Heat Transfer (27) - Heat transfer in internal flows in tubes

Summary

This script discusses the transition from studying external flow over surfaces to internal flow within tubes and pipes, focusing on fluid dynamics and heat transfer. It explains the development of velocity and temperature boundary layers, leading to fully developed regions, and introduces two common heat transfer scenarios: constant wall heat flux and constant surface temperature. The importance of understanding flow patterns for engineering applications, such as in radiators and heat exchangers, is highlighted, along with the calculation of mean temperature and Reynolds number.

Takeaways

• 📚 The lecture transitions from Chapter 7, which focused on flow over external surfaces, to Chapter 8, which discusses internal flow within tubes, pipes, and cylinders.
• 🔍 The flow inside a circular tube is characterized by a uniform velocity at the entrance, with a boundary layer that thickens until it meets at the center line at a distance called x subscript FD (fully developed).
• 🌡️ In the context of heat transfer, the tube surface is assumed to be held at a constant temperature, while the fluid enters at a uniform inlet temperature, leading to the development of a thermal boundary layer.
• 🔥 Two distinct conditions for heat transfer in tubes are presented: constant wall heat flux and constant surface temperature, each resulting in different temperature profiles along the tube.
• ⚖️ The mean temperature of the fluid (Tm) is defined and calculated by integrating the product of the fluid properties across the tube's cross-sectional area.
• 🔑 The mean temperature (Tm) is crucial for determining the heat transfer rate in the tube, as it appears in the heat transfer equation alongside the surface temperature.
• 🔍 The concept of 'x fully developed' is introduced, which is the distance required for the flow to become fully developed after entering the tube, differing for velocity and temperature.
• 🌀 The Reynolds number is pivotal for determining whether the flow is laminar or turbulent, influencing the calculation of x fully developed and the nature of the flow and heat transfer.
• 🔢 The script emphasizes the practical importance of understanding flow patterns in engineering applications, such as in the design of heat exchangers and radiators.
• 🌡️ The temperature variation with distance in the tube is linear for constant wall heat flux and exponential for constant surface temperature, highlighting the different thermal behaviors.
• 💡 The lecture also touches on real-world applications, such as condensers in thermal systems and solar collectors, where the concepts of constant surface temperature and constant wall heat flux are relevant.

Q & A

What is the main focus of Chapter 8 in the transcript?

Chapter 8 focuses on flow inside tubes, pipes, and cylinders, specifically addressing internal flows and the associated heat transfer.

What is the significance of the term 'fully developed' in the context of fluid flow?

The term 'fully developed' refers to the point in the flow where the velocity boundary layers have grown and met at the center line of the tube, resulting in a parabolic velocity profile for laminar flow. Beyond this point, the flow is considered to be in the fully developed region.

What does the subscript 'M' stand for in the term 'u sub M'?

In the term 'u sub M', the subscript 'M' stands for 'mean', indicating the mean velocity of the fluid flow inside the tube.

How does the temperature profile change as the fluid moves through the tube in a constant wall temperature scenario?

In a constant wall temperature scenario, the temperature profile starts off non-uniform near the entrance of the tube, but as the fluid moves further down the tube, the profile becomes flatter until it eventually becomes totally flat, indicating that the fluid temperature has equilibrated across the cross-section to match the tube surface temperature.

What is the difference between the fully developed distance for velocity and the fully developed distance for temperature?

The fully developed distance for velocity (x subscript FD) is the point where the velocity profile becomes parabolic and does not change further downstream. The fully developed distance for temperature (x subscript FDT) is the point where the temperature profile becomes uniform across the tube's cross-section. These distances can be different, especially in laminar flow.

What are the two conditions that can occur in a tube with respect to heat transfer as mentioned in the transcript?

The two conditions that can occur in a tube with respect to heat transfer are constant wall heat flux and constant surface temperature. These conditions are relevant for different engineering applications and affect how the temperature varies along the tube.

How is the mean temperature of a fluid defined in the context of heat transfer?

The mean temperature (Tm) of a fluid is defined as the temperature that would result if the fluid in the tube at a certain location were to be mixed in a cup and then measured with a thermocouple after stirring, also known as the mixing cup temperature.

What is the importance of understanding the mean temperature in heat transfer calculations?

