Modeling of a Thermoelectric Generator Device

2.1. Seebeck effect

Seebeck effect describes the induction of voltage, when junctions of two different conducting materials are maintained at different temperatures as shown in Figure 1. Seebeck effect increases in magnitude, when Seebeck coefficient of conducting materials and/or temperature difference between their connections increases. Voltage induced through Seebeck effect is defined as below:

where α is Seebeck coefficient and ΔTis the temperature difference between hot junction and cold junction.

2.2. Peltier effect

Peltier effect describes heat dissipation or absorption at the connection of two conducting materials, when current flows through the junction as shown in Figure 2. Depending on the direction of current flow, heat is either absorbed or dissipated at connection.

2.3. Thomson effect

Thomson effect describes the dissipation or absorption of heat, when electric current passes through a circuit composed of a single material, which has temperature variation along its length, as shown in Figure 3. ΔQrepresents heat dissipation, when electrical current flows through a homogeneous conductor. Thomson coefficient is given by second Kelvin relationship [6–9]:

where μand T,respectively, symbolize Thomson coefficient and temperature. If Seebeck coefficient, α, is temperature independent, then Thomson coefficient is equal to zero.

2.4. Joule heating

Joule‐heating effect defines heat dissipated by material with nonzero electrical resistance in the presence of electrical current, as shown in Figure 4,

3.1. Three‐dimensional representation of comprehensive operation of TEG

TEGs are composed of numerous legs (slabs) made of p‐ and n‐type semiconductors forming thermocouples, all connected electrically in series and thermally in parallel. Semiconductor legs are connected to each other through conductive copper tabs, and they are sandwiched between two ceramic plates, which conduct heat, but behave as insulators to electrical current. Schematic diagram of three‐dimensional (3‐D) multielement thermoelectric generator is shown in Figure 5.

Waste heat from various sources, such as automobile engines exhaust, industry and infrastructure‐heating activities, geothermal, and others, can be supplied to top ceramic plate of TEGs. As shown in Figure 5, heat flows through ceramic plate and copper‐conductive tabs before reaching the top surface of p‐ and n‐type legs made of proper semiconductors, which is defined as the hot side of TEG. Heat flows through both semiconductor's legs and then again through copper‐conductive tabs and bottom ceramic plate. Through heat sink, the bottom ceramic plate is maintained at significantly lower temperature than top ceramic in order to produce high‐temperature gradient, which will lead to high‐power output. Allowed temperature applied on top and bottom ceramic plates depends on materials of p‐ and n‐type legs. Also, p‐ and n‐type materials are designed to possess low thermal conductivity in order to restrict, as much as possible, heat flow through semiconductors and maintain temperature difference between hot and cold sides of TEG.

Pictorial distribution of temperature along legs of TEG at conditional difference of temperature ΔTbetween hot and cold sides is shown in Figure 6.

After temperature gradient has been induced between hot and cold sides of TEG, voltage occurred on TEG‐positive and ‐negative poles due to Seebeck effect, as depicted in Figure 7.

Voltage generated in TEG due to Seebeck effect induces the movement of charge carriers within p‐and n‐type semiconductor legs and, hence, electrical current in electrical circuit including load resistor RL connected to TEG poles, current density formed, is displayed in Figure 8.

3.2. 1‐D representation of TEG

Establishing one‐dimensional (1‐D) representation of TEG is helpful in determining analytical expressions of heat absorbed and heat rejected, as the power output of TEG is defined as the difference between heat absorbed and heat rejected. Figure 9 represents 1‐D schematic of TEG with heat source and heat sink, respectively, applied on top and bottom sides of TEG.

TH, QH, and KHare, respectively, heat source temperature, heat supplied from heat source to TEG, and thermal conductance of hot side of TEG. TL, QL, and KLare, respectively, heat sink temperature, heat rejected from TEG to heat sink, and thermal conductance of TEG cold side. Thand Qhdefine the temperature of hot junction of thermocouples and heat flow through hot junctions of TEG. Tcand Qcdescribe the temperature at cold junction of thermocouples and heat flow through cold junctions of TEG. Assuming thermoelectric properties to be temperature independent, α, k, ρcan, respectively, be defined as constant Seebeck coefficient, constant thermal conductivity, and constant electrical resistivity.

3.3. Electrical network resistance

Electrical resistance network of TEG is shown in Figure 10. P‐type and n‐type semiconductor legs are connected to each other electrically in series through copper‐conductive tabs.

