>> A correction on the
class notes last time. We were doing an
example problem on carbon steel
cylinder transient heat conduction Chapter 5, Part C was find out
how much energy had been gained by the cylinder after a
certain amount of time. This was the
equation that we had but I didn't put
the volume down, I put the surface
area down. The volume of that
cylinder, of course, is Pi r squared times L. So make that change
in your notes, and here are the
two correct answers for Part C of that problem. That concludes
the four chapters on conduction
heat transfer, Chapters 2, 3, 4 and 5. Now we're going to jump to Chapters 12 and 13, radiation heat
transfer and comeback in the last third
of the course for Chapters 6, 7, 8 and 9, convection. So we're going to start Chapter 12, radiation
heat transfer. Chapter 12 is probably the hardest chapter to
read and understand. You've been through
that already, you're smiling, I see. It's a hard chapter to
read and understand. So what I'm saying is
take good class notes. I'm going to pick and choose all that chapter, what I think is important
for this course. Besides the fact of the three modes
of heat transfer, conduction, convection,
and radiation. Radiation is probably one that you have the
least experience with. All people experience conduction all the time, and especially convection
our whole life, what we build
a lot of times revolves around
convection heat transfer. But radiation is something that people don't
have a good handle or feel for. So let's first of
all define what we mean by thermal
radiation. Thermal radiation is the heat transfer part
of radiation. Thermal radiation, we're going to
consider this to be electromagnetic
radiation emitted by a body due to
its temperature. The two important things there are electromagnetic
radiation, and it's due to the
body's temperature. Electromagnetic spectrum, figure 12-3 in Chapter
12 looks like this. We're plotting
different parts of the spectrum
versus wavelength. Wavelength down
here, microns, 10 to the minus four, it's a log scale up
to 10_10 or 100. So first of all, short wavelengths, just
as a for instance, what's some short
wavelength? X-rays. How about
ultraviolet? Typically in
this particular part of the chapter, ultraviolet from
0.01 down to 0.4. Visible spectrum
that we see with our eyes, 0.4-0.7. Infrared long
wavelengths, 0.7-100. Where does thermal
radiation fit into this? Here's the thermal
radiation part of the electromagnetic
spectrum. It goes from 0.1-100. That's what we're
looking at. Now, we go back
to our Chapter 1. We had a very
brief introduction to thermal radiation
in Chapter 1, and we talked about the concept of
a black body. So we'll go back to Chapter 1 and
expand on that. So black body. I'm going to put down
two important concepts of the black body, it emits a maximum
possible radiation. Number 2, it absorbs
all incident radiation. There's a number
3 that says, it's also a diffuse surface but I don't want to discuss a diffuse surface
right now, so I'll get back
to that later on, but essentially a
black body absorbs all incident radiation and it emits the maximum
possible radiation. Now, how is that
radiation distributed? Well, it's distributed
according to something called the spectral black body emissive of power. I'll write the
equation down first then I'll
define this. E_b Lambda. There's two subscripts, there's a b and
there is a Lambda, capital E, Chapter 1, stands for emissive power. Units are watts per square meter per
micron here. Of course, b means black
body not a problem. Lambda means at a
certain wavelength. That's called spectral.