The mean temperature is important because it is used in heat transfer equations to determine the heat transfer rate between the fluid and the tube surface. It is a key variable in calculating the convective heat transfer coefficient (h).

How does the Reynolds number differ when considering flow inside tubes as opposed to flow over external surfaces?

For flow inside tubes, the Reynolds number is calculated using the mean velocity (UM), diameter (D), and dynamic viscosity (mu) of the fluid. In contrast, for flow over external surfaces like a flat plate, the Reynolds number is based on the freestream velocity (U infinity) and a characteristic length (l) such as the distance from the leading edge.

What are some real-world applications where constant wall heat flux and constant surface temperature conditions are relevant?

Constant wall heat flux is relevant in applications like solar collectors where a uniform heat is applied to the fluid. Constant surface temperature is applicable in condensers where the tube surface temperature is maintained at the saturation temperature of the condensing steam.

How does the fully developed distance for velocity and temperature vary with the Reynolds number?

For laminar flow, the fully developed distance for velocity is approximately 0.05 times the Reynolds number times the diameter, while for turbulent flow, a 'magic number' of 10 times the diameter is used for both velocity and temperature due to the highly mixed nature of turbulent flow.

Outlines

00:00

🔄 Transition to Internal Flow Dynamics

The script begins by transitioning from Chapter 7, which covered external flow over surfaces like flat plates and cylinders, to Chapter 8, focusing on internal flow within tubes, pipes, and cylinders. The fundamental concept introduced is the circular tube with a diameter 'D' and the tube axis 'x'. The importance of understanding fluid mechanics before addressing heat transfer is emphasized. The script explains the development of a velocity boundary layer around the tube perimeter, which thickens until it meets at the centerline at a distance 'x_sub_fd', marking the fully developed flow region. For laminar flow, a parabolic velocity profile is expected in this region. The script also introduces the concept of mean velocity 'u' and contrasts it with the freestream velocity of Chapter 7.

05:00

🌡️ Heat Transfer in Internal Flows

This paragraph delves into the heat transfer aspect of internal flows, assuming a constant tube surface temperature 'T_s' and an inlet fluid temperature 'T_i'. It describes the development of a thermal boundary layer similar to the velocity boundary layer, which meets at the centerline at a distance 'x_sub_fdt'. Beyond this point, the temperature profiles change shape, becoming flatter until they are uniform across the cross-section, indicating the fully developed temperature region. The script provides an illustrative example with the tube surface at 100°C and the fluid entering at 30°C, eventually reaching a uniform temperature across the tube. The paragraph also discusses two conditions for heat transfer in tubes: constant wall heat flux and constant surface temperature, introducing the concept of energy balance on a differential fluid element to derive temperature variations in each case.

10:13

📚 Mean Temperature and Its Calculation

The script introduces the concept of mean temperature 'T_m' in the context of internal flows, explaining its importance and how it's defined mathematically. It describes the mean temperature as the energy crossing a hand's width at a given location 'x', and provides a formula for calculating it by integrating the product of the fluid's specific heat 'c_p', density 'rho', and temperature 'T' across the cross-sectional area of the tube. The script also offers a less mathematical definition, referring to the mean temperature as the 'mixing cup temperature', which is conceptualized as the temperature of a fluid in a cup after mixing the contents from a cut section of the tube.

15:51

🔧 Applications of Constant Wall Temperature and Heat Flux

This paragraph discusses real-world applications of the two heat transfer conditions previously mentioned: constant wall temperature and constant wall heat flux. It provides examples such as condensers in thermal classes where steam condenses at the saturation temperature of the condenser, and solar collectors in deserts that use oil as a circulating fluid, which is kept warm by electrical heating tapes to maintain fluid viscosity. The importance of these conditions in engineering, particularly in designing heat exchangers and solar collectors, is highlighted.

20:54

♨️ Reynolds Number and Its Relevance

The script explains the concept of the Reynolds number in the context of internal flow, contrasting it with its use in external flow scenarios. It emphasizes the practical importance of the Reynolds number in determining flow patterns within tubes and the challenges of obtaining velocity data in real-world applications. Instead, mass flow rate is commonly provided, and the script suggests using an equation that incorporates mass flow rate to calculate the Reynolds number. The paragraph also touches on the variations in temperature profiles for constant surface temperature and constant wall heat flux conditions.

