# Fluid Mechanics: Reynolds Transport Theorem, Conservation of Mass, Kinematics Examples (9 of 34)

>>All right. We’re going on to the last part

of this chapter, it’s a very important theorem that we need, we use it in multiple disciplines

if you’d have any 301 trigonamics, you’ve seen it there. It’s called a Reynolds Transport

Theorem a little background first. We’re going to be talking about system and control volume.

The first thing is the control volume, like it says here, is a fixed region in space.

So it’s a particular region in space. I mean you could analyze your hot water heater at

home in your garage maybe or in your house, as a control volume. You can analyze a boiler

as a control volume. All these pieces of equipment are typically analyzed by a control volume

appropriate. So rounding the control volume is a surface, a boundary. That surface is

called a control surface and these two guys are abbreviated CV and CS, control volume

and control surface. So we’re going to look at now mass crossing a boundary, which is

going to be maybe control surface. So we’ll start off and we’ll, here’s some flow passage.

So flow passage bounded on both sides. And we’ll identify by a dash line. The control

volume. The control surface is a boundary. We can call this for instance .1 and this

is .2. And maybe flow enters at 1 and flow will come out at 2. And we can also draw that,

look like this take that control volume out now, control it like this. Or it might look

something like this. So this is our control volume and we could have a velocity coming

out here. This would be v2, this would be v1. I’m going to hop to the side calculation

over here just, I’m going to use that calculation with derivation over here. So let’s look at

this real quick so you can get an idea for doing that. Let’s say there’s a pipe here.

And we have flow coming in and flow going out and we’ll just label that v. The pipe

is cut at an angle, the angle it makes with this is theta. We’re going to call this area

right here a and we’re going to call this inclined area a theta. A is the cross section

area of the pipe. So get you a piece of PVC pipe, cut it at a 45 degree angle and let

water run through it. The water’s going to come out just like this, but the area’s at

a diagonal like that. Thie flow rate for this q would be equal to v times a. Cross section

area, velocity. We mentioned this back away, we had q equals v times a, m dot equals row

times q equals row va so we’ve talked about that earlier in class. So the volumetric flow

by q equal the average velocity times the area. Okay let’s rewrite this thing a little

different. Write it in terms of a theta. Get rid of the a and make it a theta. So a theta

times the cosine of theta, same thing. Same thing if you do a little right triangle there.

This guy right here looks something like this, magnitude v, magnitude a times cosine theta.

That looks like the dot product. Okay so, if we want to deal with flow at q, it’s the

velocity vector dot with the area vector. Just want to make sure you know. This area

vector always points outward from the control volume. Like we mentioned, always points outward

from control volume. Okay we’ll use that over that there [inaudible]. So, if we want to

find q over here it’s equal to v dot a. Don’t forget the area vector always points outward

from the control volume. At 2 the area vector points outward from the control volume. At

1 the area vector points outward from the control volume. Okay that’s how you set it

up that way. Oh, that’s not, pardon me that’s q. If you want m dot, put on the board over

there, m dot equals row times q. Okay. I guess I better put down, let’s see here. Those vectors

point in the same direction. The angle between them is zero degrees they’re collinear. Okay

the angle between them is zero. Cosine is zero, is plus 1. So this sine here, cosine

theta equal cosine 0 equal plus 1. Over here, cosine theta equal 2 vectors pointing opposite,

the angle 180. Minus 1. For a dot product has a sine, if the flow comes in, it’s negative,

if the flow goes out it’s positive. The area vector always points outward from the control

volume. If you want to to label this again we’ll be real precise now, control surface.

All right so m dot is that. And then q net equal sum over the control surface of v dot

a and m dot net equal sum over the control surface row v dot a. Now don’t forget v dot

a is positive if stuff goes out, v dot a is negative if stuff comes in. So this is the

net out flow, m dot net out. What does net mean? Subtract something. The word net means

subtract something. The net balance in your checkbook, you got to subtract something.

What came in minus what goes out. In this case is what goes out minus what comes in.

