# Newton's Law of Cooling

on . Posted in Thermodynamics

Newton's law of cooling calculates the temperature of an object as it loses heat at any given time.  The law states that the rate of heat loss of a body is directly porportional to the difference in the temperature between the body and its surrounding provided the temperature difference is small and the nature of radiating surface remains the same.  The negative sign in the equation indicates that the object's temperature decreases over time.

Newton's law of cooling is based on several assumptions, such as the object and its surroundings having uniform temperatures, and the heat transfer occurring mainly through conduction or convection.  While it is an approximation and may not fully account for all the complexities of heat transfer, it provides a useful model for understanding and predicting the cooling behavior of objects in practice.

The law has applications in various fields, including physics, engineering, and thermodynamics, particularly in situations involving heat transfer and temperature regulation, such as cooling of electronic devices, heating and cooling systems, and climate control.

## Newton's Law of Cooling Formula

$$\large{ \frac { dT } { dt } = - \lambda \; \left( T_t - T_a \right ) }$$
Symbol English Metric
$$\large{ dT }$$ = temperature change in object $$F$$ $$C$$
$$\large{ dt }$$ = change in time $$\large{sec}$$ $$\large{s }$$
$$\large{ \lambda }$$ (Greek symbol lambda) = is a constant (decay constant) $$\large{sec^{-1}}$$ $$\large{s^{-1}}$$
$$\large{ T_t }$$ = temperature at time t $$F$$ $$C$$
$$\large{ T_a }$$ = ambient temperature (temperature of environment) $$F$$ $$C$$

## Newton's Law of Cooling Formula Solving $$T_t$$

$$\large{ T_t = T_a + \left( T_0 - T_a \right ) e ^{ -k t} }$$
Symbol English Metric
$$\large{ T_t }$$ = temperature at time t $$F$$ $$C$$
$$\large{ T_a }$$ = ambient temperature (temperature of environment) $$F$$ $$C$$
$$\large{ T_0 }$$ = initial temperature of object $$F$$ $$C$$
$$\large{ e }$$ = exponential $$e$$ implies a continious rate of cooling $$F$$ $$C$$
$$\large{ k }$$ = cooling constant temperature of object $$F$$ $$C$$
$$\large{ t }$$ = time taken to cool $$sec$$ $$s$$ 