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International Journal of Heat and Mass Transfer 54 (2011) 345–355

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

A three-dimensional numerical model of thermoelectric generators in ?uid power systems

Min Chen ?, Lasse A. Rosendahl, Thomas Condra

Institute of Energy Technology, Aalborg University, Pontoppidanstraede 101, DK-9220 Aalborg, Denmark

a r t i c l e

i n f o

a b s t r a c t

In thermoelectric generators, the heat sources are usually ?uids or ?ames. To simplify the co-design and co-optimization of the ?uid or combustion system and the thermoelectric device, which are crucial for maximizing the system performance, a three-dimensional thermoelectric generator model is proposed and implemented in a computational ?uid dynamics (CFD) simulation environment (FLUENT). This model of the thermoelectric power source accounts for all temperature dependent characteristics of the materials, and includes nonlinear ?uid-thermal-electric multi-physics coupled effects. In solid regions, the heat conduction equation is solved with ohmic heating and thermoelectric source terms, and user de?ned scalars are used to determine the electric ?eld produced by the Seebeck potential and electric current throughout the thermoelements. The current is solved in terms of the load value using user de?ned functions but not a prescribed parameter, and thus the ?eld-circuit coupled effect is included. The model is validated by simulation data from other models and experimental data from real thermoelectric devices. Within the common CFD simulator FLUENT, the thermoelectric model can be connected to various CFD models of heat sources as a continuum domain to predict and optimize the system performance. ? 2010 Elsevier Ltd. All rights reserved.

Article history: Received 31 May 2010 Received in revised form 13 August 2010 Accepted 19 August 2010 Available online 12 October 2010 Keywords: Thermoelectric generator Heat transfer Thermal ?uid System modeling

1. Introduction Thermoelectric generators are unique power sources that directly convert heat into electricity by means of semiconductor materials. They are of great interest in energy applications due to their well-known merits such as high durability and environmental friendliness, and they recover thermal energy to generate power in a simple manner. Thermoelectric devices have been installed in automobile exhaust pipes to reduce the power load on the vehicle’s alternator [1], and in biomass or gas ?red heaters to provide power for ?uorescent lights, TVs, pumps, fans or control panels [2–6]. Another kind of design has integrated thermoelectric generators in various micro-reactors and micro-combustors as miniature power sources for portable/mobile electronic devices and sensor network nodes to compete with low energy density electrochemical batteries and fuel cells with complex on-board reformers [7–15]. The concept of combustion-driven thermoelectric generation has also been implemented in sophisticated con?gurations of active counter?ow heat exchangers and reciprocating ?ow combustion in porous materials [16–18]. When thermoelectric devices are coupled with these ?uid and combustion systems, the interaction between the heat source and the generator is critical to the overall performance. Although

? Corresponding author. Tel.: +45 60902482; fax: +45 9815 1411.

E-mail address: mch@et.aau.dk (M. Chen). 0017-9310/$ - see front matter ? 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.08.024

the power generation and ef?ciency of thermoelectric generators have been discussed at length, their in?uence on the heat source can be important as well [1,3,9,12]. To identify the impact of incorporating thermoelectric generators into energy systems and to design a new generation of power applications with increased performance, modeling and design at the system level are mandatory. In practical applications, the majority of heat sources for thermoelectric generators are in ?uid form, whereas thermoelectric effects are described by solid heat transfer terms. Thus, the main challenge in the development of such system models is to simulate the ?uid-structure coupled multiphysics effects, and the temperature gradients across the top and bottom surfaces of the thermoelectric generator and across the contact surfaces of the hot and cold source ?uids should depend upon each other, i.e., using the third kind of thermal boundary conditions instead of the ?rst or second. The designer must integrate the thermoelectric device model into the ?uid model, and the system optimization (combustion control and combustor design, channel and heat exchanger shape, ?ow rate, inlet direction, etc.) must be done in conjunction with the thermoelectric generator optimization (geometry of pellets, segmentation/cascade, and deployment of modules/thermocouples, among others). Thus far, a number of system models have been proposed by coupling the analytical thermoelectric model with various ?uid models obtained from simpli?ed assumptions for speci?c applications [19–29]. These system models are useful in the rough