So q net out then, in that case, would mean the flow rate leaving minus the flow rate

entering. The flow rate leaving minus the flow rate entering. Okay so that’s just some

background that we’ll need for that. Okay so let’s go on. If an extensive property.

Leaves the control volume, the dot means with respect of time. The rate of change of b with

respect of time, b dot. Oh b can be different variables. One of the most common ones is

energy. We’re going to do that in about two days, energy. So maybe capital b means energy.

Little b, is big b divided by mass. Capital b is extensive. Little b is intensive. So

let’s just for the sake of giving it some kind of name. Let’s say capital b is energy

in si joules. Little b is then joules per kilogram. Okay so that’s what we mean by that.

And if you want to be real official and if it’s not a uniform flow across the inlet and

outlet. If it’s not uniform flow. Okay. That’s uniform flow. That’s not uniform flow. Non-uniform

flow. You can use a summation. If the velocity profile varying across the flow of area then

you have to be more official and use the interval definition. Because then v is going to depend

on a, da. Okay so two possible forms of it. One for uniform flow, one for non-uniform

flow. We’ll pretty much deal with uniform flow for a while. So it’s going to be uniform

flow that we’re looking at. All right let’s draw our picture one more time. I’ll change

it a little bit. The flow passes right. I don’t know if you want a subtle line coming

out. What’s inside the solid line? Solid line is system at time t equals 0. That’s a system,

is the control volume. Control volume, a fixed region of space. A system, a collection of

matter, a bunch of molecules, of fixed identity. So if I had. Fixed system. Like this where

the flow comes in here and the flow goes out here somewhere. If I want to do a system,

I’ll say okay, I’m going to take. These particles and I’m going to spray paint them maybe. I

don’t know green. So here is, there’s a system at time t equals 0. This water in the pipeline.

Wait 15 seconds, it’s up there now, wait 30 seconds, it’s down here now. Wait a minute

and a half, it’s up here now. Wait a minute and 3/4’s, comes out here. There it is. Identify

every particle I’m going to make. Particle a, 1, b, 2, c, 10. Everyone’s tagged, named.

Fixed identity. We don’t do that generally in engineering. If I’m analyzing my hot water

heater in my garage, I’m not going to watch the water molecule come from the city water

supply, go into the water heater, get heated and end up going out the faucet in the kitchen,

no. I’m going to draw a system boundary around that hot water heater and I’m going to analyze

what comes in cold water and what goes out hot water. Now over here I’m going to draw

a control arm, I might say I’m going to look at this pipe fitting, it’s called an elbow

a 90 degree elbow. I’m going to analyze that 90 degree elbow. That now becomes a control

arm. Okay, back to here again. That’s like this. I’m going to follow those particles.

But. Reynolds Transport Theorem is a missing link type equation. Most of the laws in science,

physics and so on are written for a system. Okay. In engineering though we do a lot of

control volume studies. So this theorem allows us to take basic laws written for a system

and convert it to one more useful to us the engineers which are in control volume form.

Okay so what I’m going to do now, is I’m going to draw and relate the control volume to the

system. All right, I’m going to call that same volume at time 0, my control volume.

Okay so here’s my control volume. Okay that is also going to be my control volume. This

is the control volume for anytime, remember we said control volume fixed in space, not

moving in space, fixed, that’s the control volume. They choose to make the coincident

at times 0. Okay, wait 30 seconds, does the control volume move? No, I’ll say it again

fixed in space. Does a system move? Yeah I just showed you. Now the system moves to a

new location, down that way, cause the velocity is here. Okay now, where’s it now? Okay that

was a solid line, now it’s going to be a dash line. Dash line, system at time t, I think

I’ll call that, just dt, yeah helta t. Okay now way back over here. All right, next step

for

the system. Db dt in the system. The amount of b, hows the amount of b change with respect

to time in the system? Don’t forget if you get confused just say to yourself energy.

