🔋Thermoelectric Materials and Devices Unit 5 – Thomson Effect & Kelvin Relationships

Fundamentals of Thermoelectric Effects

• Thermoelectric effects convert temperature differences into electric voltage and vice versa
• Three main thermoelectric effects: Seebeck effect, Peltier effect, and Thomson effect
• Seebeck effect produces an electromotive force (EMF) when two dissimilar materials are subjected to a temperature gradient
• Peltier effect describes the heating or cooling that occurs when an electric current passes through the junction of two different conductors
• Thomson effect relates to the heating or cooling of a single conductor when an electric current flows through it while a temperature gradient is present
• Thermoelectric effects are reversible, meaning they can be used for both power generation and refrigeration
• Efficiency of thermoelectric devices depends on the figure of merit (ZT), which is a function of electrical conductivity, thermal conductivity, and Seebeck coefficient

Thomson Effect Explained

• Thomson effect is the absorption or emission of heat when an electric current flows through a single conductor with a temperature gradient
• Discovered by William Thomson (Lord Kelvin) in 1851
• The amount of heat absorbed or emitted per unit volume per unit time is proportional to the product of the electric current density and the temperature gradient
  • This proportionality constant is known as the Thomson coefficient ($\mu$)
• The Thomson coefficient is a material-specific property that varies with temperature
• The sign of the Thomson coefficient determines whether heat is absorbed or emitted
  • A positive Thomson coefficient indicates heat absorption when current flows from hot to cold
  • A negative Thomson coefficient indicates heat emission when current flows from hot to cold
• The Thomson effect is reversible, meaning the direction of heat flow reverses when the direction of the electric current is reversed
• The Thomson effect is essential for understanding the overall thermoelectric behavior of materials and devices

Kelvin Relationships: Connecting Thermoelectric Phenomena

• Kelvin relationships, derived by Lord Kelvin, connect the three thermoelectric effects (Seebeck, Peltier, and Thomson)
• The first Kelvin relationship relates the Seebeck coefficient ($\alpha$) to the Peltier coefficient ($\Pi$):
  • $\Pi = \alpha T$, where $T$ is the absolute temperature
• The second Kelvin relationship relates the Thomson coefficient ($\mu$) to the temperature dependence of the Seebeck coefficient:
  • $\mu = T \frac{d\alpha}{dT}$
• These relationships demonstrate the interdependence of the thermoelectric effects and provide a framework for analyzing thermoelectric materials and devices
• Kelvin relationships are crucial for understanding the efficiency and performance of thermoelectric systems
• The relationships also help in determining the optimal operating conditions for thermoelectric devices

Mathematical Formulations and Derivations

• The mathematical formulation of the Thomson effect is given by the heat balance equation:
  • $\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (\kappa \nabla T) - \mu \mathbf{J} \cdot \nabla T$
  • where $\rho$ is the density, $c_p$ is the specific heat capacity, $\kappa$ is the thermal conductivity, $\mathbf{J}$ is the electric current density, and $\mu$ is the Thomson coefficient
• The derivation of the Kelvin relationships involves the application of thermodynamic principles, such as the first and second laws of thermodynamics
• The Onsager reciprocal relations, which describe the coupling between different transport processes, play a crucial role in the derivation of the Kelvin relationships
• The efficiency of a thermoelectric device can be expressed in terms of the figure of merit (ZT):
  • $ZT = \frac{\alpha^2 \sigma T}{\kappa}$, where $\alpha$ is the Seebeck coefficient, $\sigma$ is the electrical conductivity, and $\kappa$ is the thermal conductivity
• The maximum efficiency of a thermoelectric generator is given by the Carnot efficiency multiplied by a factor that depends on the figure of merit:
  • $\eta_{max} = \frac{T_h - T_c}{T_h} \frac{\sqrt{1 + ZT} - 1}{\sqrt{1 + ZT} + \frac{T_c}{T_h}}$, where $T_h$ and $T_c$ are the hot and cold side temperatures, respectively

Experimental Observations and Measurements

• Experimental techniques for measuring the Thomson effect include:
  • Thermoelectric power measurements, which involve measuring the voltage generated across a sample when a temperature gradient is applied
  • Thermoelectric cooling measurements, which involve measuring the temperature change of a sample when an electric current is passed through it
• The Seebeck coefficient can be measured using a setup consisting of two thermocouples connected to the sample, with one end maintained at a reference temperature
• The Peltier coefficient can be determined by measuring the heat absorbed or emitted at the junction of two materials when an electric current is passed through it
• The thermal conductivity of a material can be measured using techniques such as the laser flash method or the 3-omega method
• Accurate measurement of thermoelectric properties is crucial for characterizing materials and optimizing device performance
• Experimental data can be used to validate theoretical models and guide the development of new thermoelectric materials

Applications in Thermoelectric Devices

• Thermoelectric devices have a wide range of applications, including:
  • Power generation (thermoelectric generators) from waste heat (automotive exhaust, industrial processes)
  • Solid-state cooling (thermoelectric coolers) for electronic components, scientific instruments, and portable refrigerators
  • Temperature sensing and control (thermocouples, thermostats)
• Thermoelectric generators can be used for remote power generation in space missions, oil and gas pipelines, and wireless sensor networks
• Thermoelectric coolers offer precise temperature control and are used in applications such as laser diode cooling, DNA amplification, and infrared detectors
• Thermoelectric devices have advantages such as solid-state operation, no moving parts, compact size, and high reliability
• The efficiency of thermoelectric devices depends on the figure of merit (ZT) of the materials used, as well as the operating conditions (temperature gradient, current)

Limitations and Challenges

• The efficiency of thermoelectric devices is currently lower than that of conventional power generation and refrigeration systems
  • This is due to the relatively low figure of merit (ZT) of most thermoelectric materials
• The figure of merit is limited by the interdependence of the thermoelectric properties (Seebeck coefficient, electrical conductivity, and thermal conductivity)
  • Improving one property often adversely affects the others
• Thermoelectric materials often contain rare or expensive elements (tellurium, germanium), which limits their widespread adoption
• The performance of thermoelectric devices can degrade over time due to material stability issues, such as oxidation or sublimation
• Thermal management is a challenge in thermoelectric devices, as efficient heat transfer is required to maintain the temperature gradient
• Contact resistance between the thermoelectric materials and the electrodes can reduce device efficiency
• Optimizing the geometry and configuration of thermoelectric devices for specific applications can be complex and time-consuming

Recent Advances and Future Directions

• Nanostructured thermoelectric materials (quantum dots, superlattices, nanowires) have shown promise in enhancing the figure of merit by reducing thermal conductivity while maintaining electrical conductivity
• Novel materials, such as skutterudites, clathrates, and half-Heusler alloys, are being investigated for their high thermoelectric performance
• Strategies for band structure engineering, such as resonant doping and band convergence, are being explored to improve the power factor ($\alpha^2 \sigma$)
• Phonon engineering techniques, such as phonon scattering and coherent phonon transport, are being studied to reduce thermal conductivity without compromising electrical properties
• Flexible and wearable thermoelectric devices are being developed for applications such as body heat harvesting and personalized temperature control
• Computational methods, such as density functional theory (DFT) and molecular dynamics (MD) simulations, are being used to predict and optimize thermoelectric properties
• Machine learning and high-throughput screening are being employed to accelerate the discovery and design of new thermoelectric materials
• Hybrid thermoelectric systems, which combine thermoelectric devices with other energy conversion technologies (photovoltaics, thermophotovoltaics), are being investigated for improved overall efficiency