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title: Heat Conduction through Composite Walls
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# Heat Conduction through Composite Walls
Unit 1, Lecture 7

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## Heat Conduction through Composite Walls

- Course: Heat and Mass Transfer (DI05019071)
- Unit 1: Conduction
- Lecture 7: Heat Conduction through Composite Walls

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Welcome back. Today we'll look at walls made of multiple materials, known as composite walls.
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## Lecture Agenda

- 1. Recap of Thermal Resistance
- 2. Introduction to Composite Walls
- 3. Series Thermal Circuit
- 4. Overall Heat Transfer Rate Equation
- 5. Temperature Drop Across Layers
- 6. Thermal Contact Resistance

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We'll build on the electrical analogy to analyze heat flow through layers in series, and discuss contact resistance.
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## Introduction to Composite Walls

- Most practical walls are not made of a single material.
- Examples: house walls (brick, insulation, wood), furnace walls (firebrick, insulating brick, steel).
- Heat must flow sequentially through each layer.
- This is equivalent to electrical resistors in series.

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Real-world structures are layered to combine strength, insulation, and aesthetics.
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## Series Thermal Circuit

- Consider a wall with 3 layers: A, B, and C.
- Thicknesses: LA, LB, LC. Conductivities: kA, kB, kC.
- The same heat (qx) flows through all layers.
- Total Resistance R_total = R_A + R_B + R_C
- R_total = (LA/kAA) + (LB/kBA) + (LC/kCA)

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Because the heat has nowhere else to go, it flows through each layer in turn. We just add the resistances.
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## Overall Heat Transfer Rate

- Using the overall temperature difference:
- qx = (T_inner - T_outer) / R_total
- qx = (T1 - T4) / [ (LA/kAA) + (LB/kBA) + (LC/kCA) ]
- This assumes perfect thermal contact between layers.

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We only need the temperatures on the extreme outside and inside surfaces to find the heat rate, if we know the layer properties.
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## Temperature Drop Across Layers

- Once qx is known, intermediate temperatures can be found.
- qx = (T1 - T2) / R_A  =>  T2 = T1 - qx * R_A
- qx = (T2 - T3) / R_B  =>  T3 = T2 - qx * R_B
- Larger resistance = larger temperature drop.

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The temperature drops fastest in the layers with the highest thermal resistance—that is, the best insulators.
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## Thermal Contact Resistance

- In reality, surfaces are rough.
- When two solid surfaces are pressed together, they touch only at high spots.
- Gaps are filled with air, which is a poor conductor.
- This creates an additional resistance at the interface: Contact Resistance (R_t,c).

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Surfaces are never perfectly smooth. The air trapped between layers acts as an insulator, creating a sudden temperature drop at the interface.
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## Accounting for Contact Resistance

- Contact resistance R_t,c = 1 / (hc * A)
- where hc is the contact conductance.
- It is simply added to the series circuit.
- R_total = R_A + R_t,c + R_B

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We treat contact resistance just like another layer in our series circuit.
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## Reducing Contact Resistance

- How to minimize it:
- 1. Increase joint pressure.
- 2. Decrease surface roughness (polish).
- 3. Use thermal grease or conducting paste to displace air in gaps.
- 4. Insert a soft metallic foil (e.g., tin, silver).

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This is why you use thermal paste when installing a CPU cooler—it displaces the air and massively reduces contact resistance.
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## Example Problem Introduction

- A furnace wall consists of 200 mm firebrick (k=1.2 W/mK) and 50 mm insulating brick (k=0.15 W/mK).
- Inner temperature is 1000°C, outer is 40°C.
- Calculate the heat loss per unit area.

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Let's calculate the heat loss through a composite furnace wall.
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## Example Problem Solution

- R_firebrick = 0.2 / 1.2 = 0.167 (K m²/W)
- R_insulation = 0.05 / 0.15 = 0.333 (K m²/W)
- R_total = 0.167 + 0.333 = 0.500 (K m²/W)
- q/A = (1000 - 40) / 0.500 = 1920 W/m²

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Notice that the thinner insulating brick has twice the thermal resistance of the thicker firebrick.
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## Summary

- Composite walls are modeled as series thermal circuits.
- Total resistance is the sum of individual layer resistances.
- Intermediate temperatures depend on the resistance distribution.
- Contact resistance must be considered for rough interfaces.

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The series resistance model is an incredibly useful tool for evaluating real-world composite structures.
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## Next Lecture Preview

- Topic: Heat Conduction through Cylinders
- How does heat transfer change when the area is not constant?
- We'll explore radial heat conduction in pipes and tubes.

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Next class, we'll apply these concepts to cylindrical geometries, which are crucial for piping systems.
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