Welcome back in this lecture I shall begin
a very important topic in air conditioning
that is heating and cooling load calculations.
I shall begin this lecture with discussion
on solar radiation.
So the specific objectives of this particular
lecture are to introduce the need for load
calculations discuss the importance of solar
radiation in air conditioning discuss solar
geometry present methods for estimating solar
radiation on any surface at the end of the
lecture you should be able to.
Explain the importance of solar radiation
in air conditioning define and explain various
terms related to solar radiation calculate
various basic and derived solar angles and
finally estimate direct diffuse and reflected
radiations on any surface. So let me give
a brief introduction.
Why do we have to carry out cooling and heating
load calculations cooling and heating load
calculations are carried out. So that the
equipment design are selected serves the basic
purpose of air conditioning. So this is a
very important activity from the load calculations
one can estimate the capacity that will be
required for various air conditioning equipment.
And actually cooling and heating load calculations
is a way step wise procedure it is a systematic
step wise procedure you have to have certain
inputs like the specifications of the building
the load pattern and all that and from these
specifications you have to calculate various
energy flows across the building okay. For
example energy transferred to the building
or energy transfer from the building energy
generated inside the building etcetera okay.
So from these at the end of these this step
wise procedure what you get is the total load
on the building okay. So this total load on
the building is required in order to estimate
the capacity of the cooling or heating equipment
okay. This is the purpose of cooling and heating
load calculations okay.
So why do we have to bother about solar radiation
in air conditioning? What is the importance
of this? It is important to understand the
various aspects of solar radiation. Because
a major part of building heat gain is due
to solar radiation. So in order to estimate
cooling and heating loads one has to take
solar radiation into account and second reason
is that by proper design it is possible to
harness solar energy beneficially thereby
reduce the cost of air conditioning system
this very important reason also and the third
reason is that in certain instances it is
possible to device cooling and heating systems
that run only on solar energy okay.
So this is a very important this very important
as for as energy conservation is concerned.
Because it is possible in some cases where
you can have a cooling system. For example
using an absorption refrigerant system you
can device an air condition system which runs
only on solar energy okay it doesn't require
any other energy input. So this is a very
beneficial as far as certain remote areas
are concerned and it is also very good from
energy point of view and also from environment
point of view. Because solar energy as you
know is renewable and it is very non polluting
okay.
So you have to understand the solar radiation
properly so that you can use it properly to
reduce the load on the equipment. The load
on the heating and cooling. So that you can
go for a smaller equipment which will reduce
the initial and running cost and you can also
design systems which use solar energy okay.
First let us, let me give a very brief introduction
to solar radiation as you know very well the
sun which is a nothing but a star is treated
as a radiant energy source with surface temperature
that is equivalent to that of a black body
at six thousand Kelvin okay. So this is equivalent
to a black body of surface temperature which
is approximately equal to six thousand Kelvin
and on the spectral distribution of solar
radiation is given in the table here. So you
can see here that the solar radiation stretches
form about point two nine microns up to about
four point seven five microns and it consist
of invisible ultra violet radiation which
lies in the wave length band of point nine
microns to point four microns and it it consist
of about seven percent of the total radiation
and the visible radiation which lies between
point four to point seven microns consist
of about thirty-nine percent of total solar
radiation and the near infrared radiation
which is in fact gives rise to heating affect
lies in the range of point seven to three
point five microns and it consist of about
fifty-two percent of the total radiation and
finally you have far infrared or FIR radiation
which lies in the range four to four point
seven microns and it consist of two percent
of the total radiation so you can see that
the majority of solar radiation lies in the
range of visible radiation and near infrared
radiation okay so all most ninety-one percent
that's the reason why so sun is the major
source of light as well as heat okay
Now let me define what is known as the solar
constant. A solar constant is the flux of
solar radiation on a surface normal to the
sun's rays beyond the earth's atmosphere at
the, mean earth sun distance okay. It is important
to understand this definition it is nothing
but the solar radiation per unit surface which
is kept normal to the sun's rays and which
is kept beyond the earth's atmosphere okay
and at the mean earth's sun distance. Though
this solar constant value varies with time
from day to day the currently accepted value
is about thirteen seventy watt per meter square
okay. So you can use this value in calculations
and what happens to the solar radiation as
it enters into the environment or is as it
enters into the atmosphere in passing through
the earth earth's atmosphere the solar radiation
gets depleted due to reflections scattering
and absorption.
So we know very well that the atmosphere earth's
atmosphere consist of various gasses it consist
of water vapour it consist of several dust
particles of several sizes okay. So atmosphere
consist of all these things. So as a solar
radiation enters the atmosphere and as it
passes through the atmosphere it gets reflected
it gets absorbed it gets a diffracted okay.
So it under goes all these processes as it
passes through the earth's atmosphere okay.
So ultimately the end result is that even
though the solar radiation or the solar constant
is thirteen seventy watt per meter square
beyond the earth's atmosphere as it arrives
at the earth it gets depleted considerably
because of the presence of the atmospheric
gases atmosphere dust particles water vapour
etcetera okay.
