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 mC$dT/dt$ = hS$Tth - T$ is rearranged into the standard form dT/dt = -$hS/mC$T + $hS/mC$Tth.

Example: This is of the form y' = ay + b, where a = -hS/mC and b = $hS/mC$Tth.

The solution consists of two parts:

1. A particular solution: T = Tthermostat
2. A general solution: T(t) = 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.

Thermal Law of Newton in Terminal Physics

This page introduces the loi phénoménologique de Newton (Newton's law of cooling) in the context of terminal-level physics. It outlines the scenario where the law applies and derives the governing differential equation.

The law considers a homogeneous body at temperature T in a constant temperature environment Tthermostat. The thermal power transfer causes the body's temperature to change from T(t) to T$t + Δt$ over a small time interval.

Definition: The internal energy change is given by ΔU = mC$T(t + Δt) - T(t)$, where m is mass and C is specific heat capacity.

An energy balance shows that the heat transfer Q equals the internal energy change ΔU (assuming no work is done). Taking the limit as Δt approaches zero leads to the differential form of Newton's cooling law.

Highlight: The key equation is mC$dT/dt$ = hS$Tth - T$, where h is the heat transfer coefficient and S is the surface area.

This first-order differential equation describes the rate of temperature change of the body. It forms the basis for analyzing many thermal systems and cooling processes.

Vocabulary:

• Puissance thermique: Thermal power
• Bilan d'énergie: Energy balance
• Équation différentielle: Differential equation