Heat Transfer Key equations for Process Engineers Process Design

Heat Transfer – Conduction

Heat conduction is the process of transfer of heat energy from one point to another within a solid material or between different materials in physical contact.

Table of Content:
1. What is Heat Conduction?
2. Fourier's Law of Heat Conduction
3. Heat Transfer Through a Wall
4. Heat Conduction Through a Hollow Cylinder

What is Heat Conduction?

Conductive heat transfer occurs when heat is transferred through a material without any motion of the material itself. The heat is transferred from molecule to molecule due to lattice vibration and collision of molecules, transferring heat energy from one molecule to the next. This mode of heat transfer takes place in a stagnant medium.

Conduction is the most common mode of heat transfer in solids, and it occurs when there is a temperature gradient within a solid material. The heat flows from regions of high temperature to regions of low temperature to equalize the temperature throughout the material.

The rate of heat transfer through a material by conduction is dependent on the thermal conductivity of the material, the temperature difference across the material, and the area through which heat is being transferred. In general, materials with high thermal conductivity, such as metals, conduct heat more quickly than materials with low thermal conductivity, such as plastics.

Conductive heat transfer can be modeled using equations like Fourier's law of heat conduction.

Fourier's Law of Heat Conduction

Conduction of heat in a solid wall is expressed using Fourier's equation for heat conduction,
fourier law of heat conduction formula
T: Temperature at a point in the wall
θ: Time
k: Thermal Conductivity of wall material
A: Cross-sectional area of the element around the point
x: Distance perpendicular to the area element

For steady-state heat transfer, this equation becomes,

fourier's law of heat conduction

Q: Heat transfer rate
A: Area of the wall

This equation can be further developed to express temperature profiles in various geometries with one-dimensional heat transfer.

Heat Transfer Through a Wall

Let's take a case of heat transfer across a flat wall as shown in the figure below.

heat transfer through a wall

Assuming that thermal conductivity 'k' is independent of temperature and location and A is independent of location, as is the case for a solid wall with a constant cross-sectional area.

Heat conduction through a wall

Similarly, the case of conductive heat transfer through a composite cylinder can be studied here.

Heat Conduction Through a Hollow Cylinder

For the radial geometry of a hollow cylinder, the following equation expresses the heat transfer rate.

Heat conduction through hollow cylinder

Heat conduction through a hollow cylinder

The integral of this equation from inner radius r1 to outer radius r2 represents the total heat transfer across the cylindrical wall.

Heat conduction through hollow cylinder formula

N = length of the hollow cylinder

T1 and T2 are the inner and outer wall temperatures of the hollow cylinder.

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