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PROFESSOR: So what
I wanted to do today

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00:00:27,860 --> 00:00:31,540
was talk about thermal
properties of foams.

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00:00:31,540 --> 00:00:34,530
And foams are often used
for thermal insulation.

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00:00:34,530 --> 00:00:36,150
And that's always
closed cell foams

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00:00:36,150 --> 00:00:37,691
that are used for
thermal insulation.

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And we'll see why.

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00:00:38,780 --> 00:00:43,290
And the foams tend to have
a low thermal conductivity.

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00:00:43,290 --> 00:00:46,620
And that's largely because
gases have lower conductivity

18
00:00:46,620 --> 00:00:47,684
than solids.

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00:00:47,684 --> 00:00:49,100
And if you have
mostly gas, you're

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00:00:49,100 --> 00:00:50,559
going to have a
lower conductivity.

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00:00:50,559 --> 00:00:53,183
So they have a low conductivity
because they have a high volume

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00:00:53,183 --> 00:00:54,230
fraction of gas.

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00:00:54,230 --> 00:00:57,640
And they've got a low volume
fraction of the solid.

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00:00:57,640 --> 00:00:59,490
They also have cells.

25
00:00:59,490 --> 00:01:01,480
And the heat is
transferred partly

26
00:01:01,480 --> 00:01:03,450
by radiation and convection.

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00:01:03,450 --> 00:01:06,070
And if you have small
cells, you reduce the amount

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00:01:06,070 --> 00:01:08,070
of convection and radiation.

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00:01:08,070 --> 00:01:09,360
And we'll see that.

30
00:01:09,360 --> 00:01:12,270
So that, by having a
cellular structure,

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00:01:12,270 --> 00:01:14,230
and in particular, by
having small cells,

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00:01:14,230 --> 00:01:16,740
you can decrease
the heat transfer.

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00:01:16,740 --> 00:01:19,010
OK, so let me write
some of the stuff down.

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00:01:19,010 --> 00:01:30,160


35
00:01:30,160 --> 00:01:36,355
So closed cell foams are widely
used for thermal insulation.

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00:01:36,355 --> 00:01:44,480


37
00:01:44,480 --> 00:01:48,190
And the only materials with
lower thermal conductivity

38
00:01:48,190 --> 00:01:50,040
than closed cell foams
are aerogels gels.

39
00:01:50,040 --> 00:02:07,355


40
00:02:07,355 --> 00:02:08,729
And I'll I talk
a little bit more

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00:02:08,729 --> 00:02:11,230
about aerogels later on today.

42
00:02:11,230 --> 00:02:12,780
But the difficulty
with aerogels is

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00:02:12,780 --> 00:02:16,600
that they tend to be very weak
and brittle, like ridiculously

44
00:02:16,600 --> 00:02:17,580
weak and brittle.

45
00:02:17,580 --> 00:02:21,030
So we had a project on
aerogels a couple of years ago.

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00:02:21,030 --> 00:02:23,530
And the students who I
was collaborating with

47
00:02:23,530 --> 00:02:24,410
would make aerogels.

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00:02:24,410 --> 00:02:26,210
And they'd bring
it up to my office.

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00:02:26,210 --> 00:02:28,390
And I would pick it up and
like-- I would pick it up

50
00:02:28,390 --> 00:02:29,640
like this, and it would break.

51
00:02:29,640 --> 00:02:33,010
So they have very low
thermal conductivity,

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00:02:33,010 --> 00:02:34,770
but they're very brittle.

53
00:02:34,770 --> 00:02:37,060
And I brought a few of
our samples of aerogels,

54
00:02:37,060 --> 00:02:39,060
just so you can see
what they look like.

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00:02:39,060 --> 00:02:41,000
And I'll pass them around
in that little tube,

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00:02:41,000 --> 00:02:42,980
so you can kind
of play with them.

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00:02:42,980 --> 00:02:46,710


58
00:02:46,710 --> 00:02:49,720
OK, so we're going
to focus on foams.

59
00:02:49,720 --> 00:02:57,830
And whoops-- And we can say
the low thermal conductivity

60
00:02:57,830 --> 00:03:03,620
of foam arises mostly from the
high volume fraction of gas

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00:03:03,620 --> 00:03:05,200
and that the gas
has a low lambda,

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00:03:05,200 --> 00:03:06,325
a low thermal conductivity.

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00:03:06,325 --> 00:03:32,384


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00:03:32,384 --> 00:03:33,800
So lambda is thermal
conductivity,

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so I'm just going
to put lambda there.

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00:03:35,383 --> 00:03:38,470


67
00:03:38,470 --> 00:03:40,120
Then it has a small
volume fraction

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00:03:40,120 --> 00:03:43,494
of solid, which has a
higher thermal conductivity.

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00:03:43,494 --> 00:03:50,940


70
00:03:50,940 --> 00:03:53,679
And then the foams have a
relatively small cell size.

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So one of the things
we're going to look at

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00:03:55,470 --> 00:03:58,829
is how does the cell size
effect the thermal transfer,

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00:03:58,829 --> 00:03:59,870
the thermal conductivity.

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00:03:59,870 --> 00:04:26,771


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OK, so there's lots of
applications for foams.

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And I guess one of the
main ones is in buildings,

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insulating buildings--
also insulating

78
00:04:35,690 --> 00:04:39,340
refrigerated vehicles,
things like LNG tankers.

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00:04:39,340 --> 00:04:42,577
So there's lots of the
applications for using foams

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00:04:42,577 --> 00:04:43,535
for thermal insulation.

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00:04:43,535 --> 00:05:05,800


82
00:05:05,800 --> 00:05:08,330
Foams, in addition to having
a low thermal conductivity,

83
00:05:08,330 --> 00:05:11,210
they also have good
thermal shock resistance.

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00:05:11,210 --> 00:05:14,250
So thermal shock is if you have
a material, you heat it up,

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00:05:14,250 --> 00:05:16,750
and then you suddenly cool the
surface of it, for example.

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00:05:16,750 --> 00:05:18,958
So say you takes something
and you quench it in water

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00:05:18,958 --> 00:05:22,880
or quench it in some
fluid, then the surface,

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00:05:22,880 --> 00:05:24,922
it wants to shrink because
its temperature drops,

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00:05:24,922 --> 00:05:26,838
but it's connected to
everything underneath it

90
00:05:26,838 --> 00:05:28,120
and it can't really shrink.

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00:05:28,120 --> 00:05:31,230
And so it's constrained and you
can get cracking and spalling.

92
00:05:31,230 --> 00:05:34,060
And so, it turns out foams
have a good resistance

93
00:05:34,060 --> 00:05:37,600
to that thermal shock
kind of loading.

94
00:05:37,600 --> 00:05:38,925
And we'll see why that is, too.

95
00:05:38,925 --> 00:05:57,270


96
00:05:57,270 --> 00:06:00,710
Roughly, you can see if
the thermal expansion

97
00:06:00,710 --> 00:06:05,030
strain is the thermal expansion
coefficient times the change

98
00:06:05,030 --> 00:06:06,307
in temperature.

99
00:06:06,307 --> 00:06:07,890
And the stress that
you might generate

100
00:06:07,890 --> 00:06:11,580
is just going to be related to
the modulus times alpha times

101
00:06:11,580 --> 00:06:12,561
delta-t.

102
00:06:12,561 --> 00:06:14,810
And because we're going to
see that alpha for the foam

103
00:06:14,810 --> 00:06:16,770
is the same as alpha
for the solid, but the E

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00:06:16,770 --> 00:06:19,670
foam is going to be a lot less
that E of a solid would be.

105
00:06:19,670 --> 00:06:21,710
So because the
modulus is smaller,

106
00:06:21,710 --> 00:06:26,230
you would get a better
thermal shock resistance.

107
00:06:26,230 --> 00:06:31,010
OK, so I wanted to go
over a couple of sort

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00:06:31,010 --> 00:06:32,980
of laws of heat
conduction, so we

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00:06:32,980 --> 00:06:35,860
can talk about what
thermal conductivity is

110
00:06:35,860 --> 00:06:37,750
and how we define it.

111
00:06:37,750 --> 00:06:40,526
So the first one here--

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00:06:40,526 --> 00:06:49,300


113
00:06:49,300 --> 00:06:52,080
--first one here is for
steady state conduction.

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00:06:52,080 --> 00:06:57,860


115
00:06:57,860 --> 00:07:00,990
So when we say steady state
conduction, what we mean

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00:07:00,990 --> 00:07:03,597
is that the temperature
is constant with time,

117
00:07:03,597 --> 00:07:05,305
the temperature doesn't
change with time.

118
00:07:05,305 --> 00:07:13,360


119
00:07:13,360 --> 00:07:16,530
So time's not going to come
into the equation here.

120
00:07:16,530 --> 00:07:20,570
And heat transfer for
steady state conduction,

121
00:07:20,570 --> 00:07:23,480
where there is no change in
the temperature with time,

122
00:07:23,480 --> 00:07:24,962
described by Fourier's Law.

