Physical parameters of Bi2Te3-based materials (273 K < T < 493 K).
This chapter aims to build one-dimensional thermoelectric model for device-level thermoelectric generator (TEG) performance calculation and prediction under steady heat transfer. Model concept takes into account Seebeck, Peltier, Thomson effects, and Joule conduction heat. Thermal resistances between heat source, heat sink, and thermocouple are also considered. Then, model is simplified to analyze influences of basic thermal and electrical parameters on TEG performance, when Thomson effect is neglected. At last, an experimental setup is introduced to gauge the output power and validate the model. Meantime, TEG simulation by software ANSYS is introduced briefly.
- thermoelectric generator
- thermoelectric model
- output power
Output power Pout and energy conversion efficiency η are the primary parameters to characterize TEG performance. They are intensively influenced by such factors as temperature of heat source and sink, thermoelectric materials physical properties, thermocouple geometries, thermal and electrical contact properties, and load factor. Therefore, it is necessary to build physical model formulating these factors concisely, to conduct realistic TEG design. At present, many significant works have been undertaken for modeling device-level TEG precisely [1–3]. In addition, comprehensive three-dimensional (3D) thermoelectric model has been successfully developed in software ANSYS . In Refs. [5–7], quasi-one-dimensional thermoelectric model is established, where Thomson effect and thermal resistances between thermocouple and heat source, heat sink are neglected. In Ref. , improved one-dimensional model including Thomson coefficient and thermal resistances is used to analyze the matched load, the limit of energy conversion efficiency, and the influence of Peltier effect. It shows, that expression of matched load contains not only the inner electrical resistance of TEG, but also the terms resulting from Peltier and Joule effects. In Ref. , one-dimensional model to analyze the influence of Thomson heat is built and experimentally validated.
In this chapter, Seebeck, Peltier, Thomson effect, and Joule conduction heat are formulated in thermoelectric generation module model. By model simplification, analytical expressions of output power and energy efficiency are introduced. Essential factors for enhancing the output power are extracted. Then, an experimental setup is built to measure the output power and validate the model. And TEG simulation by software ANSYS is presented.
2. Thermoelectric model for device-level TEG
2.1. TEG cell structure
TEG cell consisting of thermocouple is shown in Figure 1, where basic thermoelectric effects including Peltier and Joule heat and a circuit with load RL are included. The p and n thermoelements are cuboids of the same thickness and bridged by an electrode in series. Practical devices usually make use of thermoelectric modules containing a number of TEG cells connected electrically in series and thermally in parallel. Cross-sectional area and thickness of thermocouple are marked as A and l. Subscripts ‘n’ and ‘p’ are used to discriminate conductivity type of thermoelements. Temperature of heat source and heat sink is T1 and T0, and that of hot and cold side of thermocouple is Th and Tc. ∆Tg = Th − Tc is temperature difference on thermoelements, and ∆T = T1 − T0 is the one of heat source and heat sink.
There are Joule heat flowing out and Peltier heat flowing in at hot end of thermoelements, and at cold end, Peltier heat flows out and Joule heat flows out. In addition, there is thermal resistance Rth,h and Rth,c between thermoelements, heat source and heat sink. Heat flow qh passes from heat source to hot side of thermocouple and the counterpart qc outflows from cold side of thermocouple to heat sink.
2.2. Basic model
It is assumed, that thermoelements are physically homogeneous and insulated from the surroundings both electrically and thermally, except at junction-reservoir contacts [8–9]. Variable x is defined as location in the thickness direction of thermoelements. According to nonequilibrium thermodynamics under steady heat transfer, energy conservative equations of temperature distributions Tn(x) and Tp(x) are:
Three terms in the above equations represent thermal conduction, Thomson and Joule heat. K, R, and τ are thermal conductance, electrical resistance and Thomson coefficient (V∙K−1), respectively. Relationship of K, R and A, l is and , where λ and ρ are thermal conductivity and electrical resistivity of thermoelectric materials. To solve Eq. (1) analytically, material parameters K, R, and τ are considered to be constant. The boundary conditions of Eq. (1) are:
Electrical current I is determined by formula:
where U0 is the voltage of thermocouple, Rg is the electrical resistance of TEG cell, which contains resistance of thermocouple and contact resistance, and α(T) = αp(T) − αn(T) is Seebeck coefficient (V∙K−1) of thermocouple.