And the extent of depletion depends on the
composition of the atmosphere and the length
of travel of sun's rays okay, obviously the
composition of atmosphere has a great effect
on the extent of depletion. And it also depends
upon the length of the travel of sun's rays
and the length of travel depend on what is
known as altitude angle. So I will define
this altitude angle a little later.
Now let us briefly look at sun earth relationship
we know that the planet earth makes one rotation
about its axis every twenty-four hours and
one revolution about the sun in a period of
about three sixty-five point two five days
earth's equatorial plane is tilted at an angle
of about twenty-three point five degrees with
respect to its orbital plane this is very
important. For example in this figure you
can see that, this is the sun okay. And this
is the earth, so earth is orbiting around
the sun not in a circular this thing is lightly
elliptical and it is tilted at about twenty-three
point five centigrade with reference to its
own axis. So it is not perpendicular you can
see that it is lightly tilted okay. This is
the pole north south pole of the sun. So it
has a tilt right. So this angle is about twenty-three
point five degrees okay. And you can see the
position of the, if you take the sun as a
reference then you can see the position of
the earth at different day.
For example at on June twenty-first its position
is like this and on September twenty-first
it is like this on December twenty-first,
it is like this and on march twenty-first,
it is like this okay. And June twenty-first
is known as summer solstice and September
twenty-first is known as autumnal equinox
and December twenty-first is known winter
solstice and march twenty-first is known as
vernal equinox. Now if you look at this figure
you will find the tilt and what is the effect
of the tilt okay. For again you can see that
is the, this angle is twenty-three point five
degree centigrade and on June twenty-first
this is your northern hemisphere. And this
is your southern hemisphere okay. So on June
twenty-first the northern hemisphere is tilted
towards the sun okay whereas the southern
hemisphere is tilted away from the sun. So
as a reason you find that the northern hemisphere
experience a summer during this season okay.
Similarly when you come to December twenty-first
December twenty-first this is the northern
hemisphere this is a southern hemisphere okay.
Northern hemisphere is tilted away from the
sun whereas southern hemisphere is tilted
towards the sun as a result this experiences
winter okay. The northern hemisphere whereas
the southern hemisphere experiences summer
okay. So the seasonal variations are mainly
because of the tilt angle okay. So as you
know very well the earth's rotation is responsible
for day and night whereas its tilt is responsible
for the change of seasons since the distance
between earth and sun is very large for all
practical purposes. It can be considered that
the sun's rays are parallel to each other
when they reach the earth okay. The distance
is about ninety-three million kilometres.
So it is very large so you can consider the
sun's ray to be parallel when they reach the
earth.
Now there are, we are interested in the solar
radiation striking a surface as far as the
air conditioning load calculations are concerned.
So let us see how to calculate this the rate
at which solar radiation is striking a surface
per unit area of the surface is called as
the total solar irradiation on the surface.
And the total solar irradiation on the surface
is given by this equation I subscript i theta
is equal to I subscript dn cos theta plus
I subscript small d theta plus I subscript
r theta. And as you can see here I subscript
i theta is the total solar irradiation of
the surface in watt per meter square and I
subscript capital DN is the direct radiation
from sun and I subscript small d theta is
the diffuse radiation from sky and I subscript
r theta is the reflected radiation and theta
is known as angle of incidence okay.
So you can see that the total radiation incident
on any surface consist of basically three
parts one is the pick of the direct radiation
the second part is pick of the diffuse radiation
from the sky and the third part is pick of
the reflected radiation okay, from the ground
and from the surrounding surfaces on a cloudless
day. That means when the sky is cloudless
you find that the contribution of the direct
radiation is about eighty-five percent. So
the of the total radiation, so the contribution
of diffuse and reflected radiation is quite
small on a clear day okay. Whereas on a cloudy
day you find that the importance of or the
contribution of diffuse radiation increases
where as the contribution of direct radiation
reduces okay. Now what is the angle of incidence
how is it defined.
You can see that the angle of incidence is
nothing but the, this is the sun's rays okay.
This is your sun ray okay. And this is the
surface on earth you can see here and this
is a normal to the surface okay. So the incident
angle is theta, theta is nothing but the angle
between the sun ray and the normal to the
surface okay. So this is this angle right.
Now the incident radiation depends on solar
geometry involving several basic and derived
angle. So it is important to understand these
angles and what are the basic solar angles
the basic solar angles are the first one is
what is known as latitude and the latitude
indicates the location on earth. So what is
this latitude?
The picture here shows northern hemisphere,
this is your northern hemisphere; this is
your equatorial plane okay. So this is your
equatorial plane and O is the centre of the
earth and O dash is the centre of the sun
and as I have mentioned for our calculation
purposes we can assume that the sun's rays
are parallel okay. So they arrive parallely
on earth okay and point P is any point on
the surface of the earth okay. Maybe it is
a point of the observer right now with references
to this we define all these solar basic solar
angles starting with the latitude. Latitude
is defined as the angle between the lines
OP okay. Line OP is nothing but the line joining
the centre with point P okay. Line OP and
the projection of OP on the equatorial plane
the projection of OP on the equatorial plane
is nothing but OA okay. So the latitude is
nothing but angle between OP and OA that is
nothing but angle AOP that is this angle oaky.
