Solution to the heat equation from a source

AI Thread Summary
The discussion focuses on solving the heat equation for a system where heat is transmitted from a source at a higher temperature to a reservoir at a lower initial temperature, with one end exposed to a fixed ambient temperature. A differential equation is proposed to model the heat flow, considering the effects of convection as heat transfers to the air. The scenario also explores variations in room temperature, such as when a heater is introduced. The complexities of heat transfer due to convection are acknowledged, emphasizing the need for precise modeling. Overall, the conversation highlights the importance of understanding heat flow dynamics in practical applications.
Adel Makram
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What is the best solution of the heat equation that described a transmission of heat from a source kept at certain temperature to a reservoir with an initial constant temperature (lower than the source) where its ends are not insulated from the surroundings and the surrounding is kept at a fixed temperature. For example, heating one end of a metallic rod while the other end is not insulated but exposed to the room of a constant temperature. What if the room temperature is allowed to rise like in case of a heater?
 
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Write down a differential equation for the heat flow in your setup, solve it.
Heat transfer to air can be complicated due to convection.
 

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