Okay, energy. If you watch a cubic inch of water come into your cold water pipe in hot

water heater and you watch it go through the hot water heater. Do you think it’s going

to change energy? Oh yeah. Why? Cause you’re adding heat from the natural gas burner. Yeah

as it moves through the hot water heater, it’s going to change its energy. Well b is

like energy, b can be various properties. Energy, momentum, mass. So this is the time

rate of change, the amount of b in the system. Start off, basic definition, what is a derivative

limit as delta t approaches 0. The amount of b in there at time t plus delta t minus

the amount of b in there at time t equals 0. T equal t, we won’t call that 0. We won’t

use a 0 just call it time t. Divided by, what was the time change, delta t. Equals. Okay

the amount of b in there at time t plus delta t. I’m going to call this region 1, this region

2, this region 3. At time t equals 0, there’s the system, black line. The amount of b in

region 1 plus the amount of b in region 2. Let’s put that here. Let’s do it this way.

I’ll put this down first and explain it. I’m going to keep this in the right order. Okay

the amount of b in region t, was the amount b1 plus b2. T plus dt, system at t plus dt.

Dash lines 2 plus 3, 2 plus 3. Okay. Keep going rearrange. Db, dt system. You can see

where the terms came from. Nothing new was added, it was just moving the terms around.

This term is this term, this term is that term, this term is that term, this term is

that term. Yeah just rearranging the two. Okay let’s look at this term first. The amount

of b in region 2, lets see what is region 2? Region 2, that’s right here, see in the

middle here. In region 2 at time t plus dt time t. Okay. All right so this guy right

here in the [inaudible] db cv dt. This guy right here, what went out minus what came

in. What went out minus what came in, the amount of b that went out minus the amount

of b that came in. Okay, so this term right here. Okay. So, I just erased it unfortunately,

maybe it’s still up here, yeah okay here right here. B row v dot a is what came out minus

what came in. So that’s what that term is. Let’s then combine these two into one equation.

We’ll put that here. Okay db system dt, d dt. Okay now we did this guy right here. This

is the amount of b in the control volume, the change of it with respect of time. This

is the amount of b in the control volume. So this is b in the control volume, integrate

over the control volume. Let’s see what the integration is. I’ll just pretend it’s si.

What is b? Okay let’s just say b is joules again to make something up, energy in joules.

So b is joules per kilogram. What is row? Kilograms per cubic meter. What’s the volume?

Cubic meters. When you’re done, what do you have? Joules. That’s the amount of b in the

control volume. That’s a function of time. That’s what the interval represents. If it’s

steady state of course, goodbye nothing but the time, that term goes out if we’re steady

state. Okay we’re almost there. This one, time rate of change of b in the system. This

one time rate of change. Of b in the control volume. This one, net outflow rate of b. Okay,

this guy here, if it’s not uniform flow, okay remember not uniform flow or if not uniform

flow. The we have the interval of b row v dot n da. This guy unit vector in the n direction.

Okay I can’t botch it in, there’s to much up there. That final equation right there,

in symbolic terms and word terms that is the Transport Theorem, Reynolds Transport Theorem.

What is it relate? Well it relates on the left hand side, so there’s the board of course.

Changes in the system, that’s the basic laws of science or generally expressed it. We engineers

want to express in control volume terms. So the right hand side is the control volume

terms. Control surface there, control volume here, give a steady state that term goes out.

Okay so that’s what you use for mass, momentum, and energy in chapter 5. Will that capital

b be mass first of all, then momentum and finally energy? And develop three equations

from that Reynolds Transport Theorem. Okay so you have an example for homework. Let me

see which one you got assigned. Yeah you have 68 and 69. I’m going to work problem 70. It’s

somewhat similar to 69. So let me, I don’t need this. I need that over there maybe, we’ll

see. Okay see what problem 470. Let me draw a picture first. Here’s the picture given

in the textbook. There’s two plates pulled in opposite directions. Let’s see here, the

speed of the plates are 1 foot per second. So this plate will be pulled this way, velocity

equal 1 foot per second. And the other one is being pulled to the left at 1 foot per

second. Okay, the oil is between the two plates. I’ll just put it there. Okay, velocity is

10y times I. Y is measured from the middle. And the distances

are 0.1 feet, [inaudible] talking about 2 feet. That’s, let’s just make sure that’s

what it is. Uh 470. No, 1/10th. Yeah 1/10th, 1/10th. Okay now let’s see if I can find.

