Worksheet on water flow in a pipe

Dan Young and Dawn Meredith 11/2014 page 1/10

Worksheet on water flow in a pipe

1. As we start this worksheet, it’s important to remind ourselves of the differences between liquids (like water) and gases.

a. Compressibility: i. Water is [choose one] more/equally/less compressible than air. ii. Do you have everyday experiences of the compressibility of gas or

water? If so, what are they?

iii. Let’s put some numbers to compressibility. Consider two “blocks” of water with the same number of molecules, one at the top of a lake at atmospheric pressure (about 105Pa) and one 10 meters down in the lake (about 2 x 105Pa). The percent change in volume is given by the following relationship:

𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑉𝑜𝑙𝑢𝑚𝑒 =Δ𝑉𝑉 =

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒2 𝑥 10!𝑃𝑎

(the number 2 x 109Pa tells us how hard it is to compress). What is the fractional change in the volume of water? Is this big enough to notice by eye?

iv. Now let’s compare that to air. Imagine a “block” of air, one with P=

105Pa and the other with P= 2.0x105Pa. If the number of molecules and the temperature is the same, what is the ratio of V1 to V2? Is this difference big enough to notice by eye?

Page 2/10 Worksheet on water flow in a pipe

v. Given your answers to the last problem, comment on whether

each of the following statements is true or false, and how you know. 1. PV=nRT describes water as well as gases.

2. The volume of water really does not change much under normal conditions, so we can consider it essentially incompressible.

3. Water is more like wood than air as far as compressibility

goes – it can exert a large force (=pressure*area) without looking like it’s doing anything.

2. Thinking about liquid motion in a pipe that has a changing cross section. a. What happens to the velocity of the water when you put your finger

over the end of a water hose, closing off some of the nozzle?


b. Now we want to build on these experiences and make them quantitative. The sketch below shows a cross section of a pipe with varying cross section. Assume that the pipe width (coming out of the page) is the same all along the pipe, and is one block wide. Water totally fills the pipe, and is moving to the right. (Idea adapted from Knight’s “Physics for Scientists and Engineers”)

Six blocks of water pass by plane A in one second. (Shade in the blocks)

i. How many blocks must pass by plane B in one second assuming that water is incompressible? (Shade in the blocks)

ii. How many blocks must pass by plane C in one second? (Shade in the blocks)

iii. Explain your reasoning.

A B C


c. If the velocity of the fluid at plane A is 2cm/s, what is the velocity at plane B? At plane C? Hint: use velocity = Δx/Δt and the information on the previous page.

d. In describing fluid flow, the flow rate Q (volume/s = cm3/s) moving past a plane is an important parameter of the system. What is the flow rate past plane A if each block is 1 cm3? Calculate Q in two ways – both as volume/s and cross sectional area*velocity.

e. What is the value of Q at plane B (calculated in two ways)?

f. What is the value of Q at plane C (calculated in two ways)?

g. Are your previous three answers consistent with the following equation:

𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑄 = 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 = 𝐴𝑟𝑒𝑎 ∗ 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

If not, what evidence or experience says it should not be constant?


h. Consider water flowing at 3m/s in a large pipe. This large pipe splits into two

smaller pipes, each with half of the cross sectional area of the first pipe. In using the continuity equation to find the velocity in b, should we consider the cross sectional area in pipe b alone, or the total cross sectional area in b and c, or something else? Explain your reasoning

i. What is the velocity in each of the smaller pipes? j. Consider water flowing at 3m/s in a large pipe. If this large pipe splits into eight

smaller pipes, each with one quarter of the cross sectional area of the first pipe, what is the velocity in each of the smaller pipes? Explain how you know.

k. Let’s apply these ideas to blood flow. We want blood to flow fairly quickly in the aorta over the long distance from your heart to the individual cells, but once the blood is in the capillaries and needs to exchange nutrients, oxygen and carbon dioxide with the cells, the blood needs to move slowly to allow the diffusion of these molecules. Given this, is the total cross section of ALL the capillaries greater than, about equal to, or less than the cross section of the aorta?