25:54

🚫 Potential Dangers of Increasing Fluid Temperature

This paragraph addresses the potential dangers associated with the increasing temperature of a fluid within a tube, such as the risk of vaporization or explosion. It reassures that the temperature will not exceed the tube surface temperature due to the exponential approach to the asymptote. The script differentiates between the characteristics of temperature increase in constant wall heat flux and constant surface temperature scenarios.

30:56

🔑 The Importance of Finding the Fully Developed Flow Region

The script discusses the importance of determining the fully developed flow region within a tube, providing equations to calculate the distance 'x_fully_developed' for both laminar and turbulent flows. It explains that for laminar flow, this distance is calculated using the Reynolds number and the tube's inner diameter, while for turbulent flow, a simplified approach using a 'magic number' of 10 is applied. The paragraph also emphasizes the significance of knowing whether the flow is in the entrance region or the fully developed region for effective heat transfer and engineering design.

36:04

📏 Calculation of Fully Developed Flow Regions for Different Fluids

This paragraph provides a practical calculation of the fully developed flow region for both water and air in a one-inch tube, using the previously introduced equations. It demonstrates that water requires a significantly longer tube length to reach the fully developed region compared to air. The script underscores the importance of these calculations for engineers in designing radiators and heat exchangers, particularly in understanding where the most heat transfer occurs within the tube.

41:04

🛠️ Final Thoughts on Flow Patterns and Engineering Design

The script concludes by reiterating the importance of understanding flow patterns and whether the flow is fully developed for effective engineering design. It provides a simplified equation for calculating the fully developed distances in turbulent flow conditions, emphasizing that these distances are independent of the Reynolds number once it exceeds the threshold for turbulence. The paragraph also mentions the textbook's recommendation to use the number 10 as a standard for these calculations in both academic and practical settings.

Keywords

💡Flow over external surfaces

This concept refers to the study of how fluids move over the outside of objects, such as flat plates and cylinders. In the video, it's contrasted with internal flows, which are the focus of Chapter 8. The script discusses how understanding flow over external surfaces is foundational before delving into more complex internal flow scenarios.

💡Internal flows

Internal flows are the movement of fluids within enclosed spaces, such as tubes, pipes, and cylinders. The script introduces Chapter 8 by shifting focus from external to internal flows, emphasizing the importance of understanding fluid dynamics within these confined spaces for applications like heat exchangers and solar collectors.

💡Mean velocity (u_m)

Mean velocity is the average speed of fluid flow within a system. In the script, it's denoted by 'u_m' and is used to describe the uniform velocity of fluid entering a tube. The concept is crucial for analyzing fluid dynamics and heat transfer within the tube, as it impacts how a boundary layer develops.

💡Boundary layer

A boundary layer is a thin layer of fluid where the velocity changes from zero at the surface of a tube to the bulk fluid velocity. The script explains how a boundary layer builds up around the perimeter of a tube and grows until it meets at the centerline, a phenomenon critical for understanding fluid behavior and heat transfer in internal flows.

💡Fully developed region

The fully developed region is the area in a flow where the velocity profile no longer changes. In the script, this region is described as being beyond a certain distance 'x_fd' from the tube entrance, where the boundary layers have merged. The concept is essential for analyzing the steady-state conditions in internal flows.

💡Constant wall heat flux

Constant wall heat flux refers to a scenario where the heat transfer rate per unit area at the wall remains constant along the tube. The script contrasts this with constant surface temperature, explaining how these conditions affect temperature profiles and heat transfer within the tube.

💡Constant surface temperature

This term describes a condition where the temperature of the tube surface is held constant. The script uses this concept to explore how temperature varies within the fluid as it flows through the tube, which is different from the scenario with constant wall heat flux.

💡Energy balance

Energy balance is a fundamental principle stating that the energy entering a system is equal to the energy leaving it, plus any energy stored. In the script, an energy balance is performed on a differential fluid element to derive how temperature changes within the tube under different heat transfer conditions.

💡Reynolds number

The Reynolds number is a dimensionless quantity used to predict flow patterns in fluid dynamics. In the script, it's used to differentiate between laminar and turbulent flow within a tube, which is crucial for understanding the behavior of internal flows and their heat transfer characteristics.

