Optimization of a thermoelectric generator for heavy-duty vehicles

Energy Conversion and Management 179 (2019) 178–191

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Optimization of a thermoelectric generator for heavy-duty vehicles André Marvão, Pedro J. Coelho , Helder C. Rodrigues ⁎

T

Mechanical Engineering Department, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal

ARTICLE INFO

ABSTRACT

Keywords: Thermoelectric generator Heat exchangers Energy recovery Heavy-duty vehicles Optimization

A thermoelectric generator used to recover energy from the exhaust gases of a heavy-duty freight vehicle is numerically simulated and optimized using two different algorithms: a gradient-based search method, and a global and local direct search method. The model uses a one-dimensional finite volume method to calculate the temperature and the convective heat transfer of the hot and cold fluids along the streamwise direction, as well as the pressure drop, and employs a thermal resistance network and a global energy balance to account for the heat transfer and the thermoelectric effect. Three different heat exchanger configurations are studied with plain, offset strip or triangular fins. The objective function is the net output power, and the geometrical dimensions of the heat exchanger and thermoelectric units are taken as design variables. The results show that the height of the fins, the spacing between them, the height and side length of the cross section of the thermoelectric units, and the distance between the legs are critical parameters in the optimization process. The thickness of the fins, wall ducts, electrical conductor and ceramic strips should be as small as possible, as well as the height of the cooling wall duct. The two optimization methods yield similar optimal solutions.

1. Introduction Fossil fuels continue to lead the market for road vehicles, despite the contribution of biofuels and the increasing use of electric vehicles. Moreover, these alternatives are mostly used in passenger cars. Since commercial vehicles and heavy-duty vehicles will continue to use fossil fuels in the near future, solutions need to be found to improve the energy efficiency and to comply with the increasingly stringent emission regulations. Only about one-third of the energy released in the combustion of fossil fuels consumed by these vehicles is used to move them, while another one third are heat losses in the exhaust gases, and the remainder is used by accessories, wasted in the coolant system, and lost as mechanical losses. If some of the energy available in the exhaust gases is recovered, the efficiency of the vehicles increases and CO2 emissions are lower. Several different technologies can be used to recover waste heat of the exhaust gases in automotive applications [1], but in this work we are concerned with thermoelectricity [2]. When two different conductor wires are joined at both ends, and the junctions are maintained at different temperatures, a potential difference is generated between the two junctions and an electrical current flows between them. This phenomenon constitutes the Seebeck effect and is the working principle of the thermoelectric generator (TEG) [3]. A TEG converts directly thermal energy into electrical energy by means of simple, reliable and environmentally friendly energy conversion

⁎

devices. The TEG has no moving parts, operates silently, it is compact, light and highly reliable, and requires little maintenance, but the cost is high and the efficiency is low. TEGs have a wide range of applications, encompassing electricity generation in remote areas and in space, decentralized domestic power, combined heat and power generation systems, waste heat recovery in industries and transports, and microgeneration for sensors and microelectronics [4,5]. Among these applications, we will focus on the recovery of energy of the exhaust gases in road vehicles [6]. Although TEGs can be used for any road vehicle, we will consider heavy-duty Diesel vehicles used for freight transportation, since road transport carried three-quarters of inland freight in the European Union in the period 2011–2016 [7]. This means that a very significant amount of energy can be recovered from the exhaust gases of these vehicles using TEGs, despite their low conversion efficiency. Since the TEG is a promising technology for heat recovery in road vehicles, it has attracted the attention of many researchers in the last few years. Many of these studies are based on mathematical models. Standard simplified models based on global energy balances and thermoelectric effects [8] are commonly used in the literature. Some of them deal with a single thermoelectric unit [9], while others model a TEG with several thermoelectric units, but assume uniform temperatures of the hot source and cold sink [10]. Other authors who also employed simplified models considered the variation of the temperature of the fluids in the streamwise direction. They divided the exhaust

Corresponding author. E-mail address: [email protected] (P.J. Coelho).

https://doi.org/10.1016/j.enconman.2018.10.045 Received 29 June 2018; Received in revised form 12 October 2018; Accepted 14 October 2018 0196-8904/ © 2018 Elsevier Ltd. All rights reserved.

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Nomenclature A A b cp d F h H I k K L Ls m n P q R Rt s t T T W x

αpn ΔT λ ρ

area (m2) matrix of coefficients vector of independent terms specific heat capacity (J/kg·K) array of search directions distance ratio fins’ height (m); convective heat transfer coefficient (W/ m2·K) height (m) electric current (A) thermal conductivity (W/m·K) thermal conductance (W/K) TEG length (m) distance between legs of a thermoelectric unit (m) mass flow rate (kg/s) number of thermoelectric units power (W) heat transfer rate (W) electrical resistance (Ω) thermal resistance (K/W) fins’ spacing (m) thickness (m) temperature (K) array of temperatures TEG width (m) array of design variables

seebeck coefficient (V/K) temperature difference (K) lagrange multiplier (adjoint state variable) electrical resistivity (Ω·m)

Subscripts Al c Cer Cu cw ex h L n p pn SS t x y z

aluminium cold ceramic copper cooling water exhaust gases hot load n-type material of thermocouple leg p-type material of thermocouple leg thermoelectric unit stainless steel thermal streamwise direction vertical direction transverse direction

Superscripts (k) T

iteration number transpose

Greek symbols αk

step size in kth iteration

gases and coolant ducts in control volumes, each one associated with a thermoelectric unit, and employed a 1D finite volume/finite difference method to calculate the fluid temperature distribution [11]. At every control volume, and given the inlet temperature of the fluids and the convective heat transfer coefficients, the outlet temperature of the fluids and the temperatures at the hot and cold junctions of the thermoelectric unit are determined by solving a system of algebraic equations. Several authors introduced improvements in simplified models, for example, to assess the relevance of the Thomson effect, which is neglected in most simplified models [12], to investigate the performance of a TEG in vehicles during typical driving cycles [13], or to account for the variation of the convective heat transfer and the pressure drop in the exhaust gases duct [14]. While most works dealing with the recovery of energy from the exhaust gases in road transports are concerned with passenger cars or light-duty vehicles, Espinosa et al. [15] modelled a TEG for a long-haul truck Diesel engine, and Vale et al. [16] considered two vehicles used for freight transportation. In both works, the heat exchanger was modelled using a 1D finite volume/finite difference method with empirical correlations to determine the convective heat transfer coefficient, but Vale et al. [16] considered two different heat exchanger configurations and investigated the influence of several fin parameters on the electrical and pumping power. Models based on a local energy balance are able to account for the temperature dependence of the material properties and are expected to provide improved accuracy in comparison with the simplified models. Examples of these models are found in Refs. [17–19]. Zhang [17] developed a non-linear 1D model based on the first law of thermodynamics and Ohm’s law, and used a control volume approach to solve analytically the nonlinear energy transport equation for a thermoelectric unit. Marchenko [18] used the perturbation method to solve the

system of equations for heat and electrical charge transfer in a thermocouple leg. Shen et al. [19] applied a 1D finite-element method and integral-averaged Seebeck coefficients to accurately account for the temperature dependence of the materials, Thomson effect, and heat transfer from the side surface of the legs of a thermoelectric unit. Several authors have developed 3D models to investigate the performance of TEGs [20–24]. Du et al. [20] studied the influence of the cooling type, coolant flow rate, baffles (length, number, and location) and flow arrangement on the performance of a TEG for an internal combustion engine. Meng et al. [21] developed steady-state and transient 3D models that solve the energy and the electric potential equations for a TEG unit, accounting for all thermoelectric effects, variable material properties, and heat losses. The model was applied to investigate the performance of a TEG for waste heat recovery in an automobile for both parallel flow and counter flow cooling patterns [22]. Niu et al. [23] developed two 3D models for TEGs based on different formulations and boundary conditions. Parametric studies based on various thermal boundary conditions were reported. Massaguer et al. [24] used a finite element method/finite volume method to simulate an automotive TEG and proposed a method to predict the fuel economy. The performance of the TEGs may be improved by using materials with higher figures of merit in the range of operating temperatures [25], enhancing the heat transfer from the heat source to the thermoelectric modules and/or from these to the heat sink [26] or optimizing the design of the TEG [27]. In this work, we will focus on the optimization of the design of the TEG. Many parametric studies have been reported to find the geometrical or operating parameters that maximize the power or the conversion efficiency, but only a few of them used optimization algorithms to achieve that goal. Among them, the conjugate gradient method and the genetic algorithm have been the most widely used ones. 179

