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title: Heat Conduction through Plain Wall
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# Heat Conduction through Plain Wall
Unit 1, Lecture 6

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## Heat Conduction through Plain Wall

- Course: Heat and Mass Transfer (DI05019071)
- Unit 1: Conduction
- Lecture 6: Heat Conduction through Plain Wall

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Welcome. Today we will apply Fourier's Law to a simple geometry: the plane wall.
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## Lecture Agenda

- 1. 1D Steady State Conduction
- 2. The Plane Wall Model
- 3. Applying Fourier's Law
- 4. Temperature Distribution
- 5. Thermal Resistance Concept
- 6. Electrical Analogy

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Here is our agenda. We'll start with the basic model and end with a very useful analogy to electrical circuits.
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## 1D Steady State Conduction

- Assumptions for our model:
- - Steady State: Temperature does not change with time.
- - One-Dimensional: Heat flows in only one direction (x).
- - No Internal Heat Generation.
- - Constant Thermal Conductivity (k).

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These assumptions simplify the general heat conduction equation significantly, allowing us to focus on the x-direction only.
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## The Plane Wall Model

- Consider a plane wall of thickness L.
- Area A is perpendicular to the direction of heat transfer.
- Temperatures at surfaces are T1 (at x=0) and T2 (at x=L).
- Assume T1 > T2, so heat flows in the positive x direction.

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Visualize a typical building wall or a metal plate. Heat moves from the hotter side to the colder side.
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## Applying Fourier's Law

- Fourier's Law: qx = -k * A * (dT/dx)
- Since conditions are steady, the heat transfer rate (qx) is constant throughout the wall.
- Separating variables: qx * dx = -k * A * dT

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We start with Fourier's Law. Because no heat is stored or generated, the heat entering equals the heat leaving.
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## Temperature Distribution (Derivation)

- Integrate from x=0 to x=L, and T=T1 to T=T2:
- qx * ∫(0 to L) dx = -k * A * ∫(T1 to T2) dT
- qx * L = -k * A * (T2 - T1)
- qx = k * A * (T1 - T2) / L

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Integrating over the wall thickness gives us the total heat transfer rate in terms of the surface temperatures.
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## Temperature Profile Characteristics

- The temperature distribution T(x) can be found by integrating to an arbitrary x:
- T(x) = T1 - (T1 - T2) * (x / L)
- The profile is linear.
- Slope dT/dx is constant.

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For a plane wall with constant thermal conductivity, the temperature drops linearly from the hot side to the cold side.
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## Thermal Resistance Concept

- Rearranging the heat rate equation:
- qx = (T1 - T2) / (L / kA)
- Let R_th = L / (k * A)
- R_th is the thermal resistance to conduction.
- Units: °C/W or K/W

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This is a very powerful concept. We can group the geometric and material properties into a single resistance term.
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## Electrical Analogy

- Ohm's Law: I = ΔV / R_elec
- Thermal Equivalent: qx = ΔT / R_th
- Current (I) <-> Heat Rate (qx)
- Voltage drop (ΔV) <-> Temperature drop (ΔT)
- Electrical Resistance (R_elec) <-> Thermal Resistance (R_th)

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Just like voltage drives electrical current through a resistor, a temperature difference drives heat through a thermal resistance.
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## Example Problem Introduction

- Problem: A glass window (k = 0.78 W/m·K) is 2 m by 1.5 m and 5 mm thick.
- Temperatures at the inner and outer surfaces are 15°C and -5°C.
- Find the rate of heat loss through the window.

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Let's apply our new formula to a real-world example: heat loss through a window in winter.
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## Example Problem Solution

- Area A = 2 * 1.5 = 3 m²
- Thickness L = 0.005 m
- ΔT = 15 - (-5) = 20 K
- R_th = L / (k*A) = 0.005 / (0.78 * 3) = 0.00214 K/W
- q = ΔT / R_th = 20 / 0.00214 = 9346 W = 9.35 kW

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We first calculate the area, then the thermal resistance, and finally the heat rate. The window loses over 9 kilowatts of heat.
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## Summary

- 1D steady-state conduction in a plane wall yields a linear temperature profile.
- Heat transfer rate is proportional to ΔT and area, inversely proportional to thickness.
- Thermal resistance (R_th = L/kA) allows us to use an electrical analogy for heat transfer.

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To summarize, the plane wall is our simplest conduction model, and the thermal resistance concept will be vital going forward.
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## Next Lecture Preview

- Topic: Heat Conduction through Composite Walls
- What happens when a wall is made of multiple layers of different materials?
- We will use the electrical analogy to build series thermal circuits.

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Next time, we extend this to composite walls, like the insulated walls of a house, using the series resistance concept.
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