An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at 65 degrees F. After 10 minutes, the beam warmed to 35 degrees F anf after another 10 minutes its temprature was 50 degrees F. Use Newton's Law of Cooling to estimate the beam's initial temperature.
The differential equation that is obeyed is
M C dT/dt = h A (65-T)
where mass M, specific heat C, film cooling coefficient h and surface area A are constants.
The solution has the form
dT/(65-T) = c dt
-ln(T-65) = c t + a
T-65 = exp (-a -ct)
You have two unknowns constants (a and c) and two equations for T at t = 10 and t = 20 minutes. Once you have a and c you can solve for T when t=0.
To estimate the beam's initial temperature using Newton's Law of Cooling, we need to find the cooling constant, also known as the cooling rate or cooling coefficient.
Newton's Law of Cooling states that the rate of change of temperature of an object is directly proportional to the difference between its temperature and the ambient temperature. Mathematically, it can be expressed as:
dT/dt = -k(T - Ta)
Where:
- dT/dt is the rate of change of temperature (how fast the temperature is changing) with respect to time.
- k is the cooling constant, which represents the rate at which the object cools.
- T is the temperature of the object at a given time.
- Ta is the ambient temperature (the temperature of the surroundings or the machine shop).
We are given two pieces of information:
1. After 10 minutes, the temperature of the beam was 35 degrees F.
2. After another 10 minutes (so a total of 20 minutes), the temperature of the beam was 50 degrees F.
Let's use this information to estimate the cooling constant (k) and then find the initial temperature (T).
First, let's set up two equations using Newton's Law of Cooling for the given data:
When t = 10 minutes:
(dT/dt)1 = -k(T1 - Ta)
When t = 20 minutes:
(dT/dt)2 = -k(T2 - Ta)
Now, let's solve for the cooling constant (k) by dividing the two equations:
[(dT/dt)1] / [(dT/dt)2] = [(T1 - Ta)] / [(T2 - Ta)]
Plugging in the given values:
(35 - Ta) / (50 - Ta) = (dT/dt)1 / (dT/dt)2
We can rearrange the equation to solve for Ta:
(35 - Ta) / (50 - Ta) = (dT/dt)1 / (dT/dt)2
Cross-multiply:
(35 - Ta) * (dT/dt)2 = (50 - Ta) * (dT/dt)1
Expand:
35*dT/dt2 - Ta*dT/dt2 = 50*dT/dt1 - Ta*dT/dt1
Collect common terms:
35*dT/dt2 - 50*dT/dt1 = -Ta*dT/dt2 + Ta*dT/dt1
Rearrange to solve for Ta:
Ta = (35*dT/dt2 - 50*dT/dt1) / (dT/dt2 - dT/dt1)
Now, insert the values provided in the question:
dT/dt1 = (35 - 65) / 10 = -3 degrees F per minute
dT/dt2 = (50 - 35) / 10 = +1.5 degrees F per minute
Ta = (35*(-3) - 50*(1.5)) / ((-3) - 1.5)
Ta = (-105 - 75) / (-4.5)
Ta = 180 / 4.5
Ta = 40 degrees F
Therefore, the estimated initial temperature of the beam would be approximately 40 degrees Fahrenheit.