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Jang and Tsai [27] used a simplified conjugate gradient method to optimize the spacing between thermoelectric units and the spreader thickness on the power density of a TEG in a waste heat recovery system. Meng et al. [28] reported a multi-objective and multi-parameter study to design the optimal structure of a TEG, using their previously developed model [21] along with a conjugate gradient algorithm. The multi-objective function is a weighted average of the output power and conversion efficiency, while the design variables are the number of thermoelectric units, the length of the legs and the ratio of the area of the cross-section of the legs to the total area of the base of the TEG. The area of the TEG module base was prescribed. The genetic algorithm was used by Bélanger and Gosselin [29] to optimize a TEG for waste heat recovery sandwiched in a crossflow heat exchanger. The design variables were the number of thermoelectric units and the electrical current in each control volume, and the objective function was the power output. The electrical topology was also optimized. This work was extended by considering a multi-objective genetic algorithm to optimize the total volume, the total number of thermoelectric modules, output and pumping powers of the TEG [30]. Ibrahim et al. [31] used an analytical model of a thermoelectric unit and three different multi-objective evolutionary algorithms, namely a genetic algorithm, a generalized differential evolution 3 algorithm and particle swarm optimization, to optimize several geometric features and operating conditions in order to maximize the efficiency and output power. Huang and Xu [32] used a model adapted from Kumar [11], a combined response surface method and a genetic algorithm to optimize a TEG for waste heat recovery. Liu et al. [33] performed a multi-objective optimization of the heat exchanger in a TEG for heat recovery in

an internal combustion engine. Five parameters that characterize the fins geometry and location were optimized and four optimization targets were defined. A set of 16 CFD simulations of the heat exchanger were carried out, and four surrogate models, one for each objective function, were obtained using a third-order response surface method. Then, an archive based micro-genetic algorithm was employed to determine the optimal combination of design variables. Arora et al. [34] developed a two-stage TEG model considering internal irreversibilities and used a non-dominated sorting genetic algorithm in order to maximize the power output and efficiency and minimize the entropy generation. They used as design variables the electrical current, the number of thermoelectric elements in the top and bottom stages and the temperatures on the hot and cold sides. Recently, multi-objective genetic algorithms were used by Chen et al. [35] and Kwan et al. [36]. In the former work, a 3D finite element method was used in the optimization of the dimensions of a thermoelectric element in order to maximize the output power and efficiency of a TEG that recovers waste heat from heat pipes. In the latter, the geometrical design of a thermoelectric device was optimized considering its operation for both electricity generation and cooling purposes. The main objective of this work is to optimize the dimensions of a TEG aimed at the recovery of energy from the exhaust gases of two heavy-duty vehicles. We will only consider thermoelectric units with two geometrically equal legs of uniform square cross section. Even though different shapes [37], asymmetrical legs [38] or segmented TEGs [39] may have some advantages, the manufacturing costs are higher. The optimization relies on a mathematical model developed in our previous work [16]. Although this a relatively simple model, it

n

p n

Exhaust gases pipe y

Cooling fluid duct

x z

p

Thermoelectric module Exhaust gases duct

n y x

(a)

z

(b)

Thermocouple

Lpn Ls Lpn

n-type semi-conductor p-type semi-conductor

Exhaust gas duct Tex tSS tcer tCu

Stainless steel Ceramic layer Thin metallic conductor (copper)

p

Th n

p

Hpn Tc

Aluminium Cooling water duct

Cooling water RL

(c)

Tcw y z

Fig. 1. (a) Thermoelectric generator; (b) Thermoelectric module; (c) Thermoelectric unit. 180

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accounts for several important features that have been neglected in many other works. In fact, it includes the calculation of the variation of the temperature and convective heat transfer coefficient along the streamwise direction, as well as the pressure drop in the heat exchanger. This is essential when considering heat exchangers with fins, since the net heat power may be significantly lower than the electrical power. Moreover, in contrast to most previous optimization studies that address either the thermoelectric units or the heat exchanger, the present work involves not only the optimization of the geometrical dimensions of the thermoelectric units, but also the height and thickness of the fins, and the spacing between them, as well as the thicknesses of the electrical connectors, ceramic strips, and walls of the ducts. In addition, although several optimization studies of TEGs have been reported, as mentioned above, we are not aware of previous applications carried out for the type of vehicles addressed in the present work. Two different optimization methods are employed, namely a gradient-based search method and a direct search method, which will be compared. The influence and the relative importance of each design variable on the net power of the TEG will be studied, as well as the importance of the constraints in the optimal solution. The partial derivatives of the net power in order to the design variables and the Lagrange multipliers provide useful insight into the analysis that has not been exploited in previous TEG optimization studies.

2. Thermoelectric generator model 2.1. TEG configuration The TEG is constituted by several thermoelectric modules, each one of them being packed between a heat source and a heat sink, as shown in Fig. 1(a). The exhaust gases of the diesel engine, which flow through a duct with fins, constitute the heat source, while the cooling water, flowing through a duct of rectangular cross section, behaves as the cold source. It is assumed that both fluids flow in the same direction. Each thermoelectric module is constituted by a rectangular array of nx × nz thermoelectric units, also commonly referred to as thermocouples, which are connected thermally in parallel and electrically in series, as shown in Fig. 1(b). The thermoelectric units consist of a pair of parallel elements: a p-type and an n-type semiconductor elements made of different materials (see Fig. 1c). The thermocouples are connected by copper strips that allow the electric current to flow across them. Ceramic strips are used to ensure electrical insulation and to provide structural support of the thermoelectric units. The temperature difference between the hot and the cold thermocouple junctions allows the direct conversion of thermal energy into electrical energy by means of the Seebeck effect. Three different fin structures were used in the heat exchanger on the hot side, namely plain fins, offset strip fins and triangular fins, as illustrated in Fig. 2.

l

y

l

y

x z

x z

Flow y

Flow

s

t

h

y

s

t

h

z

z (a)

y

(b)

x z Flow 2

y

h z

t

(c)

s

Fig. 2. Heat exchanger fin structures. (a) Plain fins; (b) Offset strip fins; (c) Triangular fins.