123
00:07:24,962 --> 00:07:27,840


124
00:07:27,840 --> 00:07:31,970
And that says that
the heat flux q

125
00:07:31,970 --> 00:07:37,720
is equal to minus lambda times
the gradient in temperature.

126
00:07:37,720 --> 00:07:41,450
And if you want to think about
just a one diversion of that,

127
00:07:41,450 --> 00:07:46,280
it's equal to minus
lambda times dt by dx.

128
00:07:46,280 --> 00:07:49,710
So here, q is our heat flux.

129
00:07:49,710 --> 00:07:54,290


130
00:07:54,290 --> 00:07:56,920
So that would have
units of joules

131
00:07:56,920 --> 00:07:59,120
per meter squared per second.

132
00:07:59,120 --> 00:08:03,622
So how much heat transfer
per unit area per unit time.

133
00:08:03,622 --> 00:08:05,080
Lambda is the
thermal conductivity.

134
00:08:05,080 --> 00:08:12,370


135
00:08:12,370 --> 00:08:17,880
And it has units of watts per
meter k, so degrees kelvin.

136
00:08:17,880 --> 00:08:22,054
And then delta
or-- and then this

137
00:08:22,054 --> 00:08:23,220
is our temperature gradient.

138
00:08:23,220 --> 00:08:31,320


139
00:08:31,320 --> 00:08:32,919
OK, so that's Fourier's
Law, and we're

140
00:08:32,919 --> 00:08:37,340
going to use that later on
when we talk about the foams.

141
00:08:37,340 --> 00:08:39,940
And then, just so that we have
things a little more complete,

142
00:08:39,940 --> 00:08:42,770
if you have a non-steady heat
conduction, if the temperature

143
00:08:42,770 --> 00:08:44,987
varies with time, then
there's a difference equation

144
00:08:44,987 --> 00:08:46,570
that involves the
thermal diffusivity.

145
00:08:46,570 --> 00:08:50,570


146
00:08:50,570 --> 00:09:00,190
So if we have non-steady
heat conduction-- so t

147
00:09:00,190 --> 00:09:03,170
varies with time.

148
00:09:03,170 --> 00:09:04,610
I'm going to call a time tau.

149
00:09:04,610 --> 00:09:08,830


150
00:09:08,830 --> 00:09:12,810
Then a partial differentiation,
the partial derivative

151
00:09:12,810 --> 00:09:16,290
of temperature with
respect to time

152
00:09:16,290 --> 00:09:17,970
is equal to the
diffusivity, that's

153
00:09:17,970 --> 00:09:21,220
given the symbol a, times
the second derivative

154
00:09:21,220 --> 00:09:24,020
of temperature with
respect of distance,

155
00:09:24,020 --> 00:09:25,870
so with respect x squared.

156
00:09:25,870 --> 00:09:27,836
So here a is the
thermal diffusivity.

157
00:09:27,836 --> 00:09:36,760


158
00:09:36,760 --> 00:09:40,420
And it is equal to the
thermal conductivity

159
00:09:40,420 --> 00:09:44,740
divided by the density and
divided by the specific heat.

160
00:09:44,740 --> 00:09:50,600
So here, rho is the density
and cp is the specific heat.

161
00:09:50,600 --> 00:09:54,700


162
00:09:54,700 --> 00:09:57,250
The specific heat
is the heat required

163
00:09:57,250 --> 00:09:59,570
to raise the temperature
of a unit mass

164
00:09:59,570 --> 00:10:03,220
by the unit temperature.

165
00:10:03,220 --> 00:10:07,700
And so, the density times cp is
the volumetric heat capacity.

166
00:10:07,700 --> 00:10:09,570
It's how much energy
you would need

167
00:10:09,570 --> 00:10:13,410
to raise a certain
volume by, say, 1 degree

168
00:10:13,410 --> 00:10:14,750
k instead of a certain mass.

169
00:10:14,750 --> 00:10:33,090


170
00:10:33,090 --> 00:10:38,110
OK, so on the table
here, on the screen,

171
00:10:38,110 --> 00:10:39,810
we have different materials.

172
00:10:39,810 --> 00:10:42,350
And we have the thermal
conductivity lambda.

173
00:10:42,350 --> 00:10:44,820
And we have the
thermal diffusivity, a.

174
00:10:44,820 --> 00:10:48,320
And I guess I should
also say a has units

175
00:10:48,320 --> 00:10:51,510
of meters squared per second.

176
00:10:51,510 --> 00:10:56,240


177
00:10:56,240 --> 00:11:00,230
So this table is arranged
in order of decreasing

178
00:11:00,230 --> 00:11:01,280
thermal conductivity.

179
00:11:01,280 --> 00:11:08,010
So here's copper at the
top, 384, watts per meter k.

180
00:11:08,010 --> 00:11:09,880
Here's, you know,
different metals.

181
00:11:09,880 --> 00:11:11,020
You've got aluminum.

182
00:11:11,020 --> 00:11:12,420
Here's a couple of ceramics.

183
00:11:12,420 --> 00:11:15,230
They're about a factor of
10 less than the metals.

184
00:11:15,230 --> 00:11:18,580
Here's the polymers, another
factor of 10 less than that.

185
00:11:18,580 --> 00:11:20,540
And here's some gases.

186
00:11:20,540 --> 00:11:23,030
Air is about 0.025.

187
00:11:23,030 --> 00:11:25,220
Carbon dioxide is
less than that.

188
00:11:25,220 --> 00:11:27,410
Triclorofluoromethane,
which used

189
00:11:27,410 --> 00:11:30,390
to be used as a gas
in foams because it's

190
00:11:30,390 --> 00:11:33,470
got such low thermal
conductivity, is 0.008.

191
00:11:33,470 --> 00:11:36,830
But it's no longer used because
it's a-- what you call it?

192
00:11:36,830 --> 00:11:39,210
A fluorocarbon.

193
00:11:39,210 --> 00:11:41,590
Anyway, it decreases
our ozone layer.

194
00:11:41,590 --> 00:11:43,100
So they don't use that anymore.

195
00:11:43,100 --> 00:11:44,220
Now here's some wood.

196
00:11:44,220 --> 00:11:46,040
So that's one sort
of cellular solid.

197
00:11:46,040 --> 00:11:48,430
And they're around 0.04--
something like that.

198
00:11:48,430 --> 00:11:51,150
And here's a group
of polymer foams.

199
00:11:51,150 --> 00:11:53,940
And they're a little over 0.025.

200
00:11:53,940 --> 00:11:56,030
So if you think of--
if you had the gas,

201
00:11:56,030 --> 00:11:58,680
air-- if the air was
the gas inside the foam,

202
00:11:58,680 --> 00:12:00,905
0.025 is lambda for the gas.

203
00:12:00,905 --> 00:12:02,850
So you're not going to
get lower than that.

204
00:12:02,850 --> 00:12:06,440
And you have to use a low
conductivity gas to get these

205
00:12:06,440 --> 00:12:10,690
values, like 0.025
here, 0.020, 0.017.

206
00:12:10,690 --> 00:12:13,360
And then, hear some other sorts
of sort of mineral fibers,

207
00:12:13,360 --> 00:12:15,140
glass foams, glass wools.

208
00:12:15,140 --> 00:12:19,280
OK, so that's just a table so
that you have some data there.

209
00:12:19,280 --> 00:12:19,780
All right?

210
00:12:19,780 --> 00:12:20,720
Yes?

211
00:12:20,720 --> 00:12:30,310
STUDENT: [INAUDIBLE] --foams,
if they are closed cell,

212
00:12:30,310 --> 00:12:31,540
with a different gas rate.

213
00:12:31,540 --> 00:12:32,480
Because if they're open cell--

214
00:12:32,480 --> 00:12:32,530
PROFESSOR: --Right.

215
00:12:32,530 --> 00:12:32,660
The gas is going to--

216
00:12:32,660 --> 00:12:33,680
STUDENT: --it would
just always be air.

217
00:12:33,680 --> 00:12:34,846
PROFESSOR: It's going to go.

218
00:12:34,846 --> 00:12:37,420
And in fact, one
of the difficulties

219
00:12:37,420 --> 00:12:39,950
with using the lower
conductivity of any gases

220
00:12:39,950 --> 00:12:42,500
is there's a phenomenon called
aging, that if, you know,

221
00:12:42,500 --> 00:12:44,720
you've got your gas
inside your foam,

222
00:12:44,720 --> 00:12:46,480
it's going to diffuse
out into the air.

223
00:12:46,480 --> 00:12:47,900
And air's going to diffuse in.

224
00:12:47,900 --> 00:12:49,980
So over time, the
thermal conductivity

225
00:12:49,980 --> 00:12:51,850
tends to increase
because you're getting

226
00:12:51,850 --> 00:12:56,440
air coming and the local--
conductivity gas going out.

227
00:12:56,440 --> 00:12:58,410
But I think,
typically, that process

228
00:12:58,410 --> 00:13:00,104
takes a number of years.

