Fluid Mechanics Laboratory

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Oct 24, 2013 (3 years and 5 months ago)

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Fluid Mechanics Laboratory


Peter Love, Physics Department
,

Haverford College © 2011




Introduction

Fluids


the generic term for substances that can flow (i.e., gases and liquids)


are
everywhere in the world around us: from the atmosphere that sustain
s us, to the
plumbing systems in our homes and workplaces, to the blood that flows through our
circulatory systems. Analyzing fluid behavior with high accuracy is a great challenge,
but a good general understanding of the behavior of fluids is possible us
ing relatively
simple theoretical constructs and experimental apparatuses. In this lab, we’ll explore
the basic properties of fluids at rest (the field of
hydrostatics
) and in motion
(
hydrodynamics
).


Experiment 1: Equations of State

In fluids, pressure,
temperature and volume are related to each other by an
equation of
state
. If you have studied basic chemistry, you may be familiar with the equation of
state of an “ideal gas”


a gas whose molecules interact only by (rare) direct collisions
rather than
through long
-
range forces. The equation of state for such a gas is known as
the
ideal gas law
, which appears as


PV = nRT = nkT.





(1)


Here
P

is pressure. The unit of pressure is Newtons per square meter (N/m
2
), and one
Newton per square meter is calle
d a Pascal, and 1000 Pascals is 1 Kilopascal, or kPa.
These are the units measured by the pressure sensor.
V

is volume (in m
3
),
n

is the
number of moles of atoms or molecules in the volume (where 1 mole = 6.02 × 10
23

molecules =
N
a
, which is Avogadro’s num
ber),
N

is the actual number of atoms or


molecules,
T

is the temperature (in Kelvin, K),
k

is “Boltzmann’s constant” (
k

= 1.38 ×
10
-
23
Joules/K) and
R

is the “gas constant.” The gas constant is related to Boltzmann’s
constant by
R

=
N
a

k
. [The Joule (J)
is a unit of energy, related to force by 1 J = 1 N ∙ 1 m
= 1 N∙ m.]


The ideal gas law shows, for example, that if a fixed amount of gas is compressed at
constant temperature then PV is a constant. Processes at constant temperature are
called
isothermal
. When you compress a gas you perform work on it and its temperature
will tend to rise (which is why bicycle pumps heat up when you use them). An
isothermal expansion or compression must be performed slowly, so that the heat
added to the gas can leak away
and the temperature of the gas remains the same as its
surroundings.

On your desk you should find the “Gas Law Apparatus” that will allow you to explore
the equation of state of air. This Apparatus is shown above. We don’t know the
number of moles or mol
ecules of air in the piston, so
n

and
N

are unknowns. But they
should remain constant during the compression process.

Start with the valve
open

and raise the plunger to near the top of its range; then close the
valve.
Slowly

turn the screw to move the
plunger down, and record the pressure and
volume at convenient points. If you change things gradually enough, the temperature


should remain essentially constant (at the ambient air temperature; variation by a few
tenths of a degree isn’t significant).


You can determine whether the air is accurately modeled by the ideal gas law by
confirming that the product
PV

is a constant. Plot P as a function of 1/V and perform a
power law fit (an Allometric fit in Origin under non
-
linear curve fitting). Print out th
is
plot showing the fit. Complete the section in the report form for Experiment 1.



Experiment 2: Hydrostatic pressure in Fluids

The hydrostatic pressure at the bottom of a column of fluid of height h is

p =
ρ
gh

where p is the pressure, g is the gravitational acceleration on the surface of the earth
and
ρ

is the fluid density. In this experiment you will verify this relationship by
measuring the pressure at varying heights in a narrow column of fluid.


The ap
paratus to do this is shown at the left. Connect this apparatus to


the fluid resevoir


first making sure that all the valves are closed
. The
valves in their open and closed configuration are shown below:













Valve open



Valve Closed


Open the valve at the very bottom of the tube to fill the column with water,
being
careful that water that flows out goes into a graduated cylinder
, then close the valve
again.

Next take six pressure mea
surements using the pressure sensor


attach the pressure
sensor to the valve, open the valve, take the pressure reading from Logger Pro (you can
just read it off, no need to collect data), then close the valve and remove the pressure
sensor.
Be careful no
t to get water in the pressure sensors
.

Measure the height from the surface of the water to the T junction in the column. Plot P
against h, attach the plot and complete the section in the report form for Experiment 2.


Experiment 3: Measuring viscosity in
Poiseuille Flow

As you have seen in the case of terminal velocity and various experiments where the
effects of air resistance are evident, fluids resist motion through them. This tendency to
resist motion varies from fluid to fluid. Oil and honey are harde
r to move through than
water. Viscosity is a material property that is a measure of the internal friction of fluids,
how much they resist flow.

For flow in a pipe, a pressure difference is required between the ends of the pipe in
order to produce a flow.
The fluid flows fastest in the center of the pipe, the flow
velocity is zero next to the walls and the flow profile is a parabola, as shown below.





