In this screencast we are going to work through a pipe flow example.
In which similar to another screencast were we had to calculate the diameter of the pipe
given a certain fluid characteristic.
Here we have to calculate the flow rate Q for our piping net work.
So lets take a look at this schematic so we can figure out what is going on.
We have a water coming in from our city supply network enter a house at 25 psi at gauge pressure.
We know that it leaves in a faucet that is in the second floor in the house.
So now we have to find a way that p2 is equal to 0 psig.
Since it is open to atmosphere.
So we are given some information in terms of lengths and heights of this pipe network.
As well as some of the connections that make up this system.
Now for those of you with some experience solving this type of problems.
You right away thinking this is going to be an iterate approach in you are right, because
we have no known velocities and we are trying to determine the volumetric flow rate.
This is going to be a solution that is reliant on both calculating the Re number as well
as the losses associated with the pipe system.
This is all based on velocity.
Since we do not have a velocity.
It is going to be an iterative approach.
So the first thing I like to do is convert all the values to the same unit system and
collect any type of information that we may need to help us along the way like the properties
of the fluid.
So we have water coming into our system and doing a quick Google search for say city water
lines.
We are going to assume this is about 60 degrees Fahrenheit.
So we are going to want the density of the water as well as the viscosity of the water
to do our calculations.
So I am going to work in English units.
Since we are starting with a pressure of psi we are looking for Q. So we want it in gal/
min.
So we are also told about the kinetic connections.
We will get there when we get to our losses.
In determining our losses for the system.
The velocity does play a large role, but we don't have that and that doesn't allow us
to also determine whether it is not laminar or turbulent flow which will help us determine
the major frictional losses in the pipe system.
So we need at least to set up an upper equation so we can get to a point were we are just
plugging in values and checking if they match.
So the first place in terms of determining if it laminar or turbulent flow that is rising
in our type of flow in our system is using the Re number.
So we have the density the diameter and the viscosity, So I can rewrite the Re number
in terms of V. When I do this 6906 times V.
Now we have a starting point in determining Re number.
You will see that number is going to come into play.
So the approach to these types of problems is we start with an overall energy balance
between the two points.
Now that energy balance is going to have the following form.
It looks like the Bernoulli equation, but it isn't because we are accounting for both
major and minor losses.
So we account for the pressure term the Kinetic term, the Potential energy term, and lastly
the loss terms.
So this almost becomes a plug and chug, but then again we do not have velocity.
So we want to simplify this equation as much as possible.
To get it down to being the only variable left.
So I like writing out in terms and putting in the values in.
Then crossing out what we know.
So we at least have a good approach to solving the problem.
So p1 was given to us as 25 psig.
P2 since it is open to atmosphere is just 0.
Now our alpha, both alpha 1 and alpha 2 is our kinetic energy for our flow.
When we have turbulent flow this is approximately 1.
So we are going to assume that is 1.
So we do not have to worry about it, but if we find out later that our flow isn't turbulent
then we should come back and reevaluate this equation.
We know our density.
We do not know our velocity.
We can use the gravitational constant, and then we need to look at our elevation.
If we look at our schemetic.
We can say 0.1 elevation is equal to 0.
As we move up the system we go up 5 and 10 feet.
So our elevation at point 2 is going to be 15 feet.
Now we got a couple of other things.
F here is the major loss term as f the frictional factor, and that is something that is going
to be based on the velocity of our fluid and that Re number that we just calculated in
terms of velocity.
So we do not have that, but based on this schematic we can calculate L.
We know D, we know rho.
Now looking at our minor losses we could look up the minor loss coefficient Kl for each
of the connections in our system.
So we are going to say that we have that.
We will write that out in a second.
So know we have everything down to velocity, and this friction factor f.
So lets go to determine the length of our pipe network.
So we have total of 45 feet of pipe.
We also know the diameter is 0.0833 ft.
The last thing we said we would calculate is our loss coefficients.
So lets go take a look at our schematic and figure out what we need to look up.
So we have threaded elbows, and we have 4 elbows for our pipe systems.
So we are going to have to look up the loss coefficient for a threaded elbow.
We are also told that it is copper pipe.
Now if you look up the loss coefficient for threaded elbows you get 1.5.