In practice, temperature of heat source T1 and heat sink T0 can be measured and determined. To acquire Th and Tc, relationship of T1, T0 and Th, Tc is necessary. That is:
In Eq. (6), heat flows qh and qc are:
wherein Rch and Rcc are contact electrical resistances at hot and cold side of the thermocouple. Thermal conduction heat, Peltier heat (the third term), and contact Joule heat are within Eqs. (7) and (8). By solving Eq. (1) with Eqs. (2)–(5), Tn(x) and Tp(x) only relating to Th, Tc, and RL can be obtained. And flows qh and qc can be formulated with Th, Tc, and RL in Eqs. (7) and (8). Then, Th and Tc can be determined for a given RL by solving Eq. (6) numerically, which is presented in detail .
Finally, the output power Pout and energy conversion efficiency η are calculated by the basic equations of thermoelectricity:
and Seebeck coefficient α becomes a constant. Equation (11) suits thermoelectric module consisting of m thermocouples, as well, where α and Kg are m times of those of a single thermocouple, but Rth,h and Rth,c are exactly on the contrary.
Generally, c1 is far larger than a1 and b1 in absolute value. Mainly, because of practical module, Seebeck coefficient α has a very small value of about 10−2 V∙K−1, which is much less than unity. For example, taking module TEG-127-150-9 in Ref. , α = 0.05 V∙K−1, Rg = 3.4 Ohm, RL = 4 Ohm, Rth,c = 6 K∙W−1, Rth,h = 0.1 K∙W−1, Kg = 2.907 W∙K−1, ε ≈ 0.5, T0 = 297 K, and T1 = 323 K, calculation result is a1 ≈ 1.423 × 10−9 K−3, b1 ≈ 5.905 × 10−5 K−2, and c1 ≈ −0.7461 K−1. So, the terms with ΔTg order higher than unity can be neglected. At last, here is:
It can be seen, that ΔTg is influenced not only by thermal resistances Rth,c and Rth,h, but also by Peltier effect, which is presented in the last term of the denominator and functions to decrease ΔTg. Because it is tantamount to accelerate heat conduction in thermocouple, Peltier heat flows in and out on two sides of thermocouple. By combing Eqs. (12) and (4), (5), (9), the output power Pout is:
RL, T1, T0, Rth,c, Rth,h, α, Rg, and Kg directly affect Pout. In those parameters, α, Rg and Kg are TEG internal factors, and RL, T1, and T0 are the external ones, and Rth,c and Rth,h originate from both the internal and external. From the form of Eq. (13), it is obvious, that reducing T1, T0, Rth,c, and Rth,h can increase Pout, if ΔT is constant, owing to influence of Peltier effect on ΔTg. On the other hand, Pout has a maximum along with Rg and Kg.
2.3. Matched load, output power and energy efficiency
First of all, influence of RL on Pout is analyzed. In Eq. (13), Pout reaches maximum, when RL is:
which is the matched load and marked as RL,m. Indeed, RL,m is slightly larger than Rg due to the very small value of α2. It means, that existence of Peltier effect increases irreversible heat in thermoelectric module. And reducing T1 and T0 helps to cut down this irreversible heat. When Kg→+∞, RL,m is equal to Rg, since at this moment heat conduction in thermocouple runs under infinitesimal temperature difference and the irreversibility of heat transfer disappears. However, this irreversibility exists with finite Kg, leading to heat loss in thermocouple, that is equivalent to increase in internal resistance. Define RL/Rg as the load factor sL. So, sL is:
when RL is equal to matched load and is the figure of merit. For thermoelectric module, the output power is:
where m is the number of thermocouples. And the corresponding matched load is
By solving Eq. (17) about the partial derivative of RL, it can be obtained, that when load factor sL is:
then η reaches maximum. Equation (18) is downright different from Eq. (15) in the expressions, so achieving maximum of output power and energy efficiency simultaneously is impossible. Actually, the corresponding load factor of the former is smaller than that of the latter. When the ideal state is considered (Rth,c = Rth,h = 0), sL = 1 is for the former and , which is larger than 1, is for the latter.
2.4. Influence of Kg on TEG performance
Kg is important internal factor that influences the output performance in TEG. When matched load is reached, the corresponding output power Pout,m is:
For a common thermoelectric module, thermoelements have the same size, ln = lp and An = Ap, so the figure of merit Z is not related to their size, but material physical parameters. From Eq. (19), we can see, that increase in Z will enhance the output power. By solving Eq. (19) regarding the partial derivative of Kg, when:
then Pout,m reaches maximum, where le and Ae are thickness and cross-sectional area of thermoelements. Since Rth,c and Rth,h are related to Ae, but not to le, there is an optimal le to maximize Pout,m:
2.5. Influence of Peltier effect on TEG performance
When Peltier effect is neglected, the relation of ΔTg and ΔT is:
and the corresponding output power with matched load RL = Rg and constant Seebeck coefficient is:
It is known, that when Rth,g ≪ Rth,c + Rth,h, ηPelt is approximately (1 + 0.5ZT0 + 0.25ZΔT)2 with Rth,c ≈ Rth,h, and even with ΔT → 0, the output power calculated without Peltier effect is more than the output power considering Peltier effect, by over 120% for a common Bi2Te3-based module with ZT ≈ 1. That means, the influence of Peltier effect must be considered. Similar status is obtained, where the difference is more than 50%, when Rth,g ≈ Rth,c + Rth,h. On the contrary, when Rth,g ≫ Rth,c + Rth,h, ηPelt is approximately equal to 1, which means the influence of Peltier effect is negligible. Hence, the smaller the thermal resistance of thermocouple Rth,g, the stronger is the influence of Peltier effect.