💡Mean temperature (T_m)

Mean temperature is the average temperature of the fluid within a system. The script explains how to calculate it using the energy equation and the integral across the cross-sectional area of the tube. It's important for understanding the thermal behavior of fluids in internal flows and is used in heat transfer equations.

💡Newton's law of cooling

Newton's law of cooling is a principle that describes the rate of heat transfer between a body and its surroundings due to a temperature difference. In the script, it's mentioned in the context of heat transfer from the tube surface to the fluid, highlighting its importance in thermal engineering applications.

Highlights

Transition from Chapter 7 to Chapter 8, focusing on flow over external surfaces to internal flows in tubes, pipes, and cylinders.

Introduction of the concept of a fully developed flow region in a tube, where boundary layers meet at the centerline.

Explanation of the difference between mean velocity (u_m) and freestream velocity (U_infinity) in the context of internal flow.

Development of the velocity boundary layer in a tube and its significance in reaching a fully developed flow.

Heat transfer analysis assuming a constant tube surface temperature and its effect on fluid temperature.

The concept of a boundary layer developing around the perimeter of a tube and its growth until it meets at the centerline.

Description of the temperature profile changes in the fully developed region of a tube.

Two conditions in a tube: constant wall heat flux and constant surface temperature, with their respective energy balances.

Linear temperature increase with x in the case of constant wall heat flux.

Exponential temperature variation with x for constant surface temperature conditions.

Definition and calculation of mean fluid temperature (T_m) in a tube using integration.

Practical applications of constant wall heat flux and constant surface temperature in engineering, such as condensers and solar collectors.

Importance of knowing the flow pattern (laminar or turbulent) and its impact on heat transfer in tubes.

Calculation of the Reynolds number for internal flow and its significance in determining flow patterns.

Determination of the fully developed distance for both velocity and temperature in laminar and turbulent flows.

The impact of the entrance region and fully developed region on heat transfer in tubes, critical for heat exchanger design.

The use of the Reynolds number to differentiate between laminar and turbulent flows and its practical implications.

The concept of x fully developed for both velocity and temperature and its calculation based on the Reynolds number.

Transcripts

All right, so let's shift gears

from Chapter 7, chapter 8.

Chapter 7 was flow over external surfaces.

We looked at flat plates

and we looked at cylinders,

circular cylinder, square cylinders.

Now we jump to Chapter 8.

Now we look at flow inside

of tubes and pipes and cylinders.

So internal flows Chapter 8.

So here's how we start the model.

We say that this picture up here,

this is a circular tube,

circular tube diameter.

This is the two diameter is

D. We major x along the tube axis.

Anytime we've got convection heat transfer,

we pretty much have to go

first to the fluid situation.

We have to understand what's going on in

fluid mechanics before we

tackle the heat transfer.

So here's the fluids picture and this might

be a review for you from your fluids class.

First of all, nomenclature

u sub m. Now, let's go back.

Chapter 7, flow over external surfaces.

The freestream velocity was

capital U subscript infinity.

Flow over a circular cylinder,

r square cylinder, capital V.

Those were their velocities.

Now we're in Chapter 8.

What's going to be the velocity now?

Well, little u,

we shift gears just a little you,

the M stands for mean, the mean velocity.

So here's the velocity coming into the tube.

It's uniform.

It comes into the circular tube.

A boundary layer builds up around

the perimeter of the circular tube.

This velocity boundary layer

grows thicker and thicker until

eventually it meets at

the center line of the two,

where it meets at

the center line of the tube,

the boundary layers meet.

That distance is called x

subscript f d. F d stands for

fully developed beyond that point.

Now, if this is laminar flow,

at that point, we would have

a parabolic velocity profile.

If we go down a distance down here,

we still have the same velocity profile.

Okay?

So this region where the velocity profiles

Don't change is called

the fully developed region.

The region where the boundary layers

develop and meet at the center line.

This distance is called

the X location where it's fully developed.

So x subscript FDI, fully developed distance.

Now, we shift gears to heat transfer. Okay?

Now look at temperatures.

Here's our situation here.

We're going to assume in this picture

that the tube surface is

held at a constant temperature. T sub S.

S stands for surface.

So the tube surface is

held at a constant value.

The fluid comes in T subscript I.

The I stands for inlet.