Okay. Fixed control volume a, b, c, d. 2/10ths, 2/10ths, okay fixed control volume. Okay see

a, b, c, d. Good. That is showing this coincides, cv coincides with system at t equals 0. So,

here’s the system at times 0. Okay system at time. And the control volume, plenty of

time. Control volume is not moving, fixed control volume. Okay. Okay let’s see that’s

still given heat reading. Indicate the system at time t equal .2 seconds. So system at time

.2 seconds. It’s going to be a solid blue line. The velocity, okay let’s draw the velocity.

At the center y equals 0 velocity equals 0. At the top y equal 1/10th, v equal 1. I already

put it up there for you. At the bottom y equal minus 1, points left yeah v is 1. Between

the two, v is the function of y only, linear. Linear. So it looks something like this. There’s

the velocity field for the oil. After 2/10ths of a second, the oil molecule in the middle

between the plates move. No, because the velocity is 0. Did the oil particle at the very top

move? Yes. Which way? To the right. How much? It’s going at 1 foot per second. For how long?

2/10ths of a second. 1 times 2/10ths, it moved over 2/10ths of a foot. By the way I didn’t

put this on here but it started out at 2/10ths and 2/10ths here. This was given. Okay so

that top molecule is right there times 0. 2/10ths of a second, it moved over 2/10ths

times 1, 2 times second time 1 foot per second, 2/10ths of a foot. We’re going to start out

at 2/10ths of a foot. Where did it end up? Another 2/10ths out. This particle moved out

here, to there. Where did the middle part move? It didn’t move. How about the one down

here? It went the other direction. This guy right here, where the big blue point is. He

went to the left. How much? 1 foot per second time how long, 2/10ths of a second. He went

2/10ths that way. He’s back here. Now the guy up here, at point b on the plate or molecule.

Which way is here going? To the right. At what rate? 1 foot per second. For how long?

2/10ths of a second. He moved 2/10ths of a foot that way. He’s up here. The guy in the

middle, did he move? No. It’s linear of course. The guy at the bottom at point a, the oil

molecule, on in that plate. Did he move? Yes he did. What’s his velocity? 1 foot per second.

Which way? To the left. For how long? 2/10ths of a second. 2/10ths times 1. Okay he’s up

here now. 2/10ths, he’s right here. Now the blue on this chapter here. Better have multicolored

pens and catching techniques because you’re going to need it. This is not, don’t make

these things postage stamp size, you’ll be crazy when you get done, make them big. So

that’s the answer. Where is they system after 2/10ths of a second? In the blue boundaries.

It got skewed like this. For the last part, the last part says, identify the amount that

was leaving and the amount that came in. Okay this is 1. This is 1, this is 2, oh let’s

make that 3 make it sensible. We’ll make that 2. This is 4. 3 left, 2 out. So 1 and 3 out.

All these fluid molecules moved out of the control volume. What came into the control

volume? These guys along the edge they came in, these guys that were on the edge they

came in. 2 and 4 came in. So some fluid came in some fluid went out carrying the property

capital b. So again what did you start out with? The control volume and the system were

coincident. Then what happened? You wait 2/10ths of a second. Draw the new system about these.

I did the solid blue line. Third part, identify the amount of material that came in and went

out of the control volume. 1 came out of the control volume, 3 came out of the control

volume, 2 came in to the control volume, 4 came in to the control volume. Your homework

is just about like that but a different geometry. Identify the system after a certain time,

show the stuff that came in, the stuff that left, sketch it. And again just trying to

explain to you about the Reynolds Transport Theorem, what it means, what it means. Okay,

now our last step today. Our last step is, we start with number 1, we’re going to let

b be a certain quantity. I told you there was 3, it would be mass, momentum and energy.

We’re going to do all 3. Today we’ll take a easy one, mass. Better need it for your

hot water heater. If you got a gallon of cold water coming in from the city, guess what’s

going out to your dishwasher in the kitchen. A gallon. Why? It’s a steady state. What comes

in equal what goes out. So yeah you know some things are pretty straightforward. So, let’s

look at our mass then. We’ll call this continuity equation or conservation mass. Let capital

b equal the mass of system. Then little b equal mass of the system divided by mass.