229
00:13:00,104 --> 00:13:01,270
It doesn't happen in a week.

230
00:13:01,270 --> 00:13:02,530
But if you're
designing a building

231
00:13:02,530 --> 00:13:04,488
and want the building to
be there for 50 years,

232
00:13:04,488 --> 00:13:05,990
it occurs faster than that.

233
00:13:05,990 --> 00:13:08,385
So it's not ideal from
that point of view.

234
00:13:08,385 --> 00:13:10,900


235
00:13:10,900 --> 00:13:12,820
All right.

236
00:13:12,820 --> 00:13:15,600
So let me talk a little bit
more about thermal diffusivity.

237
00:13:15,600 --> 00:13:17,991
Let me scoot over here.

238
00:13:17,991 --> 00:13:49,970


239
00:13:49,970 --> 00:13:54,260
So materials with a high value
of that thermal diffusivity, a.

240
00:13:54,260 --> 00:13:56,180
They rapidly adjust
their temperature

241
00:13:56,180 --> 00:13:57,680
to their surroundings.

242
00:13:57,680 --> 00:14:00,190
So if they have a high
value of a, what it really

243
00:14:00,190 --> 00:14:02,220
means they've got
a, say, a high value

244
00:14:02,220 --> 00:14:04,570
of lamda-- so high
thermal conductivity.

245
00:14:04,570 --> 00:14:07,480
And, say, a low value of this
volumetric heat capacity.

246
00:14:07,480 --> 00:14:10,160
So it doesn't take much energy
to change their temperature.

247
00:14:10,160 --> 00:14:11,570
And they also conduct heat well.

248
00:14:11,570 --> 00:14:13,940
So they tend to adjust
their temperature

249
00:14:13,940 --> 00:14:15,288
to their surroundings quickly.

250
00:14:15,288 --> 00:15:01,570


251
00:15:01,570 --> 00:15:05,110
OK, so then, let's talk about
the thermal conductivity

252
00:15:05,110 --> 00:15:05,730
of a foam.

253
00:15:05,730 --> 00:15:13,620


254
00:15:13,620 --> 00:15:15,630
So I'm going to call
that lambda star.

255
00:15:15,630 --> 00:15:16,870
So the star is the foam.

256
00:15:16,870 --> 00:15:19,700
And then we'll
talk about-- lambda

257
00:15:19,700 --> 00:15:23,410
s will be the lambda for the
solid that it's made from.

258
00:15:23,410 --> 00:15:26,730
So if you think of the thermal
conductivity of the foam,

259
00:15:26,730 --> 00:15:28,720
there's contributions
from different types

260
00:15:28,720 --> 00:15:30,370
of heat transfer.

261
00:15:30,370 --> 00:15:33,610
So you could have conduction
through the solid.

262
00:15:33,610 --> 00:15:35,380
I'm going to call that lambda s.

263
00:15:35,380 --> 00:15:38,440
You could have conduction
through the gas.

264
00:15:38,440 --> 00:15:40,420
You could have convection
within the cell.

265
00:15:40,420 --> 00:15:44,550
So convection has to do with
having, say, within the cell,

266
00:15:44,550 --> 00:15:46,390
it might be a different
temperature on one

267
00:15:46,390 --> 00:15:48,390
side of the cell to the
other side of the cell.

268
00:15:48,390 --> 00:15:50,100
And the warmer side
of the cell, the gas

269
00:15:50,100 --> 00:15:52,060
is going to tend to
rise to the warmer side

270
00:15:52,060 --> 00:15:54,160
and fall to the cooler side.

271
00:15:54,160 --> 00:15:56,536
And you get a convection
current set up.

272
00:15:56,536 --> 00:15:58,160
So you can get heat
transfer from that.

273
00:15:58,160 --> 00:16:00,530
And you can also get heat
transfer by radiation.

274
00:16:00,530 --> 00:16:03,750
So radiation can cause
heat transfer, as well.

275
00:16:03,750 --> 00:16:11,570
So we're going to have
contributions from conduction

276
00:16:11,570 --> 00:16:12,342
through the solid.

277
00:16:12,342 --> 00:16:18,850


278
00:16:18,850 --> 00:16:22,780
So the amount of conduction in
the foam from the solid-- I'm

279
00:16:22,780 --> 00:16:25,510
going to call lambda star s.

280
00:16:25,510 --> 00:16:28,670
So lambda s would be the
conductivity of the solid.

281
00:16:28,670 --> 00:16:32,530
And lambda star s is the thermal
conductivity contribution

282
00:16:32,530 --> 00:16:35,490
from the solid in the foam.

283
00:16:35,490 --> 00:16:38,830
So we get kind of--
through the solid.

284
00:16:38,830 --> 00:16:42,490
We have conductivity
through the gas.

285
00:16:42,490 --> 00:16:45,230
So it's lambda star g for gas.

286
00:16:45,230 --> 00:16:47,372
And then we could have
convection within the cells.

287
00:16:47,372 --> 00:16:55,530


288
00:16:55,530 --> 00:16:58,450
We'll call that lambda star c.

289
00:16:58,450 --> 00:17:04,590
And then we could get radiation
through the cell walls

290
00:17:04,590 --> 00:17:05,465
and across the voids.

291
00:17:05,465 --> 00:17:18,609


292
00:17:18,609 --> 00:17:21,890
We'll call that lambda star r.

293
00:17:21,890 --> 00:17:24,069
And so, the thermal
conductivity of the foam

294
00:17:24,069 --> 00:17:25,908
is just the sum of those
four contributions.

295
00:17:25,908 --> 00:17:40,447


296
00:17:40,447 --> 00:17:42,030
So we're just going
to go through each

297
00:17:42,030 --> 00:17:45,880
of those contributions,
in turn, and work out

298
00:17:45,880 --> 00:17:48,759
how much thermal conductivity
you get from each of them.

299
00:17:48,759 --> 00:17:50,800
And it turns out most of
the thermal conductivity

300
00:17:50,800 --> 00:17:53,280
comes through the gas.

301
00:17:53,280 --> 00:18:04,740
So if we first look at just
conduction through the solid,

302
00:18:04,740 --> 00:18:10,250
we've got that contribution to
the conductivity of the foam

303
00:18:10,250 --> 00:18:14,730
from the solid, it's just
equal to some efficiency factor

304
00:18:14,730 --> 00:18:17,650
times the thermal conductivity
of the solid times

305
00:18:17,650 --> 00:18:21,710
the volume fraction of the
solid or the relative density.

306
00:18:21,710 --> 00:18:24,323
And here, eta is an
efficiency factor.

307
00:18:24,323 --> 00:18:30,570


308
00:18:30,570 --> 00:18:33,320
And it accounts for that
tortuosity in the foam.

309
00:18:33,320 --> 00:18:35,704
So if you think of
the solid in the foam,

310
00:18:35,704 --> 00:18:37,370
it's not like we have
little fibers that

311
00:18:37,370 --> 00:18:39,203
just go from one side
to the other like this

312
00:18:39,203 --> 00:18:41,460
and the heat just moves
along those fibers.

313
00:18:41,460 --> 00:18:44,320
You know, the foam cells have
some complicated geometry

314
00:18:44,320 --> 00:18:46,020
and the heat has to
kind of run along

315
00:18:46,020 --> 00:18:48,390
that complicated geometry.

316
00:18:48,390 --> 00:18:52,040
And people have made
estimates of what this is.

317
00:18:52,040 --> 00:18:54,540
And it's roughly
a factor of 2/3.

318
00:18:54,540 --> 00:18:58,610
So I guess it would depend on
exactly the foam cell geometry.

319
00:18:58,610 --> 00:19:00,408
But typically it's around 2/3.

320
00:19:00,408 --> 00:19:06,990


321
00:19:06,990 --> 00:19:08,590
So that's conduction
through a solid.

322
00:19:08,590 --> 00:19:11,600
That's straight forward.

323
00:19:11,600 --> 00:19:15,900
Conduction through gas is
similarly straightforward.

324
00:19:15,900 --> 00:19:18,460
It's just the
conductivity of the gas

325
00:19:18,460 --> 00:19:21,220
times the amount of the gas.

326
00:19:21,220 --> 00:19:23,550
And the volume fraction
of the gas is just 1

327
00:19:23,550 --> 00:19:25,440
minus the volume
fraction of the solid.

328
00:19:25,440 --> 00:19:27,805
So it's just 1 minus
the relative density.

329
00:19:27,805 --> 00:19:29,180
So the conduction
through the gas

330
00:19:29,180 --> 00:19:32,760
is just lambda g times 1
minus the relative density.

331
00:19:32,760 --> 00:20:07,822


332
00:20:07,822 --> 00:20:09,280
So we can do a
little example here.

333
00:20:09,280 --> 00:20:11,030
And you can see how
much of the conduction

334
00:20:11,030 --> 00:20:12,517
comes from the solid in the gas.

