Derivation of Newton’s Law of Cooling
Newton’s Law of Cooling formula can be derived using the solution of the differential equation. Let a body of mass m, with specific heat capacity s, be at temperature T2 and T1 is the temperature of the surroundings.
If the temperature falls by a small amount dT2 in time dt, then the amount of heat lost is,
dQ = ms dT2
Rate of loss of heat is given by,
dQ/dt = ms (dT2/dt)
According to Newton’s law of cooling,
– dQ/dt = k(T2 – T1)
Comparing the above equation
– ms (dT2/dt) = k (T2 – T1)
dT2/(T2–T1) = – (k / ms) dt
dT2 /(T2 – T1) = – Kdt
where, K = k/m s
Integrating the above equation
loge (T2 – T1) = – K t + c
T2 = T1 + C’ e–Kt
where, C’ = ec
The relation between the drop in temperature of the body and the time is shown using the cooling graph. The slope of this graph shows the rate of fall of the temperature.
The cooling curve is a graph that shows the relationship between body temperature and time. The rate of temperature fall is determined by the slope of the tangent to the curve at any point. The image added below shows the Temperature drop and time relation.
In general,
T(t) = TA+(TH-TA)e-kt
where
T(t) is the Temperature at time t
TA is the Ambient temperature or temp of the surroundings
TH is the temperature of the hot object
k is the positive constant and t is the time
Newton’s Law of Cooling
Newton’s Law of Cooling is the fundamental law that describes the rate of heat transfer by a body to its surrounding through radiation. This law state that the rate at which the body radiate heats is directly proportional to the difference in the temperature of the body from its surrounding, given that the difference in temperature is low. i.e. the higher the difference between the temperature of the body and its surrounding the more heat is lost and the lower the temperature the less heat is lost. Newton’s Law of Cooling is a special case of Stefan-Boltzmann’s Law.
In this article, we will learn about, Newton’s Law of Cooling, Newton’s Law of Cooling Formula, its Derivation, Examples, and others in detail.
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