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Lecture 04 Solar Energy : Part 1 — Transcript

by IIT Roorkee July 2018 · 4,082 words · 772 segments · language en · Watch on YouTube

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0:00[music]
0:06[music]
0:11[music]
0:22[music]
0:25>> Welcome all of you.
0:26So, we have already completed the
0:28introduction part.
0:30Where we went through different basic
0:32concepts, energy scenario, Sankey
0:34diagram, and capacity factor and load
0:36factor.
0:38Now, coming to
0:39the actual part of the course, renewable
0:41energy sources.
0:43So, the first renewable energy source
0:44that we'll be looking at in detail is
0:48solar energy.
0:49So, in today's lecture, we'll study
0:52about sun and solar energy.
0:55What is sun? What are the properties?
0:58What type of radiations that we receive?
1:00What is the composition of radiations?
1:02Solar constant
1:04and the concept of air mass.
1:07So, sun,
1:09it is the largest object in our solar
1:12system. And
1:14our solar system accounts of sun and I
1:16think eight or nine planets. I think
1:19Pluto is not considered as a planet. So,
1:20right now we have eight planets moving
1:22around the sun in elliptical orbits.
1:25And then there are so many asteroids and
1:27comets which accounts for the entire
1:29solar system. So, of that solar system,
1:33of all the objects in the solar system,
1:35it's the sun which
1:38occupies or accounts for 99.86%
1:42of the total mass.
1:44So, it means this entire solar system is
1:46obviously about sun and it's
1:48much, much bigger than all the planets,
1:51each of the planets
1:52in the system.
1:54And solar energy, it is the radiant
1:57light and heat energy which comes from
1:59the sun.
2:00And it is this radiant heat and light
2:03which can be captured and converted into
2:05useful energy
2:06with respect to Earth.
2:08The characteristics of Sun
2:10it's
2:11ball of it's a sphere made of like gases
2:15and it has mass in range of 10 to the
2:17power 30 kg.
2:19Diameter is also like 1.3 million
2:21kilometers.
2:22And the temperature of the surface
2:25it's 5,700 Kelvin.
2:28And pressure is 0.0008 bar. But at the
2:31core, temperature and pressure had been
2:34million and
2:35million Kelvin and billion bar.
2:38And
2:39that's what we had studied earlier also.
2:42It is
2:43at this
2:45high temperature and pressure
2:47we have the reaction of
2:50nuclear
2:52fusion.
2:53The
2:55this high energy like high temperature
2:57and pressure it
3:00like
3:01it makes hydrogen atoms to come apart
3:03and their nuclei
3:05fuse together to form a helium atom. So,
3:08four
3:09nuclei of hydrogen atom
3:11combine together to form one
3:14helium atom and the mass of this formed
3:16helium atom is always
3:19less than
3:20mass of four hydrogen nuclei.
3:23And this difference in mass is emitted
3:24as energy and that's what is nuclear
3:26fusion.
3:28So, the solar radiation it is the
3:29electromagnetic energy emitted by the
3:31Sun created by nuclear fusion reactions
3:34which occur
3:35at its core.
3:37And it is this energy which travels
3:38through space and reaches the Earth
3:41in the form of heat and light.
3:44And this energy drives our planet's
3:48climate, weather patterns and life
3:50sustaining processes. If you remember
3:52the picture which I showed in the
3:54lecture one, where sun is responsible
3:57for directly or indirectly all other
3:58sources of energy on earth with some
4:01exceptions of nuclear and geothermal.
4:04It tells us like how all the life
4:06sustaining processes are survived or are
4:09sustained because of the energy provided
4:11by sun.
4:12Photosynthesis, for that we need
4:14sunlight.
4:15Okay, formation of like growth of
4:17biomass, fossil fuels, changes in wind
4:20patterns, weather patterns, hydro, wind,
4:23solar. So, all these are coming directly
4:25or indirectly from sun.
4:28The radiation or solar radiation, the
4:30units or the terminologies depend on
4:33like whether we are measuring it
4:36instantaneously or for a period of time.
4:38So, when we talk about instantaneous
4:40intensity of solar radiation hitting on
4:42a particular surface,
4:44the term is called irradiance. It tells
4:47us the power, watt per meter squared.
4:49Okay, this is instantaneous, so maybe
4:51for a second.
