working condition. In this section, some of these researches are reviewed. The theoretical investigation of thermoelectric cooling devices was performed by Chein and Huang [1]. In this research, the effect of gas and liquid cooling medium on the performance of the electronic cooling system was evaluated. Cheng and Lin [2], try to optimize the geometrical parameters of TES such as length, the number of modules, etc. They used the genetic algorithm optimization method for maximization of cooling performance. Lee [3] performed new dimensionless analysis for the thermoelectric cooling system and introduced some dimensionless group such as thermal conduction ratio, the convection conduction ratio, and load resistance ratio. Naphon and Wiriyasart [4] considered the effect of the rectangular fin on TEC for CPU, and the cooling liquid flow is analysis. This analysis was performed with and without thermoelectric cooling. The results show that the CPU performance was affected strongly by thermoelectric cooling performance. The irreversibility analysis of thermoelectric cooling devices performed by Wang et al. [5]. In this research, the first and second laws of thermodynamics are applied to TEC and the dimensionless entropy generation number introduced. Luo et al. [6] used a new novel TEC device for a truck cab. In this research, the performance of the TEC system is tested for ambient temperature between 46-30. The results show that the cooling per-
formance of the TEC varied between 0.4 to 0.8. Also, this paper presented that the cooling performance could be maximized by optimization of design parameters. Zhang [7], considered the optimization of the thermoelectric cooler with dimensionless analysis method. In this research, the iterative procedure method is a substitute with the straight-forward method. In the area of analytical solution, there are several types of researches that have been reviewed in the following paragraph. Dogonchi et al. [8] investigated the unsteady motion of a vertically falling non-spherical particle in the incompressible Newtonian media. The analytical results indicate that the velocity of the gold particle is higher than the copper and aluminum particles. Hosseinzadeh et al. [9] examined the effect of Brownian motion, thermophoresis phenomenon and Lewis number on MHD nanofluid flow heat transfer. In another article the heat transfer and temperature distribution for semi-spherical convective-radiative porous fins are solved analytically by Atouei et al. [10]. Babaelahi and Eshraghi [11] used the analytical method for evaluation the effect of the convex parabolic fin on the thermal behavior of medical heat sink. Amiri et al. [12] examines both the numerical and the analytical approaches to the heat transfer in blood fluid flow containing nanoparticles in a porous media affected by the magnetic field. The coupled equations of the particle motion in fluid flow are evaluated by Hatami and Ganji [13] using the differential transformation method. The temperature distribution equation for a fully wet semi-spherical porous fin is presented with a new modified fin parameter by Hatami et al. [14]. Hatami and Ganji [15] presented the temperature distribution equation and refrigeration efficiency for fully wet circular porous fins. All of the researchers show that the TEC's performance is affected strongly by heat sink's temperature distribution and performance. Thus, in the present paper, a new design for the heat sink with an exponential fin is proposed for TEC's devices. To evaluation of cooling performance of the proposed system, the energy balance equation of considered heat sinks including mass transfer and variable thermal conductivity effect is considered.
The new novel analytical method named differential transformation method (DTM) used for an analytical solution of the energy balance equation. The Range-Kutta numerical solution is used for verification of the analytical solution. Finally, the optimum design point, with the lowest volume, highest heat transfer and thermal efficiency is selected with efficient multiobjective optimization procedure so-called Ant Colony Optimization.
2. Description of Problem and Solution Method The schematic view of the proposed heat sink for TEC devices with exponential fin is shown in Figure 1.
Figure 1: Thermoelectric cooling system sink and exponential fin
The cooling performance of heat sinks, strongly affected by temperature distribution in fins. For the evaluation of temperature distribution in considered heat sinks, the energy balance equation (included of conduction, convection, and mass transfer) is presented in equation (1). In this equation, the effect of variable thermal conductivity and variable heat transfer are considered [16]. (1) In Eq. (1), T is fin temperature, is fin's cross-section area, is fin's external surface area, is ambient temperature, is latent heat, ω is
humidity ratio. The variable thermal conductivity of the used material can be calculated based on thermal conductivity at the specific temperature and material parameter [17]: (2) The exponential fin's cross-section area and convective heat transfer area in equation (1), can be evaluated as below: (3) ⇒ (4) The simplification of equation (1) can be written as bellow: (5) In Eq. (5), the heat transfer coefficients due to convection (h) and mass transfer ( are linked through the Chilton-Clumberon relationship: (6) Where Le is the ratio of thermal diffusivity to mass diffusivity and named Lewis number. Sharqawy and Zubair [18] use the linear expression for calculation of as below: (7) (8) The simplified form of equation (5) can be evaluated with equation (9): (9) Where: (10)
The boundary conditions for the above nonlinear differential equation can be presented with equation (11):
(11) The governed thermal differential equation for exponentially fin with variable heat transfer coefficient (equation 9) is nonlinear. For the solution of this nonlinear differential equation a very high accuracy analytically solution so-called differential transformation method (DTM) is applied. The basic concept and principle of differential transformation and solution procedure can be found in [17]. Finally, the heat transfer rate heat sink can be calculated as below: (12) ∫ [
]
One of the important parameter for evaluation of fin's operation is thermal efficiency. This parameter is defined as the ratio of actual heat transfer from fin and heat transfer from ideal fin (with infinite thermal conductivity): (13) ] ∫ [ [
]
4. Optimization and Decision Making In heat sinks that used in in the thermoelectric cooling system, three parameters are very important in design procedure. These three parameters are heat sinks volume, thermal efficiency and heat transfer to ambient. In this paper, the multi-objective optimization process is applied for design with the lowest volume and highest thermal efficiency and heat dissipation. For multi-objective optimization, an efficient method based on the evolutionary algorithm known as Ant Colony Optimization (ACO) is used. The details and algorithm that used in ACO can be found in [19]. The multiobjective optimization results with ACO, present many optimization points (Pareto Front) that can be selected as the final optimum design point. The selection of optimum design from available ACO solutions required the decision-making process. In this paper, the LINMAP decision-making method was employed. In the LINMAP method, after non-
dimensionalization of all objectives, the distance of each solution on the solution frontier from the ideal unreachable point denoted by d is determined as follow: (14) √∑ In LINMAP method, the design point with a minimum distance from the ideal point is selected as a final optimal solution [20].
