Newton's Law of Cooling

The rate at which an exposed body changes temperature through radiation is approximately proportional to the difference between the object's temperature and its surroundings, provided the difference is small, according to Newton's law of cooling.

The rate of heat loss from a body is directly proportional to the temperature differential between the body and its surroundings, according to Newton's law of cooling.

The rate, where a body that is exposed, changes the temperature via radiation is approximately proportional to the difference between the object’s temperature and its surroundings, and the provided difference is low. This is known as Newton's Law of cooling.

To define Newton's Law of cooling, the rate of heat loss from a body is directly proportional to the difference in body temperature, and its surroundings.

In a convective heat transfer, Newton's Law is followed for pumped fluid cooling or forced air, where the fluid properties do not strongly vary with the temperature. However, only approximately true for the buoyancy-driven convection, where the velocity of the flow increases with the difference in temperature. Finally, in the heat transfer case, by the thermal radiation, Newton's Law of cooling holds only for very slight temperature differences.

Newton's Law of cooling can be given by,

Q = h. A. (T_t – T_env)

Where, 

• Q = rate of heat transfer out of the body

• H = heat transfer coefficient

• A = heat transfer surface area

• T = temperature of the object's surface

• Tenv  = temperature of the environment

• Tt = time-dependent temperature

Formula 2:

 \[\frac {dT}{dt}\] = k (Tt – Ts) 

Where,

Tt = temperature of the body at time t and

Ts  = temperature of the surrounding,

k = Positive constant that depends on the area and nature of the surface of the body under consideration.

The Formula for Newton's Law of Cooling

The greater the temperature difference between the system and the surrounding environment, the faster heat is transmitted and the body temperature changes. The formula for Newton's law of cooling is,

T_s + (T_o – T_s) e^-kt = T(t)

Where,

t stands for time, and

T(t) is the temperature of a particular body at a given time t.

T_s denotes the temperature of the environment.

t_o = body's starting temperature,

k stands for constant.

The formula for Newton's Law of Cooling can be defined as the greater the temperature difference between the system and its surroundings; the heat is transferred more rapidly; it means the body temperature changes more rapidly. 

Methods of Newton's Law of Cooling

We can assume a constant rate of cooling, which is equal to the rate of cooling related to the average temperature of the body during the interval, when we just need approximate values from Newton's law.

i.e. \[\frac {d \theta}{dt}\] = k (q – qs) . . . . . . . (4)   

If q and q_s are the body's starting and ultimate temperatures, then

<q> = \[\frac {(q_i + q_f)}{2}\] . . . . . (5)  

Remember that equation (5) is simply an approximation, and that for exact numbers, equation (1) must be used.

If we need only approximate values sometimes from Newton's Law, we can assume a constant rate of cooling, equal to the rate of cooling corresponding to the average temperature of the body during the interval.

That is,  

\[\frac {d \theta}{dt}\] = k(q – q₀) ---- (4) 

If qi and qf are the initial, final temperature of the body respectively, then,

q =  \[\frac {(q_i + q_f)}{2}\]  ---- (5)

Note that equation (5) is only an approximation, and equation (1) must be used to exact values.

Newton's Law of Cooling's Limitations

The temperature difference between the body and the environment must be minimal.

Only radiation should be used to lose heat from the body.

The temperature of the surroundings must remain constant while the body cools, which is a critical constraint of Newton's law of cooling.

Some of the limitations of Newton's Law of Cooling are listed below.

• The loss of heat from the body should only be caused by radiation.

• The temperature differences between the body and its surroundings must be small.

• The main limitation of Newton's Law of Cooling is that the temperature of surroundings must remain constant during body cooling.

Derivation of Newton's Law of Cooling

The rate of cooling of a body is directly related to the temperature difference and the surface area exposed when the temperature difference between the body and its surroundings is modest.

\[\frac {dQ}{dt}\] (q – qs), where q and qs represent the temperature of the item and its surroundings, respectively.

According to the preceding expression, \[\frac {dQ}{dt}\] = -k[q – q_s] . . . . . . . . (1)

Newton's law of cooling is represented by this statement. Stefan's law, which gives, can be immediately deduced. 

k = \[\frac {4e \sigma \times \theta_{0}^{3}}{mc}\] A . . . . . (2)

Now, \[\frac {d \theta}{dt}\] = - k[θ – θ_o]    

\[\Rightarrow \int_{\theta_1}^{\theta_2} \frac {d \theta}{(\theta - \theta_0)} = \int_0^1 -kdt\]

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where,

qi = the object's starting temperature,

q_f is the object's final temperature.

ln \[\frac {(q_f -q_0)}{(q_{f}i - q_0)}\] = kt

(q_f – q_o) = (q_f – q_o) e^-kt

q_o + (q_i – q_o) e^-kt = q_f . . . . . . (3).

Check: Conduction heat transfer

Let us look at the derivation of Newton's law of cooling. For small temperature differences between a body and its surroundings, the rate of body cooling is directly proportional to the difference in temperature and the surface area that is exposed.

\[\frac {dQ}{dt}\] ∝ (q – qs)

Where q and q_s are temperatures corresponding to the object and its surroundings.

From the above expression, 

\[\frac {dQ}{dt}\]= -k[q – q_s] ---- (1)

This expression represents Newton's Law of cooling. It can be derived directly from the Stefan's Law, that gives,

k =  \[\frac {4eq \times \theta_0^3}{mc}\] A ---- (2)

Now,  

\[\frac {d \theta}{dt}\]= -k[θ – θo] 

⇒ \[\Rightarrow \int_{\theta_1}^{\theta_1} \frac {d \theta}{(\theta - \theta_0)} = \int_0^1 -kdt\]

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where,

• qi = initial temperature of the object

• qf = final temperature of the object

\[\frac {(q_f -q_0)}{(q_{f}i - q_0)}\] = kt

(qf – qo) = (qf – qo) e-kt

q_o + (q_i – q_o) e^-kt = q_f . . . . . . (3).

Newton's Law of Cooling Example

Consider Newton's Law of Cooling graph given below that states Newton's Law of Cooling.

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This process of cooling data can be measured and plotted, and the results are used to compute the unknown parameter 'k.' Sometimes, the parameter can also be derived mathematically.

Example of Solving a Problem Using Newton's Law of Cooling

A body with a temperature of 40º C is kept in the surroundings of a constant temperature of 20º C. If its temperature falls to 35º C in 10 minutes, find how much excess time it will take for the body to attain the temperature of 30º C.

Solution

From Newton’s law of cooling, qf = qi e-kt

Now, for the interval where the temperature falls from 40º C to 35º C,

(35 – 20) = (40 – 20) e-k.10

e-10k = \[\frac {3}{4}\]

Therefore, k = ln \[\frac {4}{3}\] ln \[\frac{\frac{4}{3}}{10}\] ---- (a) 

Now, for the next interval, 

(30 – 20) = (35 – 20)e-kt

So, e-kt = \[\frac {2}{3}\]

kt = ln\[\frac {3}{2}\] ---- (b) 

From equation (a) and (b), we get,

t = \[ 10 \times In (\frac {3}{2})\]= 14.096 min.

Applications of Newton's Law of Cooling

A few important applications of Newton's Law of Cooling are listed below.

• Used to predict how long it will take for a hot object to cool down at a certain temperature

• It also helps to indicate the death time given by the probable body temperature at the death time and the current body temperature