335
00:20:12,517 --> 00:20:17,900


336
00:20:17,900 --> 00:20:24,230
So for example, if we look at a
foam that's 2.5% dense and say

337
00:20:24,230 --> 00:20:28,670
it's a closed cell poly-- what
are we doing-- polystyrene.

338
00:20:28,670 --> 00:20:39,320


339
00:20:39,320 --> 00:20:43,380
So the total thermal
conductivity of the foam

340
00:20:43,380 --> 00:20:50,410
is about 0.04 watts per meter k.

341
00:20:50,410 --> 00:20:53,640
And the thermal
conductivity of polystyrene

342
00:20:53,640 --> 00:20:59,010
is 0.15 watts per meter k.

343
00:20:59,010 --> 00:21:07,500
And the thermal conductivity
of air is 0.025.

344
00:21:07,500 --> 00:21:10,900
So let's assume it's
just blown with air.

345
00:21:10,900 --> 00:21:12,510
And then if I just
add up, what's

346
00:21:12,510 --> 00:21:16,315
the contribution of
conduction through the solid

347
00:21:16,315 --> 00:21:18,999
and conduction through
the gas-- so I just

348
00:21:18,999 --> 00:21:20,790
use those two little
equations-- conduction

349
00:21:20,790 --> 00:21:23,720
through the solid--
it's going to be 2/3

350
00:21:23,720 --> 00:21:28,300
of this value of lambda s times
the amount of the solids--

351
00:21:28,300 --> 00:21:34,077
that's 0.025 and then
plus lambda g, which

352
00:21:34,077 --> 00:21:41,850
is 0.025 times the amount
of the gas, which is 0.975.

353
00:21:41,850 --> 00:21:46,360
And if I work those two
things out, this is 0.003

354
00:21:46,360 --> 00:21:52,160
and this is 0.024.

355
00:21:52,160 --> 00:21:54,730
So that total is 0.027
watts per meter k.

356
00:21:54,730 --> 00:21:58,240


357
00:21:58,240 --> 00:22:02,010
So you can see if the
total is 0.04, most of it's

358
00:22:02,010 --> 00:22:02,800
come from the gas.

359
00:22:02,800 --> 00:22:04,330
A little bit's come
from the solid.

360
00:22:04,330 --> 00:22:06,900
And the rest is going to be
from convection and radiation.

361
00:22:06,900 --> 00:22:09,519


362
00:22:09,519 --> 00:22:10,310
And that's typical.

363
00:22:10,310 --> 00:22:27,560


364
00:22:27,560 --> 00:22:29,810
And that's the reason
that they sometimes

365
00:22:29,810 --> 00:22:32,940
use low thermal conductivity
gases to blow foams

366
00:22:32,940 --> 00:22:35,020
for thermal insulation
because the gas makes up

367
00:22:35,020 --> 00:22:39,034
such a big fraction of
the total conductivity.

368
00:22:39,034 --> 00:22:40,450
If you can reduce
that, you reduce

369
00:22:40,450 --> 00:22:41,491
the overall conductivity.

370
00:22:41,491 --> 00:22:46,790


371
00:22:46,790 --> 00:22:49,220
So, we'll say foams for
insulation are blown

372
00:22:49,220 --> 00:22:53,247
with low conductivity gases.

373
00:22:53,247 --> 00:22:58,100


374
00:22:58,100 --> 00:23:00,350
But as I mentioned, you
have this problem with aging

375
00:23:00,350 --> 00:23:02,750
that, over time, that gas
is going to diffuse out

376
00:23:02,750 --> 00:23:04,090
and air is going to diffuse in.

377
00:23:04,090 --> 00:23:21,810


378
00:23:21,810 --> 00:23:26,135
Then the overall thermal
conductivity of the foam

379
00:23:26,135 --> 00:23:27,010
is going to increase.

380
00:23:27,010 --> 00:23:46,570


381
00:23:46,570 --> 00:23:50,190
So that's the conduction.

382
00:23:50,190 --> 00:23:54,580
And then the next contribution
is from convection.

383
00:23:54,580 --> 00:23:59,150
So imagine we have one
of our little cells here.

384
00:23:59,150 --> 00:24:03,260
And it's hotter on that side
than it is on that side.

385
00:24:03,260 --> 00:24:06,050
And hot air is going to rise.

386
00:24:06,050 --> 00:24:07,430
Cold air is going to fall.

387
00:24:07,430 --> 00:24:10,380
So you get a convection
current set up.

388
00:24:10,380 --> 00:24:13,160
And because of the
density changes,

389
00:24:13,160 --> 00:24:16,320
you get a buoyancy
force in the air.

390
00:24:16,320 --> 00:24:18,600
So that's kind of
driving the convection.

391
00:24:18,600 --> 00:24:20,520
But you also have
a viscous drag.

392
00:24:20,520 --> 00:24:24,580
So the air is moving past
the wall of the foam.

393
00:24:24,580 --> 00:24:27,580
And there's going to be some
viscous drag associated.

394
00:24:27,580 --> 00:24:29,320
And how much
convection you can get

395
00:24:29,320 --> 00:24:31,740
depends on the balance
between this buoyancy force

396
00:24:31,740 --> 00:24:32,815
and the viscous drag.

397
00:24:32,815 --> 00:24:37,700


398
00:24:37,700 --> 00:24:44,300
So we'll say the
gas rises and falls

399
00:24:44,300 --> 00:24:46,080
due to density changes
with temperature.

400
00:24:46,080 --> 00:24:54,871


401
00:24:54,871 --> 00:24:57,850
And the density changes give
rise to buoyancy forces.

402
00:24:57,850 --> 00:25:08,750


403
00:25:08,750 --> 00:25:18,490
But we also have
these viscous forces

404
00:25:18,490 --> 00:25:22,590
from the drag of the air
against the walls of the cell.

405
00:25:22,590 --> 00:25:33,070


406
00:25:33,070 --> 00:25:35,700
So air moving past
the walls-- this

407
00:25:35,700 --> 00:25:39,220
is kind of a fluid mechanics
thing-- so that air is a fluid.

408
00:25:39,220 --> 00:25:41,340
And in fluid
mechanics, they often

409
00:25:41,340 --> 00:25:42,894
use dimensionless numbers.

410
00:25:42,894 --> 00:25:44,310
And there's a
dimensionless number

411
00:25:44,310 --> 00:25:45,524
called the Rayleigh number.

412
00:25:45,524 --> 00:25:47,440
And the Rayleigh number,
you can think of it--

413
00:25:47,440 --> 00:25:50,570
it's not quite the balance
of the buoyancy force

414
00:25:50,570 --> 00:25:52,370
against the viscous forces.

415
00:25:52,370 --> 00:25:54,310
But it involves those forces.

416
00:25:54,310 --> 00:25:56,530
And convection is important
if this Raleigh number's

417
00:25:56,530 --> 00:25:57,215
over 1,000.

418
00:25:57,215 --> 00:26:17,400


419
00:26:17,400 --> 00:26:19,211
And here's what the
Rayleigh number is.

420
00:26:19,211 --> 00:26:22,150


421
00:26:22,150 --> 00:26:26,120
It's the density of the
fluid times the acceleration

422
00:26:26,120 --> 00:26:28,810
of gravity times beta.

423
00:26:28,810 --> 00:26:30,880
Beta's the volume
expansion coefficient

424
00:26:30,880 --> 00:26:34,284
for the gas-- times
the temperature change.

425
00:26:34,284 --> 00:26:35,950
And we're going to
look at a temperature

426
00:26:35,950 --> 00:26:38,200
change across a cell.

427
00:26:38,200 --> 00:26:39,480
And then, times the length.

428
00:26:39,480 --> 00:26:42,790
That's going to
be the cell size.

429
00:26:42,790 --> 00:26:46,730
And we divide that by
the fluid viscosity

430
00:26:46,730 --> 00:26:50,380
and the thermal diffusivity.

431
00:26:50,380 --> 00:26:52,910
So let me write down what
all these things are.

432
00:26:52,910 --> 00:26:54,608
So rho is the
density of the gas.

433
00:26:54,608 --> 00:26:58,910


434
00:26:58,910 --> 00:27:01,820
So the g's gravitational
acceleration.

435
00:27:01,820 --> 00:27:09,170


436
00:27:09,170 --> 00:27:11,945
Beta is the volume
expansion of the gas.

437
00:27:11,945 --> 00:27:23,010


438
00:27:23,010 --> 00:27:24,682
And for a constant
pressure that's

439
00:27:24,682 --> 00:27:26,015
equal to 1 over the temperature.

440
00:27:26,015 --> 00:27:31,390


441
00:27:31,390 --> 00:27:35,460
Then delta tc is the temperature
difference across a cell.

442
00:27:35,460 --> 00:27:43,090


443
00:27:43,090 --> 00:27:44,410
And l is the cell size.

444
00:27:44,410 --> 00:27:46,990


445
00:27:46,990 --> 00:27:51,815
Mu is the dynamics
viscosity the fluid.

446
00:27:51,815 --> 00:27:55,610


447
00:27:55,610 --> 00:27:57,270
And a is our
thermal diffusivity.