5. The Solution with the Differential Transformation Method (DTM) For the analytical solution of governed differential equation (Eq. (9)), the differential transformation of all terms in Eq. (9) is performed as below: (15) ∑ ∑∑ ∑∑ ∑∑ ∑
From boundary conditions in Eq. (11), DTM of boundary condition can be evaluated as bellow: (16)
Where , is the auxiliary parameter and is calculated at the final step. With the solution procedure, the required coefficients are calculated as bellow: (17)
(
)
(
)
. . . . Thus, the temperature distribution can be presented in: ∑
(18)
6. Result and Discussion 6-1. Validation For the evaluation of analytical results accuracy, the 4-order Range-Kuta numerical solutions are used. The operational and geometrical parameters of the case study can be found in Table 1.
Table 1: geometrical and operation variables of case study
Fin Material aluminum Length Thickness (at base plate) Base plate temperature Ambient temperature ⁄ Convection coefficient Thus, the depended parameter can be calculated as below: (19) With applying boundary condition to the analytical solution, the auxiliary parameter v is calculated the equal to 0.9219. Comparison of the analytical solution and the numerical solution can be found in Table 2. The results show that the analytical solution implies very good agreement with the numerical solution. Table 2: Comparison of analytical and numerical results X/L
DTM
Numerical
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.92198 0.92205 0.92260 0.92408 0.92695 0.93170 0.93878 0.94867 0.96185 0.97880 1.00000
0.92198 0.92205 0.92260 0.92408 0.92695 0.93170 0.93878 0.94867 0.96185 0.97880 1.00000
Error Absolute Relative 4.48E-12 4.86E-12 4.48E-12 4.86E-12 4.48E-12 4.86E-12 4.48E-12 4.85E-12 4.48E-12 4.84E-12 4.48E-12 4.81E-12 4.49E-12 4.78E-12 4.49E-12 4.73E-12 4.38E-12 4.55E-12 3.64E-12 3.72E-12 0 0
6-2. Effect of Geometrical and Operational Parameter The effect of ambient temperature on temperature distribution at the different position is shown in Figure 2. Results show that, at lowest value for the distance from the base plate (x/L=1), the temperatures are equal to the base plate temperature (Tb=25) and the temperature is decreased with increase in
length (x/L<1) because of axial conduction resistance of fin. Also, with increasing in ambient temperature, the temperature difference between heat sink and ambient, and heat transfer rate are deceased; thus the local temperatures are increased. 25
Temperature ( C )
24.9 24.8 24.7 Ta = 10 Ta = 15 Ta = 20
24.6 24.5 24.4 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Non-dimensional Length
1
Figure 2: The effect of ambient temperature on temperature distribution (H = 60, Tb = 25, tb = 0.03)
The effect of heat sink's thickness on temperature distribution is shown in Figure 3. Results show that the local temperature is increased with an increase in thickness. This increase can be related to increases in heat transfer surface and heat transfer rate from the heat sink.
Non-dimensional Temperature
1 0.99
tb = 0.02 tb = 0.025 tb = 0.03
0.98 0.97 0.96 0.95 0.94 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Non-dimensional Length
1
Figure 3: The effect of fin thickness on temperature distribution (H = 60, Tb = 25, Ta = 10)
The effect of base plate and ambient temperatures on the thermal efficiency at the specific thickness and ambient humidity ratio is presented in Figure 4. Results show that increases in ambient temperature caused decreasing in thermal efficiency and heat transfer rate. Because with increases in ambient temperature the temperature difference between the heat sink and ambient is decreased and thus the thermal efficiency and the heat transfer rate are increased.