So this is the inlet mean temperature.

It's uniform. It comes in.

Just like flow over a flat plate.

A boundary layer develops around

the perimeter of the circular tube.

Eventually, the boundary layers

meet at the center line.

That distance is called x subscript f d t.

The T stands for temperature.

Void develop temperature distance.

Beyond that point, we

call it the fully developed region.

But the temperature profiles do change shape.

Here it is. Just after they meet.

Here's the temperature profile.

Go down a little weight on a tube.

Notice is getting flatter in the middle.

Go down further, eventually

becomes totally flat. Totally flat.

The example, let's say

the tube surface is held at

a 100 degrees C and the fluid

comes in at 30 degrees C. You know,

eventually all that fluid

is going to be at a 100 degrees C.

If you go down far enough.

If you go out far enough,

these profiles, temperature profile,

it'll become uniform at Ts,

the fluid is heated and

heated and heated until

its temperature totally across

the cross section is Ts.

Okay, so there's what

happens inside the tube

as it's developing flow.

Now, there are two conditions

that can occur in a tube.

And we'll mentioned these in

little more detail in a few minutes.

But for right now, we can

have a constant wall heat flux on the tube.

Constant wall heat flux on the tube.

This picture down here is for

a constant temperature, Ts doesn't change.

Okay? This one up here,

this stands for surface

heat flux on the surface.

Okay?

So now we're putting the same heat flux in

the tube as we go down in the x-direction.

So we're going to

try and find how the temperature varies.

So we do an energy balance

on a differential element.

So here's my little

differential fluid element.

It's a distance Dx comes in,

T mean, goes out, it gets hotter.

T mean plus d t mean.

All around the perimeter.

I'll just show it here.

But all around the perimeter,

we have a constant surface heat flux.

All around the perimeter.

The fluid's going to get hotter, okay?

We run it a little energy

balance on this differential fluid element.

First of all, how

much she comes into the fluid?

Well, here it is. Q double prime times da.

Watts per square meter times

square meters is watts.

What area does the heat flux act on?

The perimeter?

Times dx. That's the area

through which the heat flux goes,

p, dx, P is perimeter.

What does that do? It raises

the temperature of the fluid.

Here it is.

Thermo m dot c sub p DTM.

That's how much the temperature changed.

The inner ear.

So energy coming in from the heat source

equals the energy added to

the fluid to raising the temperature.

We then separate variables,

d t m divided by dx equal

Q as double try and

perimeter divided by m dot c sub p.

Got it.

The right hand side is

a constant q double prime

of constant perimeter,

a constant m dot a constant CCP, a constant.

Integrate this guy, d t

or t m as a function of

x equal the temperature coming in.

That's the constant of integration.

Plus this constant times a distance x.

Notice it's linear with x.

So the temperature increases linearly with x.

Now, we look at another case,

the second case for

constant surface temperature.

Go, you can go through

the same little exercise.

Take a differential element,

write an energy balance on it.

We're not going through it, which write

the energy balance, okay?

And you set it up just like over here,

except now the temperature

of the surface remains constant.

And we, I end up with C where R here it is.

Ts minus t m as a function of x divided by Ts

minus TMI exponential

px m

dot Cp h.

So this is of course exponential.

So now we have two cases

there of how the temperature varies.

The mean temperature varies with x.

If it's constant wall heat flux

is constant surface temperature.

Let's define what that

mean temperature is though.

Tm

mean fluid temperature.

Okay, so to do that,

we're going to bring, go back to Chapter 1.

E dot.

E dot is energy joules.

Dot means per time,

joules per second or watts.

So that e dot stands

for if I put my hand here.

That stands for the energy crossing my hand.

How much energy goes

across my hand at that x location.

Okay?

So let's write the expression then for that.

So E dot m

dot c sub p t

mean integral across the cross section area.

That's how we define the mean temperature.

Don't forget thermal energy transport

across that cross section is m

dot c sub p t. What's our temperature?

The mean temperature.

How did we get it?

We integrate across

the cross sectional area of what row?

You cp T, DA.

What's the cross sectional area?

It's circular.

What's the outside radius R naught.

Now, what's the differential element?

In a circular geometry like this?

We define the differential area

as this very thin strip like this.

That's da. What's the radius

to the center are

what's the thickness of the strip?