448
00:27:57,270 --> 00:28:06,680


449
00:28:06,680 --> 00:28:09,300
So what I'm going to
do is just work out,

450
00:28:09,300 --> 00:28:16,450
for a typical example,
how big of a cell size

451
00:28:16,450 --> 00:28:19,420
do you need to get this
Rayleigh number to be 1,000.

452
00:28:19,420 --> 00:28:22,060
And we're going to
see that, typically,

453
00:28:22,060 --> 00:28:23,830
that cell size is big.

454
00:28:23,830 --> 00:28:24,970
It's like 20 millimeters.

455
00:28:24,970 --> 00:28:26,660
So in most foams,
the convection really

456
00:28:26,660 --> 00:28:27,970
isn't very important at all.

457
00:28:27,970 --> 00:28:30,510
So it's typically-- people
don't worry about convection.

458
00:28:30,510 --> 00:28:32,500
And let me just show
you how that works.

459
00:28:32,500 --> 00:28:53,540


460
00:28:53,540 --> 00:28:56,165
So for our Rayleigh
number, which

461
00:28:56,165 --> 00:28:59,030
is ra-- for the Rayleigh
number to be 1,000--

462
00:28:59,030 --> 00:29:01,760
say we had air in the cells.

463
00:29:01,760 --> 00:29:04,095
And say the temperature
was room temperature.

464
00:29:04,095 --> 00:29:07,670


465
00:29:07,670 --> 00:29:13,170
Then the volume coefficient
of expansion is just 1 over t.

466
00:29:13,170 --> 00:29:16,150
So it's 1 over 300, say.

467
00:29:16,150 --> 00:29:18,670
degrees k to the minus 1.

468
00:29:18,670 --> 00:29:22,700
Let's say our change in
temperature across one cell

469
00:29:22,700 --> 00:29:25,480
was 1 degree k.

470
00:29:25,480 --> 00:29:26,980
Bless you.

471
00:29:26,980 --> 00:29:32,400
The viscosity of air is 2
times 10 to the minus 5,

472
00:29:32,400 --> 00:29:34,280
pascal seconds.

473
00:29:34,280 --> 00:29:41,850
The density of air is 1.2
kilograms per cubic meter.

474
00:29:41,850 --> 00:29:45,680


475
00:29:45,680 --> 00:29:47,960
And the thermal
diffusivity for air

476
00:29:47,960 --> 00:29:53,840
is 2 times 10 to the minus
5 meters squared per second.

477
00:29:53,840 --> 00:29:56,220
And if you plug all of
these into that equation

478
00:29:56,220 --> 00:29:59,470
for the Rayleigh number and
you solve for the cell size,

479
00:29:59,470 --> 00:30:04,280
you find that the cell
size, l, is 20 millimeters.

480
00:30:04,280 --> 00:30:06,740
So that says convection is
only important if the cell

481
00:30:06,740 --> 00:30:08,010
size is bigger than that.

482
00:30:08,010 --> 00:30:21,200


483
00:30:21,200 --> 00:30:23,752
And so most foams have cells
much smaller than that.

484
00:30:23,752 --> 00:30:24,960
And convection is negligible.

485
00:30:24,960 --> 00:30:46,000


486
00:30:46,000 --> 00:30:48,420
So I have enclosed
cells and the heat's

487
00:30:48,420 --> 00:30:50,770
not transferred so easily
from one cell to another

488
00:30:50,770 --> 00:30:52,230
by the gas moving.

489
00:30:52,230 --> 00:30:55,280
And by having small cells
the convection drops out.

490
00:30:55,280 --> 00:30:56,930
So you don't have
to worry about that.

491
00:30:56,930 --> 00:31:00,705
So the last contribution to
heat transfer is from radiation.

492
00:31:00,705 --> 00:31:10,960


493
00:31:10,960 --> 00:31:12,830
And there's something
called Stefan's law

494
00:31:12,830 --> 00:31:16,490
that describes the heat
flux for radiated heat

495
00:31:16,490 --> 00:31:18,920
transfer from a surface
at one temperature

496
00:31:18,920 --> 00:31:21,110
to another surface at
a different temperature

497
00:31:21,110 --> 00:31:22,990
across a vacuum.

498
00:31:22,990 --> 00:31:29,350
So we can say we
have a heat flux

499
00:31:29,350 --> 00:31:34,484
qr not from a surface
of one temperature.

500
00:31:34,484 --> 00:31:37,310


501
00:31:37,310 --> 00:31:41,180
So I'm going to call that t1--
to one at a lower temperature.

502
00:31:41,180 --> 00:31:47,090


503
00:31:47,090 --> 00:31:51,730
I'm going to call tnot-- with
a vacuum in between them.

504
00:31:51,730 --> 00:32:03,340


505
00:32:03,340 --> 00:32:08,060
So this is [? Stefan's ?] law
so this is the radiative heat

506
00:32:08,060 --> 00:32:13,850
flux is equal to the emissivity
of the surfaces, which

507
00:32:13,850 --> 00:32:16,130
is beta 1 times
a constant called

508
00:32:16,130 --> 00:32:21,460
Stefan's constant-- sigma
times the fourth power

509
00:32:21,460 --> 00:32:24,800
of temperatures.

510
00:32:24,800 --> 00:32:32,830
I'm taking the difference
of the temperatures

511
00:32:32,830 --> 00:32:37,874
so here are the Stefan's
Constant-- is sigma.

512
00:32:37,874 --> 00:32:41,750


513
00:32:41,750 --> 00:32:48,760
And that's equal to 5.67
times the 10 to the minus 8.

514
00:32:48,760 --> 00:32:54,740
And that's in watts per meter
squared per k to the fourth.

515
00:32:54,740 --> 00:32:58,770
And beta is a
constant describing

516
00:32:58,770 --> 00:33:00,260
the emissivity of the surfaces.

517
00:33:00,260 --> 00:33:18,980


518
00:33:18,980 --> 00:33:21,480
So it gives the
radiant heat flux

519
00:33:21,480 --> 00:33:24,244
per unit area of the sample
relative to a black body.

520
00:33:24,244 --> 00:33:26,160
And that's a characteristic
of the emissivity.

521
00:33:26,160 --> 00:33:29,830


522
00:33:29,830 --> 00:33:33,599
All right, so then,
so if we-- yes?

523
00:33:33,599 --> 00:33:34,432
STUDENT: [INAUDIBLE]

524
00:33:34,432 --> 00:33:37,932


525
00:33:37,932 --> 00:33:39,890
PROFESSOR: Now-- so right
now, forget the foam.

526
00:33:39,890 --> 00:33:40,540
We have no foam.

527
00:33:40,540 --> 00:33:42,748
We just have two surfaces
with a vacuum between them.

528
00:33:42,748 --> 00:33:45,140
And now I'm going to stick
a foam between the surfaces.

529
00:33:45,140 --> 00:33:48,320
And we're going to see how
that changes the heat flux, OK?

530
00:33:48,320 --> 00:33:50,680
So the next step is we put
the foam between those two

531
00:33:50,680 --> 00:33:51,604
surfaces.

532
00:33:51,604 --> 00:33:53,270
And the heat flux is
going to be reduced

533
00:33:53,270 --> 00:33:55,686
because the radiation is going
to be absorbed by the solid

534
00:33:55,686 --> 00:33:57,560
and reflected by the cell walls.

535
00:33:57,560 --> 00:33:59,040
And so we're going
to characterize

536
00:33:59,040 --> 00:34:00,090
how much it's reduced.

537
00:34:00,090 --> 00:34:03,440
So there's another law called
Beer's Law, which characterizes

538
00:34:03,440 --> 00:34:07,736
the reduction in the heat flux.

539
00:34:07,736 --> 00:34:39,239


540
00:34:39,239 --> 00:34:41,480
Piece of chalk's
getting to small

541
00:34:41,480 --> 00:35:34,550


542
00:35:34,550 --> 00:35:38,810
OK, so Beer's Law gives us
the attenuation, so the sort

543
00:35:38,810 --> 00:35:42,510
of reduction in the heat flow.

544
00:35:42,510 --> 00:35:45,990
So qr is equal to qr not.

545
00:35:45,990 --> 00:35:49,100
That would be the heat flux,
if we just had the vacuum.

546
00:35:49,100 --> 00:35:51,530
And then there's
an exponential law.

547
00:35:51,530 --> 00:35:56,630
And it's the exponential
of minus k star t star.

548
00:35:56,630 --> 00:35:59,050
And here, k stars in an
extinction coefficient

549
00:35:59,050 --> 00:36:00,254
for the foam.

550
00:36:00,254 --> 00:36:02,170
Talk a little bit more
about that in a minute.

551
00:36:02,170 --> 00:36:10,090


552
00:36:10,090 --> 00:36:12,200
and t star is just the
thickness of the foam.

553
00:36:12,200 --> 00:36:18,970


554
00:36:18,970 --> 00:36:20,758
And then this thing
is called Beer's Law.