4:53And when we talk about energy
4:55in kilowatt hour per meter squared, it
4:57tells us the accumulated amount of or
4:59the collected amount of solar radiation
5:03on a area received over a specific
5:05period of time.
5:06And the term is called as insolation.
5:08Insolation term, it's actually made from
5:11the initials of three terms, which is
5:13incident solar radiation. So, you get in
5:17solation.
5:18That's how that term is formed.
5:20But often in numericals or in our daily
5:23lives, we'll treat
5:26these terms as just radiation. Many
5:28times you will just find that solar
5:30radiation of this type is
5:33100 watt per meter squared or 200 watt
5:35per meter squared. So, radiation is also
5:37commonly used term in place of
5:39irradiance or insolation.
5:41Let's look at availability of solar
5:42energy.
5:44So, all the energy which is coming from
5:45sun, if we
5:47give it a value say 100%.
5:50How much of it is actually making its
5:52way to the earth?
5:53So, most of it is reflected. So, almost
5:5730% is reflected from the atmosphere by
5:59the clouds or from the earth's surface.
6:03Okay? And 51% is absorbed by lands and
6:07oceans.
6:08Now, whatever is absorbed by lands and
6:10oceans or small amount of
6:13energy is absorbed by atmosphere or
6:15clouds.
6:17Total of this energy is later
6:19re-radiated back to space.
6:21And that's what completes the energy
6:24balance. If we look at 100% incoming
6:27energy, almost 100% goes back to the
6:30atmosphere. This is 624, that's 30, and
6:33this is 70. So, almost 100% goes back to
6:36the atmosphere later.
6:38But, during the time
6:40when this energy is absorbed by lands
6:43and oceans or other parts of the
6:45atmosphere, it is
6:47a portion of it is used
6:49for sustaining life processes on earth.
6:53So, amount-wise, the total solar energy
6:56which is absorbed by earth's atmosphere,
6:59ocean, and land masses is around 3.8
7:02million exajoule per year. Exajoule is
7:0510 to the power 18. That's a huge huge
7:08amount of energy which is absorbed by
7:11earth.
7:12And
7:13energy use or energy demand for human
7:16beings
7:17per year is just 550 exajoule. It's not
7:20even
7:220.01%
7:23of the total annual energy that is
7:26coming from the sun on the earth.
7:28And if you look at this matrix,
7:31it can help us understand like even
7:33though solar PV or solar thermal have
7:35like say low efficiencies,
7:37but
7:39if you are able to maximize the capture
7:41of solar energy to satisfy our energy
7:43needs.
7:44We have
7:46plenty of it. It is there's a huge
7:48abundance of available solar energy
7:50compared to what it is what is actually
7:52required. And that's the
7:54one of the reason why the investment and
7:57the technology and R&D in solar sector
7:59has gone
8:01like has been exponential. The growth or
8:03the rise of all these investments in the
8:06solar sector has been
8:07exponential.
8:10Now, if you look at the spectrum of
8:11solar radiation, what type of radiations
8:13they are come like composed of. So, it's
8:16mainly three types: ultraviolet,
8:18visible, and infrared based on the
8:20spectrum. And you can see from this
8:22graph also
8:24infrared radiation makes up to 49 to
8:2750%.
8:29Visible light
8:3042 to 43% and ultraviolet radiation
8:34makes up 7 to 8%.
8:37The red pictures which you can which we
8:38are seeing here like this is the graph
8:40of irradiance
8:42versus wavelength. And each of these
8:46three types of radiations have different
8:48range of wavelengths and that's how
8:49these graphs are marked.
8:52Okay.
8:53Now, in
8:54infrared radiation or in a visible
8:57light, this is the yellow part shows the
8:59sunlight without atmospheric absorption.
9:02And the red one shows the sunlight at
9:04sea level.
9:05So,
9:07this water vapor, oxygen, ozone, CO2,
9:09these are the greenhouse gases. And they
9:11mainly try to absorb the radiation. And
9:14that radiation is
9:16infrared radiation.
9:18You might have heard that UV is not good
9:21for human health.
9:23Wherever we have holes in the ozone
9:25layer, it's because of UV people are
9:27getting skin diseases, skin cancer,
9:29burning skin burns. Okay, so yeah, UV is
9:33harmful. But if you talk about global
9:35warming,
9:36it's not me because of UV or visible
9:38light,
9:39but mainly the infrared radiations
9:42because it is this radiation which is
9:44absorbed primarily by greenhouse gases.