96.4
Efficiency (%)
96.2 96 95.8 Tb = 35 Tb = 30 Tb = 25 Tb = 20 Tb = 15
95.6 95.4 95.2 95 3
6
9
12 15 18 21 24 27 30 33 36 Ambient temperature (C)
Figure 4: The variation of thermal efficiency at various ambient and base plate temperature (H = 60, tb = 0.025)
150
Heat transfer (W)
130
Tb = 35 Tb = 30 Tb = 25 Tb = 20 Tb = 15
110 90 70 50 3
6
9
12 15 18 21 24 27 30 33 36 Ambient temperature (C)
Figure 5: The variation of thermal efficiency at various ambient and base-plate temperature (tb = 0.025, H=60)
Figure 4 and Figure 5, show that increases in base plate temperature caused the rise in thermal efficiency and heat transfer. Because the heat transfer and thermal efficiency is the function of temperature difference and thermal resistance. Thus, with increases in base plate and ambient temperature, these values are improved. In general, the operation of the thermoelectric converters in high temperature conditions and lower ambient temperature will increase the thermoelectric cooling system's efficiency. The effect of ambient humidity ratio on thermal parameters is shown in Figure 6. With increasing in ambient air humidity ratio, the convective heat transfer coefficient is decreased, and thus thermal efficiency is decreased.
96.2 96.1
Efficiency (%)
96 95.9
H = 80 H = 70 H = 60
95.8 95.7 95.6 95.5 95.4 95.3 5
6
7
8 9 10 11 12 13 Ambient Temperature (C)
14
15
Figure 6: The variation of thermal efficiency at various ambient temperature and Humidity (tb = 0.025, Tb = 25)
The effect of thickness on the thermal efficiency of the considered heat sink is shown in Figure 7. Increasing in thickness causes improvement in heat transfer area and convective heat transfer. Thus the thermal efficiency is increased with an increase in this parameter. However, this increment is directly led to increasing the material costs, and a balance should be performed between increasing the efficiency of the thermoelectric generator and the associated costs.
Efficiency (%)
95.9
tb = 0.03 tb = 0.025 tb = 0.02
95.8 95.7 95.6 95.5 95.4 4
6
8
10 12 14 16 18 20 22 24 26 Ambient temperature (C)
Figure 7: The variation of thermal efficiency at various ambient temperature and fin thickness (H = 60, Tb = 25)
6-3. Multi-Objective Optimization For the multi-objective optimization with ACO, the selected objective function can be presented as below: f(1)=Volume (cm3 ), should be minimized f(2)=Heat transfer to Ambient (W), should be maximized f(3)=thermal efficiency (%), should be maximized The Pareto front from Ant Colony Optimization procedure is shown in Figure 8 and Figure 9.
Figure 8: The Pareto front with Ant Colony Multi-Objective Optimization
Figure 9: The normalized Pareto front with Ant Colony Multi-Objective Optimization
In Pareto front, the ideal point with the lowest volume (0.665 cm3), highest heat transfer to ambient (185.61 W) and highest thermal efficiency (97.37 %) is located out of the Pareto front. Thus with the principle of LINMAP method, the nearest point in Pareto front from this ideal point selected as the optimum point. The characteristics of this optimum point are presented in Table 2. Table 3: The decision variable and objective function at optimum point
Objective Function Decision Variables
Variable Volume (cm3) Heat transfer (W) Thermal efficiency (%) Thickness (m) Ambient Temperature Base Plate Temperature Humidity Ratio
Optimum Value 206.797 64.624 96.864 0.058 11.23 33.48 83.35
7. Conclusion In this paper, the problem of optimum design for special type of heat sinks that used in thermoelectric cooling devices is considered. For analysis, the proposed heat sink, the energy balance equation for the heat sink with the exponential profile is governed. The very excellent analytical method (DTM) is applied to the nonlinear differential equation. The comparison of analytical results and numerical solution show the good agreement between these results. Investigation the effect of geometrical and operation parameters show that improvements in base plate temperature, fin thickness led to increasing the thermal efficiency and heat transfer. Finally, the multiobjective optimization procedure with (ACO) algorithm is performed, and a suitable optimum point with the lowest volume and highest thermal efficiency and heat transfer are selected.
Nomenclature Constants defined in Eq.(10) kg
w
.kg a1
Fin area m 2
fin's external surface area
B
Parameter defined in Eq.(10) K Parameter defined in Eq.(10) kg
w
.kg a1
specific heat of incoming moist air stream
w
h
The heat transfer coefficient
k
The thermal conductivity of the fin w / m .k
L
Fin length m
/ m 2 .k
j .kg 1.k 1
Lewis number
m
Fin parameter defined in Eq.(10) Fin parameter defined in Eq.(10)
P
Fin perimeter m Heat transfer through fin W Efficiency (%)
H T
b V
Relative humidity of ambient air Temperature K Base temperature (K) Fin thickness at base Width of the fin Auxiliary parameter which calculates at final step Fin volume m 3
Dimensional distance parameter m X Non-dimensional distance parameter Greek letters x
Slope of the thermal conductivity-temperature curve 1/ k
Non-dimensional parameter defined in Eq.(10) Non-dimensional Temperature defined in Eq.(10) humidity ratio of air
kg
w
.kg a1
humidity ratio of base temperature
subscripts a Ambient air c Cross section Dew point s Fin surface
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