Mathematical Method Assignment Submission • Newton’s Law of Cooling (Question 2)
Mathematical Method Assignment Submission Newton’s Law of Cooling Azizi Bin Rosli 1312387 Muhamad Azwan Bin Jaafar 1316401 Mohammad Amar Bin Ridzwan 1314441 Muhammad Jundullah Bin Ismail 1318065 International Islamic University Malaysia
1.
Introduction
Hot and cold have been an improper concept until the subjective measurement of temperature around the time of Galileo came up. It was not precise since temperature was measured by various indicator, e.g. the change in colour or glow of a heated iron. The more precise measurements of temperature came during the time of the invention of Celsius and Fahrenheit by Anders Celsius and Daniel Gabriel Fahrenheit respectively in the early 18th century. Temperature differs quite a lot in our surrounding, thus resulting to an important phenomena which causes heat to flow. The second law of thermodynamics suggest that heat could only transfer from an object of higher temperature to an object of a lower temperature until they both reach the state of thermal equilibrium – a state where there is no net heat transfer between both object. In the late 17th century, the first experiment on the nature of cooling was made by Sir Isaac Newton, noting that the rate at which the heat loss will be proportional to the difference in the temperature when the difference between temperatures of two different objects is small which is approximately less than 10◦ C. The model circulate around finding the rate of change of temperature of an object with respect to time,
dθ dt ,
and around the idea that temperature would change as long as there is difference in
temperature between the object and the surrounding, (θ − S, S = Temperature of Surrounding). Past experiments suggest that the arithmetical change in the time would result in a geometrical change in the temperature.
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Mathematical Method Assignment Submission • Newton’s Law of Cooling (Question 2)
2.
Case Study
Question 2: According to Newton’s Law of Cooling, if θ is the temperature of the object at time t, we have dθ = −k(θ − S) dt Find the solution under the initial condition θ0 . Suppose that a corpse was discovered in a motel room at midnight and its temperature was 75◦ F. The temperature of the room is kept constant at 60◦ F. Two hours later the temperature of the corpse dropped to 68◦ F. Find the time of death.
2.1
Solving the Differential Equation
Given, dθ = −k(θ − S) dt By rearranging the equation we get, dθ = −k dt θ−S Integrating both side, Z
dθ = θ−S
Z
−k dt
We come to, ln |θ − S| = −kt + C =⇒ eln|θ −S| = e−kt+C Which result in, θ − S = e−kt+C
Hence solution for θ is,
θ = e−kt+C + S
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Mathematical Method Assignment Submission • Newton’s Law of Cooling (Question 2)
2.2
Initial Value Problem
Before we could solve the problem, we need to know all the constant k, C and θ. By using the initial condition θ0 and the surrounding temperature 60◦ F, we could find the constant C. Since θ0 = θ (0) = 75◦ F, thus 75◦ F = e(0)k+C + 60◦ F = eC + 60◦ F Solving for C,
C = ln|75◦ F − 60◦ F| = ln 15 ≈ 2.708050201 It was said, that after 2 hours, the temperature of the corpse was 68◦ F, hence the following applies 68◦ F = e−2k+2.708050201 + 60◦ F Solving for k,
k=−
ln|68◦ F − 60◦ F| − 2.708050201 ≈ 0.314304329 2
Hence the final solution for θ with θ (0) = 75◦ F is
θ = e−0.314304329t+2.708050201 + 60◦ F
2.3
Solving the Case
Since our reference point for θ (t) is θ (0) = 75◦ F, θ (2) = 68◦ F means that the body drop its temperature from 75◦ F to 68◦ F after 2 hours from t = 0. Remember that we were told to find the time of death, assuming that the body temperature of the dead body was 98.6◦ F when it was first killed – 98.6◦ F is the standard temperature for human body in Fahrenheit. Filling up the equation, we get
98.6◦ F = e−0.314304329t+2.708050201 + 60◦ F
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Mathematical Method Assignment Submission • Newton’s Law of Cooling (Question 2)
Solving for t,
ln|98.6◦ F − 60◦ F| − 2.708050201 t=− ≈ −3.007283032 hours −0.314304329 It shows that the body is at the temperature 98.6◦ F approximately -3 hours after midnight, which means 3 hours before midnight. It could be concluded that the time of death was 3 hours before the corpse was discovered in a motel room at midnight.
2.4
New but Similar Problem
Suppose that a pack of blood at 36◦ C that has been withdrawn from a person need to be stored in nitrogen filled cold storage room for 2 hours. The temperature in the storage room are kept at
−10◦ C. When the blood is taken out after 2 hours, the temperature has fallen to 5.5◦ C. Predict the temperature of the blood after 30 minutes it was stored. Try again after 1 hour.
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