Collaborative Simulated Annealing Genetic Algorithm for Geometric Optimization of Thermo-electric Coolers

Abstract

Thermo-electric Coolers (TECs) nowadays are applied in a wide range of thermal energy systems. This is due to its superior features where no refrigerant and dynamic parts are needed. TECs generate no electrical or acoustical noise and are environment friendly. Over the past decades, many researches were employed to improve the efficiency of TECs by enhancing the material parameters and design parameters. The material parameters are the most significant, but they are restricted by currently available materials and module fabricating technologies. Therefore, the main objective of TECs design is to determine a set of design parameters such as leg area, leg length, and the number of legs. Two elements that play an important role when considering the suitability of TECs in applications are rated of refrigeration (ROR) and coefficient of performance (COP). In this chapter, the technical issues of TECs were discussed. After that, a new method of optimizing the dimension of TECs using collaborative simulated annealing genetic algorithm (CSAGA) to maximize the rate of refrigeration (ROR) was proposed. Equality constraint and inequality constraint were taken into consideration. The results of optimization obtained by using CSAGA were validated by comparing with those obtained by using stand-alone genetic algorithm and simulated annealing optimization technique. This work revealed that CSAGA was more robust and more reliable than stand-alone genetic algorithm and simulated annealing.

Appendices

Acronyms

Symbol	Description	Unit
A	Cross-sectional area of TEC legs	mm2
L	Height of the confined volume	mm
N	Number of thermo-electric couple	–
S	Total area of STECs	mm2
ROR	Rate of refrigeration	W
COP	Coefficient of performance	–
TE	Thermo-electric element	–
TECs	Thermo-electrics coolers	–
STECs	Single-stage thermo-electrics coolers	–
TTECs	Two-stage thermo-electrics coolers	–
MWD	Measurement while drilling	–
Z	Figure of merit	–
ZT	Dimensionless figure of merit	–
I	Supplied current to TECs	A
T h	Hot side temperature	K
T c	Cold side temperature	K
T ave	Average of cold and hot side temperature	K
SOP	Single-objective optimization problem	–
MOP	Multi-objective cptimization problem	–
CSAGA	Collaborative simulated annealing genetic algorithm	–
GA	Genetic algorithm	–
SA	Simulated annealing	–
PSO	Particle swarm cptimization	–
ACO	Ant olony ptimization	–
NSGA-II	Non-dominated sorting algorithm	–
TLBO	Teaching–learning-based optimization	–
TS	Tabu search	–
DE	Differential evolution	–
α n = α p	Seebeck coefficient of n-type and p-type thermo-electric element	V/K
ρ n = ρ p	Electrical resistivity of n-type and p-type of thermo-electric element	Ωm
к n = к p	Thermal conductivity of n-type and p-type of thermo-electric element	W/mK
r c	Electrical contact resistance	Ωm2

Key Terms and Definitions

Thermodynamics: is a branch of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy, and pressure that partly describe a body of matter or radiation. It states that the behavior of those variables is subject to general constraints that are common to all materials, not the peculiar properties of particular materials. These general constraints are expressed in the four laws of thermodynamics. Thermodynamics describes the bulk behavior of the body, not the microscopic behaviors of the very large numbers of its microscopic constituents, such as molecules. Its laws are explained by statistical mechanics, in terms of the microscopic constituents. Thermodynamics apply to a wide variety of topics in science and engineering.

Thermo-electrics coolers: uses the Peltier effect to create a heat flux between the junctions of two different types of materials. A Peltier cooler, heater, or thermo-electric heat pump is a solid-state active heat pump which transfers heat from one side of the device to the other, with consumption of electrical energy, depending on the direction of the current. Such an instrument is also called a Peltier device, Peltier heat pump, solid-state refrigerator, or thermoelectric cooler (TEC). They can be used either for heating or for cooling (refrigeration), although in practice the main application is cooling. It can also be used as a temperature controller that either heats or cools.

Rate of refrigeration: ROR or can be called cooling rate is the rate at which heat loss occurs from the surface of an object.

Coefficient of performance: or COP of a heat pump is a ratio of heating or cooling provided to electrical energy consumed. Higher COPs equate to lower operating costs. The COP may exceed 1, because it is a ratio of output–loss, unlike the thermal efficiency ratio of output–Input energy. For complete systems, COP should include energy consumption of all auxiliaries. COP is highly dependent on operating conditions, especially absolute temperature and relative temperature between sink and system, and is often graphed or averaged against expected conditions.

Meta-heuristic: is a higher-level procedure or heuristic designed to find, generate, or select a lower-level procedure or heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity. Meta-heuristic may make few assumptions about the optimization problem being solved, and so they may be usable for a variety of problems.

Hybrid algorithm: is an algorithm that combines two or more other algorithms that solve the same problem, either choosing one (depending on the data), or switching between them over the course of the algorithm. This is generally done to combine desired features of each, so that the overall algorithm is better than the individual components. “Hybrid algorithm” does not refer to simply combining multiple algorithms to solve a different problem—many algorithms can be considered as combinations of simpler pieces—but only to combining algorithms that solve the same problem, but differ in other characteristics, notably performance.

Genetic Algorithm: is a search meta-heuristic that mimics the process of natural selection. This meta-heuristic was routinely used to generate useful solutions to optimization and search problems. Genetic algorithms belong to the larger class of evolutionary algorithms which generate solutions to optimization problems using techniques inspired by natural evolutions, such as inheritance, mutation, selection, and crossover.

Simulated Annealing: is a generic probabilistic meta-heuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more efficient than exhaustive enumeration—provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution.

Robust optimization: is a field of optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution. There are a number of classification criteria for robust optimization problems/models. In particular, one can distinguish between problems dealing with local and global models of robustness, and between probabilistic and non-probabilistic models of robustness. Modern robust optimization deals primarily with non-probabilistic models of robustness that are worst-case oriented and as such usually deploy Wald’s maximum models.

Nonlinear constraint optimization: is an important class of problems with a broad range of engineering, scientific, and operational applications. The form is

$${\text{Minimize}}\, f(x)\, {\text{subject to}}\, c(x) = 0 \, {\text{and}} \,x \geq 0,$$

where the objective function, f: Rn → R, and the constraint functions, $c:\,R^{n} \, \to \,R^{m}$, are twice continuously differentiable. We denote the multipliers corresponding to the equality constraints, c(x) = 0, by y and the multipliers of the inequality constraints, x ≥ 0, by z ≥ 0. An NCO may also have unbounded variables, upper bounds, or general range constraints of the form $\text{l}_{i} \, \le \,c_{i} \left( x \right)\, \le \,u_{i}$, which we omit for the sake of simplicity.