Spectral means wavelength dependent. You have to give me a wavelength right
there, Lambda, in order for me to give you an E_b Lambda is
wavelength dependent. Just a reminder, remember
anything to do with radiation every time we see a temperature has
to be at absolute. If we're given the
problem in degrees C, add 273 degree K. So
everything should be converted to absolute right away before you
start the problem. Otherwise, you
might end up making "A silly mistake." C_1 and C_2 are constants. See the text for values. So we can plot this. We can plot E_b
Lambda versus Lambda. I'm going to pick a
certain temperature. I'll say T is 1000 Kelvin. Now, if I say T
is 1000 Kelvin, I'm going to plot E_b
Lambda versus Lambda. The shape of the curve looks something like this. That's at a temperature
T_1 let's say. Now I'll choose a
different temperature. Let's say 500 Kelvin, curve looks like this. Let's say 300 Kelvin, curve looks like this, where T_1 is greater than T_2 is greater than T_3. I'm going to box this
guy in, by the way. That equations give it
a name, Planck's law. Now I noticed the
shape of those curves. Every curve seems to
go through a maximum, here, here, here. Now, I can connect
those maximums with a line, looks like that,
the dash line. If I do that then I get what's called the
displacement law, which is Lambda max T
equal a constant C_3. >> In SI 2,897.6. So that equation is the equation of that
dashed line right there. Now, don't get confused by the subscript maximum. That has nothing to do with the maximum
wavelength. You don't seem to say it. Here's what you say. This is the
wavelength at which the blackbody
spectral emissive power goes to a maximum. This is a wavelength where give me a temperature, I'll give you a wavelength where that curve
goes to a maximum. If the temperature is T_1, this equation gives me one. Goes straight down here, that gives me that value, which I call Lambda_max. If temperature is T_2, put T_2 in here. That'll give me where that equation goes
to a maximum, and so on. That's
what it means. It's the wavelength
at which the blackbody spectral
emissive power goes through a maximum. Now, we're going to integrate over
the whole curve, and also I'm going to leave that guy
up there for you. I want to find the
area under that curve. I want the emissive
power, E_b. I don't want the
Lambda on it. I want the total
emissive power. I integrate this guy. When I do that, I get a very simple expression. Integrated over
all wavelengths, the emissive power
becomes a constant Sigma times the
absolute temperature raised to the fourth power. You think, "Well,
you integrate this complex
expression here, and that's what you get?" Yeah, that's what you get. Of course, that's
Chapter 1. So now we have Chapter 1, Stefan-Boltzmann law. Yes. >> Is the C_3 from
a chart or is that one value [inaudible]? >> I'm sorry. Say again. >> Is C_3 from a chart? >> No, it's in the textbook. It's
not a chart though. >> Yeah. >> Yeah. Is that okay? >> Are there more
than that value? >> There's one in
English engineering too, which we don't care
about right now, but it's in the book. >> Okay. >> Yeah. Thank you. So
that is what we have. We integrate over
the whole area. Now, we can also say, what if we don't want to integrate over
the whole area, but we just want
to integrate over part of the area? So here's the curve for
a certain temperature T. I want to find out how much energy
goes from zero to here. That Lambda can
be any value. I don't want to
confuse things there. That area right there. So that area right there is
this integral from zero to Lambda
E_b Lambda, d Lambda. F stands for fraction. I'm going to write an expression for the
fraction of energy, this area, divided
by the total area. That's the fraction.
So integral zero to Lambda E_b Lambda, d Lambda divided by
integral zero to infinity E_b
Lambda, d Lambda. I'm going to save some
space here and put that down here. I
need more space. That gives me the fraction. Now, what do you have to integrate from zero to some Lambda to do that? Well, the book gives you the results that had
been done for you. So luckily, you don't need to do any integration. Table 12.2 gives you
one column title, Lambda T, the other column is called title,
F_0 to Lambda. That's this guy here. So left-hand column, take the wavelength
Lambda in my picture, multiply it by the
temperature of the emitting surface,
absolute, go here, find that value and
read off F_0 to Lambda. Once I know that, I know this guy right here. This guy right
here is Sigma T_4. I'll put that down here. There it is down
there. So there are tables available to
do that calculation. Now, I'll take
one more case. Somebody says, "Well,
you know what? I want to find out
how much energy is contained in this area." That's a band
between wavelengths Lambda_1 and Lambda_2.
That's a band. We call that the
band emission. So I'll write
that out then. The fraction of energy
contained between wavelengths Lambda_1
and Lambda_2. Chapter 12 and radiation in general is very
vocabulary-intensive. You have to understand
the language of radiation or you going to be in
deep trouble. This chapter contains
multiple words. The word spectral, emissive power,
irradiation, radiocity, specular,
sounds like spectral. No, it's different.
There's a ton of different words
that mean something. It's hard to solve
a problem in radiation unless you
understand the language. Our authors have
been very good. End of the chapter,
a whole page of definitions of words. Half a page of
definitions of words. So there, if you ever say, "Well, what does
that word mean?" There it is, the end of the chapter. It's there. Now, let's go back
to here again. So band emission. What is this
divided by that? By the way, let me
mention first of all. This numerator, what
am I doing here? You can see, I'm
taking the area under the curve from wavelengths of zero to Lambda_2, where my two hands are. Subtracting the area under the curve where
my two hands are. What am I left with? This, the band. That's what I'm
doing right here. Divide by this
guy. So take this divided by this minus
this divided by this. Fraction_0 Lambda_2 minus fraction_0 Lambda_1. That's what we do in the case of band emission. Sometimes band emission
could be important, like solar cells only respond to certain
wavelengths. So maybe you want to know what fraction of
the sun's energy are contained between
two wavelengths because that's where my solar cell
generates power. You're concerned about
band emission there. So there are many places in the real
Engineering world where we're concerned about band emission.