As you know the latitude varies from zero
degrees to ninety degrees when is at zero
when the point P lies on the equatorial plane
obviously the latitude is zero okay. And when
the point P lays on the pole this angle becomes
ninety degrees. So this is how the latitude
is defined. And latitude along with the longitude
completely specify the location of any point
on earth the second basic solar angle is what
is known as our angle and this indicates the
time of the day. So how is this defined?
Again referring back to the same figure oh
here the hour angle h is defined as the angle
between the projection of OP on the equatorial
plane OP is nothing but the line joining point
O that is the centre of the earth and the
position of the observer P okay. And its projection
is, as I have said is OA. So our angle is
nothing but the angle between OA and the projection
of the line joining the centre of the earth
to the centre of the sun now point O is the
centre of the earth and point O dash is the
centre of the sun okay. So the projection
of O O dash on the equatorial plane is nothing
but AB okay. So that is the projection of
O O dash on the equatorial plane. So our angle
is nothing but the angle AOB okay, that an
angle between OA and OB okay.
That is this angle right now at solar noon
when the earth rotates and when point P become
comes at this point you find that the our
angle is zero okay. And the our angle varies
from zero at solar noon to three sixty degrees
okay. Because it rotates in one day right
three sixty degrees in twenty-four hours okay.
It may makes one rotation in twenty-four hours
the third basic angle is what you known as
declination an declination indicates the day
of the year.
Again we refer to the same figure and declination
d is the angle between the line joining the
centre of the earth O and sun okay. That is
nothing but the line O O dash and its projection
on the equatorial plane the projection of
line O O dash on equatorial plane as i have
already discussed is OB okay. So declination
is nothing but angle between OB and O O dash
that is nothing but angle d okay. And since
this angle that means earth is tilted a twenty-three
point five degrees okay. You have an angle
of tilt of twenty-three point five degrees
you find that for northern hemisphere the
declination varies from plus twenty-three
point five degrees on June twenty-first to
minus twenty-three point five degrees on December
first and this will be exactly reverse for
the southern hemisphere okay. This will be
clear with the help of this picture.
As I said, this is the orbital motion of earth
around the sun and this is the summer solstice
that is on June twenty-first on June twenty-first.
As I said for northern hemisphere your angle
of declination is plus twenty-three point
five degrees and on September twenty-first
and on March twenty-first. That means at the
two equinoxes the declination is zero degrees
okay.
And on December twenty-first. That means on
winter solstice you find that the declination
d is minus twenty-three point five degrees
oaky. This as I said remember that this is
for northern hemisphere this is this will
be very clear if you look at this picture.
For example take just summer solstice and
winter solstice that is June twenty-first
and December first this is the equatorial
plane of the earth okay. And since this angle
is twenty-three point five this angle is tilted
as twenty-three point five the declination
is nothing but this angle okay. So this is
also twenty-three point five degrees okay.
So by convention we take this as plus twenty-three
point five degree centigrade and on winter
this is your equatorial plane. And this is
your angle of declination okay. The line joining
point O and this and its projection on equator
okay. So this angle is minus twenty-three
point five on December twenty-first okay.
So it varies from plus twenty-three point
five to minus twenty-three point five through
zero degrees and it is zero degrees at the
two equinoxes 
and the hour angle. So you can see now the
basic solar angles depend upon three important
parameters. That is what is the location of
the surface on earth which is indicated by
the latitude and what is the time of the day
time of the day is specified by the hour angle
and what is the day of the year okay, day
of the year is indicated by our declination.
So location time of the day and day of the
year these three define three basic solar
angles latitude hour angle and declination
okay. And the hour angle it is very important
to keep in mind that the hour angle is based
on what is known as local solar time.
What is local solar time since the earth's
orbital velocity varies throughout the year
the local civil time, which is based on our
clock say the mechanical or a, or electronic
is different from the local solar time so
local civil time is what is the time shown
by your watch whereas local solar time is,
for example time shown the time shown by let
us say a sun dial okay. So this is the time
shown by a sun dial you will find that these
two will be different okay. They are not exactly
equal and the difference between the local
civil time and the local solar time is what
is known as equation of time or EOT okay.
And empirical equations for EOT are available.
So if you know the day of the year you can
calculate what is the equation of a time for
a particular day okay.
It is taken as constant for a particular day.
So you can calculate EOT as a function of
the day of the year okay. If you know the
number of the day you can find out EOT and
the mean monthly mean values of EOT are also
available in tabular form if you look at ASHRAE
handbooks or anything you will find the values
of EOT okay. And at any local at any location
the local solar time LST is given by this
expression LST is the local solar time which
is equal to LStT. This is the local standard
time okay. EOT as you know is nothing but
equation of time plus four into LON minus
LSM LON is nothing but the local longitude
LSM is a local standard meridian okay. I will
let me give a very simple example.
For example for the city of Kolkata, is located
at twenty-two point eight two latitude north
latitude and its longitude is eighty-eight
degrees twenty minutes east okay. This is
the longitude right and the local standard
meridian for India is eighty-two degrees thirty
minutes eighty-two point five degrees okay.
So this is the data and we would like to find
out on October twenty-first if the local standard
time is nine am okay, then what is the local
solar time for Kolkata okay.
