Fluid Mechanics Lesson 02F: Manometers
Summary
TLDRIn this educational video, the presenter explores the function and operation of a manometer, a device used to measure pressure differences. They conduct a hands-on demonstration with a U-tube manometer, explaining how blowing or sucking into it creates a height difference in the fluid. The video delves into hydrostatic equations, illustrating how to calculate gauge pressure and absolute pressure using manometers filled with various fluids. It also addresses the impact of fluid densities and manometer design on pressure readings, offering practical tips for accurate measurements and emphasizing the importance of not simplifying equations when densities are significantly different.
Takeaways
- 🔍 The purpose of a manometer is to measure unknown pressure or pressure differences.
- 💧 Manometers can be filled with any fluid and can be of any shape.
- 📐 The key equation used for analyzing manometers is the hydrostatic pressure equation, ΔP = ρ g h.
- 🌀 A YouTube manometer demonstration shows that blowing or sucking into a tube can create a height difference in the fluid.
- 🌡 When calculating gauge pressure, one side of the manometer is open to the atmosphere.
- 📉 In a U-tube manometer, the manometer fluid must be denser than the fluid being measured to stay at the bottom.
- 🔄 The process of solving manometer problems involves selecting a reference point and moving around the tube, accounting for pressure changes when moving up or down.
- ⚖️ The height difference in a manometer (delta z) does not depend on the diameter or length of the U-tube, as long as capillary effects are negligible.
- 📏 Inclined manometers offer better resolution because they have more tick marks per unit of height compared to vertical ones.
- 📋 The vertical location of the manometer can affect the elevation difference if the fluid above the manometer is not air.
- 🚫 It's advised not to make approximations when the fluid being measured is a liquid, as the effect can be significant and lead to errors.
Q & A
What is the primary purpose of a manometer?
-The primary purpose of a manometer is to measure an unknown pressure or a pressure difference.
What is the key equation used for hydrostatics in the context of manometers?
-The key equation used for hydrostatics in manometers is the workhorse equation for hydrostatics, which is delta p = rho * g * h.
How does a U-tube manometer demonstrate pressure changes when blowing or sucking into a tube?
-A U-tube manometer demonstrates pressure changes by showing a height difference between the left and right legs of the manometer when blowing or sucking into a tube, which corresponds to the gauge pressure.
What is the significance of the height difference observed in the manometer when the professor blows or sucks into the tube?
-The height difference observed in the manometer when blowing or sucking into the tube signifies the gauge pressure in the mouth, which is the difference in pressure between the atmosphere and the mouth.
Why does the manometer fluid have to be denser than the fluid being measured?
-The manometer fluid has to be denser than the fluid being measured to ensure that the manometer fluid stays at the bottom of the U-tube and provides an accurate pressure reading.
What is the significance of the points labeled 1, 1', and 2 in the manometer analysis?
-The points labeled 1, 1', and 2 are used to apply the hydrostatic equation to calculate the absolute and gauge pressures in the manometer. These points represent different pressure levels within the manometer system.
Why is it important to consider the direction of fluid movement (up or down) when applying the hydrostatic equation?
-The direction of fluid movement is important because it determines whether to add or subtract pressure in the hydrostatic equation. Going down increases pressure (add), while going up decreases pressure (subtract).
What is the general formula derived for calculating the pressure difference in a manometer?
-The general formula derived for calculating the pressure difference in a manometer is delta p = (rho_m - rho_a) * g * (z2 - z1) - rho_a * g * (za - z2), where rho represents fluid density, g is the acceleration due to gravity, and z represents elevation.
Why is it advised to not simplify the equation by neglecting rho_a when rho_a is very small compared to rho_m?
-It is advised not to simplify the equation by neglecting rho_a because even if rho_a is small, it can become significant in certain situations, and neglecting it could lead to errors in pressure calculations.
How does the shape of the U-tube manometer affect the elevation difference (delta z)?
-The shape of the U-tube manometer does not affect the elevation difference (delta z) as long as the tube diameter is large enough to neglect capillary effects and the tube is long enough to include the delta z.
What is the advantage of an inclined manometer over a vertical one?
-An inclined manometer has the advantage of better resolution because it allows for more tick marks per centimeter, providing a more precise measurement of the height difference.