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Nomenclature C c I J k K1 K2 L m P q Qc Qh R Rl Rp,n capacitance speci?c heat of the element current electric current density thermal conductivity of the material interface thermal conductance between the generator and the hot ?uid source interface thermal conductance between the generator and the cold ?uid source length of the p- and n-type legs mass density of the element power on the load heat ?ow density in the leg rate of heat transfer from the cold junction to the heat sink rate of heat transfer from the heat source to the hot junction generator internal resistance load resistance resistance of the p- or n-type leg S T t Ta Tc Th Tw V x, y, z uniform cross-sectional area of the leg temperature distribution of the leg time temperature of the cold ?uid source temperature of the cold junction temperature of the hot junction temperature of the hot ?uid source electric potential directions of the 3D coordinate system

Greek symbols a Seebeck coef?cient of the device, a = ap ? an ap,n Seebeck coef?cient of the p- or n-type material DT temperature difference across the element g ef?ciency q electrical resistivity of the material Subscripts p p-type n n-type

estimation of interactions between the ?uid sources and the generators and are able to suggest general strategies for the aforementioned optimization. However, simpli?ed ?uid models are not suf?cient to accurately predict the detailed behavior of most practical ?uid systems with multidimensional construction, irregular geometry, dynamic variations in mass ?ow, or complicated combustion processes, nor can they precisely transfer nonuniform heat ?ow and temperature distributions to the thermoelectric model as the boundary condition. On the other hand, the assumption of constant material properties made in the analytical thermoelectric model is not realistic in many applications. For thermal ?uid tubes and fuel ?red combustors on which thermoelectric generators can be mounted, the temperature therein is high, ranging from hundreds to more than one thousand degree. Thus the values of the three principal properties of the p-type and n-type materials, i.e., ap,n, qp,n, and kp,n, will strongly vary with temperature. The nonlinearity in thermoelectric device modeling due to the temperature dependency of material properties in most cases necessitates a numerical approximation instead of analytical methods, whereas complicated ?uid power systems can be simulated using available computational ?uid dynamics (CFD) techniques. Particularly, a number of CFD models have been developed [15,30–34] to deal with the thermoelectric generation by FLUENT, a widely used industry code. However, these CFD models only simulate the status of heat source ?uids or convection/ radiation effects on thermoelements without an appropriate model of thermoelectric generators. In other recent works, both FLUENT and thermoelectric generator simulation have been employed to study the system thermal behavior, but coupling of the two numerical programs has not been completed, i.e., the interaction is in a single direction [35,36]. Although in principle, numerical heat transfer schemes of thermoelectricity, such as the model in the commercially available ?nite element method (FEM) program ANSYS [37] (used in [35]), the one used in [36], and those described in [38–40], can be linked to a CFD simulator through a relaxation strategy, auxiliary coding is usually required for iteration control, data transfer, and synchronization between different simulators. The processing of coupling relationships of multidimensional boundary values for the same model domain but with different simulators may be especially dif?cult [41]. Therefore, it is certainly

preferable to model both ?uid behaviors and thermoelectric power output with the same tool to minimize the number of simulators with high usability and without loss of accuracy. This work presents a three-dimensional (3D) numerical solution to the ?uid-structure coupled problem by implementing a FLUENT compatible thermoelectric model. FLUENT and its user-de?ned functions (UDFs) were selected for fuel cells to model the electrochemical reactions and the electric ?eld caused by the Nernst potential throughout the cell [42]. The most signi?cant bene?t of using FLUENT to model such power sources is that the existing knowledge/experience of this computational tool and the innumerous ?uid models already implemented by it can be ef?ciently utilized to predict the coupled system performance. Moreover, UDFs allow for a convenient customization and enhancement of FLUENT, where the thermoelectric device model can be connected to the thermal ?uid model as a continuum domain to avoid boundary-value transport problems, making the co-design of the thermoelectric generator and ?uid heat sources simpler and less time-consuming. In the following sections of this paper, 3D governing equations and the general multidimensional numerical algorithm of thermoelectric generation, model implementation in FLUENT, and model results, will be described.

2. Multidimensional multiphysics numerical scheme A typical thermoelectric power system is shown in Fig. A.1, in which many n- and p-type semiconductor legs composing the generator are connected thermally in parallel between the hot and cold ?uid sources and electrically in series to power the load circuit. Details of the general numerical algorithm used to deal with nonlinear issues such as heat production due to the Joule effect, unsteady state conduction, and temperature dependent material properties in one-dimensional (1D) transport equations for such thermoelectric generators were documented in a previous study [43], in which we introduced a combined ?nite difference and Newton–Raphson method based on the governing equation system from the energy conservation theory. The main intention of Section 2 is to extend the 1D algorithm to the multidimensional implementation of the multiphysics numerical scheme for thermoelectric generators operating in thermal ?uids.