So for this particular day that is October
twenty-first from the empirical equation for
EOT or from the table for EOT you will find
that the equation of time is given by fifteen
minutes okay. So EOT is given as fifteen minutes
so the local solar time is nothing but local
standard time plus fifteen minutes fifteen
minute is your EOT okay. Plus this is the
correction for longitude variation between
the LSM and LON okay. So our local longitude
is eighty-eight point three degrees whereas
the local standard meridian is eighty-two
point five degrees okay. So this is the correction
for that.
So if you substitute the values you find that
local solar time is nine hours thirty-eight
point three two minutes whereas the local
standard time is nine am okay. Since this
is nine hours thirty-eight point three two
minutes in the morning the corresponding hour
angle is three twenty-four point six degree
centigrade. How did I calculate the hour angle?
As I said at noon okay, noon means let us
say twelve o'clock okay. Twelve o' clock solar
time solar noon your hour angle is zero okay.
For example at one pm the hour angle is equal
to fifteen degree centigrade how is it? So
because it makes the hour angle various from
zero to three sixty degrees in twenty-four
hours okay.
So one hour is equivalent to one hour is equivalent
to fifteen degrees. So if it is, if you are
starting with noon then after noon it will
be you starting with zero at noon afternoon
hours will be let us say three pm means forty-five
degrees okay. Four pm means sixty degrees
like that and if you proceed in that manner
you find that in the morning. For example
nine am. Nine am means three hours before,
so solar noon okay. That means it i three
sixty minus forty-five that is three fifteen
degrees centigrade okay. So now the local
solar time in this example is nine hours thirty-eight
point three two minutes. So the solar angle
is three twenty-four point six degree okay.
So that is how you can calculate the hour
angle. So what you have to do to calculate
hour angle is to first find out what is the
local longitude. And you also should know
what is the local standard meridian right
and then you if you know the local standard
time and equation of time then you can calculate
first you should calculate the local solar
time. And once you know the local solar time
you can calculate the hour angle because one
hour is equivalent to fifteen degrees okay.
So the hour angle, as I said is based on local
solar time and the declination varies in an
approximately sinusoidal fashion. I have mentioned
that the declination varies from plus twenty-three
pointy five degrees to minus twenty-three
point five degrees and the variation is almost
sinusoidal okay. So you can fit an empirical
equation.
So this is an empirical equation for declination
okay. Declination d in degrees this is degrees.
It is all the angles are in degrees okay.
So this is very important keep it in mind
okay. So unit should be consistent so declination
degrees is given by this empirical equation
twenty-three point four seven into sin three
sixty into two eighty-four plus N divided
by three sixty-five what is N N is the day
of the year counted form January first. That
means January first n is equal to one January
second n is equal to two like that. So December
thirty-first n becomes three sixty-five okay.
So if you know the day of the year you can
easily calculate the declination using this
equation okay. And where from this equation
you will find that on June twenty-first okay.
On June twenty-first you find the declination
d is approximately equal to twenty-three point
five degrees and on December twenty-first
okay. This is equal to minus twenty-three
point five degrees okay. For example I want
to calculate the declination for March sixth
okay. March six happens to be the sixty fifth
day of the year. That means N is sixty-five.
So all that you have to do is substitute the
value of N here. So you will find that the
declination on March sixth is minus six point
four degrees okay. From the above equation
okay so these are the basic solar angles.
Now let us look at derived solar angles in
addition to the basic solar angles several
other angles required for solar radiation
calculations are derived first one is what
we known as altitude angle. So let us see
what is this altitude angle.
Okay, let me explain this figure. Let us say
that, this is a horizontal plane okay. So
this is the point P is the position of the
observer okay. And let this be the north and
south sides and this is the west and east
sides okay. So you have north south east west
right and this is the path of the apparent
sun path. That means this is the path of the
sun as it appears to the observer who is located
at point P. So as you can see that sun rises
at this point okay sun rises in the east and
follows this path this a this is for a particular
day okay. This path is not constant it varies
again form day to day, so for a particular
day let us say that this is the sun path so
it rises in the east and it sets in the west
okay. Now with reference to this figure what
is the altitude angle altitude angle is given
by beta is the angle between sun's rays and
its projection on a horizontal plane.
So this is your sun ray this is your sun and
this is the sun ray. So beta is nothing but
the angle between this sun ray and its projection
on the horizontal plane the projection of
the sun's ray on horizontal plane is this
okay. So altitude angle is nothing but this
angle that is beta okay. As you can see that
beta is zero at sunrise okay. So at this point
the sunray directions of the sun ray and its
projection coincides. So beta will be zero
similarly beta will be zero at sunset okay.
So at these two points beta will remain zero
and it reaches maximum at solar noon okay.
In fact that is the definition of solar noon
at solar noon you will find that the altitude
angle is maximum. So this is the first angle
the second angle is what is known as zenith
angle.
So zenith angle I am sorry this should have
been psi this okay, zenith angle is given
by this symbol is the angle between sun's
rays and the surface normal okay sun ray's.