So of course little b just becomes 1. Kilograms per kilogram, just 1. And Transport Theorem.

Says the following. Don’t forget now capital b is mass. D mass, dt of the system equal

d dt this is the control volume. Don’t forget little b is 1, there was a little b in there,

the little b was 1 plus interval over the control surface row v dot da. Okay, remember

on the board we had, still on there, yeah okay good. Our board, a system is a collection

of matter of fixed identity. I showed you the pipe. If there is 1 pound of water at

1 minute and you tag them with some kind of spray paint or something. After another 2

minutes, did their mass change? No because we’re following the same particles. So the

mass is not changing for what? A system. So this term here is 0. That’s the system approach,

it set conservation of mass. If we don’t lose particles or gain particles, we have the same

particles, then their mass didn’t change of the system. Okay, so now this becomes d dt,

interval control volume row dv. Don’t forget d slash is volume, v no slash is velocity.

Some people aren’t to good about that and they don’t make a little v very good, with

a big v. I saw an example. There’s a difference. Little v is the velocity in the, time form

of velocity, it’s in the y direction. What’s big v? It’s a velocity, it might be the average

velocity we don’t know, it’s a velocity. But this guy is a specific velocity and the v

slash is a volume. Okay anyway back to here. I’m going to assume uniform flow again to

make things simple as we go through. So our flow in whatever it is, in pipes whatever

it is. Is assumed to be uniform. It doesn’t change with the cross section area. So get

rid of these intervals and make it summation over the control surface of row v dot a. Okay

now if the flow is 1 dimensional and steady state. Okay. Okay then if that’s the case,

the steady state this guy 0, so this term here is 0, okay. Of row v dot a. All right

don’t forget those two terms at 0. I’ll put it down here. So if it’s steady and uniform

flow 1 dimensional, then we got this guy right here. Well don’t forget about there, I had

it before it’s in your notes. This guy here is a mass flow rate. Don’t forget if it leaves

it’s positive, if it comes in it’s negative. So this is m dot in equal m dot out. If you

want to do it officially and it looks like this. Outflow is positive so row out v out

a out minus, if it comes in it’s negative sign, row in v in a in equals 0. M dot out

minus m dot in equals 0. Which gives that guy right there. Okay. Next step if it’s incompressible.

Also besides b, 1d instead. Row in equal row out for steady flow. Okay so here we go cancel

out the rows, cancel out the rows. So v out a out equal v in a in. Our q out equal q in.

Okay. Okay brought some energy in. Okay those equations carry with them some assumptions

and they better be true or don’t use them. Number 1, steady flow, said it up here. Number

2, uniform flow, 1d flow. If those 2 are the assumptions you can make then m dot in equal

m dot out. Of what? The control volume. If the 2 is incompressible or can be treated

as incompressible, the density of this thing in and out. Then you can say the volumetric

flow rate in equal the volumetric flow rate out. That’s what I said if one gallon of water

goes in to your hot water heater cold water. Guess what happens? One gallon of hot water

comes out at the other end. Why? Because then your hot water heater, the flow is steady

1d and the fluid is incompressible. If those aren’t true you have to go back to the basics

like this. If it’s not steady you got to use the interval. If it’s not uniform, pardon

me, use the interval. If it’s not steady keep him in there. Okay. Okay so that is the conservation

of mass. Okay I just a good stopping point, I don’t want to do an example because I won’t

get finished with it in the time we have. So we’ll stop for today and pick it up then

on Monday.

## 29 Comments

## BEN KADİR · May 3, 2016 at 9:08 am

dont forget friends

V*A ; always outward from c.v

I'm glad to this lecture, ı dont know formula sign but I learn it.thank you

## Aditya gyawali · June 28, 2016 at 4:38 am

Best teacher really proud to have your lecture

## asdf asdf · November 16, 2016 at 3:52 am

I found this helpful and not as boring as it should be, thank you 🙂

## Loay Al Ahmad · March 11, 2017 at 7:22 pm

Better than my Prof.