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Fig. A.1. Schematic representation of a typical ?uid–thermal–electric-circuit coupled power system.

As is shown in Fig. A.1, it is clear that, for ?uid heat sources with ?owing mass, neither their temperature nor their heat ?ow pro?les along the contact interface to thermoelectric generators can be uniform. Obviously the results of the thermal analysis are now multidimensional, e.g., the temperature and heat ?ow distribution should be expressed as Tp,n(x, y, z) and qp,n(x, y, z) in a 3D model, respectively. In the case of temperature dependent material properties, neglecting the side heat loss outside the solid legs, the 3D governing equation of the thermal ?eld inside a control volume is written as,

Thomson effects. By an electro-thermal analogy, the heat transfer governed by (1) is illustrated in Fig. A.2 (a), a control volume of the 3D thermal resistor network. The electrical analysis requires these thermal analysis results, mainly the temperature pro?les, to ?nd the total Seebeck voltage generated and the temperature dependent material properties. Keeping the non-ohmic current–voltage relation in both legs of the device in mind [44], the governing equation of the electric ?eld inside a control volume under steady state is written as,

rV ? ?ap;n ?T p;n ?rT p;n ? qp;n ?T p;n ?J;

?2?

rqp;n

@T p;n ? ?r?kp;n ?T p;n ?rT p;n ? ? mp;n cp;n ? J 2 qp;n ?T p;n ? @t ? ?rap;n ?T p;n ??T p;n J;

?1?

where the ?rst term in the right side is the transient term, the second term is the temperature dependent Joule electrical energy production, and the third source term represents the contributions from Peltier (inhomogeneous or segmented materials) and

where the ?rst term in the right side of (2) is the Seebeck electromotive force (EMF) increase due to the temperature gradient, and the second term is the voltage drop due to the current ?owing through the control volume. As a result of the 3D temperature distribution, the electric current and potential distribution are also 3D, i.e., J(x, y, z) and V(x, y, z). In addition, the values of Rp,n must be obtained by a calculation of a 3D resistor network for the 3D resistivities qp,n(x, y, z), which is shown in Fig. A.2 (b), where the current

a

b

Fig. A.2. A control volume of (a) 3D thermal resistor network Eq. (1), (b) 3D electric resistor network Eq. (2).

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vector ?ows in three directions against the Seebeck voltage sources. Although the 1D analysis is fast and easy to manipulate, it lacks details about the lateral distributions of electric potential and current, and hence, it is impossible to obtain the effective electrical resistance. Therefore, the 3D electric ?eld analysis of thermoelectric generators, by which the main results of power and ef?ciency can be obtained, should be accompanied by the 3D thermal ?eld analysis. The complete solution of a multidimensional multiphysics problem relies on a clear data and algorithm ?ow chart, in which various numerical approximation methods for the nonlinear differential equation system can be identi?ed and connected as a sequential process. Fig. A.3 shows such a self-consistent ?ow chart for thermoelectric generators operating in the system mode as shown in Fig. A.1. The basic mechanism consists of four simulation modules: a 3D CFD simulation is used for the energy and mass transfer status in the hot and cold ?uid domains; a 3D thermal simulation for the temperature and heat conduction distribution in the thermoelectric domain; a 3D electric simulation for the electric ?eld distribution and electric conduction in the thermoelectric domain; and a 1D circuit simulation for the current and power performance of the actual load. The four simulation modules are coupled together and there are three direct coupling relationships among them. First, the CFD simulation is coupled with the thermoelectric thermal simulation through the contact boundary, that is, the 3D boundary temperature and heat ?ux of both the ?uid domains and the solid domain should be kept continuous with each other as a whole. Second, within the thermoelectric device, the thermal simulation and the electric simulation are coupled with each other over the entire 3D solid domain through the update of all of the temperature dependent material properties, the temperature distribution, and the current density. Third, the 3D electric simulation is coupled with the load circuit as a ?eld-circuit coupling, where the overall Seebeck voltage and effective internal resistance, calculated from the 3D electric simulation, are applied as a DC voltage source to the external load circuit. Unlike the thermoelectric power source, the load’s current– voltage relation ful?lls Ohm’s law, and there is no need to consider the power output in a 3D way. Depending on the position of the electrode pads of the actual device, the translation between the 3D current distribution from the thermoelectric electric simulation and the 1D current from the circuit simulation can be easily carried out because the current is the only boundary condition in this ?eld-circuit coupling. For the coupling between the ?uid ?eld and the thermoelectric ?eld, however, the boundary condition translation is much more dif?cult because the CFD simulation and the thermoelectric thermal simulation both output 3D heat ?ow and temperature distributions at the boundary surface. There