As I have again I said this is the sun ray
and the normal to the surface at point P is
this okay. So the zenith angle is nothing
but this angle okay. So this is zenith angle
obviously zenith angle plus altitude angle
should be equal to ninety degrees okay. So
this angle is equal to ninety degrees. That
means zenith angle is equal to ninety minus
beta. So if you know the altitude angle you
can find out the zenith angle easily next
important angle is what is known as solar
azimuth angle.
Again we refer to the same figure and the
solar azimuth angle gamma is the angle between
the north and the projection of sun's rays
on to the horizontal plane okay. So what is,
so as I said this is your north direction
okay. So azimuth angle is nothing but this
and the projection of sun's rays on to a horizontal
plane the projection of sun's rays on to a
horizontal plane is this line okay. So the
azimuth angle is the angle between this line
and this line okay. So as you can see that
is nothing but this angle okay. This angle
is given by gamma and this is measured in
counter clock wise direction of course this
varies this convention varies from text book
to text book some people define the solar
azimuth angle with reference to the south
direction okay.
Some people define with reference to the north
direction and sometimes it is defined in terms
of the clock wise direction from north and
in some books. It is defined in terms of anticlockwise
direction from the north and in some other
books it is defined for forenoon it is in
the clockwise direction and for afternoon
it is the anticlockwise direction. So the
definitions vary but basically once you have
this diagram it is easy to understand. And
it is easy to change from one convention to
the other convention okay. So in this particular
lecture I am following the convection of measuring
it from the north in the counter clockwise
direction okay. So this is the solar azimuth
angle gamma now using a analytical geometry
it is possible to derive relationships between
derived and basic solar angle. This is important
the relationships between various angles.
Okay, so altitude angle beta. So beta it can
be shown that beta. The altitude angle beta
is equal to cos l multiplied by cos h multiplied
by cos d plus sin l multiplied by sin d, where
l as I have already discussed is latitude
h is hour angle and your d is declination
okay. So you can see that the altitude angle
is simply related to the basic solar angles
of latitude hour angle and declination. That
means the altitude angle depends upon the,
a day of the year time of the day and the
location of the point okay. And as we know
at solar noon the hour angle is zero that
means cos h is equal to zero right.
So when h is equal to zero as you know cos
h is one okay. Then you find that the altitude
angle is maximum and that value is given by
beta max. Beta max is simply equal to pi by
two minus l minus d okay. This is your absolute
value of l minus where l is the latitude and
d is the declination okay. So once you know
the latitude and declination you can calculate
what is the maximum altitude angle on any
given day. For example for Kolkata again take
the example of Kolkata for Kolkata the latitude
is twenty-two degrees eighty-two minutes north.
And I want to find out what is the maximum
altitude angle on June twenty-first okay.
And as we know on June twenty-first the declination
d is twenty-three point five degrees okay.
So beta max is nothing but pi by two minus
twenty-two point eight two minus twenty-three
point five okay.
You have to take the absolute value of that
then you will find that this is equal to eighty-nine
point three degrees okay. This is the maximum
altitude angle is almost close to ninety degrees
okay. On June twenty-first for Kolkata and
you can see that the beta max is equal to
ninety degrees when l becomes d okay. So d
as we know is, that means when the latitude
becomes equal to declination the beta max
becomes ninety degrees okay. Next you, that
is another important information that you
can get from this equation. That is, that
at sunrise and sunset we know that the altitude
angle beta is zero okay. So beta is zero at
sunrise and sunset. As I have already explained
to you so we can find out what is the time
of sunrise and sunset using this equation
okay, how, if you substitute value for beta
zero here you can find out the hour angle
at which sunrises and sunsets okay. And that
hour angle is given by this expression this
is derived from this equation okay. This is
derived from this equation by taking beta
is equal to zero so h naught is the, at hour
angle at which the sunrises or sunsets. That
is given by cos inverse minus tan l into tan
d where l as you know is the altitude and
d is the declination okay let me give a small
example.
For example I would like to find out the sunrise
sunset and a total sun shine hours at IIT
Kharagpur on twenty on September ninth okay.
So I would like to find at what time the sunrises
and the sunsets and what is the total sunshine
hour. That means how many hours the sunshine
is available at this particular location which
has a latitude of twenty-two degrees okay.
That means l is equal to twenty-two degrees
twenty-two degrees north okay. So first as
you have seen h naught from the earlier this
thing is tan okay. As you can see from this
expression h naught is cos inverse tan l into
tan d. So we have to find out what is the
latitude you have to find out what is the
declination latitude is given as twenty-two
degrees. So I have to find out the declination
on this particular day September ninth. So
all that I need to know is what is the number
of the days.
So if you start counting from January first
you will find that September ninth happens
to be the two fifty second day of the year
so n is equal to two fifty-two. So if you
substitute n is two fifty-two in the expression
for declination you find that the declination
angle is equal to four point six two degrees
okay. So if substitute four point six two
degrees in this expression you find that the
hour angle at sunrise and sunset is equal
to ninety-one point eight seven degrees okay.
Since each fifteen degrees is equal to one
hour ninety-one point eight seven degrees
is equal to six hours and eight minutes okay.