9:48And when this radiation is absorbed by
9:49GHGs,
9:50with the increasing concentration of
9:52GHGs in atmosphere because of
9:54anthropogenic activities,
9:56it has led to rise in global warming.
9:59So, out of all these three radiations,
10:01it's the infrared radiation which leads
10:03to global warming.
10:06Now, with that background of sun and
10:08solar energy and composition of
10:10radiation, let's look at the first
10:13concept, that is solar constant.
10:16So,
10:18if with respect to
10:20sun,
10:24if Earth is at some distance,
10:26and the maximum energy which is received
10:29by from sun to Earth,
10:31this is going to be maximum right at
10:34this point before the atmosphere of the
10:36Earth starts. Actually, at the surface
10:39of the Earth just outside the
10:41atmosphere. And it is this value which
10:43is called as solar constant.
10:45It is a total energy received from the
10:48sun per unit time on a unit surface area
10:52which is
10:53perpendicular to the radiation in space
10:55just outside the Earth's atmosphere.
10:58Okay. So, at this point
11:00or
11:01at this point which is say perpendicular
11:04to the sun's radiation,
11:06we get the maximum radiation.
11:08And this is
11:10this happens when the Earth is at its
11:12mean distance from the sun.
11:15So, if this is sun with radius R,
11:18Earth has its own radius, and let R be
11:20the mean distance from the center of
11:22Earth to the center of the sun. Earth
11:25keeps on
11:26revolving around sun. It also keeps on
11:28rotating around itself.
11:30So, for a solar constant,
11:32we are just interested in the mean
11:34distance between Earth and Sun's
11:36distance. Now, how do we derive this
11:38value? So, we know that Earth moves
11:41around the Sun in the elliptical orbit,
11:43but for deriving the value of solar
11:45constant, we make assumption
11:48that Earth is moving in a
11:51circular path of radius R
11:54with Sun as its center. So, this radius
11:57R is nothing but the mean distance
11:59between Sun and Earth. So, let's assume
12:02that Earth is moving around Sun
12:05in circular path.
12:07Now, the second assumption here is Sun
12:10is a perfectly black body
12:12so that the energy which is radiated
12:15from Sun
12:16per unit time
12:19from the surface of the Sun as per
12:21Stefan-Boltzmann law. What is
12:23Stefan-Boltzmann law? The heat transfer
12:26flux
12:27because of radiation it is proportional
12:30to the surface area and it is also
12:33proportional to the fourth power of the
12:36temperature of the surface or
12:38temperature of the source.
12:39And the constant of that proportionality
12:41is called as
12:43Stefan-Boltzmann constant. So, the heat
12:46the energy radiated per unit time will
12:48be H equal to A sigma T to the power 4,
12:52where A is the area surface area of the
12:54Sun.
12:55T is its absolute temperature of the
12:57surface
12:58and sigma is the Stefan-Boltzmann
13:01constant which has a fixed universal
13:03value.
13:04So, till this point is clear?
13:06What we did?
13:08And the derivation starts with the
13:09assumption that Earth is moving in a
13:12circular orbit with the radius with the
13:16radius equal to mean distance that is
13:18small R.
13:19Sun is a perfectly black body. So, if we
13:22apply Stefan-Boltzmann's law, the energy
13:24radiated is given as
13:26H equal to A sigma T to the power 4.
13:30Now, how much energy is actually
13:32received by This is the energy
13:34given by the sun. Now, how much is the
13:36energy received by the earth? That's
13:38what we need to calculate.
13:41So, uh since sun is a sphere, the area
13:43of the sphere is
13:454 pi r squared, where r is the radius of
13:47the sun. So, we can write it
13:49as just 4 pi r squared sigma T to the
13:51power 4.
13:52Now,
13:54by the time this power which is emitted
13:55by sun,
13:56it reaches earth,
13:58it is spread across a sphere with a
14:01radius equal to the earth and sun's mean
14:04distance, which is r.
14:06So,
14:07while it is spread over all this
14:09distance, the energy received by earth's
14:12per unit area per second would be
14:15whatever energy is
14:16transferred upon
14:18the sphere
14:20with the radius r, that is 4 pi r
14:23squared.
14:25H value is given to us
14:274 pi r squared sigma T to the power 4.
