# Thermoelectric Generators¶

## Principles of Thermoelectricity¶

### Seebeck effect¶

Seebeck effect is a phenomenon when a temperature difference between two different metal conductors results in an induced voltage, $\Delta V_{se}$, where:

|\Delta V_{se}| = |S_{e}\Delta T|

Typical units of Seebeck coefficient, $S_{e}: \mu \text{V/K}$

• Seebeck coefficient: describes thermoelectric capability of material
• $S_{e} > 0$: p-type semiconductor (current, heat carried by positively charged holes)
• $S_{e} > 0$: Platinum (reference metal)
• $S_{e} > 0$: n-type semiconductor (current, heat carried by negatively charged electrons)

### Peltier effect¶

Peltier effect is a phenomenon when temperature difference is induced due to current flow.

\dot{Q} = S_{e}IT, \text{where} \ \pi = S_{e}T

and T is the temperature of the interface where the current flows

### Ohm's Law and Joule Heating¶

• Ohm's Law $V = IR \ \text{where} \ R= \frac{\rho l}{A}$
• Joule heating is isotropic
• 50% to top, 50% to bottom

### Thermal Conduction¶

• Due to thermal conductivity of material
\begin{align*} \dot{Q} &= \kappa A \frac{T_{h}-T_{c}}{l} \\ &= \frac{\kappa A}{l}\Delta T \\ &= \lambda \Delta T \end{align*}

## Thermoelectric Generator vs Thermomechanical Heat Engine¶

• TEG: Converts thermal energy to electrical energy to do work
• Difference in temperature creates a potential difference (Seebeck effect)

• Heat engine: Converts thermal energy to do work
• Operates between thermal reservoirs of different temperatures ($T_{hot}$, $T_{cold}$)
• Takes $Q_{h}$ from hot reservoir, converts some to work, the rest goes to cold reservoir via $Q_{c}$

## Thermoelectric Figure of Merit¶

\begin{align*} Z &= \frac{S^{2}_{e}}{R_{in}\lambda} = \frac{\text{material property}^{2}}{\text{(electrical resistance)(thermal conductance)}} \\ T &= \frac{T_{h}+T_{c}}{2} \\ \Rightarrow ZT &= \frac{S^{2}_{e}}{R_{in}\lambda}\left(\frac{T_{h}+T_{c}}{2}\right) = \frac{S^{2}_{e}\sigma}{\kappa}\left(\frac{T_{h}+T_{c}}{2}\right) \end{align*}

where:

• $R_{in}$ is internal resistance of the TEG; $R_{in}=\frac{\rho l}{A}$
• $\lambda$ is the thermal conductance of the TEG; $\lambda = \frac{\kappa A}{l}$
• $\sigma$ is the electrical conductivity of the TEG; $\sigma = \frac{1}{\rho}$
• $\kappa$ is the thermal conductivity of the TEG.

## Electrical Analysis of TEG¶

\begin{align*} \text{Voltage drop across } R_{L}, V &= \Delta V_{Se} - IR_{in} \\ &= |S_{e}\left(T_{h} - T_{c}\right)| - IR_{in}\\ \text{Power delivered by TEG, }\dot{W} &= IV \\ &= |S_{e}I\left(T_{h} - T_{c}\right)| - I^{2}R_{in} \end{align*}

Tip

The power delivered by the TEG has to match the power received by $R_{L}$, where $\dot{W} = I^{2}R_{L}$.

## Energy Flow of TEG¶

\begin{align*} \dot{Q}_{h} &= S_{e}IT_{h}-\frac{I^{2}R_{in}}{2}+\lambda\left(T_{h}-T_{c}\right) \\ \dot{Q}_{c} &= S_{e}IT_{c}+\frac{I^{2}R_{in}}{2}+\lambda\left(T_{h}-T_{c}\right) \\ \text{By First Law: }\dot{W} &= \dot{Q}_{h}-\dot{Q}_{c} \\ &= S_{e}I\left(T_{h}-T_{c}\right)-I^{2}R_{in} \end{align*}

## Efficiency of TEG¶

\begin{align*} \text{Efficiency, }\eta &= \frac{\text{desired output}}{\text{required input}} \\ &= \frac{W_{cycle}}{Q_{h}}= \frac{Q_{h}-Q_{c}}{Q_{h}}\\ &= 1-\frac{Q_{c}}{Q_{h}}\\ \end{align*}

Applying Second Law:

\frac{dS}{dt} = \frac{\dot{Q}_{in}}{T_{in}}-\frac{\dot{Q}_{out}}{T_{out}}+\dot{\sigma}_{gen} = 0

Assuming reversible process,

\dot{\sigma}_{gen} = 0 \\ \Rightarrow 0 = \frac{\dot{Q}_{h}}{T_{h}}+\frac{\left(-\dot{Q}_{c}\right)}{T_{c}}

Applying First Law:

\dot{W} = \dot{Q}_{h}-\dot{Q}_{c}\\ 0 = \frac{\dot{Q}_{h}}{T_{h}} + \frac{-\left(\dot{Q}_{h}-\dot{W}\right)}{T_{c}}\\ \Rightarrow \dot{W} = \dot{Q}_{h}\left(1-\frac{T_{c}}{T_{h}}\right)\\ \text{Carnot efficiency, }\eta_{carnot} = \frac{\dot{W}}{\dot{Q}_{h}} = 1-\frac{T_c}{T_{h}} \\ \text{(Theoretical maximum efficiency of engine with $T_{c}$ and $T_{h}$})

### Calculating Efficiency of TEG by Thermoelectric Properties and ZT¶

\begin{align*} \text{Efficiency, }\eta &= \frac{W_{cycle}}{Q_{h}}\\ &= \frac{S_{e}I\left(T_{h}-T_{c}\right)-I^{2}R_{in}}{S_{e}IT_{h}-\frac{I^{2}R_{in}}{2}+\lambda\left(T_{h}-T_{c}\right)} \end{align*}

If $\lambda \rightarrow 0, R_{in} \rightarrow 0, \eta \rightarrow \frac{T_{h}-T_{c}}{T_{h}} = 1-\frac{T_{c}}{T_{h}}$ (Carnot efficiency)

Using ZT:

\eta_{carnot} = \left(\frac{T_{h}-T_{c}}{T_{h}}\right)\left(\frac{\sqrt{1+ZT}-1}{\sqrt{1+ZT}+\frac{T_{c}}{T_{h}}}\right)

## Load Resistance for Maximum Power Output¶

V_{out} = IR_{L} = \frac{\Delta V_{se}}{(R_{in}+R_{L})}R_{L}\\ \dot{W} = \frac{V_{out}^{2}}{R_{L}} = \frac{\Delta V_{se}^{2}R_{L}}{(R_{in}+R_{L})^{2}}\\ \text{To find minimum or maximum, }\frac{d\dot{W}}{dR_{L}} = 0.\\ \Rightarrow \frac{\Delta V_{se}^{2}}{(R_{in}+R_{L})^{2}} + \frac{\Delta V_{se}^{2}R_{L}(-2)}{(R_{in}+R_{L})^{3}} = 0\\ R_{in} + R_{L} - 2R_{L} = 0 \ \mathbf{\Rightarrow R_{in} = R_{L}}.\\ \text{To check if minimum or maximum occurs when }R_{in} = R_{L}\text{, find }\frac{d\dot{W}}{dR_{L}}.\\ \frac{d\dot{W}}{dR_{L}} = -\frac{4\Delta V_{se}^{2}}{(R_{in}+R_{L})^{3}}+\frac{6\Delta V_{se}^{2}R_{L}}{(R_{in}+R_{L})^{4}}\\ \text{When }R_{L} = R_{in}: \frac{d\dot{W}}{dR_{L}} = -\frac{2\Delta V_{se}^{2}}{16R_{in}} < 0\\ \therefore \dot{W}_{max}\text{ when }R_{L} = R_{in}

## Analysing electrical power¶

• More $\dot{W}$ can be obtained with larger $\Delta T$
• $I_{@\dot{W}_{max}} > I_{@\eta_{max}}$ slightly
• Which current to use depends on what is required

## Effective Parameters of Thermoelectric Device¶

• TE device consists of multiple TE elements:
\text{Effective }S_{e} = N(S_{e,p} - S_{e,n})\\ \text{Effective }R_{in} = N(R_{p}+R_{n})\\ \text{Effective }\lambda = N(\lambda_{p}+\lambda_{n})

## Thermal Contacts¶

• Actual temperature at surface of semiconducting elements is not what is applied at the ends
• $\Delta T$ exists due to thermal conductance of ceramic plates and metal leads

## Practical Thermoelectric Devices¶

• Match $\dot{Q}_{c}$ with required cooling power
• Match $\dot{Q}_{h}$ with heat sink