Right, so the hour equivalent to six hours
and eight minutes what is the meaning of,
this means that the sunrise takes place at
twelve minus sixth six point zero eight that
is five five point five two hours in the morning
this is the solar time and sunset takes place
at twelve plus six point zero eight that is
six point zero eight pm right.
So sun rises at five point five two am and
sunset is six point zero eight pm okay. And
both are with reference to solar times and
the, so total sun shine hours are nothing
but two into h naught two into h naught by
fifteen. Let us say since it is not in degree
this will give you the total sunshine hours
okay. So you can see the usefulness of this
equation okay. This is simple and elegant
equation can be used to find out at what time
the sunrises or sunset at any location on
the earth on any particular day okay. And
you can also calculate the total sunshine
hour this information is very useful in the
design of several solar energy equipment and
it is also useful in air condition for load
calculations okay. So this is the relationship
between altitude angle and the basic angles
and the usefulness of this equation.
Now let us look at the other equation that
is the equation for solar azimuth angle in
terms of the basic angles. So solar azimuth
angle this as I have mentioned is a, as measured
from north in a counter clockwise direction
right. So that is the if you take that convention,
then this is given by gamma is equal to cos
inverse cos l into sin d minus cos d into
cos h into sin l divided by cos beta. As you
know again l is the latitude d is the declination
h is the hour angle and beta is the altitude
angle okay. So this defined in terms of again.
That means this is again a function of the
basic angles only because beta is a function
of basic angles l d and h right.
So this you also have an alternate expression.
This is also equal to sin inverse cos d into
sin h by cos beta these two expression are
equal. So that you can easily prove by using
your trigonometric relations okay. So this,
the relation between solar azimuth and other
angles and one important thing to note here
is at solar noon. That means when h is equal
to zero gamma is one eighty degrees for l
is greater than d. That means when the latitude
angle is greater than the declination angle.
Then we have to take gamma as one eighty degrees
and gamma is should be taken as zero degrees
for l is less than d okay. So this you must
keep in mind okay. So only for solar noon
you have to keep this information and a gamma
is not defined for l is equal to d at solar
noon from this equation okay.
Now let us look at incident angle of solar
radiation that is theta. So as I have already
mentioned the angle incident angle of sun's
rays theta is angle between sun's rays and
normal to the surface under consideration
the angle of incidence theta depends on solar
geometry and also on the orientation of the
surface. For example let us look at the horizontal
surface theta is equal to zenith angle this
can be shown very easily if you look at this
figure.
So this is your horizontal surface and by
definition theta is your angle of incidence.
So which as I said is defined as the angle
between the sun's rays and the surface normal
okay. Since this is the horizontal surface
this is the surface normal and this is your
angle of incidence. And it is, so happens
that the angle of incidence for horizontal
surface is all also equal to your zenith angle
okay zenith angle. And as we know psi plus
beta is psi plus beta is ninety degree centigrade
ninety degrees. So theta is equal to pi by
two minus beta okay. So if the surface is
horizontal then once you know the altitude
angle you can easily find out what is the
angle of incidence.
How this information is useful, for example
I want to calculate what is the radiation
incident on the roof on a flat roof okay.
A flat roof means it is a horizontal surface.
So all that I have to do is on a particular
day at a particular time because it is a function
of the time of a day location of the surface
and the day of the year okay. So all these
things, for these, once you know these things
all that you have do is you have to find out
what is the altitude angle okay. So calculate
the hour angle from the time of the day calculate
the declination from the day of the year.
And from the latitude information calculate
the altitude angle okay. Once you know the
altitude angle the angle of incidence is nothing
but ninety minus altitude angle. So you can
straight away get the angle of incidence for
a horizontal roof okay. Now let us look at
vertical surface the calculation of angle
of incidence for a vertical surface is little
more complicated compared to horizontal surface.
Because here in addition to the solar geometry
we also have to consider the direction the
surface is facing whether it is facing east
side or west side or south side these things
will come into picture okay. Let me explain
that with the help of this picture okay.
So in this picture this is the shaded surface
is the wall okay, or surface under consideration
okay. I want to find out what is the radiation
incident on this particular surface okay,
which is shaded in red right and this is your
south and north direction okay. And the wall
is arbitrarily oriented right. So this is
north and south this is your east and west
directions and let us say that at a particular
point on a particular day the sun is at this
position. And this is your horizontal plane
okay. And this line is nothing but the projection
of sun's rays these are the sun's rays' on
to a horizontal plane.
So this is ninety degrees so the, from definition
we know that this angle is beta okay, which
is nothing but the angle between the sun's
rays and its projection on the horizontal
plane okay. So we have already defined that
angle beta and we also have defined the incident
angle incident angle is nothing but the normal
to the surface this is your normal to the
surface okay. And the sun's rays that means
this angle. So we have defined these two angles
already that is beta and theta okay. And we
have we have also defined solar azimuth angle
gamma which is nothing but the angle between
north and the projection of sun's rays on
a horizontal plane that is this angle.
So these three angles have already been defined.