## George Popovic · March 24, 2017 at 8:37 pm

great

## Albert P. · April 12, 2017 at 2:49 am

Wow this professor is amazing! Seriously thinking about going to grad school in Pomona now

## Bougriou Cherif · August 24, 2017 at 10:45 am

استاذ لطيف

## edward · September 8, 2017 at 10:23 am

Most of the fluid is coming from McDonalds.

## jimmys fox · September 17, 2017 at 11:00 am

Very helpful , easy to understand explanations !

## Jamal Crawford · October 14, 2017 at 2:39 am

I had no clue what my professor was doing in class and the book confused me.

This was extremely helpful and I understood the topic.

## Mariana Vieira · October 23, 2017 at 12:56 am

I’m studying aerospace engineering in Portugal, and all of these videos have been my salvation for my Fluid Mechanics class! Thank you very much!

## Ahmed Qassim · October 26, 2017 at 10:15 am

4 ch what the book that I can get this question from it>>>>>>please can u answer me

## Joseph D · November 9, 2017 at 6:08 am

I should have picked CPP over CSULB 🙁

I can actually understand this professor.

## Siraj Lassoued · November 29, 2017 at 7:35 pm

I have no words to describe how glad I am to found this professor's lectures … before it I was really struggling with this subject reading from books with too many details which made the process pretty slow .. now I got the interest of following the subject again and started to actually like it … definitely, i gonna follow his lectures when I start heat transfer course… thanks is not enough for you CPPMechEngTutorials you have done a great job I hope you can upload Fluid mechanics 2

## Nahom Ghirmay · January 23, 2018 at 2:50 am

<3

## louis david Nicoue · February 22, 2018 at 6:08 am

Student from Georgia tech here and you are making my life much easier

thanks

## 雷嘉骏 · July 27, 2018 at 6:08 pm

Q=V*A=V*Aθ*cosθ=|V|*|A|*cosθ=vector(V)*vector(A), this derivation starts from numerical numbers ended up with vector form, anyone could explain step 2 to 3 in this derivation I showed at beginning? why V*Aθ*cosθ=|V|*|A|*cosθ?

## Craig Stallard · September 15, 2018 at 6:39 am

Wow, an incredible teacher, why do most professors teach such an interesting subject in such a dry way generally?

## Siddhartha Raja · November 18, 2018 at 12:58 pm

6 dislikes by those who are studying one day before their exam

## Chuck Desylva · December 21, 2018 at 7:03 am

I appreciate his effort. Hmmm. I'd have done it different. Uh, well the 2 dimensional control volume. CVs are not 2 dimensional in reality. He's leaving stuff out. All I'm saying.

## ahmet hagif · January 5, 2019 at 6:17 pm

Neydi sebep sevgilim… Senden çektiğimmm…

## demetri rhodes · March 13, 2019 at 6:22 am

thank goodness for youtube!!

## Cihat Karadağ · March 31, 2019 at 2:24 pm

Thank u so much sir

Best Regards 🙂

## Teknikokulnet E-Teknik Eğitim · May 1, 2019 at 11:28 am

Reynolds transport theorem

https://youtu.be/EgncZL5t9G4

## Teknikokulnet E-Teknik Eğitim · May 1, 2019 at 11:35 am

https://www.udemy.com/is-transferi-temelleri-heat-transfer/?couponCode=HEATTRANSFER

## Explode · May 3, 2019 at 8:58 am

we are going to ignore the legend at 2:11 ??

## João Pedro Rodolfo · May 20, 2019 at 7:52 pm

You're amazing! Thanks a lot

## Nicholas Yazzie · July 12, 2019 at 2:50 am

I have an exam in 5 days and I'm still having trouble understanding this concept.

I am writing this before watching the lecture.

Be back later to tell you if it cleared up anything for me. Toodaloo

## Necati Gören · July 19, 2019 at 6:02 pm

At 26:17 i don't understand how it equals the times rate of change of extensive property of control volume because it is only the difference of B2 property for dt and not also B1 . Can you help me please?