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2.2. Mathematical model

K = kp Ap / Hp + kn An / Hn

The mathematical model of the TEG used in this work is described in detail in Vale et al. [16]. Only a brief description is presented here for the sake of completeness. The model assumes steady state conditions, one-dimensional heat conduction in the direction normal to the thermoelectric modules (y-direction in Fig. 1), negligible heat losses due to radiation, negligible thermal contact resistances, and perfectly insulated side walls of the n and p legs of the thermocouples. The fluid domain in the exhaust gases and cooling water ducts is discretized using also a one-dimensional approximation, i.e., assuming that the temperature of the fluids only changes in streamwise direction, being uniform in y and z directions. The number of control volumes is equal to the number of rows of thermoelectric units in x-direction, and the cell faces are equidistant from neighbouring thermoelectric units in that direction. According to these assumptions, the governing equations for the jth control volume may be written as follows [3,16]:

where ρ is the electrical resistivity, k the thermal conductivity, A the area of the cross-section of a leg of a thermoelectric unit, and H its height. The properties of the materials are determined at a temperature equal to the arithmetic mean of the temperatures at the hot and cold junctions. The intensity of current in a thermoelectric unit is given by

Tex , j + 1) = (T¯ex , j

mex cp, ex (Tex , j (T¯ex , j

Th, j )/ Rt , h = nz (

mcw cp, cw (Tcw, j + 1 (Tc, j

pn

Th I

Tcw, j ) = (Tc, j

T¯cw, j )/ Rt , c = n z (

pn

Tc ))

T¯cw, j )/ Rt , c

Tc I + 0.5 R I 2 + K (Th

p Hp/ Ap

+

n Hn /An

(7)

pn

T )2 /4R

(8)

The nonlinear system of Eqs. (1)–(4) is solved for every control volume using the iterative Newton’s method, taking the exhaust gases and cooling water temperatures at the upstream cell face as input data, and obtaining the corresponding temperatures at the downstream cell face and the temperatures at the hot and cold junctions of the thermocouples as a result. The solution is carried out sweeping successively all the control volumes in the streamwise direction, and setting the fluid temperatures at the downstream cell face of the jth control volume equal to the fluid temperatures at the upstream cell face of the (j + 1)th control volume. The total electrical power output is equal to the sum of the electrical power of all thermoelectric units, accounting for all the thermoelectric modules. Part of this power is used to pump the exhaust gases through the heat exchanger, the pumping power being equal to the sum extended over all the ducts of the product of the volumetric flow rate by the pressure drop between the inlet and the outlet of the duct under consideration. The difference between the total electrical power output and the pumping power is equal to the net power of the TEG.

(2)

(4)

where m is the mass flow rate, cp the specific heat, T the temperature, Rt the thermal resistance on the hot or cold sides of a thermoelectric module (subscripts h or c, respectively), αpn the difference between the Seebeck coefficients of the p and n legs of a thermocouple, R the electrical resistance, I the electric current in the circuit, K the thermal conductance of a thermoelectric unit, and nz the number of thermoelectric units in a row in z-direction of a thermoelectric module. Subscripts ex and cw denote the exhaust gases and cooling water, respectively, while h and c denote the hot and cold junctions of a thermoelectric unit, respectively. The overbar represents a mean value over the control volume under consideration, the mean temperature being approximated by the arithmetic mean of the fluid temperatures at the entrance and at the exit of the control volume. The specific heat is evaluated at this mean temperature. Eq. (1) states that the heat loss from the exhaust gases in the jth control volume is equal to the heat transferred from the gases to the hot junction of the thermocouples, and Eq. (2) states that this is also equal to the heat transfer rate through the hot junctions of the thermocouples in that control volume. The term into brackets on the right side of Eq. (2) represents the heat transfer rate at the hot junction of a thermocouple. Eqs. (3) and (4) are the counterpart of Eqs. (1) and (2) for the cold side. The thermal resistance on each side of a thermoelectric module is equal to the sum of different thermal resistances: convection in the fluid, and conduction across the duct wall, ceramic insulation, and electrical strip. The convective heat transfer coefficients are computed using empirical correlations. On the cold side, the flow is turbulent and the Gnielinski correlation [40] is used, while on the hot side, the correlation employed depends on the type of fins and on the flow regime. In the case of plane or triangular fins, the Gnielinski correlation is still used for turbulent flow. However, if the flow is laminar, Martin’s correlation [41] is used for plain fins, while the data of Shah and London [42] are employed for triangular fins. In the case of offset strip fins, the correlation developed by Manglik and Bergles [43] is applied for both laminar and turbulent flows. The conduction thermal resistances are computed assuming that the temperature of each material is uniform and equal to the arithmetic mean of the temperatures at the two sides of the corresponding material. The electrical resistance and the thermal conductance of a thermoelectric unit in Eqs. (2) and (4) are evaluated as follows:

R=

T /(R + RL )

P = R I2 = (

(3)

Tc ))

pn

where ΔT = Th − Tc and RL is the external load resistance. The maximum electrical power output of a thermoelectric unit is achieved when R = RL [3] yielding

(1)

Th, j )/ Rt , h 0.5 R I 2 + K (Th

I=

(6)

3. Optimization methods 3.1. Objective function and constraints The purpose of the optimization process is to maximize the net power of the TEG. Although the efficiency has also been considered an objective function in several past works [28,31,34,35], it is not so relevant in the present one, because the energy of the exhaust gases is readily available when the vehicle is travelling, and the major concern is to recover as much of this energy as possible, rather than to achieve a high efficiency in the recovering process. The design variables, which are identified in Section 4.2, include the dimensions of the fins, the height of the ducts, the thickness of the various materials, the dimensions of the thermoelectric units and the distance between them. The constraints include a lower and an upper bound on every design variable, and impose an upper bound on the TEG’s height. Thus, the optimization problem is set as:

max(x, T (x)) x

(9)

where x defines the set of design variables (identified in Table 2 and explained in Section 4.2), and T is the array of temperatures (Tex, Tcw, Th and Tc). This maximization problem is subjected to the TEG geometric constraints, e.g., maximum height constraint, and to lower and upper bound constraints in each design variable. Note that the objective function dependence on design is twofold: explicitly on x and implicitly by the temperature distribution T function of design. This last dependence is implicitly controlled by the system of equations, obtained from Eqs. (1)–(4) for each thermoelectric unit, which is written here in simplified form as

(5)

A (Ti, xj ) = b (xj ) 182

(10)

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3.2. Optimization methods

d Pnet = d xj

Basically, we can divide optimization methods in two classes: gradient-based search methods and non-gradient search methods. Gradient-based search methods employ the objective and constraints functions values and gradients to progress towards the optimal solution, while non-gradient ones do not use any gradients in the optimization process, relying only on the values of the objective function [44]. Two different optimization methods, one of each class, were used in the present work: FMINCON, a gradient-based search method from the Optimization Toolbox of MATLAB [45], and the global and local optimization using direct search (GLODS) method [46]. Although computationally much less efficient, GLODS adds an interesting feature that promotes the search for global solutions. Differently, as any gradientbased method, FMINCON is more prone to identify local optimums. However, due to its mathematical structure, it provides the designer with design sensitivity information (e.g. Lagrange multipliers) very useful to support engineering design decisions.

k

Pnet + xj

Pnet Ti

Ti xj

We will first present the verification of the model by considering the TEG simulated by He et al. [13]. The geometrical configuration is similar to the present one, but there is only one thermoelectric module, along with one hot fluid duct and one cold fluid duct. There are no fins in the exhaust gases duct, and the thermal resistances due to the wall of the duct, ceramic layer and electrical connector strip are neglected. The convective heat transfer on the hot and cold sides is assumed to be constant. Despite these simplifications, this a good test for verification of the present model, because the variation of the temperature of the hot and cold fluids along the TEG is taken into account, and the solution algorithm that yields the solution of Eqs. (1)–(4) for every control volume along the streamwise direction may be checked. The calculations were performed for a thermoelectric module with a width of 0.45 m and a length of 0.14 m, for the case of coflow. The hot fluid has a mass flow rate of 0.01 kg/s and an inlet temperature of 350 °C, while the cold fluid has the same mass flow rate and its inlet temperature is 30 °C. The thermoelectric units have a height of 5 mm, and a square cross section, the length of the side being also 5 mm. The relevant properties of the materials of the thermocouple legs and of the hot and cold fluids are prescribed, and may be found in Ref. [13]. The predicted temperatures at the hot and cold junctions along the streamwise direction are plotted in Fig. 3. They closely match the temperatures computed by He et al. [13], providing confidence in the present computational model. The predicted electrical power, voltage and current intensity are 11.7 W, 5.6 V and 2.1 A. These values are also in close agreement with the results reported by He et al. [13].