Dr. What's the area?

The circumference times DR.

So da circumference, two pi

r d r circumference pi times d two pi r,

d r. That goes in here.

So solve this guy for TM.

And that da,

da, da.

So you could find

that TM at any location along a two.

But you need to know how

the temperature varies with r

and how the mean velocity varies with our,

plug it in there, integrate.

And that will give you T mean.

That's mathematically how we

define the mean temperature.

If you want to less mathematical

definition of T mean.

Sometimes some people call that

the mixing cup temperature.

So if you would

If you would cut that tube at

a certain location and

let that fluid to say it's water.

Let that fluid go into a cup of water,

and then stir that cup of water and put

a thermocouple in there that's teaming.

It's sometimes called the

mixing cup temperature.

So if you want to,

not the mathematical definition, but in it,

in the world of the tube,

you would cut the

to let the water flow into a cup,

stir the cup up,

put a thermocouple in there.

And that temperature you get

would be the mean temperature.

Okay. So why is

the mean temperature important?

Well, Q Chapter 1,

h a t surface minus t m.

Yeah. Chapter 7.

What was in the parentheses?

External flow.

T surface minus T infinity.

This is Chapter 8.

What's in the parentheses now?

T surface minus the mean fluid temperature.

So we, there's our TM

in the heat transfer equation.

This is our t, m varies with x.

This is how Tm varies with x.

This is a mathematical definition of TM.

Newton's law of cooling.

This now becomes T mean at any x location.

Okay? One thing we

didn't cover that we should have

earlier is I meant to tell you

how you get that

x fully developed over there.

So let's do that now.

So let me go back.

Okay.

Maybe what we'll do,

We could do that, I guess.

Yeah, let's let's go back and do

that. I'm going to make some room here.

I want to save while I'm here.

Let me just go ahead and describe

to you these two cases.

Why are there two cases?

Why is it constant wall heat flux

and constant surface temperature?

There are two frequently found situations

in the real engineering world.

So that's why our author

presents these two cases.

There, there's a lot of

applications I'll take maybe this one.

I'll take this 1 first.

When in the real world,

might you have

a constant surface temperature?

Okay?

Constant tube surface temperature.

Well, I'll give you one case.

In your thermal class.

You've got a condenser

whose purpose in life is to condense steam

into liquid water so

the pumps can handle it and

pump it back into the system.

The steam comes in hot in the tubes.

Water.

Pacific Ocean, cool,

Redondo Beach powerplant,

Pacific Ocean water inside condenser.

Hot steam hits the outside the surface.

Condenses. At what temperature?

The saturation temperature of

the pressure in the condenser.

So the water droplets

condense at the saturation temperature.

They turn from steam to liquid water.

They then run down the tube in

here where

they're collected and pumped around.

And of course, guess what?

This is.

The temperature everywhere along here is

the saturation temperature of

the pressure in the condenser.

Okay, that's him.

That's what I'll use.

Now let's take the case of where,

where are we going to, who's going to put

a constant surface or wall?

This pi should be constant surface heat flux.

Well, there's lots of cases.

There are lots of cases.

I'll just give you one.

Some solar collectors out in

the desert use oil as a circulating fluid,

not water, because water

can freeze at low temperatures.

So they use oil in

the solar collector and then the oils

collected and taken to a point where then

it vaporizes the steam and create steam and

Turbine and blah, blah, blah, electricity.

But the sun goes down at night

and the oils in the pipes out in the field.

And the desert can get pretty cool at night.

And then in the morning the sun comes up

and the pumps go

on who they don't like high viscosity.

They don't like that HIV it

because that oil now has really viscous.

It's been CIT know nine out there

in a desert and getting cooler.

And that pump does not like

pumping that viscous cold oil.

So what they do

sometimes is they'll wrap the pipes,

the field with electrical tape

like blankets if you wish,

and trickle electricity through it to keep

the oil a little bit warmer

so the pumps don't get that big.

Boom when it turns on a morning.

That's constant while heat flux,

every meter of that little blanket

on those tubes generate

so many watts to keep

the oil warmer than it would

have been in the desert

in the middle of the night.

So yeah, that's that's the application there.

Maybe.

If you want to stretch it and just start

with a very simplistic model.