are two boundary conditions for the same boundary domain, and one of them must be known beforehand for both simulators to start the thermal simulations. Fig. A.3 shows one iteration strategy, where initial values of the boundary heat ?ow distribution Qh,c(x, y, z) must be given before the CFD simulation starts to run. Then, the temperature distribution Tw,a(x, y, z), as the solution of the energy transfer equations solved in the CFD simulation, can be used in the numerical thermal analysis of the thermoelectric ?eld, whose analysis results are fed back to the CFD simulation to update Qh,c(x, y, z) as the new boundary values. In spite of the iteration instance, the translation of the multidimensional thermal boundary condition in practical simulation usually involves complicated mathematics and computational technique, as shown in [41] and other open reports. To avoid any theoretical dif?culties brought about by such a manipulation between CFD and heat transfer simulators, it is advantageous to implement the thermoelectric model in the CFD framework as well; thus, the solid and the ?uids are de?ned in a continuum zone and the boundary is simply solved as interior nodes by a single solver. The next section will present the detailed treatment of thermoelectric modeling in FLUENT to enable an ef?cient ?uidthermal-electric coupled simulation as the scheme displayed in Fig. A.3. The resistance of the load circuit is extracted and used in the thermoelectric modeling such that the ?eld-circuit coupling is also included to some extent.

3. Numerical model The 3D thermoelectric generator model is implemented in FLUENT 6.3, a ?nite volume method (FVM) CFD package, in which the solid thermoelectric phenomena and the algorithm described in Fig. A.3 are taken into account by developing UDF through ANSIC language. The model serves as an add on module in parallel with other power source modules [42] provided within the standard FLUENT software. Meshing of the GAMBIT pre-processor of FLUENT is applied to leg volumes in which tetrahedral elements are dominant in this work, but one may of course use other meshers and element types if the device under study has irregular leg shapes. The grid generated is read into the FLUENT solver and scaled to align the geometrical units. Both the thermal analysis and the electric analysis use the same grid, and thus, they have the same 3D coordinate system with regard to the cells and boundary walls of the leg volumes. FLUENT has the ability to compute the conduction of heat through solids when the Energy Model is activated. Temperature dependent thermal conductivities can be speci?ed by polynomial functions and assigned to p- and n-leg cell zones for p- and n-type

Fig. A.3. Energy conservation based multidimensional multiphysics simulation procedure for thermoelectric ?uid power systems.

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materials, respectively. The thermal boundary condition of the thermoelectric generator does not need to be speci?ed in the system modeling because it will be solved in terms of ?uid system boundary conditions, and all aspects of ?uid ?ow, heat and mass transfer in heat sources are handled by FLUENT. The transient source term in (1) can also be included automatically with appropriate settings in FLUENT. The values of Joule, Thomson, and Peltier, i.e., steady state source terms, however, rely on the 3D electric analysis results, as shown in (1) and Fig. A.3. Thus an electric ?eld solution is required to complete the thermal ?eld calculation. FLUENT solves generic transport equations for user de?ned scalars (UDS) which can be used to determine the electric ?eld in a CFD environment. The dif?culty herein is that the governing Eq. (2) is not in the generic transport equation form due to the Seebeck EMF. To correctly write (2) for the solver, two UDS are used to represent the two terms in the right side of (2). UDS0 is used to represent the 3D potential distribution of the ohmic voltage drop caused by the current vector,

conditions for UDS1 is the same as that for UDS0 except that a constant value is used to calculate the ?ux J instead of I. Thus, the electric ?eld calculation function in Fig. A.3 can be implemented with regard to UDS1 if the source term is appropriately included. Before proceeding to the implementation of the right side of (7), we recall the source term in the 3D thermal Eq. (1). An energy_source UDF is written to modify the heat conduction computation to include the Joule, Thomson, and Peltier heating. FLUENT calls the UDF as it performs a global loop on cells to compute the source term and returns it to the solver. Because the current distribution calculation has been done with UDS0, the Joule source term can be easily obtained. In the practical implementation, the Joule heat can be calculated as the product of the sum of the squares of vector components of the gradient of UDS0 and the temperature dependent electrical conductivity,

J 2 qp;n ?T p;n ? ?

rUDS0 ? ?qp;n ?T p;n ?J:

?3?