The flow rate is the total volume of fluid per unit time that comes out of one end of the
pipe. As the pre
ssure difference increases, the flow rate increases. For Poiseuille flow,
the flow rate is related to the pressure difference by the Hagen
-
Poiseuille equation:





(3)

Where


is the flow rate,

P is the pressure difference between the ends of the pipe, L
is the length of the pipe, r is the radius of the pipe and


is the viscosity. The units of
viscosity are Pa s, or N s /m
2
. The viscosity of water is 10
-
3
Pa s at 20
o

C. Noti
ce that for
lower viscosity you get a larger flow rate for the same pressure difference.


On your lab bench there is a Poiseuille viscometer


a pipe connected to a reservoir of
fluid (the fluid is water with food coloring in it). There are two points at t
he end of the
pipe where you can attach the pressure sensor so that you can measure

P. You will
measure flow rate by taking a video of the fluid filling a graduated cylinder and
analyzing the video in VideoPoint, just as you did for the collisions lab. Make these
measurements one after the other


don’t try and do everything at once.

Pro
cedure


Pressure measurement



1.

In Logger Pro set the data collection parameters to collect ten samples per
second for twenty seconds.

2.

Connect the gas pressure sensors to the measurement points and open the valves
to connect the pressure sensors to the flow

line.

3.

With the end of the flow line pointing in the bucket, open the valve at the end of
the flow line so that the fluid begins to flow.

4.

Collect a set of pressure data.

5.

Stop the flow by closing the valve at the end of the flow line

6.

Create a new calculated

column that gives the pressure difference.

7.

Record the mean and standard deviation of the pressure difference using the
“statistics” option under analysis in logger pro.


Procedure


Flow rate measurement


1.

Make a video of the graduated cylinder filling up
with fluid.

2.

Edit the video in Videopoint Capture so that you have the graduated cylinder
filling over a 30mL range

3.

Select one frame out of every 15 so that you have around 50 frames.

4.

Save the video and open in VideoPoint

5.

Track the height of the liquid in V
ideoPoint and use the ruler function to convert
to mL using the scale visible in the video (there is no mL unit available so just
choose meters)

6.

Import your data into Origin and perform a linear fit to find the flow rate.



Once you have your flow rate and

pressure difference and errors, measure the distance
between the pressure measurement points. The radius of the pipe is 3mm. Download
the data analysis spreadsheet from blackboard


this will propagate the errors for you
and give you a viscosity measureme
nt and standard deviation.


Complete the section of the report form for Experiment 3





Experiment 4: Low pressure experiments

Perhaps the most obvious characteristic of a fluid (besides its density) is its
pressure
,
P
,
defined as force per unit area:
P

=

F
/
A
. The unit of pressure is Newtons per square


meter, and one Newton per square meter is called a Pascal. The layer of atmosphere
above us creates a pressure at ground level of roughly 101000 (N/m
2
), or 101 kPa.
That means that each square inch of your

body’s surface essentially is supporting
approximately 14.7 pounds worth of air. So why don’t we feel the pressure?


In certain situations we
are

aware of atmospheric pressure


for example, when we use
a suction cup to hold up some object. In expellin
g the air from the inside of the cup
when it is compressed, we create an imbalance between the pressure of the atmosphere
outside and the little remaining air inside. If we could remove all of the air from inside
the cup, and it had an area of one square
inch, then it could support an object weighing
up to 14.7 pounds.




On your lab bench you should see the setup shown above


you can connect the
pressure sensor and use the syringe to pump air out of the chamber. You should be
able to get the pressure

down to less than 10 kPa (for comparison, at an altitude of
100km, the edge of space, atmospheric pressure falls to 1 Pa).



Demo A


Inflation




When you breathe in your diaphragm lowers, expanding your chest cavity and
lowering the pressure. Air is for
ced into your lungs from the atmosphere. To
familiarize yourself with the vacuum chamber, perform the following demo. Place a
balloon with only a small amount of air in, and with the end tied off, in the pressure
chamber. Then evacuate air from the chamber

using the syringe. Observe the balloon


what you are seeing is exactly the same physics that causes your lungs to inflate when
you breathe in.


Demo B


Suction cups don’t suck


Suction is simply a pressure differential between one place and another


its better
understood in terms of a higher pressure pressing onto something rather than a low
pressure ``sucking’’ onto something. To show this, lets remove the higher pressure and
o
bserve what happens. Stick the suction cup to the side of the vacuum chamber


then
evacuate some of the air. What happens to the suction cup?



Demo C


Vapor pressure of water


The temperature at which a liquid boils is dependent on pressure. In the re
port form is
a plot showing how the boiling point of water depends on pressure. In this experiment
you will find the pressure at which water that is below 100 degrees Celcius boils. Take
some hot water in the small plastic cup and measure its temperature.
Then place it in
the vacuum chamber with the pressure sensor attached and pump out the air. Record
the pressure as a function of time and mark when the water starts to boil. Complete the
section in the report form for Experiment 4.