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title: Thermal conductivity and thermal resistance
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# Thermal conductivity and thermal resistance
Unit 1, Lecture 2

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## Thermal conductivity and thermal resistance

- Course: Heat and Mass Transfer (DI05019071)
- Unit 1: Conduction
- Lecture 2: Thermal conductivity and thermal resistance

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Welcome back. Building on Fourier's law from our last session, today we will dive into the material property 'k' and introduce a concept that will simplify complex conduction problems: Thermal Resistance.
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## Lecture Agenda

- 1. Recap of Fourier's Law
- 2. Thermal Conductivity (k) Defined
- 3. Thermal Conductivity of Materials
- 4. Effect of Temperature on k
- 5. The Electrical Analogy
- 6. Concept of Thermal Resistance
- 7. Application to a Plane Wall

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Here is our agenda. We'll start by defining thermal conductivity properly, look at how it varies across materials, and then learn how to treat heat transfer problems like simple DC electrical circuits.
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## What is Thermal Conductivity (k)?

- From Fourier's Law: $k = \frac{Q/A}{-(dT/dx)}$
- Definition: Thermal conductivity is the amount of heat conducted per unit time across unit area and through unit thickness, when a temperature difference of unit degree is maintained.
- It is a thermophysical property of a material.
- High $k$ $\rightarrow$ Good heat conductor.
- Low $k$ $\rightarrow$ Good heat insulator.

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Thermal conductivity is a measure of a material's ability to conduct heat. It tells us how easily heat flows through a substance.
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## Units and Dimensions of k

- Unit of Heat Transfer Rate ($Q$): Watts (W) or J/s
- Unit of Area ($A$): $m^2$
- Unit of Temperature Gradient ($dT/dx$): K/m or °C/m
- Therefore, Unit of $k$:
- $$k = \frac{W}{m^2 \cdot (K/m)} = W/m\cdot K$$
- Dimensional Formula: $[M^1 L^1 T^{-3} \Theta^{-1}]$

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Always pay attention to units. In SI units, k is expressed in Watts per meter-Kelvin. The unit is the same whether we use Kelvin or degree Celsius for the temperature difference.
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## Thermal Conductivity of Different Materials

- Order of Thermal Conductivity:
- Pure Metals > Alloys > Non-metallic Solids > Liquids > Gases
- Examples ($W/m\cdot K$ at room temp):
- - Copper: ~400
- - Aluminum: ~237
- - Steel: ~15 to 50
- - Water: ~0.6
- - Air: ~0.026

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As you can see, metals are excellent conductors. Gases like air are very poor conductors, which is why air is often used as a trapping medium in insulators like fiberglass.
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## Why are Metals Good Conductors?

- In solids, heat is conducted by two mechanisms:
- 1. Lattice vibration ($k_l$)
- 2. Transport by free electrons ($k_e$)
- Total $k = k_l + k_e$
- In pure metals, free electrons are abundant, making $k_e$ dominant.
- In non-metals, there are no free electrons, so conduction is entirely due to lattice vibration ($k_l$).

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The abundance of free electrons in metals is the reason they are great conductors of both heat and electricity. The Wiedemann-Franz law actually relates electrical and thermal conductivity.
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## Temperature Dependence of k

- Thermal conductivity is not strictly constant; it varies with temperature.
- Generally modeled as a linear function: $k = k_0(1 + \beta T)$
- Where:
- - $k_0$: Conductivity at reference temp (e.g., $0^\circ C$)
- - $\beta$: Temperature coefficient of thermal conductivity
- For most pure metals, $k$ decreases with increasing $T$ ($\beta < 0$).
- For gases and insulating materials, $k$ increases with increasing $T$ ($\beta > 0$).

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While we often assume k is constant for simple problems, in real engineering applications with large temperature differences, we must account for its variation.
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## The Electrical Analogy

- Heat flow can be modeled similarly to electrical current flow.
- Ohm's Law (Electrical): $I = \frac{\Delta V}{R_e}$
- - $I$ = Current (Flow)
- - $\Delta V$ = Voltage Difference (Driving Potential)
- - $R_e$ = Electrical Resistance
- Fourier's Law (Thermal): $Q = \frac{\Delta T}{R_{th}}$
- - $Q$ = Heat Flow Rate (Flow)
- - $\Delta T$ = Temperature Difference (Driving Potential)
- - $R_{th}$ = Thermal Resistance

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This is one of the most powerful concepts in heat transfer. By mapping thermal problems to electrical circuits, we can solve complex composite wall problems using simple series and parallel circuit rules.
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## Concept of Thermal Resistance

- Rearranging Fourier's Law for a plane wall of thickness $L$:
- $$Q = kA \frac{T_1 - T_2}{L}$$
- $$Q = \frac{T_1 - T_2}{L / (kA)}$$
- Comparing with $Q = \frac{\Delta T}{R_{th}}$, we get:
- $$R_{th} = \frac{L}{kA}$$
- Thermal Resistance to Conduction ($R_{cond}$) for a plane wall.

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So, the thermal resistance of a plane wall depends directly on its thickness and inversely on its thermal conductivity and cross-sectional area. Thicker walls resist more; higher conductivity resists less.
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## Thermal Circuit for a Plane Wall

- Drawing a thermal circuit helps visualize the problem.
- Nodes represent Temperatures ($T_1, T_2$).
- Resistors represent Thermal Resistance ($R_{th}$).
- Current represents Heat Flow ($Q$).

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Always draw a thermal circuit before solving a conduction problem. It will prevent mistakes, especially when we start stacking multiple walls together.
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## Significance of Thermal Resistance

- - Simplifies complex heat transfer problems (e.g., composite walls, cylinders).
- - Allows analysis of heat transfer networks in series and parallel.
- - Identifies the 'bottleneck' in heat transfer (the layer with the highest resistance dictates the overall heat flow).
- - Essential for designing insulation systems.

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If you want to minimize heat loss, you add a layer with very high thermal resistance. The thermal circuit makes it immediately obvious how that layer affects the total heat transfer.
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## Summary

- Thermal conductivity ($k$) is a material property defining heat conduction capability.
- Metals have high $k$ due to free electrons; gases have low $k$.
- The electrical analogy maps Heat Flow ($Q$) to Current ($I$) and Temperature Difference ($\Delta T$) to Voltage ($\Delta V$).
- Conduction thermal resistance for a plane wall is $R = L / (kA)$.

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Remember the formula for thermal resistance. It is the building block for analyzing composite systems, which we will see soon.
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## Next Lecture Preview

- Topic: Derivation of General Heat Conduction Equation
- Key questions to ponder:
- - How do we model heat transfer in 3-dimensions?
- - What if the material is generating heat internally?
- - How is the general equation reduced to 1D steady state?

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Next lecture is highly analytical. We will derive the general heat conduction equation from first principles using an energy balance on a control volume.
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