555
00:36:20,758 --> 00:36:28,800


556
00:36:28,800 --> 00:36:31,120
So we have very thin
walls and struts.

557
00:36:31,120 --> 00:36:33,890
And we're just going to consider
optically thin walls and struts

558
00:36:33,890 --> 00:36:35,070
to make life easy.

559
00:36:35,070 --> 00:36:37,070
Then we can say that, if
they're optically thin,

560
00:36:37,070 --> 00:36:38,906
they're transparent
to radiation.

561
00:36:38,906 --> 00:36:40,280
They're optically
thin if they're

562
00:36:40,280 --> 00:36:42,170
less than about 10 microns.

563
00:36:42,170 --> 00:36:44,950
Then this extinction
coefficient is just the amount

564
00:36:44,950 --> 00:36:47,420
of solid times the extension
coefficient for the solids.

565
00:36:47,420 --> 00:36:49,670
So it's just the relative
density times the extinction

566
00:36:49,670 --> 00:36:50,936
coefficient for the solid.

567
00:36:50,936 --> 00:37:20,500


568
00:37:20,500 --> 00:37:25,390
OK, and then I can say,
the heat flux by radiation.

569
00:37:25,390 --> 00:37:28,320


570
00:37:28,320 --> 00:37:31,148
I can use two equations
to write that down now.

571
00:37:31,148 --> 00:37:34,890


572
00:37:34,890 --> 00:37:37,060
And then I'm going to
let them be equal to get

573
00:37:37,060 --> 00:37:38,870
the thermal conductivity.

574
00:37:38,870 --> 00:37:44,920
I can say qr is going equal
to lambda r times dt by dx.

575
00:37:44,920 --> 00:37:49,800
So that's the Fourier's Law
that we started out with.

576
00:37:49,800 --> 00:37:52,160
And then I've also
got the qr that I'm

577
00:37:52,160 --> 00:37:55,780
going to get by combining the
Stefan's Law with the Beer's

578
00:37:55,780 --> 00:37:57,440
Law up there.

579
00:37:57,440 --> 00:38:00,130
So if I do that, I
get that qr is beta

580
00:38:00,130 --> 00:38:09,630
1 times sigma times t1 to the
fourth minus t not the fourth.

581
00:38:09,630 --> 00:38:14,730
So that's the qr not up
there from down there.

582
00:38:14,730 --> 00:38:18,370
And then I've got an
exponential for the attenuation.

583
00:38:18,370 --> 00:38:21,500
And instead of k star, I'm going
to put the relative density

584
00:38:21,500 --> 00:38:22,268
of times ks.

585
00:38:22,268 --> 00:38:27,610


586
00:38:27,610 --> 00:38:32,330
and then I've got the thickness
of the foam, t star, as well.

587
00:38:32,330 --> 00:38:39,900
OK, so that's qr, but that has
to equal lambda times dt by dx.

588
00:38:39,900 --> 00:38:44,700


589
00:38:44,700 --> 00:38:46,402
So I'm going to use
some approximations.

590
00:38:46,402 --> 00:38:48,360
Here and I'm going to
end up with an expression

591
00:38:48,360 --> 00:38:51,162
for the contribution
from radiation

592
00:38:51,162 --> 00:38:52,370
to heat transfer in the foam.

593
00:38:52,370 --> 00:38:53,134
Yeah?

594
00:38:53,134 --> 00:38:55,300
STUDENT: So when you say
optically thin walls, where

595
00:38:55,300 --> 00:38:57,050
t is less than 10
microns, you mean

596
00:38:57,050 --> 00:38:59,585
like the walls of the foam?

597
00:38:59,585 --> 00:39:00,460
PROFESSOR: Yeah, yea.

598
00:39:00,460 --> 00:39:02,774
STUDENT: So it's
different t than the--

599
00:39:02,774 --> 00:39:04,190
PROFESSOR: t star
is the thickness

600
00:39:04,190 --> 00:39:05,850
of the whole thing, yeah.

601
00:39:05,850 --> 00:39:07,800
So imagine we had
our two surfaces.

602
00:39:07,800 --> 00:39:10,320
And they might be like
100 millimeters apart

603
00:39:10,320 --> 00:39:12,650
or something. t star is
the sort of thickness

604
00:39:12,650 --> 00:39:15,170
of the foam in between
the two surfaces.

605
00:39:15,170 --> 00:39:18,250
And the optically thin
is the cell walls, which

606
00:39:18,250 --> 00:39:21,687
are microns kind of thickness.

607
00:39:21,687 --> 00:39:22,186
OK.

608
00:39:22,186 --> 00:39:30,570


609
00:39:30,570 --> 00:39:32,840
So I'm going to make
some approximations here.

610
00:39:32,840 --> 00:39:36,560


611
00:39:36,560 --> 00:39:39,140
And that's going to allow
me to solve for t star.

612
00:39:39,140 --> 00:39:43,060
So I'm going to
say that dt by dx

613
00:39:43,060 --> 00:39:47,100
x is approximately
equal to just t1 minus t

614
00:39:47,100 --> 00:39:51,570
not over the
thickness of the foam

615
00:39:51,570 --> 00:39:58,220
or I'll call that
delta t over t star.

616
00:39:58,220 --> 00:40:00,440
And then, the other
approximation I'm going to use

617
00:40:00,440 --> 00:40:05,420
is that t1 to the 4th
minus t not to the fourth

618
00:40:05,420 --> 00:40:10,860
is equal to 4
times delta t times

619
00:40:10,860 --> 00:40:13,240
the average temperature cubed.

620
00:40:13,240 --> 00:40:18,350
So here t bar is the average
temperature, t1 plus t

621
00:40:18,350 --> 00:40:19,090
not over 2.

622
00:40:19,090 --> 00:40:44,070


623
00:40:44,070 --> 00:40:47,220
So then, if I use those
two approximations,

624
00:40:47,220 --> 00:40:52,470
I can write that qr, our heat
flux from radiative transfer.

625
00:40:52,470 --> 00:40:53,500
I got the beta 1.

626
00:40:53,500 --> 00:40:54,710
I've got the sigma.

627
00:40:54,710 --> 00:40:59,280
And instead of the difference
of the fourth power,

628
00:40:59,280 --> 00:41:05,250
I'm going to write 4
delta t t bar cubed.

629
00:41:05,250 --> 00:41:06,675
And then I've got
my exponential.

630
00:41:06,675 --> 00:41:09,780


631
00:41:09,780 --> 00:41:12,130
Blah, blah, blah, blah, blah.

632
00:41:12,130 --> 00:41:17,990
So then, here's the relative
density times ks times

633
00:41:17,990 --> 00:41:20,780
t star, the overall thickness.

634
00:41:20,780 --> 00:41:23,720
That's going to equal the
radiative contribution

635
00:41:23,720 --> 00:41:26,300
to the thermal
conductivity of the foam.

636
00:41:26,300 --> 00:41:29,230
And instead of dt
by dx, I'm going

637
00:41:29,230 --> 00:41:34,160
to have delta t
over t star here.

638
00:41:34,160 --> 00:41:35,940
So part of the reason
for doing these

639
00:41:35,940 --> 00:41:40,490
approximations I end up with
a delta t term on both sides.

640
00:41:40,490 --> 00:41:42,150
Now I can cancel that out.

641
00:41:42,150 --> 00:41:45,540
And if I just take this mess
here and multiply it by t star,

642
00:41:45,540 --> 00:41:47,670
then I've got lambda r star.

643
00:41:47,670 --> 00:42:03,560


644
00:42:03,560 --> 00:42:08,790
That's our thermal conductivity
contribution from radiation.

645
00:42:08,790 --> 00:42:10,840
So one of the things
to notice here

646
00:42:10,840 --> 00:42:15,820
is that, as the relative
density goes down,

647
00:42:15,820 --> 00:42:19,690
then the contribution
from radiation

648
00:42:19,690 --> 00:42:22,802
to the thermal conductivity
of the foam goes up.

649
00:42:22,802 --> 00:42:49,920


650
00:42:49,920 --> 00:42:53,900
OK, so this chart here
shows thermal conductivity

651
00:42:53,900 --> 00:42:55,330
as a function of
relative density.

652
00:42:55,330 --> 00:42:56,830
And it breaks down
the contributions

653
00:42:56,830 --> 00:43:00,770
from the gas, g, the solid,
s, and the radiation, r.

654
00:43:00,770 --> 00:43:03,070
And you kind of see
the gas contribution

655
00:43:03,070 --> 00:43:04,230
doesn't change that much.

656
00:43:04,230 --> 00:43:06,790
These are relative densities
between a little over 2

657
00:43:06,790 --> 00:43:08,330
and a little less than 5%.

658
00:43:08,330 --> 00:43:10,570
So the amount of
gas-- it's mostly

659
00:43:10,570 --> 00:43:14,050
gas in all of these things.

660
00:43:14,050 --> 00:43:18,670
The solid contribution increases
as the relative density

661
00:43:18,670 --> 00:43:19,260
increases.