So I introduce here two new angles that is
one is what is known as wall solar azimuth
angle alpha and the other angle is surface
azimuth angle zeta okay. Now first let us
look at wall solar azimuth angle alpha wall
solar azimuth angle alpha is defined as a
angle between the normal to the wall. That
means this line this thick line and the projection
of sun's rays on to a horizontal plane the
projection of sun's ray on to a horizontal
plane is this okay. So alpha which is called
as wall solar azimuth angle is nothing but
this angle that is the angle between the normal
and the projection of sun's rays on to a horizontal
plane. So that is alpha and I said we have
also defined another angle called zeta that
is what is known as surface azimuth angle
zeta okay. Surface azimuth angle zeta is nothing
but this is defined as a angle between the
normal. That mean this line this thick line
normal to the wall and the south is this okay.
So for this particular picture the picture
I have taken in this is your zeta if the normal
is somewhere here. Let us say then the zeta
becomes this. That means zeta is always defined
with reference to the south direction and
the normal to the surface okay. The normal
to the surface obviously depends upon the
orientation of the surface and the direction
in which the surface is facing okay. And the
convention here is that it is taken as positive
for west of south. That means all for all
these direction this is the south so of on
all these directions you take zeta as positive
okay.
That means zeta varies from zero degrees when
it the normal coincides with the south to
one eighty degrees when it coincides with
the north okay. That means, for this one it
is one eighty degrees and a in this direction
if it is facing in this direction it varies
from again zero to one eighty degrees through
a negative this thing. That means this become
for example if you take east is minus ninety
degrees okay, whereas west is plus ninety
degrees okay. Similarly by if you consider
let us say southeast okay south east is minus
forty five degrees whereas southwest is plus
forty-five degrees okay. Like that, so if
you know the orientation of the surface you
can easily find out what is the value of zeta
and remember the convention followed that
is positive for west of south and negative
for east of south.
Now you can find a relationship easily between
alpha that is surface solar azimuth or wall
solar azimuth angle alpha. And the solar azimuth
angle gamma and surface azimuth angle zeta
okay. So if you look at this picture this
is one eighty degrees this total angle okay,
is one eighty degrees. So one eighty degrees
is equal to gamma that means this angle plus
alpha that mean this angle plus zeta okay,
for this particular case. So for this particular
case which happens to be the afternoon because
the sun is towards the west okay. It has crossed
the solar noon okay. So this example is for
after noon right for afternoon you find that
gamma plus alpha plus zeta is equal to one
eighty degrees okay. That means alpha is equal
to one eighty minus gamma plus zeta right.
Now for forenoon the sun will be on this side
towards the south okay. Sun will be somewhere
on this side then you will find that gamma
is equal to one eighty plus alpha plus zeta
okay. In such case you'll find that alpha
is given by one eighty minus gamma plus zeta
into minus one okay. So that is why you have
given this equation. Now this is the relationship
alpha is equal to F into pi minus with in
brackets gamma plus zeta where F takes the
value of minus one for forenoon and plus one
for afternoon okay.
Now it can be shown that the angle of incidence
because ultimately this is what is important
to us on a vertical surface is given by this
relation theta vertical is equal to cos in
inverse within brackets cos beta into cos
alpha cos beta. As you know is, altitude angle
and alpha is your surface azimuth surface
solar azimuth angle okay. And the expression
as you, this you have seen is the function
of your gamma and zeta right. So if you know
the orientation that means oriental surface
zeta and I should know beta and other angles
you can find out gamma. So from gamma and
zeta you can find out alpha and from your
basic latitude hour angle and declination
you can find out beta. And from this information
you can find out what is the angle of incidence
on a vertical surface theta vertical and where
do you use this information. This information
is required to find out what is the solar
radiation incident on a vertical wall okay.
So both horizontal orientation and vertical
orientation are very frequently encountered
in air conditioning calculation. Because many
most of the buildings will have horizontal
roofs and all the buildings generally have
vertical walls okay. So you have vertical
walls and horizontal roofs. So you have to
use these two expressions for finding the
angle of incidence of solar rays okay. Now
let me give a very general expression for
arbitrarily tilted surface okay. Any surface
it can be horizontal vertical or tilted for
this arbitrarily tilted surface.
It can be shown okay. First this is arbitrarily
tilted surface and here we define another
angle called as tilt angle okay. That is sigma
this is a tilt angle okay. And this is the
surface which is exposed to your solar radiation
okay. So solar radiation is falling on this
surface right, so tilt angle is nothing but
the angle between the horizontal and the surface
okay. That you can see here and it can be
shown that the angle of incidence on any arbitrarily
orientated surface is given by theta is equal
to cos inverse sin beta into cos sigma plus
cos beta into cos alpha into sin sigma okay.
So in addition to your beta and alpha you
also should know what is the angle of tilt
in order to calculate the angle of incidence
of sun's rays on any surface okay. As I said
this is a general equation and they can be
used for any arbitrarily oriented surface
for example for a horizontal surface this
tilt angle becomes zero right. Because this
coincides with this it comes here so sigma
is equal to zero once sigma is equal to zero
this vanishes and this becomes one. So you
will find that theta is nothing but ninety
minus beta okay. So this as simplifies down
to this expression and for a vertical surface
sigma is equal to ninety degrees for vertical
surface. Because this surface becomes like
this okay, ninety degrees. So when sigma becomes
ninety degrees cos sigma becomes zero. So
you will not have this term and sin sigma
is one. So it can be you can easily find that
theta vertical is equal to cos inverse cos
beta into cos alpha okay. So if you remember
the expression for the tilt angle on any arbitrary
oriented surface then you can simplify it
to other surfaces like vertical or horizontal
surfaces okay.