Eventually, basic factors for enhancing TEG output power are summarized as:
Enhancing ZT of thermoelectric materials and ΔT, decreasing Rth,c and Rth,h.
When ΔT is fixed, lower T0 and T1 can reduce irreversible heat to elevate output power.
Matched load is a little larger than the inner electrical resistance of TEG.
There exists an optimal thermocouple thickness to maximize output power.
3. Test validation
3.1. Materials property
P-type and n-type Bi2Te3-based materials are, respectively, Bi0.5Sb1.5Te3 and Bi2Te2.85Se0.15, which are prepared by mechanical alloy + spark plasma sintering method. Seebeck coefficient and resistivity of the materials are tested by HGTE-II thermoelectric material performance test system (Chinese patent no. ZL200510018806.4) with test temperature up to 1073 K, relative error of not more than 6%. Thermal conductivity of the materials is measured by laser perturbation method (Type TC-7000 of ULVAC RIKO®). As shown in Table 1, parameters are obtained by polynomial fitting of the experimental data. In temperature range 273 K < T < 493 K, Seebeck coefficient value α is between 170 × 10−6 V∙K−1 and 220 × 10−6 V∙K−1, decreasing with rising temperature. Electrical resistivity ρ is (8.3–20.0) × 10−6 Ohm∙m and thermal conductivity λ is 1.4–2.1 W∙m−1∙K−1, which both show obvious increase with temperature rise.
|p-Bi0.5Sb1.5Te3||α/V∙K−1||−1.791 × 10−11T3 + 1.763 × 10−8T2 − 5.714 × 10−6T + 8.304 × 10−4|
|ρ/Ohm∙m||−7.929 × 10−13T3 + 7.992 × 10−10T2 −1.947 × 10−7T + 1.728 × 10−5|
|ƛ/W∙m−1∙K−1||3.342 × 10−5T2 − 2.24 × 10−2T + 5.118|
|n-Bi2Te2.85Se0.15||α/V∙K−1||1.321 × 10−11T3 − 1.383 × 10−8T2 + 4.81 × 10−6T − 7.774 × 10−4|
|ρ/Ohm∙m||−7.618 × 10−13T3 + 8.098 × 10−10T2 − 2.537 × 10−7T + 3.207 × 10−5|
|ƛ/W∙m−1∙K−1||3.264 × 10−5T2 − 2.228 × 10−2T + 5.302|
In practice, it is difficult to measure thermal resistances Rth,c and Rth,h, and contact electrical resistances rcc and rch. Their values are determined according to empirical formulas. Contact electrical resistivity ρc (Ohm∙m2) at leg-strap junctions and thermal conductivity λc (W m−1 K−1) of thermal conductive layer (≈1.2 mm thick) are according to the empirical formulas given by Rowe et al. :
Here ρ and λ are electrical resistivity and thermal conductivity of thermoelements, respectively. In our calculation, the mean values of ρ and λ over the temperature range are taken as references. As ρ and λ vary with temperature, values of ρc and λ c are also different as the temperature varies. Experiments under four temperature conditions are carried out and the corresponding parameter values are shown in Table 2.
|Temperatures, °C Parameters||T0 = 23|
T1 = 81
|T0 = 23|
T1 = 111
|T0 = 27|
T1 = 147
|T0 = 27|
T1 = 177
3.2. Test setup
System for measuring output performance of thermoelectric modules was established, mainly including electric heating plate controlled by PID, adjustable load, circulatory cooling unit, thermal imaging device, temperature and voltage data acquisition units, etc., with its basic structure as shown in Figure 2. Electric heating plate is used as heat source, with temperature control precision of ±0.1 K and temperature ranging from room temperature to 773 K. Cooling unit, which consists of heat sink, water tank, flow meter and flow valve, etc., takes cold water as the coolant. Heat sink is made of red-copper and its temperature could be adjusted by controlling flowrate of cooling water. In addition, some thermal conductive filler is pasted on both sides of module to reduce thermal resistance between module, heat source and heat sink. Electrical current in the circuit is obtained via measuring voltage on both ends of sampling resistor (metal film precision resistor: 0.2 Ohm, precision of ±1%).