(12)

A xj

(13)

4.2. Case study

We define now the adjoint (state) variable solution of the adjoint problem (linear system of equations)

AT Ti

=

Pnet Ti

Multiplying both sides of this equation by

obtain

Pnet d Ti = Ti d x j

Two different vehicles were considered in the present study: vehicle A with 3.5 tonnes and travelling at a constant speed of 120 km/h, and vehicle B with 40 tonnes and a constant speed of 90 km/h. The frontal area, the engine displacement, the nominal power and the maximum torque of the two vehicles are given in Table 1. Based on these data, and on the chemical composition of the exhaust gases, which is determined by assuming complete combustion of the fuel, the software ADVISOR [48] calculates the mass flow rate and the temperature of the exhaust gases immediately after the diesel particle filter, as well as the temperature of the cooling water. These results are summarized in Table 1, and are used as input data for the mathematical model of the TEG. The

(14)

T

db d xj

A xj

d Ti d xj

(16)

4.1. Model verification

The first term of the right side of the previous equation is easily obtained by simple differentiation. To obtain the second term, that regulates the implicit dependence on design, we differentiate the system of equations characterizing the temperature distribution,

A d Ti db = Ti d xj d xj

A xj

4. Results and discussion

In this basic generic step k, d(k) is the search direction in the design space, αk the step size which is a positive scalar obtained by a onedimensional search, and x(k) the vector of the design variables at kth iteration. FMINCON relies on a sequential quadratic programming algorithm requiring objective function and constraints gradient information to compute d(k). Since TEG geometric constraints and bound constraints only depend explicitly on the design variables, the respective gradients are obtained by simple differentiation. This is not the case of the objective function since it has an implicit design dependence via the state variable solution of the system of Eq. (10). In this case, analytical derivatives can be obtained using a direct differentiation or adjoint variable methods (see, e.g., [47]). In our case we have more design variables than functions (objective and constraints) dependent on the state variable T, thus it is computationally more efficient to use the adjoint variable method, as discussed in Haftka and Gurdal [47]. To obtain the gradient of the objective function (net power) with respect to (w.r.t.) the design variables, we start deriving the following equation

d Pnet = d xj

db d xj

3.2.2. GLODS The GLODS method [46] is a direct search method that is organized around a search step and a poll step, and is aimed at solving bound constrained minimization problems. The main goal of the search step is to explore the feasible region and locate good promising subdomains. The poll step is responsible for the local exploration of the subdomains found by the search step, in order to ensure the convergence of the method. A list that stores the feasible points is kept during the optimization process. Points generated at search and poll steps may be added to this list. Each new point x is stored along with the corresponding step size parameter, α, the comparison radius, r, and an index i that is equal 1 or 0, depending on whether the point is active or inactive. A point is active when it has a lower value of the objective function than the neighbouring ones, in a region whose size is defined from the comparison radius (see [46] for further details).

(11)

d(k )

T

with λ, the adjoint state variable, satisfying the linear system of equations (14).

3.2.1. FMINCON Gradient-based search methods are based on the following iterative process to update the design variables and progress towards the optimum:

x (k + 1) = x (k ) +

Pnet + xj

and using Eq. (13) we

(15)

Finally substituting Eq. (15) in Eq. (12) the gradient is given by 183

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260 240 220 T (°C)

Table 2 Design variables default values and allowed range of variation (bounds).

Th data [13] Tc data [13] Th, present model Tc, present model

200 180 160 140

0.0

0.2

0.4

x/L

0.6

0.8

1.0

Design variable

Default value

Minimum value

Maximum value

s (mm) h (mm) t (mm) l (mm) Hcw (mm) tCu(mm) tCer (mm) tAl (mm) tSS (mm) Hpn (mm) Ls(mm) Lpn (mm) Fz Fx

− − − − 10.0 0.2 0.7 1.0 1.0 5.0 0.5 5.0 1.2 1.2

2.0 5.0 1.0 1.0 5.0 0.1 0.6 0.8 0.8 0.1 0.5 1.0 1.1 1.1

99.0/199.0a 50.0 8.0 75.0 50.0 1.0 1.0 1.2 1.2 5.0 4.0 5.0 2.0 2.0

a

The first value stands for vehicle A and the second one for vehicle B.

initial guess for the design variables that are present in ith case, but were not optimized in (i − 1)th case, was set equal to the respective default values listed in Table 2. In case 1, only the size of the fins is optimized. Therefore, the design variables are the height, h, and the thickness, t, of the fins, the distance between them, s, and their length, l, when applicable, i.e., for offset strip fins. All the remaining geometrical variables listed in Table 2 are fixed and set to their default values given in the table. In case 2, the height of the cooling water duct, Hcw, is added to the design variables, while in case 3 the thickness of the walls of the exhaust gases duct, tSS, and cooling water duct, tAl, and the thickness of the ceramic strips, tCer, and electrical conductor strips, tCu, are also included. The height of the legs of the thermoelectric units, Hpn, is included in the list of design variables in case 4, while the distance between the p and n legs of a thermoelectric unit, Ls, as well as Fz are considered in case 5. The Fz variable is defined as Fz = (W/nz)/Lcer, where Lcer is the length of the insulation ceramic strip in the transverse direction. Finally, the length of the side of the square cross section of the legs of a thermoelectric unit, Lpn, and Fx are also accounted for in case 6, yielding 13 or 14 design variables, depending on the type of fins. The Fx variable is defined as Fx = (L/nx)/Lpn. The size of the fins (height, thickness, and length) and the distance between them influence the convective heat transfer coefficient, hex, and the heat transfer area on the hot fluid side. Similarly, the height of the cooling water duct influences the convective heat transfer coefficient on the cold fluid side. Moreover, the height of the exhaust gases duct, which is related to the height of the fins, and the height of the cooling water duct are related to the height of the TEG, which cannot exceed the maximum prescribed value. The thickness of the walls of these two ducts, as well as the thickness of the ceramic insulator and electrical strip, affect the thermal resistance of the materials, as well as the total height of the TEG. The height and the length of the side of the

Fig. 3. Predicted temperatures at the hot and cold junctions of the thermoelectric units of the TEG simulated by He et al. [14] for the case of coflow and a module length of 0.14 m.

