The governing equation for internal flow, where we neglect viscous dissipation, assume
the fluid is incompressible, we neglect heat transfer in the axial direction, and the tube
is of a finite length, is that the heat transfer rate is equal to the mass flow rate of the
fluid, times the heat capacity of the fluid, the mean temperature out minus the mean temperature
in.
And this applies irrespective of the nature of the surface or tube flow conditions.
However, due to the exponential decay of the temperature difference, Ts minus Tm, when
there's a constant surface temperature, we need to use a temperature difference such
that q equals m dot, Cp, delta Ti minus delta To, where delta T is equal to the surface
temperature minus the mean temperature at any point.
So now, when we apply Newton's Law of Cooling, we use that q equals an average heat transfer
coefficient, times the area, times delta T log mean, where that delta T log mean is equal
to delta T out minus delta T in, where again these are with respect to the surface temperature,
divided by the natural log of delta T out over delta T in.
So let's look at an example with a constant surface temperature.
So we have water.
It flows through a pipe that has a length of 3 meters and a diameter of 5 centimeters,
the mean temperature of the fluid at the inlet is 10 degrees C, and that at the outlet is
50 degrees C.
We're given an average convective heat transfer coefficient.
So if our surface temperature, which is constant, is equal to 75 degrees C, what is the mass
flow rate of the fluid?
We're going to start with our governing equation.
So our q dot equals our mass flow rate times that heat capacity, delta Ti minus delta To.
So our mass flow rate is going to equal our heat transfer rate, divided by that heat capacity
times those temperature differences.
So we're going to have to use Newton's Law of Cooling, q equals our heat transfer coefficient,
times the area, times the delta T log mean.
With that we're going to find our q and then use that to find our mass flow rate.
So let's start by calculating all those delta T's.
So our delta Ti is equal to 75 degrees C minus 10 degrees C, or 65 degrees C. Our delta To,
or out, is 75 degrees C minus 50 degrees C, which is 25 degrees C. So our delta Ti minus
delta To is equal to 40 degrees C. Now let's look at our delta T log mean.
So that's going to equal our delta T out, 25 degrees C, minus our delta T in, which
is 65 degrees C, divided by the natural log of 25 over 65.
And when you calculate that delta T log mean, we come up with 41.9 degrees Celsius.
So our q is equal to our average heat transfer coefficient, times our area, which is pi,
times our diameter, times the length of the pipe, times our delta T log mean.
Remember that 1 degree Celsius equals 1 Kelvin.
And this is going to equal 1973.5 watts.
And now finally, we can put it back into this equation so that we can find our mass flow
rate, which is going to equal 0.0118 kilograms per second.
And remember, if you're wondering about the units, a watt is equal to joules per second.
You can also look at a similar screencast that looks at internal flow, however, instead
of a constant surface temperature, it has a constant surface heat flux.