Rpand Rn are electrical resistance associated, respectively, with p‐ and n‐type semiconductor legs. Rcpeh, Rcpec, and RLare, respectively, electrical resistance of copper‐conductive strips on the hot side, electrical resistance of copper‐conductive strips on the cold side, and external load resistance.

3.4. Thermal network resistance

Thermal resistance of TEG is shown in Figure 11 and it assists in determining heat transfer rate through ceramic plates, copper strips, and p‐ and n‐type semiconductor legs. The number of thermocouples is N.

Teceh, Ticeh, and Rcehare, respectively, external temperature of hot ceramic plate, internal temperature of hot ceramic plate, and thermal resistance associated with ceramic plate on the hot side. Th, Rcph, and Rtegare, respectively, the temperature at the hot junction of p‐ and n‐type semiconductor legs, thermal resistance of copper strip on the hot side, and thermal resistance of both p‐ and n‐type semiconductor legs. Tc, Rcpc, and Ticec, are, respectively, the temperature at cold junction of p‐ and n‐type semiconductor legs, thermal resistance of ceramic plate on the cold side, and internal temperature of cold ceramic plate. Rcecand Tececare, respectively, thermal resistance of cold ceramic plate and external temperature of cold ceramic plate.

4.1. Analysis of thermoelectric material properties and geometry of TEG

Thermoelectric materials of TEG legs, p‐ and n‐type semiconductors, are characterized by parameter called the figure of merit Z,which measures the ability of thermoelectric materials to convert heat into electrical power. The figure of merit is expressed as follows:

where α, ρ, and kare, respectively, Seebeck coefficient, electrical resistivity, and thermal conductivity of thermoelectric materials. Great thermoelectric materials possess high Seebeck coefficient, low electrical resistivity, and low thermal conductivity [10].

In order to obtain maximum figure of merit, when designing TEG, the geometry of semiconductor legs and properties of thermoelectric materials need to satisfy the following equation [1, 11]:

where Ap, An, Lp, Ln,kp, kn, ρp, and ρnare, respectively, the cross‐sectional area, length, thermal conductivity, and electrical resistivity of p‐ and n‐type semiconductor legs.

To reduce manufacturing costs, p‐ and n‐type semiconductor legs are fabricated with the same geometry, that is, Ap=An=A, and  Lp=Ln=L. Similarly, p‐ and n‐type semiconductor legs are made of doped alloys to produce the same thermoelectric properties, that is, ρp=ρn, kp=kn, and  αp=−αn[12].

4.2. TEG performance analysis

In order to obtain expressions describing TEG performance, thermocouple built by legs of p‐ and n‐type semiconductors is extracted from Figure 9 and represented in Figure 12. Figure 12 represents heat transfer within single thermocouple. The length and cross‐sectional area of both p‐ and n‐type semiconductor legs are equal and symbolize as Land A, respectively. The junction of thermocouple is fixed at thermal conducting and electrical‐insulating ceramic plate.

Qh, Qc,  Qkin, Qkout, Qj, Lp, Ln, and δcuare, respectively, heat absorbed at hot junction, heat rejected at cold junction, Fourier heat conduction transferred inside of control volume, Fourier heat conduction transferred out of control volume, Joule heating generated within control volume, the length of p‐ and n‐type legs, and the thickness of copper electrical‐conducting strips.

Employing the conservation of energy and assuming one‐dimensional steady‐state condition, the energy equation of differential control volume inside of p‐type semiconductor leg can be expressed as follows:

Using Taylor expansion:

Irepresents electrical current induced within TEG device:

Fourier's law of conduction for one‐dimensional heat conduction states:

Substituting Eq. (9) into Eq. (8):

Provided that thermoelectric properties are temperature independent, kpcan be taken out of derivative and Eq. (10) can be expressed as follows:

Integrating Eq. (11):

where Qp(0)is Fourier heat conduction transferred inside of top p‐type leg:

Considering Peltier effect happening at the hot junction of p‐type leg:

where Qphis the total heat absorbed at the hot junction of p‐type leg.

