Seebeck Effect

The Seebeck Effect is a fundamental thermoelectric phenomenon where a temperature difference across two different conductive materials generates an electric voltage. Discovered by Thomas Johann Seebeck, this effect forms the basis of thermocouples, thermoelectric generators, and many modern energy-harvesting technologies. Understanding the Seebeck Effect is essential in fields like electronics, renewable energy, and materials science because it explains how heat can be directly converted into usable electrical power.

1.0Introduction of Thermoelectricity

Thermoelectricity describes the generation of an electromotive force (EMF) when a temperature difference exists. This phenomenon is observed in two main scenarios:

• Within a single conductor (between points at different temperatures).
• At the contact point between two dissimilar conductors.

2.0Definition of Seebeck Effect

Seebeck Effect: This is the phenomenon where a voltage, known as the thermo-EMF (electromotive force), is produced in a circuit made of two different conductors when their two junctions are kept at different temperatures. If the circuit is closed, this EMF will drive a continuous electric current.

Thermocouple: The device used to demonstrate this effect, consisting of two dissimilar conductors (like copper and iron) joined together at two separate junctions, is called a thermocouple.

Factors Affecting Thermo-EMF: The magnitude and direction of the generated thermo-EMF depend on two main factors:

• The specific materials (e.g., the pair of metals) used to create the thermocouple.
• The temperature difference between the hot and cold junctions.

Reversibility: The Seebeck effect is reversible. If the hot and cold junctions are swapped, the direction of the thermo-EMF and the resulting current will also reverse.

Magnitude: The voltage produced by a thermocouple is typically very small. Its magnitude is usually in the range of microvolts ($\mu V$)to millivolts $(mV)$ for each degree Celsius of temperature difference between the junctions.

3.0Seebeck Series

• The Seebeck series is an experimentally determined ranking of conductive materials (primarily metals). Its purpose is to predict the magnitude and direction of the thermo-EMF (and resulting current) generated when two different materials are used to form a thermocouple.

Predicting Magnitude: The magnitude of the thermo-EMF produced by a thermocouple is directly related to how far apart its two constituent metals are in the Seebeck series.

• For a fixed temperature difference between the junctions, a wider gap between the metals in the series results in a larger thermo-EMF.
• Example: A thermocouple made from Antimony (Sb) and Bismuth (Bi) will produce the maximum thermo-EMF for a given temperature difference because they are at the opposite ends of the series.

Predicting Direction: The series determines the direction of the conventional electric current. When two metals from the series form a closed circuit:

• At the cold junction, the current flows from the metal that appears earlier in the series to the metal that appears later (e.g., from Sb to Bi).
• At the hot junction, the current flows from the metal that appears later in the series to the metal that appears earlier (e.g., from Copper (Cu) to Iron (Fe), since Fe comes before Cu in the series).

Origin of Thermo-EMF

• Electron Density: Every conductive material has a unique electron density, which is the number of free electrons (charge carriers) available per unit volume. This property depends on the specific nature of the material.
• Contact Potential: When two different metals are brought into contact, they form a junction. Because their electron densities are different, free electrons will diffuse across this junction, typically moving from the material of higher density to the one of lower density. This migration of charge establishes a small, stable potential difference at the junction, known as the contact potential.
• Equilibrium at Same Temperature: In a thermocouple, there are two junctions (e.g., Metal A to Metal B, and Metal B back to Metal A). If both of these junctions are at the exact same temperature, the contact potential at each junction will be identical. These potentials oppose each other and cancel out, resulting in zero net EMF and no current flow.
• Effect of Temperature Difference: The magnitude of the contact potential is dependent on temperature. If one junction is heated (the "hot" junction) and the other is kept cool (the "cold" junction), the contact potentials at these two junctions will no longer be the same. This imbalance creates a net potential difference across the entire circuit.
• Generation of Thermo-EMF: This net potential difference, created by the difference in the contact potentials of the hot and cold junctions, is the thermo-EMF. This EMF is what drives the current through the thermocouple. It can be expressed as:

$V_{EMF}=V_{Hot}-V_{Cold}$

Where $V_{Hot}$ and $V_{Cold}$ are the contact potentials at the hot and cold junctions, respectively.

4.0Variation of Thermo EMF With Temperature

5.0Relationship Between Temperature and Thermo-EMF

Zero EMF Condition: If both junctions of a thermocouple are at the same temperature (e.g., $T_{Hot}=T_{Cold}$), there is no temperature difference, and the net thermo-EMF is zero.

EMF Behaviour with Heating: As the temperature of the hot junction ($T_{Hot}$) is increased (while keeping the cold junction ($T_{Cold}$), at a constant low temperature):

• The thermo-EMF first increases.
• It reaches a maximum value.
• It then decreases, eventually becoming zero and even reversing its direction.

