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title: Heat Conduction through Plane and Composite Walls
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# Heat Conduction through Plane and Composite Walls
Unit 1, Lecture 4

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## Heat Conduction through Plane and Composite Walls

- Course: Heat and Mass Transfer (DI05019071)
- Unit 1: Conduction
- Lecture 4: Heat Conduction through Plane and Composite Walls

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Welcome to week 2. Today we apply the math we learned last week to practical engineering problems: heat flow through walls, particularly walls made of multiple different materials.
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## Lecture Agenda

- 1. Temperature Profile in a Plane Wall
- 2. Thermal Resistance Review
- 3. Concept of Composite Walls
- 4. Thermal Circuit for Composite Walls
- 5. Convection at Boundaries
- 6. Overall Heat Transfer Coefficient (U)
- 7. Thermal Contact Resistance

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We will start by proving why the temperature profile in a simple wall is linear, then expand our analysis to multiple layers and incorporate convective boundaries.
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## Conduction in a Plane Wall

- Governing Equation: $\frac{d^2 T}{dx^2} = 0$
- Integrating once: $\frac{dT}{dx} = C_1$
- Integrating twice: $T(x) = C_1 x + C_2$
- Boundary Conditions:
- 1. At $x = 0$, $T = T_1$
- 2. At $x = L$, $T = T_2$

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As we established last time, the second derivative of temperature is zero. Integrating this twice yields a linear equation for temperature as a function of position x.
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## Temperature Profile

- Applying boundary conditions to $T(x) = C_1 x + C_2$:
- - From BC 1: $C_2 = T_1$
- - From BC 2: $T_2 = C_1 L + T_1 \Rightarrow C_1 = \frac{T_2 - T_1}{L}$
- Resulting Temperature Profile:
- $$T(x) = T_1 - (T_1 - T_2)\frac{x}{L}$$
- Conclusion: The temperature distribution in a plane wall at steady state is linear.

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The constants of integration are found using the surface temperatures. The result proves that temperature drops linearly across a homogeneous plane wall.
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## Heat Rate through Plane Wall

- Using Fourier's Law: $Q = -kA \frac{dT}{dx}$
- Since $\frac{dT}{dx} = \frac{T_2 - T_1}{L}$, then:
- $$Q = -kA \frac{T_2 - T_1}{L} = kA \frac{T_1 - T_2}{L}$$
- In terms of Thermal Resistance ($R_{th} = L/kA$):
- $$Q = \frac{T_1 - T_2}{R_{th}}$$

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We recover the resistance formula. This resistance form is what we will use heavily for composite walls.
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## Composite Walls

- Most real-world structures are composite (multiple layers in series).
- Example: A building wall consisting of brick, insulation, and plasterboard.
- Key Principle: At steady state, the rate of heat transfer ($Q$) is the SAME through all layers in series.
- $$Q = Q_A = Q_B = Q_C$$

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Just like current in a series electrical circuit is constant everywhere, the heat flow rate through steady-state series layers is constant.
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## Thermal Circuit for Composite Walls

- For a 3-layer wall (A, B, C) with interface temperatures $T_1, T_2, T_3, T_4$:
- Total Resistance $R_{total} = R_A + R_B + R_C$
- $$R_{total} = \frac{L_A}{k_A A} + \frac{L_B}{k_B A} + \frac{L_C}{k_C A}$$
- $$Q = \frac{T_1 - T_4}{R_{total}}$$

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By identifying the total temperature drop across the entire wall and dividing by the total series resistance, we can find the heat flow rate without needing to know the intermediate interface temperatures.
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## Adding Convection Boundaries

- Surfaces are usually exposed to fluids (air, water) at temperatures $T_{\infty 1}$ and $T_{\infty 2}$.
- Convection Heat Transfer (Newton's Law of Cooling):
- $$Q = hA(T_{surface} - T_{fluid})$$
- Convection Resistance:
- $$R_{conv} = \frac{1}{hA}$$
- Where $h$ is the convective heat transfer coefficient ($W/m^2\cdot K$).

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In reality, we rarely know the exact surface temperature of a wall. We know the indoor air and outdoor air temperatures. So we must include convection resistances at the boundaries.
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## Comprehensive Thermal Circuit

- Wall with fluid on both sides:
- $R_{total} = R_{conv,in} + R_{wall} + R_{conv,out}$
- $$R_{total} = \frac{1}{h_{in}A} + \frac{L}{kA} + \frac{1}{h_{out}A}$$
- $$Q = \frac{T_{\infty,in} - T_{\infty,out}}{R_{total}}$$

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This is the complete thermal circuit for a plane wall exposed to convection on both sides. This is exactly how building heat loss is calculated.
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## Overall Heat Transfer Coefficient (U)

- It is convenient to express heat transfer as:
- $$Q = UA \Delta T_{overall}$$
- Where $U$ is the Overall Heat Transfer Coefficient ($W/m^2\cdot K$).
- Comparing with $Q = \Delta T / R_{total}$, we get:
- $$UA = \frac{1}{R_{total}} \Rightarrow U = \frac{1}{A \cdot R_{total}}$$
- For a plane wall: $U = \frac{1}{\frac{1}{h_{in}} + \frac{L}{k} + \frac{1}{h_{out}}}$

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The U-value is heavily used in the construction and HVAC industries. A low U-value means a highly insulated wall. It's essentially the inverse of total thermal resistance per unit area.
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## Thermal Contact Resistance

- When two solid surfaces are pressed together, they only touch at discrete high spots due to microscopic roughness.
- The gaps are filled with air (a poor conductor).
- This creates a resistance to heat flow at the interface, called Thermal Contact Resistance ($R_{t,c}$).
- Results in a sudden temperature drop ($\Delta T_{interface}$) across the junction.

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We usually assume perfect contact (no temperature drop at the interface). But in reality, surface roughness creates air pockets. Since air is an insulator, this creates a contact resistance.
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## Summary

- Temperature profile in a 1D steady-state plane wall is linear.
- Composite walls are analyzed using series thermal resistance networks.
- Convection at boundaries adds a resistance of $1/(hA)$.
- The Overall Heat Transfer Coefficient (U) represents the total thermal conductance.
- Contact resistance accounts for imperfect mating of surfaces.

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To summarize, the electrical analogy makes analyzing any plane wall system straightforward. Just map the resistances, calculate total R, and apply the temperature difference.
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## Next Lecture Preview

- Topic: Critical radius of insulation for a cylinder
- Key questions to ponder:
- - Does adding insulation always reduce heat loss?
- - How is conduction through a cylinder different from a plane wall?
- - What happens to the surface area as you add layers to a pipe?

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Next lecture, we shift to cylindrical coordinates. We will discover a very counter-intuitive phenomenon: sometimes adding insulation to a pipe actually increases heat loss!
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