So our kL for the threaded elbows is 4 times 1.5, which gives us 6.
We also have a loss coefficient associated with this open globe valve.
if you look this up you get a kL of 10.
Then lastly we are told that the loss coefficient for our valve at the end is 2.5.
It is based on the fluid velocity that we do not know.
So now we have the loss coefficients for our pipe network.
Calculate the sums of of the loss coefficients.
So the sums was equal to 18.5.
So lets start plugging in numbers and simplifying the energy equation.
So I am going to plug in all the values in the equation.
I get the following.
Now I can start simplifying each component of this equation.
Now we have this equation that we thought that we had down to 1 or 2 variables the velocity
and the friction factor, but you will see that we have a velocity at point 2, a velocity
at point 1, and these undefined velocities.
Now our undefined velocities are losses for our pipe network, and so really we are evaluating
the velocity through our pipe network, which is the same as 0.1, and the velocity at point
2 is our exit velocity, which is slightly different because it is smaller diameter.
So in simplifying this equation now the thing you want to do is substitute our velocity
at point 1 into our velocity at point 2 here, and so using continuity we know that the flow
rate at both points have to be the same, which means that V1A1 has to equal V2A2.
So now we can calculate an expression for V2^2 this is going to be equal to V1 times
A1 over A2 squared.
So in knowing our area at point 1 and our area at point 2.
We can get a relationship for this and plug in for V2, and I get 3.16V1^2.
So now plug in this in here, and simplify our minor equation.
Now we have our equation were we can determine a friction factor and our velocity plug it
in to satisfy what we know here.
Plus we have our Re number equation at the top.
Where we set the Re number was equal to 6906 times our velocity.
So the only thing left now is an equation to calculate our friction factor.
Now one way to do this is using they mody diagram, and if you did it this way.
You would be guessing and reading, guessing and reading until you got it right.
A more analytically accurate method maybe in this case would be to use the Colebrook
equation.
Now this the Colebrook formula, but as you can see this also requires the iterative approach.
Since we have the friction factor f on both sides of the equation.
Another one that does't is the Haalond equation, and that is what we are going to use here.
So you have your choice of going about it, but we have our 3 relationships.
We have our first equation, which is based on our energy balance.
We have our second equation, which is based on a relationship for our dimensionless parameters
the Re number, and now that is used here in our Haaland equation, and I am going to put
all of three of these into excel and use goal seek to make our lives much easier.
For Goal Seek to work properly we are going to have to rewrite this equation in terms
of f.
So now we have f equals based on the velocity.
So by guessing a velocity we can calculate f and the Re numbers.
Plug these into our Haaland Equation, and as long as it matches up we have guessed the
correct velocity, but excel and goal seek will do that for us.
So I have excel open here on the right side.
So I can show you how I set this up.
Epsilon our surface roughness.
Something that we need in both equations and the moody diagram.
That is something we would look up for copper piping.
In this case it is 5 times ten to the minis 6 feet.
So I have that in our equation sheet.
Plus I am guessing I am guessing a velocity of 5 ft/s. and I am just starting there, using
it as a reference.
This energy f is my equation at the bottom to calculate the friction factor based on
the velocity.
I also have the Re number which is 6906 times the velocity.
Now both of these goes into the Haanland equation, and the Haaland equation marked as 3 on my
sheet.
You see I have it typed up here in our excel equation bar, and I brought that negative
1 over the square root of our friction factor onto the right side.
So all of this should equal 0.
So now what I do is the what if analysis, and I do goal seek.
I want to set this value to 0, because that should be satisfied by changing the velocity.
When I do this I find that my velocity is 9.36 ft/s.
So we are almost there.
So we have found V1.
We want to solve for Q. So Q is just going to be equal to V1 times A1.
This gives me flow rate of 0.051 ft^3/s.
Now that does seem really low, except we are looking it for terms of gal/min.
So multiply by 60 seconds, and use the conversion from ft^3 to gallons, and I get 22.9 gal/min.
That seems like a fairly high flow rate.
Hopefully this gives you an idea on how to use the iterative approach in pipe problems
to solve for things we don't necessarily have.
In this case we did not have velocity needed.
So we had to guess and check until we got our equations to check out.