Outlines
📏 Introduction to Manometers
This paragraph introduces the concept of a manometer, a device used to measure unknown pressure or pressure differences. The manometer can be of any shape and contain various fluids. The lesson includes a demonstration of a simple manometer, known as a U-tube manometer, which is used to show the effect of blowing or sucking air into a tube connected to a fluid column. The demonstration illustrates how the height difference in the fluid can be used to calculate gauge pressure, which is the pressure relative to atmospheric pressure. The formula used for this calculation is the hydrostatic equation, which is the fundamental equation for hydrostatics. The lesson also includes an example problem that demonstrates how to use the hydrostatic equation to calculate absolute and gauge pressures in a U-tube manometer with a high-pressure tank connected to one leg of the manometer.
🔍 Analyzing Manometers with Different Fluids
This paragraph delves into the analysis of manometers containing different fluids, specifically when the manometer fluid is denser than the fluid in the tank. The paragraph explains how to use the hydrostatic equation to calculate the pressure difference between two points in a system with two tanks and three different fluids. It emphasizes the importance of not approximating the densities of the fluids, especially when the difference is significant, as it can lead to errors. The paragraph also discusses the general case where the densities of the two fluids are not equal and provides a formula for calculating the pressure difference. Additionally, it highlights the importance of including all terms in calculations, even if they seem negligible, as they may become significant in different scenarios. The paragraph concludes with a note on the importance of using the full equation for accurate results, especially when using software like Excel or MATLAB.
📐 Factors Affecting Manometer Readings
This paragraph discusses various factors that can affect the readings of a manometer, such as the type of manometer fluid, the vertical location of the manometer, and the densities of the fluids involved. It explains that the elevation difference (delta z) in a U-tube manometer does not depend on the diameter or length of the U-tube, as long as capillary effects are negligible. The paragraph also points out that the shape of the manometer does not affect the readings, and that an inclined manometer can provide better resolution due to a higher number of tick marks per centimeter. However, it notes that the vertical location of the manometer can affect the readings, as moving the manometer to a different vertical position can change the pressure experienced by the manometer fluid, thus altering the elevation difference. The paragraph also cautions against approximating that the elevation differences are the same when one fluid is a gas and the other is a liquid, as this can lead to significant errors if both fluids are liquids. The paragraph concludes with advice to avoid approximations to prevent future errors and encourages viewers to subscribe to the YouTube channel for more educational content.
Mindmap
Keywords
💡Manometer
💡Hydrostatics
💡Gauge Pressure
💡U-tube Manometer
💡Density
💡Absolute Pressure
💡Hydrostatic Equation
💡Resolution
💡Inclination
💡Capillary Effects
💡Reservoir
Highlights
A manometer is used to measure unknown pressure or pressure differences.
The fundamental equation for hydrostatics is essential for understanding manometers.
Manometers can be of any shape and contain various types of fluids.
Demonstration of a simple U-tube manometer with a ruler.
Blowing into the tube results in a height difference due to pressure.
Suction into the tube also creates a height difference, indicating negative pressure.
Gauge pressure in the mouth can be calculated using the equation delta p = rho g h.
Conversion factors are necessary when calculating pressure in different units.
The manometer fluid's density must be greater than the fluid being measured to ensure stability.
Labeling points in the manometer helps in solving hydrostatic problems.
Pressure changes are calculated by moving through the manometer, adding for downward movement and subtracting for upward.
The general case calculation for manometers does not require approximations about density differences.
In some cases, the density of fluid a can be neglected if it's much smaller than fluid m.
The elevation difference in a U-tube manometer does not depend on the tube's diameter.
The height difference and pressure difference in manometers will be the same regardless of their length.
The shape of the U-tube manometer does not affect the elevation difference.
An inclined manometer offers better resolution due to more tick marks per centimeter.
The vertical location of the manometer affects the elevation difference due to additional pressure from the fluid.
When fluid 1 is a gas, the elevation difference is often approximated to be the same as with fluid 2, but this can be significant if both are liquids.
It's advised not to make approximations to avoid future errors in manometer readings.
