A Differential Equation for Heat Transfer According to Newton's Law of Cooling
A Differential Equation for Heat Transfer According to Newton's Law of Cooling
Let be the temperature of a building (with neither heat nor air conditioning running) at time and let be the temperature of the surrounding air. Newton's law of cooling states that
u(t)
t
T(t)
du
dt
where is a constant independent of time.
k
In this example, is plotted in red; is the initial temperature of the air surrounding the building. The value of depends on a number of factors including the insulation of the building. In this example, you can modify the proportionality constant and mean temperature.
T(t)=T(0)+15sin(2πt)
T(0)
k
Five specific solutions around the initial temperature are plotted as solid blue curves through the dashed blue lines that represent the direction fields.