Voltage and temperature signal are acquired by 9207 and 9214 acquisition card of National Instruments (NI) Company, with precision of ±0.5%. Data to be acquired include as follows: (1) temperature of heat source and heat sink; (2) temperature of the coolant (water) inside heat sink and water tank; (3) voltage on adjustable load and sampling resistor. K-type thermocouples with diameter of 1 mm are inserted in heat source and heat sink to measure temperature values. Actually, even though electric heating plate is controlled by PID, heat source temperature still fluctuates during the change of load resistance. In order to eliminate impacts of such transient effect, data shall not be acquired until the heat source and heat sink temperatures are stable.
Test of energy efficiency is not undertaken due to its complexity, where the heat flow into the hot side of the module must be measured or evaluated. An effective way is to adopt heat flux sensor and bury it just under the module. But that would impact heat conduction between heat source and the module, leading to higher thermal resistance. And heat flux sensors of high temperature enduring are really costly. Another useful method is by calculating electrically generated heat in heat source, and at the same time, radiation and convective heat loss must be subtracted, as is introduced in Ref. .
3.3. Comparison of test results with calculation
Figures 3 and 4 show variations of output power Pout with load RL at four temperature conditions, acquired by physical model calculation, ANSYS simulation and experiment. ANSYS method will be introduced in the next part. Figure 3 shows data at heat sink temperature T0 = 300 K, while Figure 4—at T0 = 296 K. Figures 5 and 6 are the corresponding current-voltage (I-V) characteristics. RL results are disposed in the same way. From the results, it is found, that the output power has maximum value with the increase of load. And current is linearly related to voltage. Calculation results are well coincident with ANSYS results, and they are both a little higher than experimental data. Under the four temperature conditions, values of maximum output power are 2.5, 2.6, 2.8 and 1.1% higher than experimental results with T1 changing from high to low. They are especially coincident well, when temperature difference ΔT is small. From the analysis follows, deviation of calculated results is caused mainly by taking thermal conductivity and electrical resistivity as constant (i.e., using the mean values), when solving Eq. (1). When ΔT is small, then material physical parameters vary within a narrow range. So, values of parameters are close to the real values. Otherwise, when ΔT is large, material physical parameters change within a large scale, leading to a great deviation of calculations.
4. Introduction to TEG simulation in ANSYS
4.1. TEG cell model
By software simulation, TEG performance can be achieved both in thermal and in electrical aspects. But it is not direct to cognize and understand the influence of thermoelectric effects, when compared with the above physical model. In this part, TEG cell model is set up by ANSYS, and geometry and meshing methods are illustrated in Figure 7. Thickness and cross-sectional area of thermoelements are 1.6 mm and 1.4 mm × 1.4 mm, respectively. Other geometry parameters are shown in Figure 7. Thermoelectric module consists mainly of p-n thermoelements, current-conducting copper straps and ceramic substrates for heat conducting and electric insulation. Thermoelements and copper strap are meshed by element SOLID226 in ANSYS. This type of element contains 20 nodes with voltage and temperature as the degrees of freedom. It can simulate 3D thermal-electrical coupling field. Element SOLID90 is used to mesh ceramic substrate. It has 20 nodes with temperature as the degree of freedom. Load resistance is simulated by element CIRCU124.
Contact properties of the leg-strap junction are implemented with element pairs CONTACT174/TARGET170. Detailed finite element formulations in ANSYS are introduced in , and the range of contact thermal conductivity and electrical resistivity is explicated in .
4.2. APDL codes for TEG simulation
ANSYS Parametric Design Language (APDL) is widely used for programed simulation. The following APDL codes have taken temperature variation of materials properties, thermal contact and thermal radiation (although its influence is very weak) into consideration. According to the practical requirements, the readers could use the code more concisely by neglecting certain physical effects. The unit referring to length is meter and the temperature unit is Celsius.