Besides what I just mentioned,

you can take a parabolic trough collector

carrying a fluid at the focal point,

they're fluid and here's

the two runs this way.

Here's the parabolic trough collector.

They're out in the Mojave Desert.

We drive by sometimes going to mammoth,

Highway 395 and Highway 58.

There's a big solar field out there.

They've got parabolic trough

collectors out there.

And the troughs track

the sun sunrise to sunset.

And here's the sun's rays coming in.

Who it almost looks like.

It's a uniform heat flux on

the tube from the parabolic trough collector.

Yeah, you could start

out your analysis that way.

It's not perfectly true,

but not a bad starting point maybe.

So, yeah, there are many applications

of both of these conditions.

That's why we engineers

are interested in both these guys.

They'll appear a lot in our studies.

Designing heat exchangers.

You're looking at solar collector

feels like that.

So yeah, big, big time situations.

So that's why we study both these guys.

Let's take a look.

These are just little bits

of odds and ends here, right, novel.

Let's look at another thing that I mentioned.

How do I, how do I get the Reynolds number?

Reynolds number d rho U M D

over mu i Chapter 7, what was it?

Flat plate.

U infinity l over

new circular R-square tubes.

D stands for distance, okay?

U infinity D over Nu.

Now we go to Chapter 8.

What's the Renaissance?

Chapter 80, flow inside

tubes from your fluids class.

Row, UM, D over mu.

And your fluids class, you probably

call this v. That's okay.

We call it, UM, in the heat transfer class.

You've had thermo and you know,

they in the second part of thermally,

they give you a big power plant and

there's a turbine owners

or condensed or nerves,

a boiler nerves, blah, blah, blah.

And they'll say, Okay,

fine, the temperature leaving whatever,

I don't care, 5'10 output of the turbine.

I guarantee you that nobody in

that no problem in that class with that.

It could be a gas turbine,

it could be a steam turbine.

They didn't say, Oh,

by the way, the velocity in that,

in that two is ten meters per second.

They never gave me the velocity and tubes.

Nobody in the real-world speaks like

that as yet has been people.

Yeah.

As a San Diego Gas and Electric people,

they don't speak that language.

They don't give you velocities in tubes.

What did they give you? You know what

they give you the mass flow rate.

Every problem and thermal with those guys,

they always give you a mass flow rate.

So we don't like that guy in tubes Chapter 8.

We don't like him. I want to

see mass flow rate net equation.

So go ahead. You can convert it and you

can end up with the mass flow rate in there.

And let's see, we got here.

For m dot.

Yeah. For m dot

over I think it was row.

Let me see.

I've got that. I'm better get that guy right.

Oh, I know where it is.

Here is not up there it is right

here for m dot pi mu d pi mu d.

So this is what you want to use pretty much.

You check out the homework.

Every homework problem gives

you the mass flow rate.

Don't waste your time converting that

to a velocity to put in the top equation,

that's a waste of valuable time on a midterm.

Use that equation to find out what

the Reynolds number is.

Okay, so let's take a look then.

At, let's see,

I think I'll put that over here.

Let's look at how the temperature varies.

Let's take,

I'll take that

C constant surface temperature.

Let's take him for constant wall heat flux.

Q double prime equal constant. Okay?

Okay.

And let's look

at, let's look at him.

T mean in, comes in T mean in.

How does a temperature vary with x linearly?

T-sql constant.

Cost of surface temperature.

Fluid comes in. Mean inlet temperature.

Hazard vary exponentially.

So there's our two temperature profiles that

go along with

those two boxed equations over there.

The author also throws in,

for interest's sake,

how the surface temperature varies here.

It comes up like this

than it is parallel to this guy here.

This is T surface x.

We don't have that equation

on the board here.

But he does mention that in the textbook.

We're more interested in how

the mean fluid temperature varies with x,

the two boxed equations over there.

So you gotta be a little bit careful here.

If you have a fluid in here

and what happens as it goes on the tube,

it keeps getting hotter and

hotter until something bad may happen.

You, you might have

vaporization of the fluid in there.

You may have an explosion

of steam or fluid in there.

This could be danger

because it keeps getting hotter

either until the two bins or something

bad happens down here.

Now, don't worry about

it too much because it's never going

to get hotter than

the tube surface temperature is exponential.

It approaches asymptotically.

