This problem is going to use the energy equation to find head loss.
In a non-circular duct.
Usually when we talk about head loss.
We talk about it in a pipe, which is round, but here we are going to use a duct that is
6 inches by 12 inches on the side.
So basically what it looks like is 6 by 12.
So lets start with the governing equation for head loss, and head loss is found by multiplying
the friction factor times the length of that pipe over the diameter times the average velocity
squared, over 2 times gravity.
So in this case we have galvanized iron pipe and we have standard air.
That flows at a volumetric flow rate of 5 ft^3/s.
The length of the pipe that we are interested in is 20 ft.
Lets put together our plan.
So the first thing looking at our governing equation is we need to figure out our average
velocity, which is Q over A. What makes this different, because it is a non-circular duct
is that we have to use a hydraulic diameter, which is defined as 4 times the area divided
by the perimeter, and in our case it is 2 times the width, times the length, over the
width plus the length.
Once we find the velocity we have the length, we have the hydraulic diameter.
We are going to have to find the friction factor, and the friction factor is a function
of the roughness of the time, divided by the hydraulic diameter, as well as the Re number.
So in order to find the friction factor we are going to have to look up the roughness
of galvanized iron, and we are going to have to calculate our Re number, which is the velocity
times the hydraulic diameter, divided by the kinematic viscosity.
First we will find our hydraulic diameter, which is 2 times 0.5ft, which was our 6 inches.
Times 1 ft, which is our 12 inches.
Divided by 0.5 plus 1 or 1.5 ft, and that ends up being 0.67 ft.
Next we need to find our average velocity, which is our Q, 5.0 ft^3/s divided by our
area, which is 0.5 ft times 1 ft, and that equals 10 ft/s.
Now there are a number of charts that give roughness factors, and in this case we look
up the roughness factor for galvanized iron and we find that equals 0.0005 ft.
Again it is a function of this roughness factor our friction factor is divided by the hydraulic
diameter.
So this is 0.00075.
The next thing we need to do is calculate our Re number, and so that is our average
velocity, which is 10 ft/s.
Again we have to use instead of diameter our hydraulic diameter, which is 0.67 ft, and
we look up the kinematic viscosity in standard conditions, and that is 1.57x10^-4 ft^2/s,
and when we calculate this we get a Re number of 4.3x10^4.
So what does it mean that the friction factor is a function of this roughness factor and
the Re number.
Well in order to find the friction factor we use what is known as the moody chart, and
what the moody chart gives us is based on both these roughness factor and the Re number.
We are able to us the chart and look up a friction factor, and in this case our friction
factor is 0.024.
Now we are ready to calculate the head loss.
Again it is the friction factor times the length of the pipe divided by the hydraulic
diameter, times the average velocity squared, divided by 2 times gravity.
So that is 0.024, times 20 ft, over 0.67 ft, times 10 ft/s squared, divided by 2 times
32.2 ft/s^2.
When we do the calculations we find out that the head loss for this 20 ft of our non-circular
duct is 1.1 ft.