mass flow rate of the cooling water was assumed to be 2.1 kg/s. The thermoelectric materials were selected according to the working temperature range of the thermoelectric units of the TEG. In the case of vehicle A, the n-type material is Bi2Se0.3Te2.7 [49], and the p-type material is Bi3Se1.5Te3 [50]. In the case of vehicle B, (Bi0.001Pb0.999Te)0.88(PbS)0.12 [51] and AgSbTe2 [52] were used for the n- and p-type materials, respectively. 4.3. Optimization methodology The geometry of the TEG was optimized for both vehicles under consideration and for the three different fins configuration using the two optimization methods described in Section 3.2. In order to perform the optimization of the geometry of the TEG, the design variables and their constraints need to be defined for each vehicle. The total length, L, and width, W, of the TEG were fixed. The length was set to 0.15 m for both vehicles, while the width was set to 0.10 m for vehicle A and 0.20 m for vehicle B. A maximum height of 0.15 m and 0.20 m was allowed for vehicles A and B, respectively. Table 2 lists all the geometrical variables whose values were allowed to vary, along with their range of variation. This range was based on typical lowest and highest values of these variables found in the literature. The length of the fins, l, is only considered in the case of offset strip fins. The optimization process was carried out for 6 cases, with the number of design variables increasing from case 1 to case 6. The initial guess for the ith case is equal to the optimal solution found for the (i − 1)th case, since a good initial guess is important to obtain a global rather than a local optimum when a gradient based method is used. The Table 1 Characteristics of the heavy-duty vehicles and output results from ADVISOR. Vehicle specifications

Vehicle Weight (tonnes) Speed (km/h) Frontal area (m2) Engine displacement (L) Power (kW) Torque (N·m) Transmission

A 3.5 120 3.288 2.14 95 at 3800 rpm 305 at 1200–2400 rpm Manual 5-speed

B 40 90 8.348 15.6 380 at 1600 rpm 2600 at 1100 rpm Manual 5-speed

Output results from ADVISOR

Exhaust gases mass flow rate (g/s) Exhaust gases temperature at Diesel particle filter (K) Coolant fluid temperature (K) Engine brake power (kW) Fuel consumption (L/100 km)

80.12 568.93 368.15 25.5 7.8

201.48 710.86 368.15 127.0 39

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cross section of the legs of the thermoelectric units influence their thermal and electrical resistances, and the height of the TEG. The distance between the two legs of a thermoelectric unit, and the distance between two neighbouring thermoelectric units in streamwise and transverse directions, which are related to Fx and Fz, define the number of thermoelectric units in a thermoelectric module. The results of the optimization process for vehicle A in cases 1 and 6 are summarized in Tables 3 and 4, respectively, while the corresponding results for vehicle B are shown in Tables 5 and 6. The two optimization methods yield similar results, no matter the vehicle under consideration or the type of fins. Therefore, the optimization method will be discussed only in Section 4.4 and is not relevant for the analysis of the results presented in the following section.

long, because that would decrease the convective heat transfer coefficient and the electrical power, and too short, due to the increase of the pressure drop and the pumping power. According to the results shown in Table 3, the optimal number of thermoelectric units in both streamwise and transverse directions, and the optimal number of thermoelectric modules are independent of the type of fins in case 1. A similar conclusion is obtained for the height of the fins, and therefore the height of the TEG. However, the optimum distance between fins differs for different fins, and the maximum net power is obtained for the triangular fins. In case 6, the optimal value of the thickness of the fins remains equal to its minimum allowed value, as discussed above for case 1 (see Table 4). A similar result is obtained for the thicknesses of the wall ducts, ceramic insulator strip, and electrical conductor strip. In fact, these materials behave as a thermal resistance to heat transfer from the hot gases to the cooling fluid, and therefore the thinner they are, the better the heat transfer rate, and therefore the higher the electrical and net powers. The optimal height of the cooling water duct is also equal to the minimum allowed value, because the smaller the height is, the greater are the velocity of the cooling water and the convective heat transfer coefficient, yielding an increase of the heat transfer rate and electrical power, and more space is available in the y-direction. The variables Fx and Fz, which are related to the distance between neighbouring thermoelectric units, should also be as small as possible. Hence, in addition to the height and length of the fins, and the distance between them, the three geometrical variables that play an important role in the performance of the TEG are the dimensions of the thermoelectric units, namely the height of the legs, the length of the side of the cross section, and the distance between the two legs. When the height of the legs of a thermoelectric unit is reduced, two opposite effects occur, as discussed in Vale et al. [16]. First, the electrical resistance decreases, leading to an increase of the electrical power. Second, the thermal resistance is smaller, and so the difference between the temperatures at the hot and cold junctions decreases, implying a decrease of the electrical power. Owing to these two opposite effects, the optimal value of the height of the legs lies within the prescribed range. Similarly, the variation of the cross-section of the legs of a thermoelectric unit produces two opposite effects. If the length of the side, and therefore the area of the cross section, decreases, the electrical resistance becomes higher, thus lowering the electrical power. However, the thermal resistance increases too, and therefore the difference between the temperatures at the hot and cold junctions tends to increase, as well as the electrical power. As far as the distance between the legs is concerned, the smaller it is, the higher the number of thermoelectric units that can be placed in a thermoelectric module, but the smaller the area available for heat transfer in every unit. The former factor contributes to increase the electrical power, while the latter causes a reduction of that power. Table 4 further shows that when all the design variables are considered in the optimization algorithm, the optimal number of

4.4. Physical analysis Table 3 shows that the thickness of the fins should be as small as possible, since its optimal value is equal to the minimum allowed value. This is not surprising, since the smaller the thickness of the fins, the higher the area available for the flow of the exhaust gases, and the lower the pumping power. Furthermore, the heat exchange area increases with the reduction of the thickness of the fins, and so does the heat transfer rate, yielding an increase of the electrical power. In contrast, the optimal values of the height and length of the fins, and of the distance between them, lie between the minimum and maximum prescribed values. The reduction of the distance between neighbouring fins allows an increase of the heat exchange area and a reduction of the flow area, which yield an increase of both the electrical power and the pumping power. The reduction of the height of the fins also leads to a reduction of the flow area, which implies an increase of the pumping power. However, the lower the flow area is, the higher the mean velocity and convective heat transfer coefficient are. This yields an increase of the electrical power with the reduction of the height of the fins, as shown in Vale et al. [16]. The reduction of this height may also leave space for more thermoelectric modules, which would further increase the electrical power. Therefore, both the decrease of the distance between fins and the decrease of their height yield an increase of the pumping and electrical powers. When the distance between the fins or their height is large enough, the pumping power is small, and it is advantageous to reduce that distance or height, due to the higher electrical power that overwhelms the pumping power. However, below a certain distance/height, the pumping power increases faster than the electrical power, and therefore the net power decreases, as discussed in Vale et al. [16]. Hence, there are optimal values of the distance/height that maximize the net power. In the case of offset strip fins, both the convective heat transfer coefficient and the pressure drop decrease with the increase of the length of the fins, and so do the electrical and the pumping powers. Therefore, the optimum length is dictated by the need to avoid fins too Table 3 Optimized results for vehicle A in case 1. Plain fins

Number Number Number Number Number s (mm) h (mm) t (mm) l (mm) Pnet (W)

of of of of of

iterations evaluations of objective function thermoelectric modules thermoelectric units in streamwise direction thermoelectric units in transverse direction

Offset strip fins

Triangular fins

GLODS

FMINCON

GLODS

FMINCON

GLODS

FMINCON

31 274 5 25 7 3.76 24.33 1.00 − 54.44

3 10 5 25 7 3.77 24.33 1.00 − 54.45

22 261 5 25 7 5.13 24.32 1.00 50.00 55.38

3 23 5 25 7 5.13 24.33 1.00 50.00 55.42

29 258 5 25 7 6.50 24.33 1.00 − 60.50

7 34 5 25 7 6.50 24.33 1.00 − 60.50

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Table 4 Optimized results for vehicle A in case 6. Plain fins

Number of Number of Number of Number of Number of s (mm) h (mm) t (mm) l (mm) Hcw (mm) tCu(mm) tCer (mm) tAl (mm) tSS (mm) Hpn (mm) Ls(mm) Lpn (mm) Fz Fx Pnet (W)

iterations evaluations of objective function thermoelectric modules thermoelectric units in streamwise direction thermoelectric units in transverse direction

