Another Simple Heat Loss Question

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SUMMARY

This discussion focuses on accurately sizing a heater to heat a glass of water from an initial temperature (Ti) to a final temperature (Tf) over a specified time period, such as one hour. The user notes that heat loss from the glass is zero at Ti and increases to Y watts at Tf, with the assumption that heat loss increases linearly. The energy required to heat the water without heat loss is denoted as X watts. The challenge is to find a method to size the heater more accurately than simply using the sum of Y and X watts, especially when the relationship between X and Y varies significantly.

PREREQUISITES
  • Understanding of thermal dynamics and heat transfer principles
  • Familiarity with wattage calculations for heating elements
  • Knowledge of linear functions and their application in real-world scenarios
  • Basic concepts of heat loss in materials, particularly glass
NEXT STEPS
  • Research methods for calculating heat loss in materials, focusing on glass
  • Explore thermal conductivity and its impact on heating efficiency
  • Learn about PID controllers for maintaining temperature in heating applications
  • Investigate the use of thermal insulation to minimize heat loss
USEFUL FOR

Engineers, physicists, and anyone involved in thermal management or heating system design will benefit from this discussion, particularly those looking to optimize heater sizing for specific applications.

TheH
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I'm trying to size (in watts) a heater to heat a glass of water from ambient (Ti) to a certain temperature (Tf) in a given time period (say, 1 hour).

The heat-loss from the glass at Ti is 0 watts.
The heat-loss from the glass at Tf is Y watts.
I'm assuming that the heat loss increases linearly between Ti and Tf (I hope this is reasonable).
The energy needed (assuming no heat-loss) to heat the water from Ti->Tf in 1 hour is X watts.

Naturally, I can size the heater to be (Y + X) watts. However, this will heat the water up faster than necessary because initially there will be no heat-loss from the glass.

Is there a clever way to more accurately size this heater?

Thanks for your time.
 
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Note that here it is possible for X to be much larger than Y (when I want to heat the water up rapidly) and Y to be much larger than X (when I want to heat the water up slowly).
 

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