Sample Assignment on Newton's law of cooling

Mathematical model of Newton's law of cooling and experimental verification

Introduction


With the time a hot object cools down and it temperature slowly goes down till it reaches the surrounding temperature. Similarly a cold object temperature rises to the surrounding temperature with the time. In both cases object exchanges its heat with the surrounding to reach the final temperature or so called equilibrium temperate. Further we observer that more the difference in the temperature between the surrounding and the objects faster is the heat exchange under the similar condition of the other parameters.


Sir Newtown’ proposed a law called “Newton’s Law of cooling” for cooling of objects. This law states that the rate of change of the temperature of an object is proportional to the temperature difference between object’s temperature and the ambient temperature that is surroundings temperature of the objects. Such behavior has been observed many times in laboratory experiments and it is also verified, Hence Newton’s Law of cooling widely accepted. We know that object can transfer the heat to its surrounding by any of the three method of heat exchange not only due to conduction and convection but also due to radiation. The latter does not vary linearly with temperature difference, which leads to deviations from Newton's law.


Sample Assignment on Newtons law of cooling

The Mathematical Model

Newtown’s law of cooling - The rate of temperature change of a substance is proportional to the difference between its temperature and the ambient room temperature.


Mathematically it can be rewritten as

dT/dt = k(T - Ta)
Where T(t ) = Temperature of the object at any time t.
Ta = Ambient temperature.
T(0) = Initial Temperature of the object
K = Constant of proportionality that depends
on the nature of object and surrounding


Solving this differential equation we get and using the initial condition that temperature of objects and ambient temperature we get the equation as follows that is in the most general form.


T(t) = Ta + (To - Ta)ekt


From this equation we can drive the linear form of the plot and final formula would be


kt = in

Newton's law of cooling image 1

The experiment

Equipment Quantity Purpose
Styrofoam Cup 1 Hot was container
Hot water NA Experimental fluid
Temperature probe 1 To measure the temperature
Stop watch 1 For measurement time
Microwave 1 T o rise the water temperature

Aim: Our aim is to verify that under our experimental condition that is mentioned in the above table if Newtown’s law of cooling is applicable.


Procedure: A cup of hot water at 194 degree Centigrade was taken in the Styrofoam cup with the decision based upon the availability of the cup. Putting the temperature probe in the water half dipped and then we allowed the water to cool down and we measure the temperature as following.


Observation table

Time(min) Temp(F)
   
O 194
15 183
30 162
45 144
60 132
75 122
90 115
105 109
120 103
135 99
150 97
165 94
180 90

Experimental condition:

Initial water temperature = 194 F
Final water temperature = 90 F
Room temperature = 75 F
Number of observation point= 15

The MatLab Analysis

MatLab Analysis image 1
MatLab Analysis image 2

From the derivation we know that transformed equation for linear plot can be written as

Newton's law of cooling image 1 MatLab Analysis image 3

The Model

MatLab Analysis image 4

From the plot slope = Delta Y / Delta X = 0.0119 / 113 = 1.05 x 10-4

MatLab Analysis image 6

Conclusion


We see that under the laboratory condition Newton’s law of cooling holds. However various plots show the deviation from ideal behavior that is due to experiment condition and measurement errors. Accuracy of the experiment can be increase by removing the source of various avoidable and more precise time sampling

References

http://www.nsa.gov/academia/_files/collected_learning/high_school/calculus/newtons_law.pdf
http://archives.math.utk.edu/ICTCM/VOL07/C023/paper.pdf
http://www.instruction.greenriver.edu/projecttime/documents/aim/newtons_law_of_cooling-computer.pdf
http://www.instruction.greenriver.edu/projecttime/documents/aim/newtons_law_cooling_ez.pdf

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