In this screencast, we're going to show you
how to calculate the mass flow rate of a fluid
inside a pipe that has a constant heat flux.
So the governing equation that we're going
to use is that the heat transfer rate is equal
to this mass flow rate, times the heat capacity
of the fluid, times the difference between
the mean temperature of the fluid at the outlet
and the mean temperature of the fluid at the
inlet.
To use this, we have to neglect viscous dissipation,
we have to assume that the fluid is incompressible,
we have to neglect heat transfer in the axial
direction, and the tube needs to be of finite
length.
We can use other expressions for certain situations
such as constant surface heat flux, and we
use Newton's Law of Cooling, which is h*A*delta
T equals q, to relate q to the surface temperature
at the outlet and the inlet.
So we rewrite this as q dot equals (now because
the temperatures are changing, we have local
heat transfer coefficients) so this is at
the outlet, times the area, the surface temperature
at the outlet minus this mean temperature
at the outlet.
And we can do the same thing at the inlet.
Again, these are local heat transfer coefficients,
so the h's will be different.
Let's take a look at an example.
So we have water and it flows through a pipe
that has a length of 10 meters, a diameter
of 50 centimeters, the surface temperature
at the inlet is 50 degrees C, and that at
the outlet is 95 degrees C. And if we have
a constant heat flux that's equal 1000 watts
per meter squared, we want to know what this
mass flow rate of the water is.
If you note, our equation that we're going
to use that could tell us the mass flow rate
has both the outlet and the inlet mean temperatures,
neither of which we have.
However, we can use Newton's Law of Cooling
to calculate both the mean outlet temperature
and the mean inlet temperature.
So we can use a form of Newton's Law of Cooling
to find our mean temperatures.
Since we have a heat flux, we don't include
our area, so at the outlet, this is just the
surface temperature minus the mean temperature,
and we rewrite that as Tso minus our flux
divided by our heat transfer coefficient.
And we can rewrite it the exact same way at
the inlet.
So, if we calculate these quantities, we find
that at the outlet, so our mean outlet temperature
is going to be equal to 80.7 degrees C.
We do the exact same thing for the inlet,
just making sure we use the surface temperature
at the inlet as well as the heat transfer
coefficient, and we come out with 30 degrees
C. Now, our governing equation is written
in terms of our heat transfer rate, so we're
going to have to change our heat flux into
a heat transfer rate.
And that's just that our q dot, heat transfer
rate, is our flux times the surface area that
the fluid touches.
So this is equal to, again, our 1000 watts
per meter squared, times pi, times our diameter,
times the length of the pipe, which equals
15,707 watts.
Now we can put it into our equation to solve
for the mass flow rate, which equals q over
our heat capacity, times the difference in
our mean temperatures.
And when we calculate those numbers, we end
up with a mass flow rate of 0.074 kilograms
per second.
So this is a good example of having to use
different expressions for the heat transfer
rate to calculate mean temperatures, in and
out, and finally, a mass flow rate.
