In this screencast we are going to preform an integral momentum balance on a specific
system shown here.
We have a tee, attached to a pipe.
That splits at incoming water stream, and the 2 free jets.
The exit speed of both streams is 30 ft/s.
We are going to neglect any viscous effects with in the pipe and the tee, and also neglect
gravity.
What we are interested in determining is the x and y components of the force that the pipe
exerts on this tee.
So the first thing to start with any problem is with a figure.
We have one here.
So we want to double check that everything is labeled appropriately, and we see that
our velocity coming out of both parts of the tee is labeled as 30 ft/s.
We are given areas of both ends, and we are given the area of the in coming in.
So the second thing in preforming a momentum balance is to determine what our control volume
will be for this system.
Lets go ahead and draw that out.
Now I have drawn this line here to indicate that we do have a tee attached to a pipe.
So we are going to choose the tee as our control volume.
So I am going to just draw in red the control volume of interest, and we need to rewrite
what we are looking for.
So I am going to say find; we want the force in both the x and y direction of the pipe
on the tee.
Alright, so, good place to start this problem is to write our general equation for preforming
a linear momentum balance on the tee.
So we know that the differential with respect to time of the control volume is the velocity
times the density, times the change in volume, plus whatever enters that control volume through
its control surfaces.
This is the velocity times the density times the velocity dot product with the normal.
Over the area that enters, and this must be equal to the sum of all forces acting within
the control volume or on the control volume.
This must equal the sum of the forces acting on the content of the control volume that
we have designated here in red.
At this point we would list our assumptions.
We are going to say it is steady state flow.
The flow is steady state so this allows us to get rid of that first term.
The rate of change of the momentum of the content of the control volume.
So we are left with just that second part of the equation.
So lets work with that type first.
So if we are interested in only the x component of the momentum equation at this time we could
rewrite this.
So lets go up to our figure as we do this, and label a couple more things.
First of all we are going to say this into this control surface.
This is control surface 1.
We will call this control system 2, and we will call this control surface 3.
So for the control surface at point, for control surface 1 we have momentum entering our control
volume.
So we will have negative V1, times the density of the fluid, again times V1, and area of
1.
So that covers this region here, and for momentum leaving out on the other side is going to
be positive.
We will do the same thing but this time we will use the velocity at 2, rho, V2, A2.
We don't have to worry about the control surface at 3, because that is only in the y direction
and right now we are only concern with the x components.
Now lets work on the right side of our momentum balance equation.
The sum of the forces.
We have our pressure force acting on section 1, and if we work in gauge pressures.
We can say this is Pgauge at 1 times the area at 1, and since at section 3 it is open to
atmosphere.
As a free jet there is going to be no pressure force at that point.
So the only other force would be that of interest.
Which we will say is Fxp for the force on the tee, and this force if we looked at up
at our diagram would pretty much be the anchoring force that the pipe is exerting on to hold
on to that tee.
So all said and done.
If we put these two together we get the following equation.
So to determine the force we need to look at our diagram and see what we have.
Since we are working with water we know the density.
We are given the velocity at 2, and the area at 2.
So we would know that first term.
We have the area at 1, by we don't have the velocity at one.
So V1 would be an unknown.
As we said we have the are at one, but we are not given the gauge pressure of the pipe.
So we would also have to determine that.
We don't have the gauge pressure of the water, at point one.
So we don't have to determine that.
So we could use the continuity equation to determine V1.
This basically says for an incompressible fluid that the volumetric flow rate coming
into the tee must be the volumetric flow rate leaving the tee.
So we could rewrite this as the area 1 times the velocity at 1.
Equals the area at 2 times the velocity at 2 plus the area at 3 times the velocity at
three.
Since we are interested in V1.
We can rewrite this as the following.
So now we have a relationship for V1.
So the other variables that we are intersted in is this gauge pressure at 1.
So we need a relationship between the pressure and the velocity of the fluids, and we can
go back and estimate this gauge pressure using the Bernoulli Equation for flow between point
1 and point 3, point 2.
So writing the Bernoulli equation.
We see that the pressure at the control surface at one divided by the density of the fluid
plus the square of the velocity at that point divided by 2 must equal the same at another
point, which we are going to say point 2.
So that we can get rid of the height term, and we know all the values here except for
the gauge pressure at point 1.
So we can solve for this.
So know we have three equations.
Our first one, the one that gives us the velocity at point 1, and the one that gives us the
gauge pressure at point one.
So we can now solve for the x-component in the force in the x-direction.
So plugging in numbers.
Now if I plug all these numbers in.
I can get a velocity V1 equal to 18 ft/s, and a final gauge pressure of point 1 equal
to 558 lb/ft^2.
One thing that I want to point out is the unit that we are using.
Since the density reported is in slugs per feet cubed.
We need to convert this to our lbf.
So just a reminder here that is why I have this arrow.
Our lb/ft^2 is the same thing as 1 slug/ft*s^2, because the conversion is that one slug times
1 ft/s^2 is the same thing as a lbf.
So just a reminder when we are using these types of units.
So know we can plug these values in to our first equation that we had up here to calculate
the force in the x-direction.
So let me rewrite that.
So here I have our final equation to calculate our force.
I have plugged in all our numbers.
When I calculate this I get a force of negative 4880 lbf.
So since our force is negative what does that imply.
We have our water flowing into our tee from left to right.
When we declared our nomenclature for this system we said that positive y was up, and
positive x was to the right.
So a negative force means that the force acting on the fluid is the the left.
Meaning the force of the fluid acting on the tee is to the right.
If the inquiring force was not here.
So if this force was not here to hold that tee in place.
The force of the water acting on this tee would just shoot it off to the right, So that
is what our negative force implies.
The force required to anchor that tee in place.
Now lets work through the y-component of the force.
Now for the y-component of the force exerted by the pipe to the tee.
We use the y-component of the linear momentum equation, which is pretty straight forward.
We look back at our diagram and the only component we have moving in the y-direction is at the
control surface 3.
So pretty much the force in the y-direction is going to be equal to the momentum leaving
the system.
We don't have any pressure force in the y-direction, because at control surface 3 it is atmosphere.
There is no pressure acting against this control volume in the y-direction.
So we could write this as the velocity at 3.
Times the density of the fluid times the velocity, times the area, must be the force in the y-direction.
We can plug in some numbers.
We have 30 ft/s.
We square this, times our density of 1.94 slugs/ft^3, and our area of 2 feet squared.
This gives us a force of 3500 roughly lbf, and because it is positive it is acting in
the y-direction up, which means that the force the pipe is acting on the tee in the y-direction
is to, is upwards.
So hopefully this gives you an a good idea on how to calculate a momentum balance using
the integral method.
So hopefully this gives you a good idea on how to use the integral method of the momentum
balance to determine the forces on a system.
To determine the forces in a control volume.