Transcripts
welcome to lesson 2f manometers
in this lesson we'll describe the
purpose of a manometer and we'll
demonstrate how it works we'll discuss a
simple way to analyze manometers they
can be of any shape and have any kind of
fluids in them we'll also do some
example problems along the way the
purpose of a manometer is to measure an
unknown pressure or a pressure
difference the only equation we need is
our workhorse equation for hydrostatics
here's a quick demonstration of a
youtube manometer
this is a simple youtube manometer with
a ruler in inches one side is connected
to a tube watch what happens when i blow
in the tube
i can maintain a height difference
between the left and right legs of about
eight inches i can also suck into the
tube
i still obtain the height difference of
about eight inches but this time in
suction this is the only time i let my
students say that professor symbala
sucks
a couple notes from the demonstration
when i was blowing i got eight inches of
water difference between the two legs
from this we can calculate the gauge
pressure in my mouth it's a gauge
pressure because one side of the
manometer is open to the atmosphere and
the other side is connected to my mouth
with a different pressure so delta p is
rho g h and this is a gage pressure we
plug in our values rho g and h along
with some unity conversion factors
converting from inches to meters and
then a newton is a kilogram meter per
second squared and a kpa is a thousand
newton per meter squared this gives us
1.993 kpa or to 2 significant digits 2.0
kpa when we apply suction we get the
same result except with the negative
sign since the pressure in my mouth was
less than atmospheric we can also write
this 2.0 kpa vacuum let's learn by
example suppose we have a u-tube
manometer where the right leg is exposed
to atmospheric pressure i drew the
little triangle to indicate that the
left leg is exposed to high pressure in
tank a through a tube liquid or gas that
is yellow here has density rho a the
manometer fluid is rho m rho m has to be
bigger than rho a or else the manometer
fluid wouldn't stay on the bottom as
shown here let's label some points 1 1
prime and 2. to solve this we use our
equation of hydrostatics for part a we
want to calculate the absolute and gage
pressures for the general case where rho
a is not small compared to rho m first
we know that p2 is equal to p atmosphere
since this surface is exposed to
atmospheric pressure p1 is equal to p1
prime since we can draw a curve from one
to one prime through the same fluid and
one and one primer at the same elevation
i'm going to show you an easy way to do
these kinds of problems namely you pick
a point and then work around the
manometer tube in this case i'm going to
start at point pa here and work around
counterclockwise you could start at 2
and work clockwise if you want from our
hydrostatic equation anytime you go down
you add pressure and any time you go up
you subtract pressure so starting at pa
i first go down to 0.1 so pa plus rho a
yellow fluid here g and then delta z is
za minus z2 that gets me to this point
now i'll go from here to here again i
add since we're going down we're still
in fluid a rho a g z two minus z one now
i'm at this point i go around to one
prime which is at the same pressure now
i'm going to go up to 0.2 when you go up
you subtract in this case row m since
it's the blue manometer fluid g i'm
still using absolute value of z
from this equation but i'm subtracting
since we're going up so this is z2 minus
z1 now we're at point 2 which is
atmospheric pressure as we stated here
we used plus signs when we were going
down pressure is increasing and we used
a negative sign when going up since
pressure is decreasing so we've worked
around from pa counterclockwise all the
way up to 2. we can simplify this
equation and solve for pa pa is p
atmosphere plus rho m minus rho a g
times the quantity z2 minus z1 minus rho
ag times the quantity za minus z2 this
is our answer for part a and it's the
general case where we haven't made any
approximations about the density
differences in some cases for example if
a is air and m is mercury the difference
between these two densities is huge and
you can neglect row a i actually don't
advise doing this solve for the general
case and then it works for any fluids
here regardless of how far apart the
densities are but if you want to
simplify when rho a is very small
compared to rho m we can neglect rho a
in this term we cannot neglect rho a in
this term since we don't know how z a
minus z 2 compares with the z2 minus z1
our approximate answer is then pa is
approximately p atmosphere plus rho mg
z2 minus z1 minus rho ag za minus z2 and
that's our answer to part b as i said i
would not advise you to do that just
leave this alone it doesn't hurt to keep
this term in a quick comment especially
if you're plugging this into some
software like excel or matlab include
this term even if it may be negligible
because somewhere down the line you may
have two fluids where row a is not very
small compared to row m and then you
would need that term it might save you
some time in the future so it is best to
keep all terms note that the answer is
in variable form you can plug in some
numbers yourself be careful with units
let me do another more complicated
example here we have two tanks tank a
and tank b three fluids row a row m the
manometer fluid and row b again we'll
calculate the general case and then we
can simplify this time for the case
where row a and row b are the same fluid
in general they are not and i colored
them different colors again we use our
hydrostatic equation again i'll start at
tank a and work around counterclockwise
pa plus row a g delta z
that gets us from here to this point now
let's go down distance h to this point
the fluid is now rho m and delta z is h
again we used plus signs since we're
going down and increasing pressure now
we're at this point which is the same as
this point and we go up from there i'll
