Optimum analytical design of medical heat sink with convex parabolic fin including variable thermal conductivity and mass transfer
Introduction
Electronic medical devices have become more sophisticated and powerful in recent years. These systems contain large arrays of power consuming electronic components and thus the temperature is increased continuously. It is required; the high-performance heat sinks to prevent medical devices from overheating, damaging or destroying. The electronic devices perform a variety of functions and thus the levels of heat generation and power dissipation vary widely. The range of dissipated heat changes from semiconductors with low dissipated heat (about 1-watt heat dissipation) to laser and radio-frequency devices with high dissipated heat (about 1000 watts heat dissipation). The choice of a proper medical heat sink can eliminate (or reduce) the need for a fan and thus create a suitable medical heat sink with the highest reliability and lowest noise. In addition, high-performance heat sinks can efficiently use for cooling medical components without taking up much space.
Selection and design of medical heat sinks are related to thermal properties and manufacturing cost of them. These parameters can be calculated based on geometrical characteristics and temperature distribution in convex parabolic fin. Thus, in this section, the thermal differential equation for convex parabolic convective fin is presented. Furthermore, the effect of variable thermal conductivity and mass transfer is considered.
Bartas and Sellers [1] presented effectiveness of fin for the system consisting of parallel tubes, which joined by web plates. Wilkins [2] presented some observation on the possible reduction in the mass of fins used in space applications by taking advantage of the freedom to use the fin profile. Actually, he predicted the efficiency of triangular fins radiating space at absolute zero temperature. Karlekar and Chao [3] introduced a new method for attaining maximum heat flux from a longitudinal fin which arranged symmetrically around a small base cylinder of uniform temperature. Cockfield [4] discussed the effects of the different configuration in the optimization of radiator fins. Heat transfer and temperature distribution for various shapes and materials of the circular convective–radiative porous fin were studied by Hatami et al. [5]. With the aim of the least square method (LSM) Hatami et al. [6] in addition, predicted temperature distribution in a porous fin by three highly accurate analytical methods: Differential transmission method (DTM), Collocation Method (CM) and Least Square Method (LSM) also fin’s material is Si3N4 and heat generation varies linearly with temperature. Arslanturk [7] applied adomian decomposition method to estimate the efficiency of the straight fin with variable thermal conductivity. Singla and Das [8] solved a non-linear heat equation for a rectangular fin with adomian decomposition method and generic algorithm. Ahmadi et al. [9] investigated unsteady heat transfer caused by the linear motion of a horizontal flat plate over a nanofluid by the differential transmission method. Hatami et al. [10] studied the availability of using extended surface in a heat exchanger to recover waste heat in a diesel engine. Then numerical modeling was done by same author [11]. Ghasemi et al. [12] used DTM for solving the nonlinear temperature distribution equation in a longitudinal fin with temperature-dependent internal heat generation and conductivity. Bhowmik et al. [13] applied Adomian decomposition method in conjunction with the differential evaluation for estimating the annular fin dimension with the rectangular and hyperbolic profile in order to satisfy a prescribed temperature. Hatami et al. [14] presented fin efficiency and temperature distribution for a semispherical fin at fully wet condition by LSM and fourth order Runge–Kutta. They used Darcy’s model in order to simulate the heat transfer through porous media. Hatami and Ganji [15] used new fin parameter, which presented by Sharqawy and Zubair to drive expression for temperature distribution and refrigeration efficiency for fully wet circular porous fins. Hatami and Ganji [16] considered a porous fin and heat transfer through the fin simulated (by) using passage velocity from Darcy’s model also for predicting heat transfer and temperature distribution the least square method and Runge–Kutta is applied. Heidarzadeh et al. [17] employed adomian decomposition method for the solution of the unsteady convective–radiative fin. Also, VIM is included for comparison. ADM method was applied by Chiu and Chen [18] to analyzing convective–radiative fin with nonlinear boundary conditions. The assumption of constant physical properties and uniform heat transfer coefficient reduces the mathematical complexity of the energy equation and allows us to use standard mathematical functions and some cases are documented by Kraus et al. [19]. DTM was applied to steady-state fin with triangular-profile by Bert [20] and nonlinear terms were neglected. Kundu [21] presented the thermal analysis and optimization of longitudinal and pin fins of uniform thickness subjected to fully wet, partially wet and completely dry surface conditions were carried out analytically, and also a comparative study were made between the longitudinal and pin fin for a wide range of design parameters. An analysis was performed to study the efficiency of straight fins of different configurations when subjected to simultaneous heat and mass transfer mechanism by Sharqawy and Zubair [22]. Domairry and Fazeli [23] applied homotopy analysis method for the convective fin with variable thermo-physical properties. Arslanturk [24] derived a nonlinear fin equation which is associated with variable thermal conductivity solved by Adomian decomposition method and also presented correlation for the optimum shape of the annular fin. Residue minimization technique was used to solve nonlinear energy equation for a straight convective fin with temperature-dependent thermal conductivity by Kulkarni and Joglekar [25]. Khani et al. [26] developed an analytical solution for thermal performance straight trapezoidal fin when both thermal conductivity and heat transfer are variable respect to temperature. The variational iteration method was employed as an approximate analytical method by Fouladi et al. [27] to overcome some inherent limitation arising uncontrollability to the nonzero end point boundary condition. Then this method was used to solve some examples in the heat transfer field. Aziz