Question? Yes, sir. >> What can't you
do or can you do the integral from Lambda_1 to Lambda_2
[inaudible] >> You can. Do you
want to do it by hand or you want
to go to a table? >> [inaudible] >> Don't want to
do it by hand. Only this table
is done for you. This is the easy way out. Of course, its
integral period here. But where do you
find the integral? Nowhere in the book,
but can you find it? Yeah, you can
integrate it, but it's much easier
to do it that way, where all you do is subtract two things
from the table rather than do the real
calculus integration. So band emission
using Table 12.2. So let's look at
a problem then. We're going to take
the solar spectrum outside the Earth's
atmosphere, for instance. So example. Find the fractions of solar energy in
ultraviolet, visible, and
infrared regions of the spectrum down here. Well, we're going to assume sun behaves like
a blackbody at 5,800 K. Thermal UV,
visible, infrared. Now, any problem for
homework or exam, we treat solar radiation as if it came from the sun, which behaves like
a blackbody at an equivalent temperature
of 5,800 Kelvin. You say, "Well, did
I miss something? They send a probe
into this on the sun's surface and record the temperature?" I don't recall that.
No, I don't think so. So how did they get
that temperature? Let's just use this.
One way would be, they send a probe up on a spacecraft
or satellite, point a thermal detector to the sun with a certain wavelength
filter on it so only certain
wavelengths are allowed to come
into this sensor. So they use different
wavelength filters and they say, "Okay, that's what we get when we put different
filters in here." So I'm going to
connect this guy and say, "Hey,
you know what? That looks a lot
like the shape of that blackbody curve that I saw in the textbook. I think the sun
might behave like a blackbody." Okay, good. I'm going to assume
the sun is like its equivalent
temperatures 300 Kelvin room
temperature, no. Room temperature,
no, not even close. I'm going to guess
the sun's temperature is 1,000 Kelvin. Getting there. I'm
going to guess the sun's temperature
is 10,000 Kelvin. No. I'm going to guess the sun's temperature
is 5,800. Say, "Whoa, I got it." My best guess now is
the sun is behaving as a blackbody whose
temperature is 5,800. Good. That's what
we're going to use in this class
from now on. Back over here.
So here we go. That's where
that came from. Use that table, Table 12.2. Lambda_1 T equal, Lambda_1, one-tenth, temperature, 5,800, 580. Lambda_2 T, I'll
put them all down. Lambda_3 T, Lambda_4 T, four-tenths, 5,800,
seven-tenths 5,800, 100 times 5,800, 580,000, 2,320, and 4,060. Now go to table and my first Lambda T, 580, second one, 2,320, and 4,060, and then way
down here, 580,000. Fraction, 580,
0, 2,320, 4,060. Almost there. Table is really
long in the book. There it is. Starts
on one page, ends up over
here, big table. The last table
entry, 100,000? Make sure, yeah, 100,000. The last table
entry is 100,000. So that's not
going to work. I need 580,000. The table doesn't give me that answer or
even close to it. Yes, it does. A
hundred thousand, I'm going to extrapolate. Sometimes
extrapolation is good, sometimes it's bad. In this case,
extrapolation is fine. I'm going to round
it off to 1.00. Of course, you
will. Why not? If it's 0.9999, guess what? Five hundred and
eighty thousand is 0.999999, round off one. Just so you know,
anything above a 100,000, round off to one,
that's the rule. Go to table. So let's
get our fractions. Fraction in UV.
There it is, right there. There's
the equation. Fraction zero to a Lambda 0.120
minus 0, 0.120. I take this guy
minus this guy. Fraction visible
495 and 120. Fraction infrared. So the answers are; from the solar spectrum
about 12 percent of the energy is in the UV portion
of the spectrum, the visible portion
37.5 percent, and the infrared a
real big 50.5 percent. So I found the
fractions of energy contained within those particular
wavelengths. So we're going
to stop today.