662
00:43:19,260 --> 00:43:21,670
So you'd expect that.

663
00:43:21,670 --> 00:43:26,030
And then, as I just said, as
the relative density goes down,

664
00:43:26,030 --> 00:43:30,030
the amount of radiation
contribution goes up.

665
00:43:30,030 --> 00:43:32,800
And so you can kind of see
how that all fits together.

666
00:43:32,800 --> 00:43:35,654
Another plot that shows
the thermal conductivity

667
00:43:35,654 --> 00:43:36,820
versus the relative density.

668
00:43:36,820 --> 00:43:39,590
These are for a few
different types of foams.

669
00:43:39,590 --> 00:43:42,450
You can see for
this plot here, you

670
00:43:42,450 --> 00:43:45,067
reach a minimum in the
thermal conductivity.

671
00:43:45,067 --> 00:43:46,900
And that's because
you've got this trade off

672
00:43:46,900 --> 00:43:49,399
between the contribution from
the solid and the contribution

673
00:43:49,399 --> 00:43:50,760
from the radiation.

674
00:43:50,760 --> 00:43:54,750
And those two kind of trade
off and you get to a minimum.

675
00:43:54,750 --> 00:43:56,545
So let me write
some of this down.

676
00:43:56,545 --> 00:43:59,950


677
00:43:59,950 --> 00:44:03,968
So I'll just say that-- hang on.

678
00:44:03,968 --> 00:44:05,610
Write this over again.

679
00:44:05,610 --> 00:44:13,897


680
00:44:13,897 --> 00:44:16,063
This is looking at the
overall thermal conductivity.

681
00:44:16,063 --> 00:44:19,600


682
00:44:19,600 --> 00:44:24,650
And we can see the
relative contributions

683
00:44:24,650 --> 00:44:30,180
of lambda, solid, lambda,
gas, lambda, radiation.

684
00:44:30,180 --> 00:44:31,942
I'll just say this
shown in the figure.

685
00:44:31,942 --> 00:44:38,490


686
00:44:38,490 --> 00:44:49,389
I'm going to say the next
figure shows a minimum

687
00:44:49,389 --> 00:44:50,555
in the thermal conductivity.

688
00:44:50,555 --> 00:44:53,990


689
00:44:53,990 --> 00:45:00,630
Then I'll just say
there's a trade off

690
00:45:00,630 --> 00:45:03,300
between the conduction
through the solid

691
00:45:03,300 --> 00:45:05,593
and I can direction
from the radiation.

692
00:45:05,593 --> 00:45:10,610


693
00:45:10,610 --> 00:45:12,640
And then we also
have a plot here

694
00:45:12,640 --> 00:45:16,637
that shows the conductivity
versus the cell size.

695
00:45:16,637 --> 00:45:27,620


696
00:45:27,620 --> 00:45:30,710
And you can see that the
conductivity increases

697
00:45:30,710 --> 00:45:31,775
with cell size.

698
00:45:31,775 --> 00:45:39,580


699
00:45:39,580 --> 00:45:41,960
And the reason for that is
the bigger the cells get,

700
00:45:41,960 --> 00:45:45,040
the radiation is
reflected less often.

701
00:45:45,040 --> 00:45:58,120


702
00:45:58,120 --> 00:46:04,800
And one thing I wanted to
mention with the cell size

703
00:46:04,800 --> 00:46:08,920
is that if you look at aerogels,
the way aerogels shells work

704
00:46:08,920 --> 00:46:11,790
is that they have a very small
cell size, a very small pore

705
00:46:11,790 --> 00:46:12,610
size.

706
00:46:12,610 --> 00:46:15,200
So typically, it's less
than 100 nanometers.

707
00:46:15,200 --> 00:46:20,230
And the mean free path
of air is 68 nanometers.

708
00:46:20,230 --> 00:46:22,140
So the mean free path
is the average distance

709
00:46:22,140 --> 00:46:25,400
the molecules move before they
collide with another molecule.

710
00:46:25,400 --> 00:46:29,130
And if your pore size is
less than the mean free path,

711
00:46:29,130 --> 00:46:32,360
then that reduces the
thermal conductivity.

712
00:46:32,360 --> 00:46:34,200
It reduces the
ability of the atoms

713
00:46:34,200 --> 00:46:37,810
to pass the heat along
between one another.

714
00:46:37,810 --> 00:46:39,310
So the way the
aerogels work is they

715
00:46:39,310 --> 00:46:40,830
have a very small pore size.

716
00:46:40,830 --> 00:46:46,690


717
00:46:46,690 --> 00:46:49,120
And what's important
is how big the pores

718
00:46:49,120 --> 00:46:50,900
are relative to the
mean free path of air.

719
00:46:50,900 --> 00:48:04,380


720
00:48:04,380 --> 00:48:06,790
OK, so that's the
thermal conductivity.

721
00:48:06,790 --> 00:48:09,820
I wanted to talk about a
few other thermal properties

722
00:48:09,820 --> 00:48:11,205
of foams, as well, today.

723
00:48:11,205 --> 00:48:43,100


724
00:48:43,100 --> 00:48:45,100
So one is the specific heat.

725
00:48:45,100 --> 00:48:47,760
And since the specific
heat is the energy required

726
00:48:47,760 --> 00:48:51,990
to raise the temperature
by a unit mass,

727
00:48:51,990 --> 00:48:57,794
then the mass is
the same-- you know,

728
00:48:57,794 --> 00:48:59,210
if you have a
certain mass of foam

729
00:48:59,210 --> 00:49:01,440
or a certain mass of solid--
the specific heat from the foam

730
00:49:01,440 --> 00:49:02,481
is the same as the solid.

731
00:49:02,481 --> 00:49:32,390


732
00:49:32,390 --> 00:49:36,030
So the specific
heat for the foam

733
00:49:36,030 --> 00:49:40,080
is the same as the specific
heat for the solid.

734
00:49:40,080 --> 00:49:41,680
So that would have
units of joules

735
00:49:41,680 --> 00:49:43,610
per kilogram per degree k.

736
00:49:43,610 --> 00:49:48,560


737
00:49:48,560 --> 00:49:51,490
And the next property is the
thermal expansion coefficient.

738
00:49:51,490 --> 00:50:02,380


739
00:50:02,380 --> 00:50:03,460
And it's a similar thing.

740
00:50:03,460 --> 00:50:06,150
The thermal expansion
coefficient for the foam

741
00:50:06,150 --> 00:50:07,850
is equal to the
thermal coefficient

742
00:50:07,850 --> 00:50:10,840
of expansion for the solid.

743
00:50:10,840 --> 00:50:14,250
So imagine you have-- say you
had something like a honeycomb.

744
00:50:14,250 --> 00:50:15,910
If you heat it up
a certain amount,

745
00:50:15,910 --> 00:50:17,879
every member is going
to expand by alpha.

746
00:50:17,879 --> 00:50:19,420
And if every member
expands by alpha,

747
00:50:19,420 --> 00:50:20,794
the whole thing
expands by alpha.

748
00:50:20,794 --> 00:50:22,460
And this is the same.

749
00:50:22,460 --> 00:50:24,090
And it's the same
idea with the foam.

750
00:50:24,090 --> 00:50:26,590
So if every member just
gets longer by alpha,

751
00:50:26,590 --> 00:50:28,340
then the whole thing
gets bigger by alpha.

752
00:50:28,340 --> 00:50:44,010


753
00:50:44,010 --> 00:50:46,730
OK, so the last topic
I wanted to talk about

754
00:50:46,730 --> 00:50:48,612
was the thermal
shock resistance.

755
00:50:48,612 --> 00:50:51,070
And thermal shock is the idea
is that if you have something

756
00:50:51,070 --> 00:50:54,140
that's hot, and say you
quench it in a liquid--

757
00:50:54,140 --> 00:50:56,220
so you put it
suddenly in a liquid--

758
00:50:56,220 --> 00:50:57,630
the surface is
going to cool down

759
00:50:57,630 --> 00:50:59,560
faster than the bulk of it.

760
00:50:59,560 --> 00:51:00,970
And because the
surface is trying

761
00:51:00,970 --> 00:51:02,785
to contract because
it's cooling down,

762
00:51:02,785 --> 00:51:05,160
but it's attached to the bulk
of it and it's constrained,

763
00:51:05,160 --> 00:51:08,692
it can't really cool down,
then you generate stresses.

764
00:51:08,692 --> 00:51:10,150
And if the stresses
are big enough,

765
00:51:10,150 --> 00:51:12,566
you can cause fracture and
have the thing crack and spall.

766
00:51:12,566 --> 00:51:17,530


767
00:51:17,530 --> 00:51:20,340
So we'll say if the
materials is subjected

768
00:51:20,340 --> 00:51:34,880
to a sudden change in the
surface temperature, that

769
00:51:34,880 --> 00:51:37,060
induces thermal
stresses at the surface

770
00:51:37,060 --> 00:51:39,080
and can induce
spalling and cracking.

