Equation of rate of flow of heat| Definition and Examples

The equation of flow of heat represents Fourier’s Law of Heat Conduction, describing the heat transfer through a solid material. It states that the rate of heat flow is directly proportional to the thermal conductivity of the material, the cross-sectional area through which heat is flowing, and the temperature gradient across the material.
The rate of flow of heat, often denoted as Q˙(pronounced as “dot Q”), is expressed by the following equation:

Q˙​=−kdT/dx

where:

• Q˙ is the heat flow rate (in watts or joules per second),
• k is the thermal conductivity of the material through which heat is flowing (in watts per meter-kelvin),
• A is the cross-sectional area through which heat is flowing (in square meters),
• dT/dx is the temperature gradient along the direction of heat flow (in kelvins per meter).

The negative sign indicates that heat flows from regions of higher temperature to lower temperature.

Examples:

1. Metal Rod Conducting Heat:
   • Scenario: Consider a metal rod of length L with a cross-sectional area A. One end of the rod is in contact with a hot reservoir at temperature T1, and the other end is in contact with a cold reservoir at temperature T2.
   • Application of Equation: Using the heat flow equation, you can determine the rate at which heat flows through the rod. The temperature gradient dT/dx is (T1−T2)/L​.
2. Insulating Material in a Wall:
   • Scenario: In a building, there’s an insulating material with thermal conductivity kk between the inner and outer walls. The inner wall is at a higher temperature T1, and the outer wall is at a lower temperature T2​.
   • Application of Equation: The heat flow equation helps in calculating the rate of heat transfer through the insulating material, crucial for maintaining a comfortable temperature inside the building.

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