Neutral Temperature ($T_n$):

• This is the specific temperature of the hot junction at which the thermo-EMF reaches its maximum value.
• The value of $T_n$ is a fixed property that depends only on the nature of the two materials forming the thermocouple.
• It is independent of the temperature of the cold junction ($T_{Cold}$).
• Example: For a copper-iron (Cu-Fe) thermocouple, $T_n$ is approximately 270°C.

Temperature of Inversion ($T_i$):

• This is the temperature of the hot junction at which the thermo-EMF becomes zero again (after passing its maximum) and is about to reverse its polarity.
• The value of $T_i$ depends on two factors:
• 1. The nature of the materials.
  2. The temperature of the cold junction ($T_{Cold}$).

Relationship between Temperatures:

• The neutral temperature is always exactly halfway between the cold junction temperature and the temperature of inversion.
• In other words, the neutral temperature ($T_n$) is the arithmetic mean (average) of the cold temperature ($T_{Cold}$) and the inversion temperature ($T_i$).
• This is expressed by the formula: $T_n=\frac{T_{\text{cold }}+T_i}{2}$

Thermo-EMF Equation:

• The relationship between the thermo-EMF (E) and the hot junction temperature (T, in Celsius) can be approximated by a quadratic equation, assuming the cold junction is kept at 0°C:

$E=\alpha T+\frac{1}{2}\beta T^2$

• Here, (alpha) and (beta) are thermoelectric constants that depend on the specific pair of materials used.

6.0Thermoelectric Power

Thermoelectric Power (Seebeck Coefficient)

1. Definition: Thermoelectric power, also known as the Seebeck Coefficient (S), is defined as the rate of change of the thermo-EMF (E) with respect to the temperature difference (T). It measures how much voltage is produced for each degree of temperature change

$S=\frac{dE}{dT}$

 2. Derivation from EMF Equation: If we use the standard equation for thermo-EMF (where the cold junction is at T=0°C):

$E=\alpha T+\frac{1}{2}\beta T^2$

We can find S by differentiating E with respect to T

$S=\frac{d}{dt}\left(\alpha T+\frac{1}{2}\beta T^2\right)=\alpha +\beta T$

Here $\alpha$ is the Seebeck coefficient at T=0°C

3. At the neutral temperature ($T_n$)

• By definition, the neutral temperature ($T_n$) is the point where the thermo-EMF (E) is at its maximum.
• In calculus, the maximum of a function occurs when its first derivative is zero. Therefore, at ($T_n$), the Seebeck Coefficient (S)must be zero.

$S=\frac{dE}{dT}=0$ using the formula for S

$\alpha +\beta T_n=0\Rightarrow T_n=-\frac{\alpha }{\beta }$

4. At the Temperature of Inversion ($T_i$):

• By definition, the temperature of inversion ($T_i$) is the point (other than T=0°C) where the thermo-EMF (E)becomes zero again.
• We set the original EMF Equation to zero

$E=\alpha T_i+\frac{1}{2}\beta T_i^2=0$

Factoring out $T_i$

$\left(\alpha +\frac{1}{2}\beta T_i\right)T_i=0$

• This gives two solutions $T_i=0$ the cold junction and the inversion temperature

$\alpha +\frac{1}{2}\beta T_i=0\Rightarrow T_i=-\frac{2\alpha }{\beta }$

5. Relation Between ($T_n$) and ($T_i$)

$T_n=-\frac{\alpha }{\beta }$ and $T_i=-\frac{2\alpha }{\beta }$. This confirms the general relationship from the previous section $\left(T_n=\frac{T_{\text{cold }}+T_t}{2}\right)$ for the specific case where the cold junction $T_{\text{cold }}=0^{\circ }C$.

6. Sign of Thermoelectric Power:

• The sign of S tells us if the EMF is increasing or decreasing as the temperature rises.
• S is positive: When the hot junction temperature T is between $T_{\text{cold }}(0)$and Tn. This means the total EMF is increasing.
• S is negative: When T is between $T_n$ and $T_i$. This means the total EMF is decreasing from its maximum value.

7.0Law of Thermoelectricity

1. Law of Successive Temperatures (Law of Intermediate Temperatures)

• In a thermocouple, the EMF generated for a specific temperature difference can be broken into smaller temperature intervals.
• The total EMF across the full temperature range is equal to the sum of EMFs across each smaller interval.

2. Law of Successive Metals (Law of Intermediate Metals)

• When multiple metals are connected in series at the same temperature, the total EMF between the first and last metal equals the sum of EMFs between each pair of adjacent metals.
• This applies only when all junctions are maintained at the same temperature.