771
00:51:39,080 --> 00:52:00,642


772
00:52:00,642 --> 00:52:03,100
So we're going to think about
a material at one temperature

773
00:52:03,100 --> 00:52:04,570
that's dropped
into, say, a liquid

774
00:52:04,570 --> 00:52:05,695
at a different temperature.

775
00:52:05,695 --> 00:52:27,912


776
00:52:27,912 --> 00:52:29,370
So the surface
temperature is going

777
00:52:29,370 --> 00:52:31,564
to drop to the cooler
liquid temperature

778
00:52:31,564 --> 00:52:33,480
and it's going to contract
the surface layers.

779
00:52:33,480 --> 00:52:52,560


780
00:52:52,560 --> 00:52:55,510
And the fact that they're bound
to the layers underneath that

781
00:52:55,510 --> 00:52:57,710
are not contracting
as quickly, it

782
00:52:57,710 --> 00:53:00,860
means that you generate
a thermal strain.

783
00:53:00,860 --> 00:53:11,821


784
00:53:11,821 --> 00:53:13,320
So the thermal
strain is going to be

785
00:53:13,320 --> 00:53:15,904
the coefficient of thermal
expansion times the change

786
00:53:15,904 --> 00:53:16,528
in temperature.

787
00:53:16,528 --> 00:53:52,300


788
00:53:52,300 --> 00:53:54,560
So you're going to
constrain the surface

789
00:53:54,560 --> 00:53:56,300
to the original dimensions.

790
00:53:56,300 --> 00:53:58,105
And then you're going
to induce the stress.

791
00:53:58,105 --> 00:54:19,450


792
00:54:19,450 --> 00:54:23,310
So if it's a plane or
thing, it's e alpha delta t.

793
00:54:23,310 --> 00:54:25,970
And then, there's a
factor of 1 minus nu,

794
00:54:25,970 --> 00:54:29,850
just because it's a
plane, in a plane.

795
00:54:29,850 --> 00:54:31,590
And then you'll get
cracking or spalling

796
00:54:31,590 --> 00:54:34,210
when that stress equals
some failure, stress.

797
00:54:34,210 --> 00:54:43,120


798
00:54:43,120 --> 00:54:47,490
So I can rearrange this and
solve for the critical delta t

799
00:54:47,490 --> 00:54:50,580
that you can withstand
without getting cracking.

800
00:54:50,580 --> 00:54:54,106
So I just rearranged this and
say sigma's equal to sigma f.

801
00:54:54,106 --> 00:55:00,600
That would be sigma f times 1
minus nu over e and over alpha.

802
00:55:00,600 --> 00:55:02,450
So that's the critical
change in temperature

803
00:55:02,450 --> 00:55:03,660
to just cause cracking.

804
00:55:03,660 --> 00:55:13,210


805
00:55:13,210 --> 00:55:15,864
So now what I can do is I can
substitute in there for what

806
00:55:15,864 --> 00:55:17,030
you would have for the foam.

807
00:55:17,030 --> 00:55:20,294


808
00:55:20,294 --> 00:55:22,210
And I'm going to do it
just for the open cells

809
00:55:22,210 --> 00:55:24,210
just because it's easier
to write the equations.

810
00:55:24,210 --> 00:55:27,580


811
00:55:27,580 --> 00:55:32,230
So for the foam, I would have
some sort of fracture strength.

812
00:55:32,230 --> 00:55:34,180
So when we did the
modeling of the foams,

813
00:55:34,180 --> 00:55:37,140
we said that was equal to
about 0.2 times the modulus

814
00:55:37,140 --> 00:55:42,980
of rupture times the relative
density to the 3/2's power

815
00:55:42,980 --> 00:55:46,730
and 1 minus nu.

816
00:55:46,730 --> 00:55:49,460
And if I divide by the
modulus of the foam,

817
00:55:49,460 --> 00:55:54,460
that's es times the
relative density squared.

818
00:55:54,460 --> 00:55:57,150
And then we just had
alpha for the foam

819
00:55:57,150 --> 00:55:58,575
was the same as alpha s.

820
00:55:58,575 --> 00:56:01,130


821
00:56:01,130 --> 00:56:03,760
So then, I can
rearrange this slightly

822
00:56:03,760 --> 00:56:07,640
and say it's equal to 0.2
over the relative density

823
00:56:07,640 --> 00:56:09,240
to the 1/2 power.

824
00:56:09,240 --> 00:56:12,950
So I'm canceling out these
relative densities here.

825
00:56:12,950 --> 00:56:17,431
And then I can combine all
the solid properties together.

826
00:56:17,431 --> 00:56:19,180
And I'm going to say
that nu for the solid

827
00:56:19,180 --> 00:56:21,263
is about equal to the
same as nu for the foam.

828
00:56:21,263 --> 00:56:34,650


829
00:56:34,650 --> 00:56:37,951
So what I can do here is I can
group all the solid properties

830
00:56:37,951 --> 00:56:38,450
together.

831
00:56:38,450 --> 00:56:41,880
And this just is delta t
critical for the solid, right?

832
00:56:41,880 --> 00:56:45,930
So this is saying that the
critical temperature range

833
00:56:45,930 --> 00:56:48,890
before you get
cracking in the foam

834
00:56:48,890 --> 00:56:52,050
is equal to the
range for the solid,

835
00:56:52,050 --> 00:56:54,879
but multiplied by
this factor of 0.2

836
00:56:54,879 --> 00:56:57,170
and divided by the square
root of the relative density.

837
00:56:57,170 --> 00:56:58,919
So if the square of--
the relative density

838
00:56:58,919 --> 00:56:59,959
is going to less than 1.

839
00:56:59,959 --> 00:57:02,000
So this number here is
going to be bigger than 1.

840
00:57:02,000 --> 00:57:04,830
So it's saying that the
temperature range that

841
00:57:04,830 --> 00:57:06,700
will give you
spalling in the foam

842
00:57:06,700 --> 00:57:09,780
is going to be bigger than the
temperature range in the solid.

843
00:57:09,780 --> 00:57:14,440
So the foam's going to be
better than the solid, OK?

844
00:57:14,440 --> 00:57:16,420
And that uses our little
models from before.

845
00:57:16,420 --> 00:57:21,217


846
00:57:21,217 --> 00:57:23,050
So I think I'm going
to stop there, probably

847
00:57:23,050 --> 00:57:27,030
cause my throat is
starting to get too sore.

848
00:57:27,030 --> 00:57:29,370
There's a little case
study in the notes.

849
00:57:29,370 --> 00:57:31,770
And I'll just put that on the
notes on the Stellar site.

850
00:57:31,770 --> 00:57:34,130
It's like one page and it's
really straightforward.

851
00:57:34,130 --> 00:57:36,171
You can just read that, OK?

852
00:57:36,171 --> 00:57:38,420
So this is the end of the
bit on thermal conductivity.

853
00:57:38,420 --> 00:57:40,530
That's just this one lecture.

854
00:57:40,530 --> 00:57:43,200
And this is really the
end of the whole section

855
00:57:43,200 --> 00:57:45,300
on modeling of the honey
combs and the foams.

856
00:57:45,300 --> 00:57:47,633
So that's kind of the first
half of the term is modeling

857
00:57:47,633 --> 00:57:49,317
the honey combs and the foams.

858
00:57:49,317 --> 00:57:50,900
And the second half
of the term, we're

859
00:57:50,900 --> 00:57:53,850
kind of applying those models
to different situations.

860
00:57:53,850 --> 00:57:56,720
So next week, we'll have
the review on Monday,

861
00:57:56,720 --> 00:57:59,730
have a test on Wednesday, week
after that is Spring break.

862
00:57:59,730 --> 00:58:02,339
I can't believe we're
at Spring break already.

863
00:58:02,339 --> 00:58:04,630
And then after that we'll
start we'll do the trabecular

864
00:58:04,630 --> 00:58:05,850
bone for a week.

865
00:58:05,850 --> 00:58:08,670
We'll do tissue engineering
scaffolds and cell mechanics

866
00:58:08,670 --> 00:58:10,490
for two or three lectures.

867
00:58:10,490 --> 00:58:12,830
We'll look at some
other applications

868
00:58:12,830 --> 00:58:15,910
to engineering design, look at
energy absorption and sandwich

869
00:58:15,910 --> 00:58:16,880
panels.

870
00:58:16,880 --> 00:58:19,088
And then, I'm going to talk
about plants a little bit

871
00:58:19,088 --> 00:58:19,970
at the very end, OK?

872
00:58:19,970 --> 00:58:23,200
So we've already covered a
lot of the kind of deriving

873
00:58:23,200 --> 00:58:24,910
equations part of the course.

874
00:58:24,910 --> 00:58:27,340
The rest of the course is
more applying the equations

875
00:58:27,340 --> 00:58:29,894
to lots of different
situations, OK?

876
00:58:29,894 --> 00:58:32,060
So I'm going to stop there
just because my throat is

877
00:58:32,060 --> 00:58:33,860
giving out.

878
00:58:33,860 --> 00:58:37,706