14:30Now, this 4 pi can be strike off.
14:34What we would get is r upon r squared
14:36sigma T to the power 4.
14:39Now, in this, where S is the
14:42energy received by earth's unit area per
14:45second. And this is what is solar
14:47constant also.
14:49So, R, that is radius of sun,
14:53small r, mean distance,
14:55sigma, Stefan-Boltzmann constant, and T
14:58is the temperature. All four are
15:00universally
15:01known quantities.
15:03If you substitute these numbers in this
15:05formula, you should be able to get the
15:07value of solar constant as
15:101366 or 1367 W per m squared.
15:14Okay?
15:15Depending on because certain books refer
15:17to 5778 K or 5777 K,
15:21it may change slightly, but it will be
15:23within this range only. It will be
15:24between 1361 to 1369 for sure.
15:28Now, that is the value of solar
15:30constant. So,
15:31at a point
15:34just outside the Earth's atmosphere,
15:38since Earth is called as
15:40terrestrial territory, this can be
15:43called as extraterrestrial point or
15:45extraterrestrial radiation. This the
15:47radiation received at this point is
15:49called as solar constant. But now,
15:52coming back to the reality,
15:54which tells us Earth's orbit is
15:57elliptical around the Sun and not
16:00circular.
16:02So, this if it if Earth is moving around
16:06the Sun in this elliptical orbit,
16:09it varies by plus minus 3%
16:12within the same value. Okay. And
16:16this radiation, like I told you earlier,
16:18solar radiation is in just outside the
16:21Earth's atmosphere is called
16:22extraterrestrial radiation.
16:24And it is the radiation striking the
16:26surface of Earth assuming that there is
16:29no atmosphere. Suppose you assume that
16:31there is no atmosphere, so whatever
16:32point we are talking at the surface of
16:35Earth outside the atmosphere, the same
16:37amount of radiation would strike the
16:38surface of Earth.
16:40>> [snorts]
16:40>> And solar constant is nothing but the
16:42average value of this extra
16:43extraterrestrial radiation.
16:46And if you account for that plus minus
16:483%
16:49with the time of the year,
16:51the solar constant or solar irradiance
16:53varies from 1400
16:56uh I guess 20 something
16:58to
16:591300
17:0121.
17:02Okay. So, that's the range within which
17:04solar constant
17:06can vary or extraterrestrial radiation
17:08can vary and the average of this comes
17:10out to be the solar constant, which is
17:121367.
17:13The second concept that we have to cover
17:15today is air mass.
17:18So, solar constant, it gave us the
17:20maximum
17:22solar radiation which can reach at a
17:24point just outside the Earth's
17:27atmosphere or Earth's
17:29uh if we assume atmosphere as part of
17:31Earth. So, that's the Earth's surface.
17:33>> [snorts]
17:34>> Now, airmass,
17:35it tells us it is a parameter which
17:38determines the solar irradiance under
17:40clear sky.
17:42Okay? And it is the distance that
17:44sunlight has to travel through the
17:47atmosphere. So, if sun is here and
17:49radiation has to travel through the
17:50atmosphere,
17:52so this proportion, this distance is
17:54actually the
17:56parameter determining the airmass. The
17:58ratio of actual path of sunlight, so
18:02this is actual path of sunlight,
18:04to the minimal distance, the minimal
18:07distance, it's known as the
18:10airmass. Minimal distance will be at
18:12the zenith point or the noon point.
18:14>> [snorts]
18:14>> So, airmass represents a portion of
18:16atmosphere
18:17that the light has
18:19light has to pass through before
18:21striking the Earth
18:23related to its overhead path length and
18:25is equal to Y by X. So, when sun is at
18:28the
18:29zenith position or the noon position,
18:31angle
18:32Z or
18:35theta Z is equal to zero.
18:37Now, if sun is at some position,
18:40let's say here and the distance is Y.
18:43So, the angle measured from the vertical
18:45is called zenith angle.
18:47So,
18:48the airmass is defined as
18:51actual length,
18:53that is Y,
18:54by the shortest length,
18:56that is X. So, this can be
18:59derived by the cosine of this angle. So,
19:02if I just calculate
19:04cos
19:05Z,
19:07it will be It's a right angle triangle
19:08here.
19:09So, we can apply trigonometry. Cos Z is
19:12equal to adjacent side
19:15by
19:16hypotenuse.