1 1 ?rUDS0?2 ? qp;n ?T p;n ? qp;n ?T p;n ? 2 2 # @UDS0 @UDS0 : ? ? @y @z

"

2 @UDS0 @x ?8?

Solving for 3D electrical conduction is directly analogous to the computation of heat transfer. The ?eld throughout the conductive regions is calculated based on ?ux (charge) conservation in each cell,

rJ ? 0;

so we have

?4?

The source terms of Thomson and Peltier (in the case of inhomogeneous or segmented materials) in each cell involve the gradient of the Seebeck coef?cient ap,n. To express rap,n(Tp,n) for the source term, UDS2 is used to represent the scalar of ap,n(Tp,n). We constitute a transport equation for UDS2,

r??rUDS2? ? 0;

?5?

?9?

"

r ?

# 1 rUDS0 ? 0: qp;n ?T p;n ?

FLUENT solves this Laplace equation for the potential ?eld by enforcing a speci?c ?ux J on one boundary wall of both p- and ntype solid regions and zero potential on the other boundary wall to represent the ground, where the I value of the present iteration is the basis on which the speci?c J distribution on the UDS0 boundary is calculated. The temperature dependent electrical conductivities 1/qp,n(Tp,n) are speci?ed as the diffusivities of UDS0 by polynomial functions for p- and n-type materials. Referring to Fig. A.3, the functions of the resistor network calculation and current distribution calculation are both implemented in terms of UDS0. UDS1 is used to represent the 3D Seebeck EMF distribution produced by the temperature ?eld,

where a unit diffusivity for UDS2 is taken. It should be remarked that the result for UDS2 from the solver is not interesting because (9) does not hold any physical meaning. Instead, once per iteration, an at_end UDF is executed to ?ll UDS2 with the local Seebeck coef?cient ap,n(Tp,n) according to the temperature Tp,n(x, y, z) for each cell. The solved scalar ?eld for UDS2 is replaced therein, where temperature dependent Seebeck coef?cients are speci?ed in the UDF by polynomial functions for p- and n-type materials, respectively. If we substitute rUDS2 into the source term of Thomson and Peltier in (1), we obtain the thermoelectric source term,

?rap;n ?T p;n ??T p;n J ? ?rUDS2

T p;n

qp;n ?T p;n ?

rUDS0:

?10?

It consists of three components,

?rUDS1 ? ap;n ?T p;n ?rT p;n ;

whose divergence is

?6?

r??rUDS1? ? r?ap;n ?T p;n ?rT p;n ?:

?7?

@ ap;n ?T p;n ? @UDS2 T p;n @UDS0 T p;n J x ? ? ; @x @x qp;n ?T p;n ? @x @ ap;n ?T p;n ? @UDS2 T p;n @UDS0 T p;n J y ? ? ; @y @y qp;n ?T p;n ? @y @ ap;n ?T p;n ? @UDS2 T p;n @UDS0 T p;n J z ? ? ; @z @z qp;n ?T p;n ? @z

?11a? ?11b? ?11c?

If we take an unit diffusivity for UDS1, (7) becomes a generic transport equation where the right side can be assumed to be the source term, although it does not have a real physical meaning as a source in the thermoelectric modeling. The setting of the boundary

all of which are included in the sum of the energy_source UDF. Now we proceed to the source term in (7) for UDS1, implemented by an UDS1_source UDF,

Table A.1 Transport equation parameters of scalar ?elds. . UDS0. UDS1. UDS2. UDS3. UDS4. UDS5. Temperature. Diffusivity

1 qp;n ?T p;n ?

Source term 0 UDS1_source UDF 0 0 0 0 energy_source UDF

Top wall boundary speci?ed ?ux (I) speci?ed ?ux speci?ed ?ux speci?ed ?ux speci?ed ?ux speci?ed ?ux from CFD (0) (0) (0) (0) (0)

Bottom wall boundary speci?ed potential (0) speci?ed potential speci?ed potential speci?ed potential speci?ed potential speci?ed potential from CFD (0) (0) (0) (0) (0)

1 1 1 1 1 kp,n(Tp,n)

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r?ap;n ?T p;n ?rT p;n ? ?

@ ap;n ?T p;n ? @x ?

@T p;n @x

?

