00:01
Hi there, so for this problem, we are told that the heat transfer coefficient for air flowing over a sphere is to be determined by observing the temperature time in history of a sphere fabricated from pure cooper.
00:16
So the sphere, which is, which has a diameter that we're going to call the diameter capital d, is 12 .5 millimeters.
00:28
Meters, 30 meters we can write it as 12 .5 .7 times 10 to the minus 3 meters.
00:40
So is at a temperature of 60 celsius degrees before it is inserted into an air stream having a temperature of 27 degrees.
00:55
So we have 6 celsius degrees, and the temperature that we are going to say, the temperature of the surroundings is equal to 27 celsius degrees.
01:22
Now, a thermocopal on the outer surface of the sphere indicates a temperature of a temperature that is 55 celsius degrees after a time that is equal to 69 seconds after the sphere is inserted into the airstream.
01:56
So assume and then justify that the sphere as a space -wise isothermal object and calculate the heat transfer coefficient.
02:09
So, in this case, we first start with the time temperature history that is given by the following equation.
02:23
Theta and function of the time over titan the initial theta is equal to the exponential of minus the time divided by the product between rt and ct, where rt is defined as 1 over the product between the heat transfer coefficient h and the surface area, where the surface area is defined as pi times the distance square and c t is defined as the density times the volume times the capacitive cp so where the volume is defined as pi times the distance to the three divided by sets now with this set and we know that in this case theta is the difference between the temperador minus the temperature of the surroundings.
03:52
So in this case, recognizing that the time is equal to 69 seconds, we're going to have that theta, the time t, divided by the initial theta, is equal to.
04:06
So we will have 55.
04:09
This is at the temperature at the time 69 seconds.
04:14
So it is 55 celsius degrees minus the temperature of the surroundings, that is 27, and this divided by 66 minus 27 celsius degrees.
04:33
And from this, we obtain a value of 0 .718, and this is equal to the exponential of minus the time, divided by tau, where tau is a time constant that is a measure of how fast the system responds to a sudden change in ambient temperature.
05:09
So if we solve for that value in this equation right here, we will obtain that tau is equal to 208 seconds.
05:23
So with this information, we can obtain the heat transfer coefficient because that is the product between the density, the volume, the capacity, heat capacity, divided by the surface area, times tau, the one that we just have calculated.
05:52
So with this, we know that the density of cooper is equal to 8 ,8 ,933 in units of kilograms per cubic meter...