In this article, we will learn Newton’s law of cooling along with the basic statement, definition, explanation, differential equations, formula, and many examples.
The cooling of numerous objects is often narrated by a theory termed Newton’s Law of Cooling, assigned by the great physicist Sir Isaac Newton.
This Newton’s Law is invoked in an extensive range of situations in Applied Science, but the law is also applied in other important areas such as,
So without any further ado, let’s get into a greater aspect of what Newton’s Law of Cooling actually is!
We all may have observed that when a hot cup of tea is placed in a room in the summer season, the tea inside the cup slowly starts to get cold, but why does it happen in the first place?
Because the temperature of the tea inside is dissimilar to the temperature outside the cup, so there is not much temperature difference.
This rate of loss of heat of a particular body directly proportional to the temperature difference between the body and the surrounding fluid medium is called Newton’s Law of Cooling.
To understand Newton’s Law of Cooling, we all should be familiar with the first and the second Laws of thermodynamics.
Heat transfer is also an important aspect in understanding Newton’s Law of Cooling as it is beneficial in every side of day-to-day life such as technology, hygiene, cooking, transportation, etc.
Heat transfer is a very extensive subject used in many fields of engineering, and the notion of heating and cooling is crucial in the building of every design and machine having moving parts.
The types of methods through which the heat transfers are conduction, convection, and radiation.
Let’s see conduction, convection & radiation at a glance,
But it was in the early 1740s that Scientists found out that Newton’s law does not apply to all contexts, and it is still much of a debate throughout the 18th and 19th Centuries, particularly with radiation.
Therefore, scientists discovered three well-defined models (laws of Cooling) that aim to relate the cooling of different thermal systems in laboratory environments.
Since all the above models vary on the basis of different bodies, surroundings, and systems, hence in this article we would only be looking into the general Newton’s Law of Cooling.
For the above example of Tea, the following formula can be used according to Newton’s Law of Cooling.
The mathematical equation is,
Rate of cooling ∝ ΔT
This equation can also be written as, dT/dt = – k (T-Ts)
(Proportionality constant also depends on the specific heat of the object, how much surface area is exposed to the object, etc.)
dT/dt = – k (T-Ts)
Now let’s divide both sides by (T-Ts) and multiplying by dt
By variable separable and Integrating,
Taking log on the other side,
T = Ts + c e-kt
This Newton’s law of cooling formula (Equation 2) can give us the temperature of the body when there is a great difference in the temperature between the body and the surrounding.
Now, let’s see Newton’s law of cooling derivation according to Stefan’s law,
According to Stefan’s law, the rate of loss of heat can be calculated as below,
E = ε σ [T4 – To4]
(Here the condition is that the T is slightly greater than To )
Let T – To = ΔT
T = To + ΔT
Now comparing equations 1 and 2 we get,
(1 + ΔT/To)4≈ 1 + 4ΔT/To [ Here, the higher order of ΔT/To is neglected]
So, we can write,
Now after opening the brackets,
Hence, we can say that 4 ε σ To3 is constant.
So we prove that,
The rate of loss of heat is directly proportional to the temperature difference.
Important applications of Newton’s Law of Cooling are as follows:
One of the crucial applications of Newton’s law of cooling can be noticed in forensic science where the detective and the forensic scientists jot down the temperature of the dead body and the surrounding room temperature of the crime scene.
These temperature values are then put in the mathematical formula obtained by Newton’s law of cooling because of which we can determine the actual time of the victim’s death.
Different types of Fluid Cooling
Newton’s law of cooling is applied to calculate the cooling of different types of fluids as it can assist us in predicting how much time it will take for hot fluids or beverages to cool down at a certain temperature.
Correspondingly, when the temperature difference is low, the rate of cooling is decreased. Finally, to calculate the time involved in such processes can be calculated through Newton’s Law of Cooling.
Canned food Packaging
The food or the liquid packed in tin cans or plastic jars is always checked with the temperature in the production line. This is to prevent the growth of microorganisms that damage the food or contaminate it.
Calculating temperatures of high degree metals
Let’s consider the case of a blacksmith when he heats a piece of metal to a substantially higher temperature, but at the time of noting down the temperature of the heated metal, he notices that there is no such measuring equipment or thermometer to record such high-temperature values.
Thus, with Newton’s law of cooling, we can measure the rate of cooling of the metal thereafter estimating the earliest temperature of the metal.
In an ice cream factory, the ice cream made and stored out of the freezer immediately starts to melt.
However, this rate of melting can be pre-planned with the assistance of Newton’s law of cooling on the condition that the difference between the temperature of ice cream and the surrounding is studied earlier.
This assists the ice cream distributors and vendors to calculate sufficient temperature and accurately organize their storage facilities.
Several limitations are given below,
Based on the above equations, theories and formulas, we can be able to find different temperature aspects of various bodies and their surrounding area with the help of Newton’s law of cooling in our day-to-day life.
Refer to our few other articles,