@ ap;n ?T p;n ? @y :

@T p;n @y

respectively, as three scalar ?elds. Similarly to UDS2, we constitute transport equations for these scalars,

@ ap;n ?T p;n ? @z

@T p;n @z

?12?

r??rUDS3? ? 0; r??rUDS4? ? 0; r??rUDS5? ? 0;

?13a? ?13b? ?13c?

To express the above equation in the UDF, UDS3, UDS4, and UDS5 @T p;n @T p;n @T p;n are used to represent ap;n ?T p;n ? @x ; ap;n ?T p;n ? @y , and ap;n ?T p;n ? @z ,

Table A.2 Comparison of 1D simulation results, Th = 423 K, Tc = 303 K. Quantity Qh, W P, W g% I, A Analytical 81.3 3.98 4.89 1.08 ANSYS 83.29 3.76 4.51 1.05 FLUENT coarse grid 80 3.08 3.85 0.952

not for their solutions from FLUENT but to ?ll them with products of the temperature dependent Seebeck coef?cient and the three

FLUENT medium grid 81.6 3.57 4.38 1.024

FLUENT re?ned grid 81.8 3.62 4.43 1.032

Measurement 70 2.51 3.6 0.86

Fig. A.4. Simulation of surface heat ?ux (W/m2) distributions. Th and Tc are 423 K and 303 K, respectively. (a) using the coarse grid. (b) using the re?ned grid.

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351

components of the temperature gradient in the at_end UDF. Because the gradients of UDS3, UDS4, and UDS5 can be accessed, the source term (12) for UDS1 is appropriately implemented. At the end of every iteration (at_end UDF), both the ohmic voltage drop represented by UDS0 and the Seebeck EMF represented by UDS1 are determined. Their spatial addition in terms of (2) in the full solid region gives the 3D potential distribution V(x, y, z), and the effective internal resistance of the legs, Rp and Rn, can also be calculated. Including the load Rl, the new value of I is

in the adjacent cells of both sides. The iteration is automatically done by FLUENT until convergence is achieved. 4. Model results First, a comparison of the 1D steady-state simulation between the FVM FLUENT and FEM ANSYS [37] is carried out to theoretically validate the proposed model. The example considered is the performance of the thermoelectric generator described in the previous study [43], the TEC1-12706 thermoelectric module made by Tianjin Institute of Power Sources, China, which was chosen for the experimental validation therein. The element length is L = 1.6 mm for both the p-type and n-type semiconductors, and the element cross-sectional areas are Sp = Sn = 1.4 ? 1.4 mm2. The generator has 127 thermocouples and is connected to a linear load resistance Rl = 3.4X. Second, third, or fourth order polynomial functions are used to ?t the temperature dependency of the p-type and n-type material properties. Initially, the constant temperature boundary condition is set on the top and bottom surfaces for the 1D simulation, where zero heat loss from the other four sides of the legs is assumed. With the temperature polynomial functions used for varying material properties, the FLUENT thermoelectric model is performed to analyze Qh, P, g, and I for the generator operating between Tc = 303 K and Th = 423 K. A gird sensitivity analysis is performed for the presented FLUENT model to check its gird independency, where three grid schemes used in the computation are designated as a coarse grid, a medium grid, and a re?ned grid, respectively. The parameters computed by FLUENT with the three grids and ANSYS as well as the analysis using the material properties evaluated at an

I?

UDS1p ? UDS1n ; Rp ? Rn ? Rl

?14?

which will be used in the next iteration as the boundary condition for UDS0. UDS1p,n is the total built-in Seebeck EMF of the leg, which should be equivalent to the open circuit voltage at no load. This calculation is relevant to the position of the actual electrodes at which I is applied, but in this work, the UDS1 potential of the boundary wall is simply used. The previous 1D study [43] aims for the co-design in which the thermoelectric generator system is associated with complicated electrical systems. For the 3D FLUENT model, if Rl can emulate the input impedance of the load circuit, the aforementioned ?eld-circuit coupling is also included. As far as can be observed, all thermoelectric function modules in Fig. A.3 are implemented by UDF and UDS. The boundary condition, diffusivity, and source term of the UDS and temperature ?elds are summarized in Table A.1. The model can be easily implemented for practical devices with multiple thermocouples. When the boundary walls of the legs are coupled with the ?uid ?ows, thermal boundary conditions are no longer required on them because FLUENT will calculate the heat transfer directly from the solution

a

90 80 70 Qh (W) 60 50 40 30 20 340 Measurement ANSYS FLUENT FLUENT with soldering bridges

b

4 3.5 3 Measurement ANSYS FLUENT FLUENT with soldering bridges

Power output: P (W)