and Torabi [28] considered all nonlinear terms. They assumed that heat transfer and surface emissivity varied with temperature. These studies consider fins with the constant cross-sectional area or tapered fins. Aziz [29] studied optimum design for rectangular stepped convecting fin by using Lagrange’s multipliers method. Kundu and Das [30] presented temperature distribution in a concentric annular fin with a step change in thickness (AFST) and demonstrated that an AFST transfer more heat compared to optimum annular fin. Differential quadrature element method as a simple and accurate method was proposed to obtain the heat transfer rate for optimization convective–radiative fin by M.alekzadeh et al. [31]. Kundu utilized an analytical method described for temperature and heat transfer of an annularly stepped fin with the Simultaneous heat and mass transfer. Kundu and Aziz [32] developed an analytical solution for a pure convective–radiative fin. Then the effects of geometrical and physical parameters were illustrated, and fin efficiency of a stepped fin was compared with uniformly thick fin. Yaghoobi and Torabi [33] solved two nonlinear heat transfer equations by DTM method with considering variable specific heat coefficient. Zhou et al. [34] used combined RSM and FVM method to the analysis of micro channel heat sink. In this research, the wavy channels are used in the micro heat sink. Hatami et al. [35], analyzed the heat transfer characteristic in the internally heated cylinder. In this research, the enclosure constructed with wavy-wall and the nanofluid flow is considered. In another research, Hatami et al. [36] considered optimization of a circular-wavy cavity. It is assumed that the cavity filled with nanofluid and dominant heat transfer term is natural convection. Rahimi-Gorji et al. [37] analyzed the heat transfer characteristics of nanofluid flow over the micro heat sink. In this study effect of various nanofluid flows on thermal performance of heat sink are examined by RSM method.
In the current paper, the thermal analysis of the medical heat sink with a convex parabolic fin with variable thermal conductivity and mass transfer is considered. Due to the nature of the equations, the concept of GDTM is employed to derive analytical solutions of nonlinear fractional differential equations and the Runge–Kutta numerical solution used for validity. The exclusive part of this article is to find the closed-form analytical solution for considering medical heat sink for the first time. Because the electronic medical systems differ in many aspects, the majority of medical applications share a common need for high reliability and performance. Also, these systems require small size and quiet operation. Thus, In the next step, multiobjective optimization is performed for minimum volume and maximum thermal efficiency. In the final step, the results of multiobjective optimization in various environmental conditions are applied to the genetic programming tool and suitable analytical correlations are created for optimum geometrical design variables in considered medical heat sink.
Section snippets
Analysis of the differential transformation method
In this section, the generalized differential transformation method that developed for the analytical solution of the fractional differential equation. The generalized differential transformation of the th derivative of function is as follows:
And the generalized differential inverse transformation of is defined as:
The fundamental mathematical operations used in generalized differential transformation method are listed
Description of problem
In this section, the heat transfer analysis of heat sink with convex parabolic fin is analyzed. These types of heat sink widely used for cooling proposes in medical application specially the implantable devices. These devices need tiny size and precise temperature change coefficients () to protect human organs. One of heat transfer device that used for this thermal management is heat pipe technology that classified as two-phase thermal devices. The suitable transition from vapor to liquid
Solution with Generalized Differential Transformation Method (GDTM)
Now we apply GDTM into Eq. (12). Choosing as unity and Taking the differential transformation of Eq. (12) with respect to :
From boundary conditions in Eq. (13), GDTM of boundary condition can be evaluated as below: where , is auxiliary parameter and is calculated at final step. With solution procedure, the required coefficients are calculated as below:
Optimization
Multi-objective optimization problems usually exhibit a probably uncountable set of solutions to assess the status of vectors showing the best possible trade-offs in the objective function space [41]. Pareto optimality is the key concept to express the relationship between multi-objective optimization results in order to determine a solution which is actually one of the best possible trades-offs [41]. In multi-objective optimization, a process of decision-making is required for selection of the
Genetic programming multivariate fitting
In current paper, the genetic programming method was used for multivariable regression. Genetic programming is a biologically inspired machine learning method that evolves computer programs to perform a task. It does this by randomly generating a population of computer programs (represented by tree structures) and then mutating and crossing over the best performing trees to create a new population. This process is iterated until the population contains programs that (hopefully) solve the task
Result and Discussion
For evaluation of analytical results accuracy, the 4-order Runge–Kuttanumerical solutions are used. The operational and geometrical parameters of case study can be found in Table 2.
Thus, depended parameter can be calculated as below:
With applying boundary condition to analytical solution the auxiliary parameter is calculated equal to 0.9219.
Comparison of analytical solution and numerical solution can be found in Table 3. Results show that the
Conclusion
In this Paper, the heat transfer characteristic in the medical heat sink with the convex parabolic fin with heat mass transfer and variable thermal conductivity is considered. Because of fractional nature of governing differential equation in the considerable problem the Generalized Differential Transformation Method (GDTM) is applied. Thus, the concept of GDTM is employed to derive analytical solutions of considerable nonlinear fractional differential equations. For verification of the
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