19:17So, 1 by cos Z
19:20will be Y by
19:22X.
19:23So, Y by X is our air mass. So, 1 by cos
19:26Z is the air mass. So, if we know the
19:29angle from the vertical, if we know the
19:30zenith angle,
19:31that which tells us the position of the
19:33sun with respect to the vertical, we can
19:35calculate the air mass.
19:38Now, there is another way
19:40to calculate air mass.
19:42So,
19:43when the sun is at zenith,
19:45optical air mass will be unity because
19:48cos zero is
19:50one. So, one by one would be one. So,
19:53air mass will be one
19:55when the sun is at its zenith, the
19:57topmost position,
19:59the overhead position.
20:01And air mass is commonly used to
20:03characterize the performance of solar
20:05cells under standardized conditions. If
20:07you look if you take any solar cell, the
20:09real solar cell, there you will see some
20:11specifications mentioned there. The
20:13radiation, it would be 1,000 W per m
20:15squared tested for that. Then air mass,
20:18it will say one, 1.5 or air mass two.
20:20Some kind of specifications in terms of
20:23air mass would be there.
20:25Another method,
20:27if you don't know the zenith angle, the
20:29easy method
20:31using Pythagoras theorem.
20:33So, if we have object or a pole of
20:35height H,
20:36and if sun is behind that pole, so it
20:39will have some shadow. And if we
20:42calculate the length of the shadow,
20:44based on that, we can calculate the air
20:46mass. Air mass is
20:491 plus S by H squared. So, if we know S,
20:53if we know H, we can calculate the air
20:55mass just by using the Pythagoras
20:57theorem, which is this value.
21:00Now,
21:01above expressions, both the zenith angle
21:04one
21:05and this Pythagoras theorem one,
21:07they assume that the atmosphere is a
21:10flat horizontal layer. But in reality,
21:12Earth is not flat. Earth is not horiz-
21:14it's a spherical in shape. So,
21:17with respect to the sun's position
21:20the surface of the Earth is like a
21:22curvature and not a flat horizontal
21:24layer. So, a lot of researchers have
21:26done
21:27and have come up with some correlations
21:29to account for the curvature of the
21:31atmosphere.
21:33And in that
21:35it was observed that air mass is not
21:37quite equal to the atmospheric path
21:39length. Whenever the sun is close to the
21:41horizon, it's not equal to the path
21:44length.
21:44One such equation is given by Kasten
21:47Kasten and Young in 1989 where you can
21:49see it's 1 upon cos Z plus some
21:53zenith angle based mathematical
21:55correlation which takes into account the
21:58curvature of the Earth's atmosphere.
22:00But
22:01commonly we use those simple approaches
22:04because for most of the angles like at
22:08least till 75°.
22:10The values given by
22:13uh both approaches is almost similar.
22:15Only at very high zenith angles
22:19uh like when sun is like almost towards
22:22like very close to sunrise or sunset
22:25where we won't be much interested in the
22:27in tapping the radiations at that point
22:30like after 85° there is a huge variation
22:33between the two approaches, but for
22:37the majority of the
22:39time when we are most concerned about
22:41the radiations, it gives us the similar
22:43values. So, that's why
22:46for both for air mass, we can use the
22:49approximate formula.
22:52Let's take one numerical before we close
22:56this lecture. So, we had studied the
22:57concept of solar constant and solar and
23:01air mass.
23:03Now, extraterrestrial radiation
23:06for solar irradiance or solar constant
23:08is 1367. It's a universally known value.
23:12At a airmass of two,
23:15if atmosphere is able to transmit only
23:1770% of that radiation,
23:20so if this is
23:24and this is the atmosphere just at the
23:26surface.
23:27Let me make it a little bigger.
23:30So, at this point
23:32we have the radiation as 1367.
23:35Now, when it passes through the
23:37atmosphere and that
23:40aerial atmosphere,
23:43it governs the property of airmass. So,
23:45if airmass is two,
23:48only 70% of the radiation is able to
23:52reach the Earth's like land surface.
23:55Now, what do you have to calculate?
23:57Calculate the solar irradiance reaching
23:59the Earth's surface, that means the
24:01land,
24:02and the corresponding zenith angle.
