Solution to Newton's Cooling Law
This page presents the solution to the differential equation derived from the loi phénoménologique de Newton. It shows how to solve the equation and interpret the results.
The differential equation mCdT/dt = hSTth−T is rearranged into the standard form dT/dt = -hS/mCT + hS/mCTth.
Example: This is of the form y' = ay + b, where a = -hS/mC and b = hS/mCTth.
The solution consists of two parts:
- A particular solution: T = Tthermostat
- A general solution: Tt = Ae^−t/τ + Tth, where τ = mC/hS is the characteristic time constant
Highlight: The solution shows that the temperature approaches the thermostat temperature exponentially over time.
The limits of the solution are discussed, showing how the temperature homogenizes with the surroundings over time.
Vocabulary:
- Solution particulière: Particular solution
- Solution générale: General solution
- Refroidissement: Cooling
This mathematical treatment of Newton's cooling law provides a powerful tool for analyzing and predicting temperature changes in various thermal systems, from simple cooling processes to complex heat transfer scenarios in engineering and physics applications.