Okay, so calculation of a angle of incidence
is one major part of a solar radiation calculations
okay. So once you calculate that is required
because you want to calculate what is the
radiation incident on a surface but in addition
to the if you remember your earlier expression
in addition to the angle of incidence. We
also have to find out what is the direct radiation
what is the diffuse radiation and what is
the reflected radiation several models are
available several solar radiation models using
which you can find out direct diffuse and
deflected radiations in air conditioning calculations.
Normally we use what is known as an ASHRAE
model this is the model suggested by ASHRAE
based on their data okay. The data collected
by ASHRAE and they are fitted empirical equations
to their data and they have suggested certain
correlations for estimating these radiations
okay. So this is a very popularly used model
as far as air conditioning calculations are
concerned okay. So in this lecture I will
discuss ASHRAE model. So first look at direct
radiation direct radiation from sun is given
by I subscript capital DN that is equal to
A into exponential within brackets minus B
by sin beta and the units are watt per meter
square here A is what is known as apparent
solar irradiation. And it takes the value
of twelve thirty watt per meter square for
the months of December and January. That means
basically for winter in winter air takes a
value of twelve thirty watt per meter square
where as during summer.
That means during say may June July air takes
a value of about ten eighty watt per meter
square okay. So if you want to do the solar
radiation calculations for winter you can
take a value of twelve thirty for air whereas
for summer you can take a value of ten eighty
and the constant B is known as atmospheric
extinction coefficient and it take a value
of point one four in winter and point two
one in summer. So if you know the values of
A and B then you can calculate I direct radiation
of course you also have to know the altitude
angle beta okay. Because here sin beta is
there so beta is the altitude angle. So from
the knowledge of altitude angle and A and
B you can calculate the direct radiation on
a surface and the values of A and B are available
in tabular form when ASHRAE gives the values
of A and B for each month. And they calculated
this fro twenty first day of each month and
these values are tabulated and they are available
on ASHRAE handbooks okay.
Next comes your diffuse radiation from sky
diffuse radiation is I subscript d and according
to the ASHRAE model the diffuse radiation
form a cloudless sky is given by id they relate
this to the direct radiation. And what is
known as an angle factor. So id is equal to
C multiplied by I subscript DN multiplied
by F subscript WS where C is a constant. And
it can be taken as point one three five for
midsummer and point zero five eight for winter
okay. And F subscript WS is called as view
factor or configuration factor and this is
nothing but the fraction of diffuse radiation
incident on the surface. Here I would like
to tell one thing this ASHRAE model assumes
that the sky is cloudless okay.
The, that means the calculations are strictly
valid for a cloudless sky this is this assumption
is required. Because this model is based on
the assumption that the diffuse radiation
arrives at the surface uniformly okay. It
you it arrives at the surface uniformly this
is valid only when the sky is cloudless if
the clouds are there in the sky then the diffuse
radiation will not be uniform okay. And this
factor as I said the view factor is nothing
but the fraction of diffuse radiation incident
on the surface. And it can be very easily
shown that F subscript WS that is the view
factor is equal to one plus cos sigma by two
where sigma is the tilt angle defined earlier.
So for a horizontal surface the tilt angle
is zero so the view factor is equal to one
that means it sees all the radiation it receives
all the diffuse radiation whereas for a vertical
surface sigma is equal to ninety degrees.
So cos sigma becomes zero and the view factor
becomes point five. That means the vertical
wall receives only fifty percent of the diffuse
radiation from the sky whereas the horizontal
surface receives all the radiation from the
diffuse radiation of the sky okay.
And the values of cr again available in tabular
form finally we have this reflected short
wave radiations from the ground or from the
surrounding surfaces. The amount of solar
radiation deflected from the ground on to
a surface is given by Ir is equal to IDN plus
Id. This is nothing but direct radiation plus
diffuse radiation multiplied by rho g multiplied
by F subscript WG where rho g is the reflective
of the ground which depends upon the surface
nature of the ground. Or a horizontal surface
and F subscript G WG is what is known as view
factor or configuration factor from ground
to the surface okay. Again by definition this
view factor is nothing but how much fraction
or what fraction of the reflected radiation
is received by the surface okay.
If the surface does not receive any of the
reflected radiation then its view factor is
zero if it receives all the radiation then
its view factor is one right and it can be
easily shown that FWG is equal to one minus
cos sigma by two where sigma is equal to tilt
angle. As you know and for a horizontal surface
facing upwards sigma is equal to zero okay.
So sigma is zero means your view factor is
zero whereas for a vertical surface sigma
is equal to ninety degrees. That means view
factor is point five okay. So this is how
using the ASHRAE model you can calculate direct
diffuse and reflected radiations from the
known data okay. Once you know these things
and once you know the angle of incidence you
can calculate the total radiation incident
on any surface okay. At this point I conclude
this lecture and I will give an example on
how to calculate the radiation in the next
lecture.
Thank you.