first go from here up distance h the
fluid is now rho b so we subtract rho b
g h then we go up delta z and we're now
at pressure pb again we have negative
signs here since we're going up we want
to calculate the difference between pb
and pa so solving for pb minus pa and
rearranging we get rho m minus rho b gh
plus rho a minus rho b times g delta z
this is our general answer if fluids a
and b are the same these would both be
yellow for example same fluid and
obviously this term would go to zero and
so pb minus pa is approximately rho m
minus rho b or rho a times g h notice
that delta z dropped out of the equation
in part b since these are the same
fluids we can move these tanks up or
down as much as we want and it will not
change the result but in the general
case where rho b is not equal to row a
the delta z term may be important again
i advise not to approximate use this
full equation or general equation if
you're putting this into any kind of
software now i want to give some notes
about manometry the elevation difference
delta z in the u-tube manometer does not
depend on the following number one
u-tube diameter we have a caveat that
the tube diameter must be large enough
so that capillary effects are negligible
as long as that's true it doesn't matter
if the tube is small diameter or large
diameter comparing manometers a and b
the height difference and the pressure
difference that we calculate will be the
same by the way these t's ensure that
all the manometers experience the same
pressure from this pressure chamber a at
height z1 we can call that 0.1 and p
equal p1 so p is p1 there we're there
there we're there there we're there
we're there we're there since these are
all at the same elevation and they all
have the same fluids our workhorse
equation tells us that p is not a
function of x or y that would be the
horizontal direction only z therefore
diameter does not matter again as long
as it's not too small so that surface
tension effects are important elevation
difference delta z also does not depend
on u-tube length provided they're long
enough to include this delta z for
example compare a and c all we did was
have a shorter manometer why do we get
the same result below interface one
which is here nothing matters as long as
we have the same fluid this could be
some kind of a weird shape down here and
it still wouldn't matter because you can
connect curve from one point to the
other on opposite sides finally delta z
does not depend on youtube shape i
showed this already at the bottom of the
manometer tube but now if we compare a
and d d is drawn with this large portion
which we call a reservoir some
commercial manometers are built this way
so that we have a large volume of
manometer fluid and this level which we
called z1 does not change very much and
it can make it easier because you just
measure from there to the height of the
manometer fluid on the other side we
also have an inclined tube on the right
in manometer d what is the advantage of
the so-called inclined manometer well
let's think about it if we have a
vertical tube compared to an inclined
tube of the same diameter and we look at
some height difference let's say it's
one centimeter in both cases either you
have a ruler or you have tick marks on
the tube itself if i draw the tick marks
evenly spaced in both cases i didn't
draw this to scale but you can see that
i have approximately twice as many tick
marks here as i do here so if i'm
reading this height i have better
resolution with the inclined manometer
in other words there's more tick marks
per centimeter in this case compared to
this case which gives me better
resolution now let's look at some cases
where the elevation difference delta z
does depend on some properties first
manometer fluid here i use a different
manometer fluid suppose this one's water
and this one's mercury well these are
not to scale but the mercury is much
heavier than the water and so for a
certain rho gh we would have a bigger
height for the water than we would for
the mercury i'll label these positions 1
and 2. by the way which manometer a or e
would have better resolution well again
if you think of a tube with tick marks
and we have the same tick marks on our
tubes we have a much smaller height for
e than we have for a in other words we
have less tick marks to read so again we
have better resolution with manometer a
for the second case i drew manometer a
again except the whole manometer is
moved down i call this manometer a prime
same manometer fluids but now we have a
bigger elevation difference down to the
manometer delta z a prime will not be
the same as delta z a so vertical
location of the manometer does matter
note that i'm ignoring changes in
atmospheric pressure so it turns out
that delta z a prime is greater than
delta za this height is greater than
this height y well the yellow liquid has
some density we labeled that density rho
one and since we moved our manometer
down some distance i'll call that h we
have an additional rho one g h pressure
on the left leg of manometer a prime all
else being equal and ignoring changes in
atmospheric pressure from here to here
this higher pressure causes the
manometer fluid to rise higher therefore
delta z a prime is greater than delta z
a i can label these 1 prime and 2 prime
we approximate that p2 prime is equal to
p2 is equal to p atmosphere i summarize
by saying that the higher pressure and
manometer a prime pushes the blue
manometer fluid higher on the right side
compared to manometer a as i've sketched
here finally if fluid 1 is air for
example and fluid 2 is mercury then
these two delta z's would be
approximately the same we typically make
this approximation when fluid 1 is a gas
but the effect can be significant if
both of the fluids are liquids so again
i advise you not to make such
approximations to avoid future errors
thank you for watching this video please
subscribe to my youtube channel for more
videos
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