Offset strip fins

Triangular fins

GLODS

FMINCON

GLODS

FMINCON

GLODS

FMINCON

35 512 7 34 7 2.00 21.87 1.00 − 5.00 0.10 0.60 0.80 0.80 2.50 3.73 4.00 1.10 1.10 96.17

4 59 7 34 7 2.00 21.88 1.00 − 5.00 0.10 0.60 0.80 0.80 2.49 3.73 4.00 1.10 1.10 96.19

26 951 9 68 11 5.13 14.10 1.00 50.00 5.00 0.10 0.60 0.80 0.80 2.50 3.80 2.00 1.10 1.10 102.33

13 410 9 64 11 5.13 14.14 1.00 50.00 5.00 0.10 0.60 0.80 0.80 2.42 3.89 2.10 1.10 1.12 102.12

29 1016 9 34 7 5.08 13.20 1.00 − 5.00 0.10 0.60 0.80 0.80 3.00 3.68 4.00 1.10 1.10 102.08

4 59 9 34 7 5.09 13.21 1.00 − 5.00 0.10 0.60 0.80 0.80 2.99 3.50 4.00 1.10 1.10 102.07

Table 5 Optimized results for vehicle B in case 1. Plain fins

Number Number Number Number Number s (mm) h (mm) t (mm) l (mm) Pnet (W)

of of of of of

iterations evaluations of objective function thermoelectric modules thermoelectric units in streamwise direction thermoelectric units in transverse direction

Offset strip fins

Triangular fins

GLODS

FMINCON

GLODS

FMINCON

GLODS

FMINCON

29 259 9 25 15 4.61 13.16 1.00 − 371.59

2 15 9 25 15 4.69 13.16 1.00 − 371.57

23 271 7 25 15 5.13 23.60 1.00 50.00 383.45

3 23 7 25 15 5.13 23.60 1.00 50.00 383.45

33 297 9 25 15 8.20 13.16 1.00 − 399.32

3 19 9 25 15 8.19 13.16 1.00 − 399.35

Table 6 Optimized results for vehicle B in case 6. Plain fins

Number of Number of Number of Number of Number of s (mm) h (mm) t (mm) l (mm) Hcw (mm) tCu(mm) tCer (mm) tAl (mm) tSS (mm) Hpn (mm) Ls(mm) Lpn (mm) Fz Fx Pnet (W)

iterations evaluations of objective function thermoelectric modules thermoelectric units in streamwise direction thermoelectric units in transverse direction

Offset strip fins

Triangular fins

GLODS

FMINCON

GLODS

FMINCON

GLODS

FMINCON

36 842 11 68 25 2.00 19.08 1.00 − 5.00 0.10 0.60 0.80 0.80 1.50 3.15 2.00 1.10 1.10 799.60

6 102 11 68 25 2.00 19.00 1.00 − 5.00 0.10 0.60 0.80 0.80 1.50 3.00 2.00 1.10 1.10 798.44

29 1032 13 45 18 5.13 13.25 1.00 30.12 5.00 0.10 0.60 0.80 0.80 2.00 3.62 3.00 1.10 1.11 786.10

15 500 13 45 19 5.13 13.27 1.00 30.04 5.00 0.10 0.60 0.80 0.80 1.99 3.50 3.01 1.10 1.11 787.37

29 1002 15 34 17 5.75 8.12 1.00 − 5.00 0.10 0.60 0.80 0.80 2.80 2.22 4.00 1.10 1.10 694.59

8 137 15 34 18 5.74 8.13 1.00 − 5.00 0.10 0.60 0.80 0.80 2.79 2.00 4.00 1.10 1.10 694.74

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720

374

700

373

Tcw (oC)

680 660 640

371 370

(b)

3500

120

3000

100

2500

80

(c)

T=Th - Tc (oC)

0.024 0.020 0.016

qc (W) (g) 0.0

240 220 200 180 160

(e)

0.012 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8

(d)

2000 260

0.028

R( )

372

368 4000

140

60 0.032

qh (W)

Case 1 - Plain fins Case 1 - Offset strip fins Case 1 - Triangular fins Case 6 - Plain fins Case 6 - Offset strip fins Case 6 - Triangular fins

369

(a)

620 160

hex (W.m-2 K-1)

The optimum height of the fins and the distance between them is smaller in case 6 than in case 1, except the distance between the offset strip fins, which remained unchanged. This reveals that it is important to optimize simultaneously the dimensions of the heat exchanger and thermoelectric units, rather than to optimize first the heat exchanger and then the thermoelectric units. The maximum net power is obtained for the offset strip fins, but it is only marginally higher than for the triangular fins. The maximum net power is much higher than in case 1, which demonstrates the importance of the dimensions of the thermoelectric units on the performance of the TEG. The increase of the net power from case 1 to case 6 ranges from about 69% for triangular fins to almost 85% for offset strip fins. The results for vehicle B reported in Tables 5 and 6 show that the

hcw (W.m-2 K-1)

Tex (oC)

thermoelectric units in both streamwise and transverse directions, and the optimal number of thermoelectric modules is no longer independent of the type of fins, in contrast to case 1. It is worth to note that there is a close relationship between the number of thermoelectric units in x and z directions and the length of the side of the cross section of the thermocouple legs, Lpn, as a consequence of the restriction on the length and width of the TEG. In fact, the number of thermoelectric units in those two directions is about twice as large for offset strip fins than for the other fins, while Lpn is about one half, i.e., the size of the TEG, which was fixed, is the same for all types of fins. There is also a relation between the number of thermoelectric modules and the height of the fins, i.e., the smaller that number, the greater the height of the fins, as a consequence of the maximum allowed height of the TEG.

0.2

0.4

0.6

x/L

0.8

1.0

140 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.0

(f)

(h) 0.2

0.4

0.6

x/L

0.8

1.0

Fig. 4. Evolution of several quantities along the streamwise direction of the TEG of vehicle B. (a) Temperature of the exhaust gases; (b) Temperature of the cooling water; (c) Convective heat transfer coefficient on the hot side; (d) Convective heat transfer coefficient on the cold side; (e) Electrical resistance of a thermocouple; (f) Difference between the temperatures on the hot and cold junctions of a thermocouple; (g) Heat transfer rate on the hot junction of a thermocouple; (h) Heat transfer rate on the cold junction of a thermocouple. 187