a

b

c


3. The last question focused on why the velocity must change if the pipe cross section changes. In this question we will think about how the velocity changes in a pipe of varying cross section. On the next page you will see a sketch of the cross section of the pipe filled with water and flowing right, as well as several sets of axes. For this simple situation, we will assume there is no friction or viscosity in the pipe.

a. Pressure will be important to this discussion of how the velocity changes, so let’s begin with a quick review of pressure.

i. Considering yourself submerged in a swimming pool, where was the pressure exerted on you? (e.g. just on the top of your head? Just on your sides? All over?)

ii. What object exerts this pressure?

b. Now let’s return to velocity. Using what you learned in the last question about velocity, qualitatively sketch the velocity along the pipe (see the axes ahead 2 pages).. (Don’t worry about numbers, just note if velocity increases, decreases or stays the same.)

c. Using the definition of acceleration, and your velocity plots, sketch the acceleration along the pipe. (Don’t worry about numbers, just note if acceleration is positive, negative or zero.)

d. Is it pressure or pressure gradients that cause acceleration? Use the

example of water moving up a straw to justify your answer.


e. Given your last few answers, fill in the following table that asks you about pressure gradient between the points labeled on the horizontal axis.

Pressure gradient (subscript refers to position label)

Circle correct answer for sign of pressure gradient

P1-‐P2 Positive/zero/negative









f. Use the information about pressure gradients in your table, and the given initial P0 on the plot (the big black dot) to sketch the pressure in the pipe. (Don’t worry about numbers, just note if pressure increases, decreases or stays the same.)

g. Summarize your findings: Pressure is the largest in a pipe of moving liquid when the cross sectional area is the [choose one] largest/smallest.


velocity

acceleration

Pressure

Pipe

0

0

0

1 2 3 4 5 6 15 14 13 12 11 10 9 8 7


4. In this last section, we want to check if the results in the previous section are

consistent with other ideas we have seen before. a. Bernoulli’s equation (which you will see in lecture) given below

describes changes in pressure and velocity. (The version we give here is accurate only if the pipe is horizontal.)

𝑃! + 12𝜌!𝑣!

! = 𝑃! + 12𝜌!𝑣!

!

Here P is the pressure, ρ (rho) is the density of the fluid and v is the velocity of the fluid. The subscripts one and two indicate these values at two different locations in the pipe. In this question we will verify that this equation agrees with what you already determined qualitatively.

Given Bernoulli’s equation above, if v1 increases, what happens to P1

in order to keep the left side of the equation constant?

Does this agree with what you have on your sketches? Note that if we rewrite Bernoulli’s equation, it says that the work done by the pressure gradient changes the kinetic energy of the water. This means that Bernoulli’s equation can be looked at from the energy point of view as well as the forces point of view.

𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑏𝑦 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑃! − 𝑃! = 12𝜌!𝑣!

! −12𝜌!𝑣!

!


a. Another student says, “Using PV=nRT, the pressure should be lower in regions with larger volume, assuming that temperature and number of molecules are constant, because the water has more room to spread out.” Is this consistent with what you found? If not, what is the flaw in their argument?

b. A fellow student says “The pressure must be larger in regions of smaller pipe radius because pressure is force/area. Since force is the same, and area is smaller, the pressure must be bigger.” Is this consistent with what you found? If not, what is the flaw in their argument? Hint: can we accurately compare the force exerted by two different blocks of water just by looking at the water? (This one is tough! Spend no more than five minutes here, and at least write down a few ideas even if you don’t feel you have a complete answer.)

5. Putting it altogether. Look back over this worksheet and summarize the big

ideas you have been exploring. Write this summary for an imaginary classmate who missed lab this week.

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Worksheet on water flow in a pipe