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Changing the size of the Pipe
Changing the size of the Pipe
How does changing the pipe size effect flow rate?
This GIF shows the flow of RBC through capillaries.
Flow Rate Exploration
Flow Rate Exploration
This simulation has a pipe with two sections: a first section, which we will not change during our experiment, and a second section, whose radius we will change.
Change the height of the pipes so that they both are at a height of 3 m. See the diagram below for what this looks like.
Turn on “Show Pressure & Velocity values.”
Record the radius of the second section and the water’s velocity in the second section.
Change the radius of the second section. Record the new velocity.
Repeat 5 until you have filled the data table.
On the Desmos graph below, plot the velocity of the water (y_1) in the second section as a function of the radius of the pipe (x_1) in the second section.
What kind of relationship does your graph show? Select one.
___ Linear ___ Quadratic ___ Inverse ___ Inverse square
Using both the graph and specific data you collected, justify your selection.
2. A student decides to try and linearize their data. Which example best linearizes the data?
Briefly justify your selection and discuss whether your selection agrees with the relationship you identified in question (1).
What kind of relationship does the radius of the pipe and velocity of water within it have?
If the radius of the pipe is doubled, what will happen to the velocity of water traveling through it?
If we increase the velocity of the water in section 1, what happens to the velocity of the water in section 2?
Does the height of the pipe change the velocity of the water?
Differences in Pressure Gives Rise to Fluid Flow
Differences in Pressure Gives Rise to Fluid Flow
One example of this pressure difference is a pipe releasing water from a large tank
The end of the pipe at the bottom of the tank experiences a large gauge pressure
The end of the pipe open to the air only experiences atmospheric pressure
The pressure difference causes fluid to flow
A pressure difference is required for a fluid to flow
Consider a large tank of water with an outlet pipe at the bottom, the water flows out of it because:
the end of the pipe attached to the tank experiences a higher gauge pressure due to the large height of water
the end of the pipe open to the air experiences atmospheric pressure only
If pressure was equal on each end, no net force would be exerted on the water, and it would not flow
Mass Flow Rate & Volume Flow Rate
Mass Flow Rate & Volume Flow Rate
Mass flow rate is a measurement of how much mass of water is traveling through pipe or system over time.
Volume flow rate is a measurement of how much volume of water is traveling through a pipe or system over time.
Water is non-compressible. That means that it does not change its density even as the pressure or temperature changes. This means that if the mass flow rate of the water is conserved (due to conservation of mass) then the volume flow rate of the water is also conserved.
The law of conservation of mass states that mass cannot be created or destroyed.
Volume Flow Rate
Volume Flow Rate
The volume flow rate, Q, is a measurement of how much volume of water is traveling through a location at a certain time. The units are m3/s.
If a pipe is full of flowing water, due to the law of conservation of mass & water’s non-compressible nature, this rate is conserved throughout a pipe. This means that if the area of a pipe decreases, the velocity of the flow must increase.
Continuity Equation
Continuity Equation
The equation to the right is called the continuity equation. It allows you to calculate the relative velocities in two sequential sections of pipes, based on their areas.
A is the cross-sectional area
v is the velocity of the water’s flow
Conservation of Mass
Conservation of Mass
The fluid in an open-ended tube is incompressible and flows steadily
This means the total mass of fluid in the pipe at any one time is constant
Mass is conserved, so the rate at which mass passes any point is constant
If 15 kg of fluid enters the pipe in 2 s, 15 kg must exit the pipe in the same 2 s
Density is constant, so this means the rate at which volume passes a given point is also constant
The Cross-sectional Area of the Pipe
The Cross-sectional Area of the Pipe
We can use geometry to solve for the cross-sectional area.
Most pipes have cross sectional areas that are circular. This means we can calculate cross-sectional area with:
This means we can modify our continuity equation thusly:
Did you know there is a geometry section of your formula sheet?
Flow Rate Problems
Flow Rate Problems
Water flows through a tube with a diameter of 2m at a rate of 800 kg/s. What is the velocity of the water? Ans: 0.25m/s
What is the volumetric flow rate of oil in a 12m diameter pipe? The velocity of the oil is 5 m/s. Ans: 565.2 m^3/s
Suppose that water flows from a pipe with a diameter of 1m into another pipe of diameter 0.5m. If the speed of water in the first pipe is 5 m/s, what is the speed in the second pipe? Ans: 20 m/s
An incompressible fluid flows through a pipe. At location 1 along the pipe, the volume flow rate is 10 m^3/s. At location 2 along the pipe, the area halves. What is the volume flow rate at location 2? When the area halves, the velocity of the fluid will double. However, the volume flow rate (the product of these two quantities) will remain the same. In other words, the volume of water flowing through location 1 per second is the same as the volume of water flowing through location 2 per second.
A human will have an average blood velocity of 40 cm/s in their aorta. The diameter of their aorta is about 22 mm. What is the flow rate of the blood through the aorta in L/min? If there is a blockage in the aorta that limits the diameter to 15 mm, what is the new velocity needed to keep the flow rate constant?
A student is studying a continuous piece of round pipe that has two sections. The first section has a known radius and the second section has an unknown radius. The pipes are full of running water.
The student measures the velocity of the water in the first section, v1, and the velocity of the water in the second section, v2. Then, they increase the velocity of the incoming water, and remeasure v1 and v2.
They plot v1 as a function of v2 and draw a linear best-fit line based on their data. How can they use their best-fit line to calculate the missing radius?
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