Heat guide
Newton's Law of Cooling
Newton’s law of cooling models convective heat transfer as proportional to the temperature difference between a surface and its surroundings.
The convection relation
The basic heat-transfer relation is Q̇ = hA(T - T∞). For an object whose internal temperature can be treated as uniform, this combines with energy storage mc dT/dt to produce an exponential temperature response.
The lumped-capacitance solution
Under the lumped assumption, T(t) - T∞ = [T0 - T∞] exp[-hAt/(mc)]. The time constant is τ = mc/(hA).
When the object can be treated as uniform
The lumped model works best when internal conduction is much faster than heat transfer across the surface. The Biot number is commonly used to assess this assumption.
Cooling and heating use the same form
The model approaches the ambient temperature from either direction. A hot object cools toward ambient, while a cold object warms toward ambient.
Real systems may change h over time
Natural convection, radiation, evaporation and changing fluid flow can make the effective heat-transfer coefficient vary as temperatures change.
The law describes a first-order temperature approach
Newton’s law of cooling assumes that the rate of heat loss is proportional to the temperature difference between an object and its surroundings. When the object can be treated as nearly uniform in temperature and the heat-transfer coefficient remains approximately constant, the temperature difference decays exponentially with time.
The model is widely useful because it turns a complicated cooling process into a simple time constant, but it depends on the lumped-temperature assumption being reasonable.
The time constant combines storage and heat transfer
For a lumped object, the characteristic time constant can be written roughly as ρVc/(hA), where ρVc represents thermal capacitance and hA represents convective conductance. Larger heat capacity slows the response, while larger area or stronger convection speeds it up.
This relationship explains why a massive object cools more slowly than a thin lightweight object under otherwise comparable conditions.
Internal temperature gradients can break the model
If heat cannot conduct through the object quickly compared with the rate at which the surface exchanges heat, the interior and surface temperatures can differ substantially. The lumped model then loses accuracy.
The Biot number is commonly used to judge whether internal conduction resistance is small enough for a nearly uniform-temperature approximation. More detailed transient conduction solutions are needed when that condition is not satisfied.
The time constant connects cooling rate to thermal mass and heat transfer
For a lumped body with approximately uniform internal temperature, combining Newton’s law of cooling with an energy balance gives an exponential temperature response. The characteristic time constant is proportional to the body’s thermal capacitance, mcp, and inversely proportional to hA. A larger thermal mass therefore changes temperature more slowly, while a larger heat-transfer coefficient or exposed area speeds the response.
This time constant is useful because it describes the shape of the cooling curve without requiring a new calculation at every instant. After one time constant, the remaining temperature difference from the surroundings is about 37% of its initial value in the ideal model.
The lumped model requires small internal temperature gradients
Newton’s law of cooling is often paired with a lumped-capacitance assumption in which the object is treated as one temperature. That approximation works best when internal conduction is fast compared with heat transfer at the surface. The Biot number is commonly used to judge whether this condition is reasonable.
Thick, poorly conducting objects can develop strong internal gradients, so their centre and surface cool at different rates. In those cases a transient conduction solution is more appropriate than a single-temperature exponential model.