24:05So, let's see. So, obviously for
24:08airmass, we will be needing zenith angle
24:11or for the uh zenith angle, we need
24:12airmass. It is given to us, so we should
24:14be able to calculate from 1 by cos z.
24:18>> [snorts]
24:18>> Given data is solar constant 1367,
24:22transmission
24:2470%.
24:26So, solar irradiance reaching the
24:27Earth's surface would be 70% of
24:311367.
24:33Right?
24:35It comes out to be 957 W/m².
24:39And for
24:41zenith angle, what we need is airmass.
24:45Airmass equal to 1 upon
24:48cos zenith angle.
24:50So, for airmass two,
24:52uh
24:53it will be 1 by
24:55cos z or cos z is equal to 1 by 2. It is
24:59represented as Z or theta Z, okay?
25:02So, cos inverse of 1 by 2 is
25:0560°. So, that's the zenith angle for
25:07this given numerical.
25:09So, if we know the air mass, we can
25:11calculate zenith angle and the vice
25:13versa. If we know zenith angle, we can
25:14calculate air mass.
25:16And from the transmission and solar
25:18constant, we are able to calculate the
25:21actual
25:22solar irradiance which made its way to
25:24the surface of the earth.
25:27So, with that I think we can close
25:30today's lecture. Just to summarize, we
25:32started with
25:33sun,
25:35how it constitutes
25:37major portion of the mass of solar
25:38system, and what are its
25:41like key parameters, like pressure,
25:43temperature, diameter, mass.
25:45And it clearly shows that it is
25:47significantly huge than any of the
25:49planets in the solar system.
25:51Then we looked at
25:52solar energy or solar radiation.
25:55What's the composition of solar
25:56radiation in terms of
25:58UV,
26:00visible,
26:01and infrared. And we also discussed how
26:04infrared radiation is the one which is
26:06responsible for greenhouse gases, or
26:09which is which is which can be absorbed
26:11by the greenhouse gases, and then which
26:13is responsible for global warming or
26:15climate change.
26:16Then
26:18the next concept we studied was of solar
26:19constant. It tells us the maximum amount
26:23of radiation that can reach on the
26:24earth's surface right at a point just
26:27outside the atmosphere.
26:29Okay? Or it can be also defined as the
26:32extraterrestrial radiation, or the
26:34average value of the extraterrestrial
26:36radiation with respect to earth. And
26:38that number comes out to be 1367
26:41W/m².
26:43We looked at the derivation, how to get
26:46this value
26:47using Stefan-Boltzmann's law, and the
26:50swept area in the form of sphere with
26:52respect to Earth and Sun mean distance,
26:54right?
26:55But then in the assumption we had made
26:57circular orbit. If we take into account
26:59elliptical orbit, it varies by plus
27:01minus
27:033%. Right? And that's what we saw the
27:06variation of
27:07solar constant over a period of year.
27:10And it this is a cyclic variation. Okay.
27:14And then finally we looked at the
27:16concept of air mass which tells us the
27:19distance
27:20through the atmosphere. Like the the
27:22distance traveled by the sun rays
27:24through the atmosphere before it made
27:26its way to the surface of the Earth.
27:29And it is given by like Y by X
27:33or
27:34cos
27:35theta Z. So it is it depends on
27:38the
27:39zenith angle. That is the position of
27:41the sun in the sky with respect to the
27:43vertical. And we looked at different
27:46ways to calculate this air mass. One is
27:49using the zenith angle, one is using the
27:52length of the
27:54shadow of the pole
27:56based on the position of the sun.
27:58But the both of these assumptions were
28:00made using Earth's atmosphere as a flat
28:02horizontal layer. But actually to take
28:05into account the curvature then there
28:06are some complicated sums of like
28:09mathematical correlations developed
28:10which can also be used. But so far as
28:130 to 70 or 80°
28:16zenith angle is concerned,
28:19both of the approaches give the same
28:21value. So that's where we stopped and
28:23then we took one numerical to apply the
28:26concepts of solar constant and air mass.
28:28And this will be used
28:31further like whenever we have to
28:32calculate the radiations. If radiations
28:35are not given to us,
28:37the maximum possible value is solar
28:39constant and cannot exceed that for
28:42Earth. So with that let's stop here and
28:45we'll continue with the remaining part
28:46of solar energy in the next lecture.
28:48Thank you.
28:50>> [music]
28:55[music]
29:08[music]
29:18[music]

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