BlogNCERTNewton’s Law of Cooling

Newton’s Law of Cooling

The rate of heat loss from a body is directly proportional to the difference in temperatures between the body and its surroundings, according to Newton’s law of cooling. It is common to qualify the law with the condition that the temperature difference is small and the heat transfer mechanism is unchanged. It is analogous to saying that the heat transfer coefficient, which mediates between temperature differences and heat losses, is constant.

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    As the thermal conductivity of most materials is only weakly temperature-dependent, this condition is generally met in heat conduction (where Fourier’s law ensures it). Forced air or pumped fluid cooling is Newton’s Law when the properties of the fluid do not vary significantly with temperature; buoyancy-driven convection, on the other hand, is only roughly true since velocity generally increases with temperature difference.

    Furthermore, Newton’s law of cooling only holds for very small temperature differences in heat transfer by thermal radiation.

    A simple differential equation expressing temperature difference as a function of time can be derived from Newton’s law (with several simplifying assumptions, such as a low Biot number) when it is expressed as a function of temperature difference. The solution to that equation describes the temperature difference decreasing exponentially with time. Newton’s law of cooling is also associated with this characteristic decay of the temperature difference.

    Historical Background

    Newton published his work on cooling anonymously in 1701 in Philosophical Transactions, volume 22, issue 270, as “Scala graduum Calories. Calorum Descriptiones & Signs.”

    Newton’s law, as originally written in 1701, was not presented in this form. Newton noted, rather, that the rate at which a body’s temperature changes depends on the difference in temperatures between it and its surroundings after a mathematical manipulation. The simplest version of the law, given by Newton himself, was based on confusion between heat and temperature that existed in Newton’s day, which would not be resolved for many years.

    The 2020 experiments were repeated by Maruyama and Moriya using modern apparatus and with modern data reduction techniques[4]. These researchers took into account the thermal effects of high temperatures (as for the molten metals used by Newton), as well as the effects of buoyancy on airflow. In comparison to Newton’s original data, it was revealed that the measurements made from 1692 to 1693 were “quite accurate”.

    Relationship to the mechanism of cooling

    Newton’s law of cooling is said to govern the cooling of convection when the heat transfer coefficient is independent, or relatively independent, of the difference in temperature between the object and the environment. When the fluid velocity does not increase with increasing temperature differences, the law applies to forced air and pumped liquid cooling.

    Newton’s law is most closely obeyed in cooling systems that use only conduction. Heat transfer coefficients in natural convective (buoyancy-driven) heat transfer, however, are influenced by temperature differences. It is only when the temperature difference is relatively small that Newton’s law approaches the result. Despite this limitation, Newton himself understood it.

    In 1817, Dulong and Petit corrected Newton’s law concerning convection to include an exponent for larger temperature differentials. Known for developing the Dulong-Petit law concerning the molar specific heat capacity of crystals, these two men developed the Dulong-Petit law.

    A third situation where Newton’s law does not apply is radiative heat transfer. Radiative cooling may be described by the Stefan-Boltzmann law, whereby the heat transfer rate varies based on the difference in the 4th powers of the absolute temperatures of the object and its surroundings.

    Newton’s Law of Cooling Formula

    Newton’s Law of Cooling can be outlined as follows: the greater the difference in temperature between the system and its surroundings, the faster the heat can be transferred, which leads to a faster change in body temperature. As an expression of Newton’s Law of cooling, the formula below is used.

    T(t) = Ts + (To – Ts) e-kt

    Where,

    1. T(t) = body’s temperature at time ‘t’,
    2. Ts = surrounding temperature,
    3. To = initial temperature of the body,
    4. t = time
    5. k = constant

    Newton’s Law of Cooling: Its Limitations

    The following are a few limitations of Newton’s Law of Cooling.

    1. Radiation should be the only process that causes heat to leave the body.
    2. There shouldn’t be a wide temperature difference between the body and its environment.
    3. Newton’s Law of Cooling has a major limitation in that the temperature of the surrounding environment must remain constant during the cooling of the body.

    Newton’s law of cooling can be cited as an example

    Take a look at Newton’s Law of Cooling graph below, which states the law.

    By measuring and plotting these cooling characteristics, we can calculate the unknown parameter ‘k.’ Sometimes, we can also derive the parameter mathematically.

    Example of solving a problem using Newton’s Law of Cooling

    1. In a temperature-controlled environment, a body with a temperature of 40°C is kept at a temperature of 20°C. Calculate how long it will take for the body to reach the temperature of 30o C if its temperature drops to 35o C in 10 minutes.

    Solution

    As a result of Newton’s law of cooling, qf = qi e-kt

    Let’s examine the period of temperatures ranging from 40o C to 35º C,

    (35 – 20) = (40 – 20) e-k.10

    e-10k = 3/4

    Therefore, k =

    ln4/3

    ln4/3/10 —- (a)

    Now, for the next interval,

    (30 – 20) = (35 – 20)e-kt

    So, e-kt = 2/3

    kt = ln 3/2 —- (b)

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

    t = 10 ×

    ln(3/2)/ln(4/3)

    ln(3/2)/ln(4/3)= 14.096 min.

    The application of Newton’s Law of Cooling

    The following are just a few of the important applications of Newton’s Law of Cooling.

    1. When soda is placed in a refrigerator for a certain period of time, Newton’s Law of Cooling allows us to calculate the temperature.
    2. When a hot object is being cooled down at a specific temperature, it can be predicted how long it will take.
    3. A possible body temperature at the time of death and the current body temperature can also be used to estimate the death time.

    Also read: Zeroth Law of Thermodynamics

    FAQs

    Q. What is the physical meaning of Newton’s Law of Cooling?

    Ans: A body’s temperature is determined by the temperature of its surroundings, so it cools to the extent that it is hotter. Hot bodies cool faster than warm bodies due to this phenomenon. At first, the same body cools faster but then progressively cools more slowly.

    Suppose a brick is at 100 degrees. It cools to 20 degrees at room temperature in 5 minutes (for example), reaches 60 degrees (halfway) in another 5 minutes, reaches 40 degrees in another 5 minutes, and in another 5 minutes is at 30 degrees, and again reaches 25 degrees in another 5 minutes. Therefore, you need to take 5 minutes for every half-step towards 20 degrees.

    When Newton's law of cooling is applied, what area is considered?

    Newton’s Law of Cooling is useful when studying water heating because we can use it to calculate the rate at which water in pipes cools down. When you turn off the breaker during your vacation, it can tell you how fast the water heater will cool down.

    Q. What is the relationship between Newton’s Law of Cooling and Stefan-Boltzmann Law?

    Ans: Researchers have concluded that they are both related based on various pieces of research.

    Stefan’s Law: In a perfectly black body, radiant energy per unit surface area is always directly proportional to its absolute temperature to the fourth power.

     

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