Heat Transfer Concept Page - 8

Example

Thermos flask

Thermos flasks keep hot liquids hot and cold liquids cold for long hours. It consists of a glass vessel with double walls. The glass vessel is enclosed by a metal or plastic cover for protection against damage. The space between the walls is vacuum which reduces the heat loss due to conduction and convection. The outer surface of the inner wall and the inner surface of the outer wall are silvered (shining). The shining surfaces reduce heat loss due to radiation. An insulating stopper (cork) is used to close the mouth of the vessel. The vessel is kept over an insulating (cork ) pad. It is therefore thermally insulated. Hence, heat does not leave or enter easily and hot liquids remain hot while cold liquids remain cold inside the flask.

Law

Newton's Law of cooling from Stefan Boltzmann Law

According to Stefan-Boltzmann Law, rate of heat transfer is given by:
dQdt=−σeA(θ4−θ4o)${\displaystyle \frac{dQ}{dt}}=-\sigma eA({\theta }^4-{\theta }_o^4)$
θ=θo+Δθ$\theta ={\theta }_o+\mathrm{\Delta }\theta$
Assuming Δθθ<<1${\displaystyle \frac{\mathrm{\Delta }\theta }{\theta }}<<1$
Then, θ4−θ4o≈=4θ3oΔθ${\theta }^4-{\theta }_o^4\approx =4{\theta }_o^3\mathrm{\Delta }\theta$
∴dQdt=−4eAσθ3oΔθ$\therefore {\displaystyle \frac{dQ}{dt}}=-4eA\sigma {\theta }_o^3\mathrm{\Delta }\theta$

Now, heat transferred to a material is given by:
dQdt=mcdθdt${\displaystyle \frac{dQ}{dt}}=mc{\displaystyle \frac{d\theta }{dt}}$

From above, dθdt=−4eAσθ3omc(θ−θo)${\displaystyle \frac{d\theta }{dt}}={\displaystyle \frac{-4eA\sigma {\theta }_o^3}{mc}}(\theta -{\theta }_o)$
dθdt=−k(θ−θo)${\displaystyle \frac{d\theta }{dt}}=-k(\theta -{\theta }_o)$
This is the expression of Newton's Law of Cooling.

Definition

Experimental verification of Newton's Law of cooling

Newton's Law of cooling can be verified with the experimental set-up shown in the figure. The set-up consists of a double walled vessel (V) containing water in between the walls. A copper calorimeter (C) containing hot water is placed inside the double walled vessel. Two thermometers through the corks are used to measure the temperatures T2$T_2$ of water in calorimeter and T1$T_1$ of hot water in between the double walls respectively. Temperature of hot water in the calorimeter is noted after fixed intervals of time. A graph is plotted between log(T2−T1)$log(T_2-T_1)$ and time(t). The nature of graph is observed to be a straight line having a negative slope which is in accordance of the equation of Newton's Law of cooling.

Definition

Newton's Law of Cooling

Here, t$t$ is time, T$T$ is temperature of the object, and Ts$T_s$ is the surrounding temperature. 
Newton's Law of Cooling :Newton's law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surrounding.

Definition

Assumptions of Newton's Law of Cooling

Assumptions of Newton's Law of Cooling are as follow :
1. Temperature difference between the object and surrounding should be small.
2. The nature of radiating surface should remain same.

Diagram

Plot of temperature vs time for Newton's Law of cooling

Diagram

Plot of log of temperature difference vs time for Newton's Law of cooling

Definition

Describe the limitations of Newton's Law of Cooling

The limitations of Newton's Law of cooling are as follows:
1. It can be used only at lower temperatures.
2. It can only be use when difference between initial and final temperatures is less than 50-80 degree Celsius.
3. It holds true only if the temperature of the surroundings remains constant throughout the cooling of the body.

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