Fluid Mechanics Lesson 02F: Manometers

Summary

In 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

00:00

📏 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.

05:00

🔍 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.

10:01

📐 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.

Keywords

💡Manometer

A manometer is an instrument used to measure pressure, often the difference between two pressures. In the context of the video, it is used to demonstrate the principles of hydrostatic pressure. The video explains that manometers can be of any shape and contain various types of fluids. For example, the script describes a YouTube manometer and its operation when the presenter blows or sucks air into a tube, creating a height difference in the fluid.

💡Hydrostatics

Hydrostatics is the branch of fluid mechanics that deals with the behavior of fluids at rest. The video uses hydrostatics to explain how manometers work, specifically through the equation 'delta p is rho g h', which relates pressure difference (delta p) to fluid density (rho), gravitational acceleration (g), and height difference (h). This principle is used to calculate gauge pressure in the manometer.

💡Gauge Pressure

Gauge pressure is the pressure relative to the atmospheric pressure. The video uses the term when discussing the pressure in the presenter's mouth during the demonstration, which is different from atmospheric pressure. The calculation of gauge pressure is demonstrated using the height difference in the manometer fluid.

💡U-tube Manometer

A U-tube manometer is a type of manometer that has a U-shaped tube filled with a fluid. The video describes a U-tube manometer setup where one leg is exposed to atmospheric pressure and the other to a high-pressure tank. This setup is used to illustrate the calculation of absolute and gauge pressures.

💡Density

Density (rho) is a measure of mass per unit volume of a substance. In the video, the density of the manometer fluid and the fluid in the tank are crucial for calculating pressure differences. The script mentions that the manometer fluid's density must be greater than that of the fluid being measured to ensure the fluid stays at the bottom of the U-tube.

💡Absolute Pressure

Absolute pressure is the total pressure relative to a perfect vacuum. The video discusses calculating absolute pressure using the hydrostatic equation, starting from a reference point and moving around the manometer tube, considering the pressure changes due to the fluid's density and the height differences.

💡Hydrostatic Equation

The hydrostatic equation is a fundamental principle used to calculate pressure changes in fluids at rest due to gravity. The video explains how to apply this equation to manometers by moving around the tube and considering the changes in pressure when moving up or down in the fluid column.

💡Resolution

Resolution in the context of manometers refers to the ability to accurately read small changes in pressure. The video mentions that inclined manometers offer better resolution because they allow for more tick marks per unit length, making small changes in fluid height easier to measure.

💡Inclination

Inclination in the script refers to the angle at which a manometer tube is positioned. An inclined manometer is shown to provide better resolution due to the increased number of tick marks per unit length compared to a vertical manometer.

💡Capillary Effects

Capillary effects are the result of surface tension causing a liquid to rise or fall in a narrow tube. The video mentions that the U-tube diameter must be large enough to ensure that capillary effects are negligible, which is important for accurate pressure measurements.

💡Reservoir

A reservoir in the context of manometers refers to a large volume of manometer fluid that helps stabilize the fluid level and makes readings easier. The video describes a manometer with a reservoir, which allows for a more stable measurement due to the reduced fluctuation in fluid level.

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

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