2.5 2 1.5 1 0.5 0 340

350

360

370

380

390

400

410

420

430

350

360

370

380

390

400

410

420

430

hot junction/device top temperature (K)

hot junction/device top temperature (K)

c

0.05 0.045 Measurement ANSYS FLUENT FLUENT with soldering bridges

d 1.1

1 0.9 0.8 I (A) 0.7 0.6 Measurement ANSYS FLUENT FLUENT with soldering bridges

Convertion efficiency

0.04 0.035 0.03 0.025

0.5 0.02 0.015 340 0.4 350 360 370 380 390 400 410 420 430 340 350 360 370 380 390 400 410 420 430

hot junction/device top temperature (K)

hot junction/device top temperature (K)

Fig. A.5. Comparison between simulation and measured results for various temperature difference. Cold junction/device bottom temperature is ?xed at 303 K, Rl = 3.4X. (a) Qh, (b) P, (c) g, and (d) I.

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average temperature of 363 K are summarized in Table A.2. The experimental data collected in [43] are also provided for comparison. Clearly, the analytical model is essentially less realistic than the numerical models and yields a higher g if average material properties are used rather than the temperature dependent material properties. This result has been pointed out in [37], where the cause of the ef?ciency decrease is mainly the heat evolution of the Thomson effect. When the grid is suf?ciently dense, FLUENT can output almost exactly the same results as those achieved by ANSYS. The discrepancy caused by the relatively coarse grid attributes to the inherent multidimensional features of FLUENT. In Fig. A.4, it can be seen that the nonuniformity of the surface heat transfer contributing to the interior conduction in the legs of a single couple is appreciable. When the grid becomes dense enough, the discreteness error is minimized, as shown in Fig. A.4 (b). For the re?ned grid, the convergent simulation results of FLUENT and ANSYS under different temperature spans are shown in Fig. A.5,

where numerical results of the performance parameters from both models display almost the same characteristics in this 1D case. To illustrate how the surface heat loss can produce multidimensional effects, the 3D temperature and electric potential distributions in the thermocouple by the FLUENT model are depicted in Figs. A.6, A.7 (a), where all leg surfaces except the top and bottom junctions are assumed to be exposed to heat transfer with a coef?cient of 500 W/m2 K, representing contributions from both natural convection and radiation heat transfer. The bulk mean temperature in the module cavity is set to be the arithmetic average of Th and Tc. i.e., 363 K. With the same model parameters, in Figs. A.6, A.7 (b) 3D simulation results of ANSYS show a qualitative agreement with the contours from the FLUENT model. For a quantitative comparison, the performance parameters are calculated for both models. Due to the prescribed temperature boundary condition on the legs, it is found that the in?uence of surface heat loss on P and I is negligible in this case. However, with the surface heat

Fig. A.6. Simulation of temperature (K) distributions for a convection heat transfer coef?cient of 500 W/m2 K. Th and Tc are 423 K and 303 K, respectively. (a) FLUENT, (b) ANSYS.

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353

Fig. A.7. Simulation of electric potential (addition of Ohm’s and Seebeck potential, V) distributions for a convection heat transfer coef?cient of 500 W/m2 K. Th and Tc are 423 K and 303 K, respectively. The cold junction of the p-type (left) leg of the thermocouple is prescribed as ground in this postprocessing, and a scaled load (Rl/127) is assumed connected between the cold junctions of the p-type and n-type legs. (a) FLUENT, (b) ANSYS.

loss, Qh is signi?cantly increased to maintain the original temperature difference, and hence g is decreased. The changes in Qh and g with different heat transfer coef?cients are plotted in the close-up of Fig. A.8, where both numerical models display almost the same characteristics again in the case of surface heat loss. For the device example modeled, the in?uence of the non-ideal effects of the thermal and electrical interface resistances on performance parameters turns out to be obvious in the 1D study [43]. These interface resistances are re-modeled three-dimensionally in conjunction with the nonuniform temperature boundary condition across the intermediate components of the soldering bridges. The detailed pro?le of the vertical thermal power of the solid model is shown in Fig. A.9. The thermal and electrical interface resistances, and the Joule heat of the electrical interface resistance are all included in the top and bottom bridges, although the last has only tiny effects in this case. At the hot and cold faces of the thermoelements the heat conduction has a sudden change in the vertical direction, re?ecting the Peltier heat absorbed and evolved at the junctions of dissimilar materials. This reversible heat is

