In this screencast, we're going to figure out the heat generation that is needed to
heat water in a pipe. So, we're going to consider a pipe with a length of 10 meters. It has
an inside diameter of 15 millimeters and an outside diameter of 30 millimeters, and this
pipe is heated by uniform heat generation. This outer surface of the pipe is insulated,
which means all the heat is going to go to heating the water within the pipe. So, we
have water that flows through this. It has a mass flow rate of 0.3 kilograms per second.
The inlet temperature is 25 degrees C, and what we want to know is, what is the heat
generation that's needed to heat the water so that the outlet temperature is 45 degrees
C? So let's start with our governing equation, so we have that q, heat transfer rate, equals
the mass flow rate times the heat capacity of the water, T out minus T in. We can easily
solve for q, but how to we relate it to heat generation? So heat generation is heat transfer
per volume, so if we solve for q and we solve for volume, we can determine the necessary
heat generation. So our q is going to equal our mass flow rate, 0.3 kilograms per second,
times the heat capacity of water, which is 4178 joules per kilogram K, times the difference
in temperature, and this is going to equal 25,068 watts. So a couple of things to remember:
first of all, remember that a watt is equal to a joule per second, and also remember that
when you're subtracting temperatures, you'll get the same answer regardless of whether
you use Kelvin or Celsius. So now let's figure out our volume. So the volume of the pipe
wall is pi over 4, our outside diameter squared, minus our inside diameter squared, all times
the length of the pipe. And the reason we look at the volume of the pipe wall is because
that is where the heat is being generated. So let's figure out what this number is, and
be careful to put these diameters in meters. So that's equal to 5.3 times 10 to the minus
3 meters cubed. And finally, we can calculate our heat generation, which is that 25,068
watts divided by our 5.3 times 10 to the minus 3 meters cubed. And that's going to equal
4.73 times 10 to the 6 watts per meter cubed. So in other words, it's going to take 4.73
times 10 to the 6 watts per meter cubed in order to heat the water from an inlet temperature
of 25 degrees to an outlet temperature of 45 degrees C in a pipe that is 10 meters long.