So, yeah, there's different characteristics

of what happens in these two situations.

Okay. Now, here we go again,

Chapter 8. What's the whole point?

We gotta find h.

H is the key.

So that again is our, our big goal.

Chapters 6 and 7, it was the same thing.

Find H, chapter eight's the same thing.

Find h, which we're going to do.

But before we do that, let me,

I think I want to I didn't mentioned,

but I want to mention over there,

how do you find x fully developed?

So let's see if

we can get the equation there.

Here's my equation.

I'm going to put that I think right here.

Okay, Let's put that right here.

To find x fully developed.

Okay, let's again Reynolds number.

If the diameter is less than 2300,

laminar, Reynolds number greater than 2300.

Turbulent.

And x, this is for laminar flow.

Okay?

X fully developed divided by diameter.

The ID, the two is approximately

0.05, Reynolds number D.

And for the thermal boundary layer,

There's two fully developed regions.

Once for velocity, That's a guy up here.

And there's one for temperature,

that's the guy down here.

They can be different. For turbulent flow.

X fully developed over d is much easier.

X fully developed temperature over D.

Much easier because

turbulent flow is so mixed

up that there's no number in there.

It'll print a number in there.

The textbook says, we're going to

use the magic number 10 for both.

So same equation for

both because of the nature of turbulent flow.

Okay?

So let's play a little game here.

Let's say laminar flow.

Let's get x fully developed. 0.05.

Reynolds number d times the diameter.

0.5 times 1000.

D is 1,

2, 3, 50 diameters.

If diameter is one inch,

x fully developed, is equal to

50 inches, about four feet.

So there it is, a one-inch tube.

The tube is 1000 homes.

It take before the velocity boundary layers

to the center line and combined for

the fully developed region, about four feet.

Four feet.

Okay.

Let's do the temperature

1 x fully developed temperature.

Far what does it take?

Water. So I took water parentals six about.

So x fully developed.

For water is 6 times 50 is 300 D.

If D equal one inch x fully developed,

temperature is equal to 300 inches,

which is about 25 feet.

So water in a one inch

to laminar flow at 1000,

it would take 25 feet of a 102

before the boundary layers

developed to meet at the center line.

And then let's do air. For air.

Crandell, I took about,

I think 0.70 up.

There should be a temperature here too.

Oh, yeah, for air is not 25 feet,

like water is three feet.

What, why these guys Important?

0 because you'll things

depend on how rapidly

the flow develops in the tube.

The heat transfer, obviously.

Where do you think the biggest heat transfer

in the tube is right now on there?

Well, you know, on a flat plate,

I'll tell you where was the most

heat transfer in a flat plate?

Right at the front,

or x equals 0.

Big heat transfer here.

As boundary layer gets thicker and thicker,

the heat transfer goes

down because the boundary layer

serves as an insulating blanket of sorts.

So yeah, we engineers need to know

that in your automotive radiator,

you've got vertical tubes carrying water.

I guarantee you, it's

critically important to know if that flow

in those radiator tubes

is in the entrance region?

Totally.

Maybe a quickly develops where

most of the tube is

in the fully developed region.

Oh, that's critical for the designers of

the radiator and heat exchangers in

general to know what's going on there.

There's two things going on here.

Here and here. Velocity, temperature.

Assume Reynolds number as turbulent.

X fully developed, equal 10 times diameter.

X fully developed temperature,

ten times diameter.

If it's a one inch tube.

Both those fully developed distances

are 10 inches less than a foot.

Less than a foot?

At what Reynolds number?

I don't care.

There's no Reynolds number in the equation.

Here's the equation right here.

Do you see a Reynolds number? No.

But of course the Reynolds number has to

be greater than 2300.

That's the, that's the limiting

when we have for

any Reynolds number greater than 2300.

Though they are very, very simple.

The textbook, by the way,

says that this guy can vary between 1060.

He says, but we're going to

use 10 as the magic number.

So the textbook said use 10.

We're going to use tendon in our,

in our example problems

and homework and so on.

Okay, so yeah, it's critical for

the engineer to know about

the flow pattern and if

the flow is fully developed or not.

All right, this is

a good introduction stopping point,

so we'll stop for today if it came in late,

look at the board on the right for

our changes in homework and exam due dates.

And so then we'll see you on Wednesday.