Employing the same procedure with the same boundary conditions to derive heat flow through n‐type leg leads to the expression of Qnhas follows:

where Qnhis the total heat absorbed at the hot junction of n‐type leg. The total heat absorbed at the hot junction of both p‐ and n‐type semiconductor legs is, therefore:

We use the same method to derive expression for heat rejected at the cold junction of p‐type and n‐type legs. Consequently, the following expression is obtained:

4.3. TEG performance expressions

TEG is characterized by numerous performance expressions, including heat absorbed on the hot side, heat rejected on the cold side, power output, voltage induced, and current flowing in the electrical circuit with load resistor. Defining symbols below from Eqs. (24) and (25):

Expressions of heat flow through the hot and cold junctions for Nsemiconductor thermocouples can therefore be expressed as follows:

As stated previously, the power generated by TEG is defined as the difference between heat absorbed at the hot junction and heat rejected at the cold junction:

Optimal current generated in TEG is obtained by first deriving Eq. (31) with respect to current as follows:

Eq. (32) is equated to zero to determine the following expression of optimal current:

Generally speaking, voltage, current, and output power induced in TEG consisting of set of thermocouples similar to the one represented in Figure 9 are, respectively, defined as:

where RL, is the external resistance load. To get optimum electrical current induced, and output power generated in the electrical circuit with TEG consisting of set of thermocouples, external resistance needs to be equal to the total internal electrical resistance of p‐ and n‐type semiconductor legs. The efficiency of TEG is given by:

In actual TEG, two thermoelectric materials are used, that is, p‐ and n‐type semiconductors. The maximum efficiency provided by TEG is expressed as follows:

where Zand T¯are, respectively, the figure of merit of p‐ and n‐type semiconductors and averaged temperature between temperatures at the hot and cold sides.

4.4. Performance simulation example of a TEG

Numerical example is adopted in order to optimize and analyze effects of heat transfer governing equations on output power, efficiency, and induced voltage of TEG.

In numerical analysis, the following geometry is adopted (Table 1).

The following thermoelectric properties are adopted (Table 2).

All obtained performance curves are computed at the hot‐side temperature up to Th=673Kand the cold‐side temperature of Tc=373K.

4.4.1. Power and efficiency as function of electrical current

By fixing the cold side at temperature Tc=373Kand varying the hot‐side temperature from 473 to 673 K with increment of 100 K, the power generated behaves as follow:

One can observe that the power as a function of current behaves as a parabola with optimum power value at specific current. Figure 13 shows the existence of maximal current value, which corresponds to optimum power. Any current higher or lower than the maximum current value generates power output less than optimum power. Also, as temperature at the hot side increases, then power produced increases as well.

Efficiency curves shown in Figure 14 behave as parabola as well, with specific current value maximizing efficiency for each temperature difference. In real devices, TEGs are always operated at an optimal current. One thing to note is that the efficiency of TEG is still low compared to other energy‐conversion techniques. A lot of effort has been made to enhance efficiency [13, 14]. Given that heat sources are plenty and free, TEGs could be promising solutions, when they are employed to harvest waste heat from industry activities and central‐heating systems.

4.4.2. I‐V dependences

Employing various temperature differences, while maintaining the cold‐side temperature at 373 K, voltage induced as a function of current behaves as shown in Figure 15.

One can observe from Figure 15 that voltage induced for each temperature difference is decreasing and the linear function of output electrical current. Slopes of I–V dependences are the same.

4.4.3. Power and efficiency as a function of hot‐side temperature

While still maintaining the cold side at a temperature of 373 K and replacing current in output power equation (Eq. (31)) by optimal current expression (Eq. (33)), power expression becomes a function of temperature at the hot side, and Figure 16 shows the behavior of output power as a function of the hot‐side temperature.

Output power as a function of hot‐side temperature behaves as nonlinear curve increasing as the hot‐side temperature increases.

The efficiency of TEG as a function of hot‐side temperature is shown in Figure 17.

4.4.4. Power as a function of external load resistance

Figure 18 depicts variations of output power as a function of external load resistance. Eq. (35) is used to obtain dependences shown in Figure 18.

Optimal output power occurs when load resistance equates to internal electrical resistance of the total number of p‐ and n‐type semiconductor legs.

4.4.5. Efficiency as a function of the figure of merit (ZT)

ZTvalue is modified figure of merit, where Trepresents averaged temperature between the hot‐side and cold‐side temperatures. For each temperature difference, efficiency increases as ZTvalue increases. Therefore, employing thermoelectric materials possessing high ZTvalues leads to great TEG efficiency (Figure 19).