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optimal solution for case 6, corresponding to the maximum net power, is now obtained for plain fins. However, in case 1, the maximum net power is achieved for triangular fins. As expected, the net power for vehicle B is much higher than for vehicle A and remains much higher in case 6 than in case 1. It can also be concluded that the optimum number of thermoelectric modules is greater in vehicle B than in vehicle A. This is partly due to a higher value of the maximum allowed height of the TEG of vehicle B, but also due to a reduction of the optimum height of the fins and optimum height of the legs for this vehicle, no matter the case (1 or 6) or the type of fins. The increase of the net power from case 1 to case 6 is greater than in the case of vehicle A, ranging from about 74% for triangular fins to 115% for plain fins. Further insight into the optimal solutions is provided by the results displayed in Fig. 4, which shows the evolution of several quantities along the streamwise direction of the TEG of vehicle B for both cases 1 and 6 and for all types of fins. The temperature of the exhaust gases decreases and the temperature of the cooling water increases along the streamwise direction, as expected, due to the heat transferred from the hot gases to the water. The temperature variation is approximately linear, in agreement with the results of Yu and Zhao [53]. The variation of the temperature of the gases is significantly higher than that of the water, due to the much lower heat capacity rate of the gases. The variation is greater in case 6 than in case 1, since the optimization performed over more variables yields an increase of the heat transfer, leading to an increase of the net power of the TEG. The variation of the temperature of the two fluids along the streamwise direction is almost insensitive to the type of fins. Therefore, the temperature profiles almost overlap in case 6, as illustrated in Fig. 4(a) and (b), while in case 1 the temperature profiles for plain and triangular fins are coincident and only marginally differ from those for offset strip fins. The convective heat transfer coefficient on the hot side, hex, depends mainly on the Reynolds number and hydraulic diameter of the channels formed by the fins and through which the exhaust gases flow. The Reynolds number depends on the mean flow velocity in these channels and on the hydraulic diameter, which is determined by the dimensions of the fins and the distance between them. Therefore, on the hot side, the changes of the convective heat transfer coefficient from case 1 to case 6 are a consequence of the variation of all those variables, whereas on the cold side only the Reynolds number and the hydraulic diameter play a role. The offset strip fins yield a higher convective heat transfer coefficient than the plain ones, as expected, since their configuration promotes turbulence and enhances heat transfer. The convective heat transfer coefficients exhibit a small variation along the streamwise direction, as a result of the dependence of the thermophysical properties of the fluids on the temperature. The electrical resistance of the thermoelectric units is directly proportional to the ratio of the height to the area of the cross section of the legs. In case 1, these values do not change, and therefore the electrical resistance is approximately the same for all types of fins, while in case 6 the electrical resistance changes according to that ratio. The slight variation observed along the streamwise direction is due to the dependence of the electrical resistivity of the materials on the temperature. The difference between the temperatures at the hot and cold junctions of the thermoelectric units and the heat transfer rates at those junctions are related according to Eqs. (1)–(4). The key parameters that influence the values of these quantities are the thermal resistances on the hot and cold sides, the thermal conductance and the electrical resistance of the n and p materials of the thermoelectric units, and the Seebeck coefficients of these materials. Therefore, the variation of ΔT, qh and qc with the type of fins or the number of design variables is related in a non-trivial way to the variation of those key parameters. In case 1, Fig. 4 shows that the values of ΔT are lower for plain fins and higher for offset strip fins, and so are qh and qc, as well as qh − qc. Even though the latter trend is hardly visible in Fig. 4, qh − qc is directly proportional to the square of both the Seebeck coefficient and ΔT, and

inversely proportional to the electrical resistance. The influence of ΔT on qh − qc is dominant because the electrical resistance is approximately the same for all types of fins in case 1, and the Seebeck coefficient is determined at the average temperature in a thermoelectric unit, which exhibits little changes with the type of fins. In case 6, the optimal solution yields values of ΔT that are close to each other for the different types of fins, while qh and qc vary similarly with the change of the fins. The results plotted in Fig. 4 further show that ΔT, qh and qc decrease along the streamwise direction, due to the decrease of the difference between the temperature of the gases and the temperature of the water along that direction. 4.5. Optimization analysis An appropriate initial guess is a fundamental step in the success of the optimization algorithm. However, this is more critical in FMINCON, since gradient-based search methods have a greater tendency to converge to a local maximum of the objective function. If the initial guess is far from the optimum point, FMINCON may stop at a local maximum, while GLODS, with its in-built global search strategy, is less sensitive to local optima and may be able to identify the global solution. However, if the initial guess is “near” the global optimum point or the problem is quasi-convex, both algorithms converge to similar points, and FMINCON requires a significantly smaller number of iterations to reach the optimum point. Yet, we should note that there are strategies, namely “intelligent” multi-starting, that may overcome these difficulties but they will not be addressed here. In the present work, both algorithms were used with the same initial guess near the optimum point for the reasons stated above, allowing their comparison in terms of performance. The two optimization methods yield similar solutions in terms of optimum values of the design variables and net power obtained. In some cases, GLODS yields a net power slightly higher, while FMINCON leads to a higher net power in other cases, but the net power differences never exceed 0.2%, and are generally much lower than that, as shown in Tables 3–6. However, when the initial guess is the same, and close to the optimal solution, FMINCON finds the optimal solution with fewer iterations and objective function evaluations, and thus less computing time, than GLODS. In fact, the gradient of a function provides additional information about its deviation, and thus accelerates the convergence of the search algorithm. Fig. 5(a) illustrates the evolution of the net power with the iteration number for vehicle A, case 6, and triangular fins. In general, the GLODS algorithm performs better when the user does not have an idea about the optimal solution, because it allows finding optimal points with poor initial guesses. This is illustrated in Fig. 5(b) for the same case considered in Fig. 5(a), but starting from a poor initial guess, which corresponds to a net power of 48.24 W. The number of iterations required to obtain convergence for the two optimization methods is now much closer to each other than formerly. Even though none of the methods was able to find the optimal solution previously obtained (102.08 W for GLODS and 102.07 W for FMINCON, as shown in Table 4), GLODS determined a better solution (92.5 W) than FMINCON (85.9 W). Useful design information about the behaviour of the TEG can be obtained from the gradients of the objective function (net power) in order to the design variables, which reveal the importance of each design variable in the net power, and from the Lagrange multipliers, which give information about the influence of each restriction on the objective function. Table 7 summarizes the gradients calculated at the optimal point for both vehicles and for the three types of heat exchangers in case 6. Only the derivative of the objective function in order to the height of the fins is positive, i.e., an increase of this height, while keeping all the other variables unchanged, would increase the objective function value. However, this would imply an increase of the height of the TEG, whose maximum value is constrained. The greater the absolute value of the derivative of the objective function, the greater is the 188

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a small relief in these constraints bounds can lead to significant gains in TEG performance. The Lagrange multipliers of the upper bound constraints of all the design variables are equal to zero, thus those constraints do not influence the optimal solution. The optimal thicknesses of the copper and ceramic strips, and of the walls of the exhaust gases and cooling water ducts are equal to their minimum allowed values, and therefore the Lagrange multipliers for the restrictions on those values are different from zero. This is not surprising, since the lower their values, the lower the thermal resistance on the hot and cold sides. Therefore, from the thermal point of view, they should be as thin as possible. The thickness of the electrical conductor strip is very small (0.1 mm), and the copper has a high thermal conductivity. Therefore, its contribution to the thermal resistance on the hot side is marginal, which explains the small value of the derivative of the net power in order to that thickness (see Table 7). A similar situation occurs in the thickness of the cooling water duct, because of the high thermal conductivity of the aluminium. The partial derivatives of the net power w.r.t. the thicknesses of the ceramic strip and exhaust gases wall duct are higher, but these thicknesses also have a small influence in the optimization process, since the previous discussion reveals that their optimal values are equal to the minimum allowed values. The dimensions of the fins of the heat exchanger and the height of the cooling water duct have an important influence on the convective heat transfer coefficients, and therefore on the thermal resistance, on the hot and cold sides of the thermoelectric modules, respectively. The optimal thickness of the fins and the optimal height of the cooling water duct are also equal to the minimum allowed values, yielding a non-zero Lagrange multiplier, in contrast to the fins’ height and spacing. In general, all these variables have a significant influence on the net power, as shown by the relatively high values of the derivative of the net power in order to those variables. The height of the legs and, to a lesser extent, the length of the side of the cross section and the distance between the legs of the thermoelectric units are other variables that have a key role on the net power (see Table 7). Finally, Fx and Fz variables, which determine the distance between neighbouring thermoelectric units in streamwise and transverse directions, should be as small as possible, since the Lagrange multipliers are different from zero, except Fx for offset strip fins, whose optimal value is slightly larger than the minimum allowed one. Both factors have a minor influence on the net power of the TEG for vehicle B, as proved by the results reported in Table 7. However, in the case of vehicle A, although Fz still has a minor role, Fx is more relevant. Fig. 6 shows the computed electrical, pumping and net power for the three types of heat exchangers, the two vehicles, and cases 1–6. It is clear that the largest increases in net power occur from cases 1 to 2, and