automatically taken into account by the thermoelectric source term described in Section 3. The key performance parameters calculated by the model with soldering bridges are also shown in Fig. A.5 for comparison. Although there are still some differences between the simulated and measured results in the high temperature range after the true 3D interface is modeled, it can be seen in Fig. A.5 that the simulation data match the real measurement closely, with a good correlation. The acceptable deviation has been analyzed and can be mainly attributed to the non-ideal accuracy of the heat ?ow determination in the device test rig and the inherent uncertainties of the commercial thermoelectric module [43]. However, the negligible discrepancy between the numerical models validates the proposed CFD model as a thermoelectric simulator equivalent to ANSYS, and to the other numerical model introduced in [43] in 1D cases. Given that the experimental determination of thermal quantities (such as conductivity) in most measurements has errors of a similar level, the FLUENT model has been able to predict the ef?ciency and output power of thermoelectric generators operating in ?uid systems

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Fig. A.8. Comparison of input heat ?ow and conversion ef?ciency between FLUENT and ANSYS simulations for various convection heat transfer coef?cients. Th and Tc are 423 K and 303 K, respectively.

with suf?cient accuracy. Further re?ning work will focus on improving the accuracy of parameter acquisition in the test rig for the model, where more advanced modules will be measured under hot sources of higher temperature. 5. Conclusions (i) The thermoelectric processes of Seebeck, Peltier, and Thomson effects are integrated with Joule source terms through a FVM numerical scheme into a CFD simulator. The proposed FLUENT model includes the temperature dependence of all properties of the p- and n-type materials composing the thermocouple. The 3D modeling results provide detailed pro?les of temperature, Seebeck potential, and current density as well as the values of power and ef?ciency. Comparisons to other modeling and experimental results validate the accuracy of the numerical model.

(ii) The functionality of solving scalar ?elds in the CFD simulator has been extended for the potential ?eld in solid zones of thermoelectric generators. The power source model is comprised of only several UDF and UDS, and hence, it is scalable and ?exible for loading in FLUENT to interact with CFD submodels, and especially useful in the design of entire power systems with 3D temperature and heat ?ux pro?les [45]. The numerous existing CFD models of ?uid ?ow and combustion in FLUENT can be immediately connected to the thermoelectric model as a continuum domain, where the dif?culties encountered in the implementation of the multidimensional boundary condition translation of the ?uidstructure coupling are avoided. To handle the ?eld-circuit interface, a general ?ux computation is de?ned in terms of the load and incorporated into the UDS boundary. (iii) One important feature of the present model is the incorporation of the reversible contributions from inhomogeneous materials and the Thomson effect into the source term of heat conduction. In particular, the thermal ?ux in the computational results expresses the usual conduction heat ?ow, and thus, it has a direct connection with the temperature ?eld and can be studied separately from the reversible Peltier and Thomson heat. This function is not available in ANSYS, in which the reversible heat and the irreversible conduction are inextricable in the total heat ?ow. Such treatment of the present model makes relevant simulation results easier to analyze and to truly understand as compared to other numerical models in which irreversible and reversible heat are de?ned together [37,40], and offers a new option for studying the effects of the Thomson heat. (iv) Not only can 3D calculations in the thermoelectric generator model aid in optimizing the selection of the 3D shape and geometry of a device, the effects of various convection and radiation conditions on power performance, especially in the case of porous medium, can also be easily studied by the model due to the inherent CFD advantages of FLUENT in modeling and implementing such source terms. Besides, the 3D FLUENT model is able to treat both isotropic and

Fig. A.9. Simulation of vertical heat ?ux (W/m2 in Z axis) with soldering bridges. Device top and bottom are 423 K and 303 K, respectively. Peltier heat’s contribution to the interior conduction is included automatically by the model.

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orthotropic physical and thermoelectric properties provided that such a study becomes mandatory to the overall system behavior.

Acknowledgments This work is funded in part by the Danish Council for Strategic Research, Programme Commission on Energy and Environment, under Grant No 2104-07-0053, and is carried out in the Center for Energy Materials in collaboration with Aarhus University, Denmark. The authors are grateful to Peter Naamansen who helped them with constructive discussions. Appendix A. Supplementary materials A computational case and the UDF library associated with the proposed model can be found in the on-line version as the supplementary data to operate in Fluent 12 directly. References

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