! defining the TEG cell dimensions
ln=1.6e-3 ! n-type thermoelement thickness
lp=1.6e-3 ! p-type thermoelement thickness
wn=1.4e-3 ! p-type thermoelement width
wp=1.4e-3 ! p-type thermoelement width
d=1.0e-3 ! Distance between the thermoelements
hs=0.2e-3 ! copper strap thickness
hc=1e-3 !substrate thickness
! definition of several physical parameters
rsvx=1.8e-8 ! copper electrical resistivity
kx=200 ! copper thermal conductivity
kxs=24 !substrate thermal conductivity
T1=250 ! temperature of heat source
T0=30 ! temperature of heat sink
Toffst=273 ! temperature offset
! defining TEG output parameters and the load
*dim,P0,array,1 ! defining P0 as the output power
*dim,R0,array,1 ! defining R0 as the load
*dim,Qh,array,1 ! defining Qh as the heat flow into the TEG cell
*dim,I,array,1 ! defining I as the current
*dim,enta,array,1 ! defining enta as the energy efficiency
*vfill,R0(1),ramp,0.025 ! setting the load (Ohm)
! pre-processing before calculation, defining element type, building the structure and meshing
toffst,Toffst ! set temperature offset
et,1,226,110 ! 20-node thermoelectric brick element
et,2,shell57 ! shell57 element for radiation simulation
et,3,conta174 ! conta174 element for contact simulation
et,4,targe170 ! target170 element for contact simulation
keyopt,3,1,4 ! taking temperature and voltage as the degree of freedom
! Temperature data points
! Seebeck coefficient of the n-type material (V·K−1)
! electrical resistivity of the n-type material (Ohm*m)
! thermal conductivity of the n-type material (m* K−1)
! Seebeck coefficient of the p-type material (V·K−1)
! electrical resistivity of the p-type material (Ohm*m)
! thermal conductivity of the p-type material (m* K−1)
! material property for cooper strap
! material property for the substrate
!radiation property for the p-n materials
! contact friction coefficient
! build the TEG cell structure
! glue the copper strap and the substrate
! meshing the TEG cell structure
! defining the contact parameters
r,5 ! selecting the thermal contact conductivity and resistivity
rmore,,7e5 ! setting the thermal contact conductivity
rmore,0.67e8,0.5 ! setting the thermal contact resistivity
! defining the contact layer between p-leg and upper copper strap
! defining the target layer between p-leg and upper copper strap
! defining the contact layer between n-leg and upper copper strap
! defining the target layer between n-leg and upper copper strap
! defining the contact layer between p-leg and bottom copper strap
! defining the target layer between p-leg and bottom copper strap
! defining the contact layer between n-leg and bottom copper strap
! defining the target layer between n-leg and bottom copper strap
! defining the shell element for radiation simulation, outputting radiation matrix
! defining the shell element for copper strap
! defining the shell element for p-n thermoelements
! defining the space node for radiation simulation
! using radiation matrix method
emis,3,1 ! setting the emissivity
stef,5.68e-8 ! setting the Stefan-Boltzmann constant
write,teg,sub ! outputting the radiation super element
! deleting the shell elements and the corresponding mesh
et,5,matrix50,1 ! defining radiation matrix element
! defining boundary conditions and the load
nsel,s,loc,y,hs+hc ! TEG cell hot side
cp,1,temp,all ! coupling of temperature degree of freedom
nh=ndnext(0) ! getting the master node
d,nh,temp,Th ! setting the temperature constraint to the hot side
nsel,s,loc,y,-(ln+hs+hc) ! selecting the TEG cell cold side
d,all,temp,Tc ! setting the temperature constraint to the cold side
cp,3,volt,all ! electrical coupling
nn=ndnext(0) ! getting the master node
d,nn,volt,0 ! setting the ground connection node
cp,4,volt,all ! ! electrical coupling
np=ndnext(0) ! getting the master node
d,10000,temp,300 ! setting the temperature of the space node
se,teg,sub ! reading the radiation super element
et,6,CIRCU124,0 ! setting the load resistor element
! setting the load value and property
circu_num=elnext(0) !getting circuit element number
! starting the calculation
antype,static ! solution type
cnvtol,heat,1,1.e-3 ! setting the converging value for heat condition
cnvtol,amps,1,1.e-3 ! setting the converging value for the current
neqit,50 ! calculation iteration step
solve ! starting solving
*get,P0(1),elem,circu_num,nmisc,1 ! getting the output power of the TEG cell
*get,Qh(1),node,nh,rf,heat ! getting the heat flow into the TEG cell
*get,I(1),elem,circu_num,smisc,2 ! getting the current
*voper,enta,P0,div,Qh ! calculating the energy efficiency of the TEG cell
The built one-dimensional model, which is validated by test results, can calculate TEG output power and energy efficiency accurately. By simplifying this model, it is convenient to analyze influences of different thermal and electrical parameters on TEG performance. And basic factors to enhance TEG output power and energy efficiency are extracted. At last, ANSYS simulation considering thermal contact and radiation effects for TEGs is introduced briefly, and basic APDL codes are shared.