102.10

Net power (W)

102.05 102.00 101.95 FMINCON

101.90 101.85

GLODS 0

10 20 Iteration number

30

(a) 100.00

Net power (W)

90.00 80.00 70.00 60.00

FMINCON

50.00 40.00

GLODS

0

5

10 15 20 25 Iteration number

30

35

(b) Fig. 5. Evolution of the net power during the optimization process for vehicle A with triangular fins in case 6. (a) Initial guess close to the optimal solution; (b) Poor initial guess.

influence of that variable on the net power. The Lagrange multipliers for the lower bound constraints of each design variable, and for the maximum height constraint, are given in Table 8. Constraints with Lagrange multiplier equal to zero do not influence the optimum (i.e. are inactive). While a non-zero Lagrange multiplier identifies active constraints and its value suggests the respective influence on the optimum result. This informs the designer that

Table 7 Partial derivatives of the net power w.r.t. each design variable at the optimal solution in case 6. Vehicle A

Pnet / Pnet / Pnet / Pnet / Pnet / Pnet / Pnet / Pnet / Pnet / Pnet /

s (W/mm) h (W/mm) t (W/mm) l (W/mm) Hcw (W/mm) tCu (W/mm) tcer (W/mm) tAl (W/mm) tSS (W/mm) Hpn (W/mm)

Pnet / Ls (W/mm) Pnet / Lpn (W/mm)

Pnet / Fz (W) Pnet / Fx (W)

Vehicle B

Plain fins

Offset strip fins

Triangular fins

Plain fins

Offset strip fins

Triangular fins

−0.24 1.07 −0.53 − −2.59 −0.14 −2.26 −0.18 −1.08 −3.05

−2.21 1.83 −18.78 −0.26 −2.66 −0.16 −2.48 −0.20 −1.18 −4.16

−0.66 0.90 −0.24 − −2.67 −0.16 −2.52 −0.20 −1.20 −4.31

−10.21 12.58 −37.78 − −7.22 −0.64 −11.33 −0.71 −4.94 −32.75

−11.59 7.74 −75.98 1.08 −5.81 −0.63 −11.58 −0.70 −4.70 −37.40

−1.77 13.46 −16.62 − −7.38 −0.65 −11.78 −0.72 −4.90 −32.96

−0.03 −1.41

−0.03 −1.95

−0.03 −2.14

−0.21 −0.60

−0.27 −0.03

−0.27 −0.44

−1.14 −1.19

−1.24 −1.16

−1.01 −1.15

189

−3.55 −3.62

−3.15 −3.77

−3.66 −3.41

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Table 8 Lagrange multipliers for the lower bound constraint of each design variable, and for the maximum height constraint, in case 6. Vehicle A

λ(s) λ(h) λ(t) λ(l) λ(Hcw) λ(tCu) λ(tCer) λ(tAl) λ(tSS) λ(Hpn) λ(Ls) λ(Lpn) λ(Fz) λ(Fx) λ(Maximum height)

Vehicle B

Plain fins

Offset strip fins

Triangular fins

Plain fins

Offset strip fins

Triangular fins

1826 0 21,613 − 4242 11,035 15,364 5684 8273 0 0 0 25 28 766

0 0 73,662 0 2452 4710 7991 2484 4419 0 0 0 24 0 249

0 0 1399 − 2835 5309 9158 2802 5112 0 0 0 28 26 280

120,370 0 102,030 − 19,003 1645 28,480 1549 15,751 0 0 0 217 204 0

0 0 529,210 0 43,241 87,652 112,300 44,487 57,605 0 0 0 279 0 3313

0 0 12,462 − 25,955 24,198 45,698 12,673 24,313 0 0 0 258 212 762

5. Conclusions

160 140

Plain fins - Electrical power Plain fins - Net power Plain fins - Pumping power Offset strip fins - Electrical power Offset strip fins - Net power Offset strip fins - Pumping power Triangular fins - Electrical power Triangular fins - Net power Triangular fins - Pumping power

Power (W)

120 100 80 60 40 20 0

0

1

2 3 4 5 Optimization case

6

A gradient-based search method, FMINCON, and a direct search method, GLODS, were used to optimize the dimensions of a TEG in order to maximize the energy recovery from the exhaust gases of two freight transportation vehicles with 3.5 and 40 tonnes. Three different heat exchanger configurations were studied, with plain, offset strip or triangular fins. A mathematical model formerly developed was used to simulate the TEG. The following conclusions may be drawn from the analysis carried out:

7

(i) The height of the fins, the distance between them, the height and the side length of the cross section of the legs of a thermoelectric unit, and the distance between those legs, as well as the length of the fins in the case of offset strip fins, are the most important parameters in the optimization process. (ii) The thickness of the fins, the thickness of the exhaust gases and cooling water duct walls, and the thicknesses of the electrical conductor and ceramic strips should be as small as possible from the thermal point of view. Further reduction of these thicknesses would have an important impact on the net power of the TEG. (iii) The optimum type of fins depends on the vehicle under consideration, i.e., on the mass flow rate and temperature of the exhaust gases and cooling water. The offset strip fins and the plain fins yield the maximum net power for the studied vehicles with 3.5 and 40 tonnes, respectively. (iv) The optimization of the heat exchanger alone, or the independent optimization of the heat exchanger and thermoelectric units, does not guarantee that the maximum net power is achieved. It is necessary to simultaneously consider the optimization of the heat exchanger and thermoelectric units. (v) The FMINCON algorithm required fewer iterations to converge in comparison with the GLODS algorithm, and requires less computational time, provided that the initial guess is close enough to the optimum solution. However, it is more sensitive to the initial guess, requiring a good one to converge to the global optimal solution. (vi) The gradients of the objective function w.r.t. the design variables provide information about the importance of those variables in the optimization process, while the Lagrange multipliers values suggest the expected benefit that each constraint relaxation can have on the TEG performance. The values of the gradients and Lagrange multipliers support the conclusions mentioned in items (i) and (ii).

Power (W)

(a)

1000 900 800 700 600 500 400 300 200 100 0

0

1

2 3 4 5 Optimization case

6

7

(b)

Fig. 6. Electrical, pumping and net power of the TEG for cases 1–6. (a) Vehicle A; (b) Vehicle B.

from 3 to 4. The difference between cases 1 and 2 is that the height of the cooling power was included as a design variable. That height is smaller in case 2, yielding a higher convective heat transfer coefficient and lower thermal resistance on the cold side. Moreover, more space becomes available in y-direction, allowing an increase of the number of thermoelectric modules. Both factors contribute to increase the electrical power, leading to the increase of the net power. The increase of the pumping power from case 1 to case 2 that is visible for vehicle A (Fig. 6) is a consequence of the change of the dimensions of the fins. The marginal variation of the net power from case 2 to case 3 confirms the previously mentioned minor role of the thicknesses of the electrical and ceramic strips, and walls of the ducts, on the net power, provided that their values are equal or close to their minimum allowed values. In contrast, the significant increase of the net power from case 3 to case 4 reveals the impact of the height of the legs of the thermocouple on the electrical and net powers. The inclusion of the remaining design variables from case 4 to case 6 has a comparatively smaller influence on the net power, in agreement with the previous discussion. 190

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Acknowledgements

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