Solar Calculations

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Finding the wall U value in problem 7.8

Component x k U Rth,stud Rth,insthermal conductivity unit conductance unit thermal

resistanceunit thermalresistance

inches BTU in / hr ft 2 oF BTU / hr ft 2 oF hr ft 2 oF / BTU hr ft 2 oF / BTU

outdoor surface 6.000 0.17 0.17siding 1.230 0.81 0.81plywood 1.600 0.63 0.63framing 3.5 0.82 0.234 4.27insulation 3.5 0.25 0.0714 14.00gypsum board 2.220 0.45 0.45indoor surface 1.460 0.68 0.68

7.01 16.74

Thermal conductivity or unit conductance values from "thermal properties ofbuilding materials" table in ASHRAE Fundamentals Handbook


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Topic06 Solar2012

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ME414/514 HVAC Systems Topic 6

Solar Energy

Why worry about solar radiation in HVAC? The cooling load increases dramaticallywith sunshine.

In the electromagnetic spectrum, the visible spectrum spans wavelengths from 0.4 to 0.7m, and the thermal band spans wavelengths from 0.1 to 100 m. The maximumemission of the sun occurs at a wavelength very close to the center of the visiblespectrum (0.55 m, green).

Total (global) irradiation G : This is the total thermal radiation (incoming from the sun).It includes all angles, all wavelengths, and all sources. The units are BTUh/ft 2 or W/m 2.Different fractions of the irradiation are reflected, absorbed, and/or transmitted by a soliddepending on the material properties: reflectivity, absorptivity, and transmissivity.From the 1 st Law of Thermodynamics,

+ + = 1


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Topic06 Solar2012

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Blackbody : a blackbody is a perfect absorber and a perfect emitter = 1 = 0, and = 0

Emissive power is the rate of energy emitted (the rate of energy release because of theradiation temperature of the body) over all wavelengths, over all directions. This is atotal, multidirectional value.

1 10 7

1 10 6

1 10 50

5 1012

1 1013

1.5 1013

2 1013

Wavelength (m)

S p e c

t r a l

B l a c k

b o d y

E m

i s s i v e

F l u x

1.71392e+013

0.0029193

e , i T s

e , i T 1

1e-0051e-007 i

The figure above shows the spectral (wavelength) dependence of blackbody emissive power. Objects at room temperature do glow; but our eyes cant see them because theyare emitting energy in the infrared range. The spectra that our eyes perceive (0.4 to 0.7m) are reflected from surfaces receiving thermal radiation emitted at high temperature(leaves are green because they reflect green light). Exceptions are, for example, thecherry-red glow of cast iron at temperatures high enough to permit brazing.


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Topic06 Solar2012

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Wiens Displacement Law: The maximum emissive power along any isotherm in the previous plot can be calculated by:

(m) T(K) = 2898 mK

For an example, our skin temperature is ~80 F, which corresponds to a peak emissionswavelength of about 10 m (which we cannot see). Hence, objects at room temperaturein buildings are still emitting thermal radiation, even though we cannot see it.

Solar Geometry

The earth revolves around the sun. It also rotates as it revolves. In solar radiationanalysis, it is convenient to consider the earth as fixed with the sun moving through thesky. In HVAC analysis, it will be important to locate the sun at any given time of theday.

Path of the sun on December 21 st, the winter solstice. Solar position is shown for noon. Note the length of the shadowon the N side of the house. Compare with the sketch belowon June 21 st.


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Topic06 Solar2012

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Path of the sun on June 21 st, the summer solstice. Solar position is shown for noon. Note that it is possible to havesolar gain on N facing walls in the summer when the sunrises N of due E and sets N of due W. Is the solar gain onthe S side of the house more direct in winter or in summer?

Solar time is the time used in all of the sun-angle relationships. It is related to the localtime of day by the following equation (equation 6.3 in the text):

Solar time standard time = 4 (L st L loc) + E

Where:Lst is the standard meridian for the local time zoneLloc is the longitude of the location in question (longitudes are in degrees west)E = equation of time (in minutes) determined from the figure below or from the

following equation:

E = 229.2 (0.000075+ 0.001868 cos(B) 0.032077 sin(B) 0.014615 cos(2B) 0.04089 sin 2B))

WhereB = (n-1) 360/365

n = day of the year

The equation of time is plotted below. It is also graphed as the analemma on the very last page of these notes for Topic 6. There are other Equations of Time (including one in the book; results from that expression are in the table at the end of the notes).


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Topic06 Solar2012

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Plot of the Equation of Time.

Solar geometry : angles of your location

Latitude , the angular location north or south of the equator, north positive,-90 lat 90 (47 latitude N in Moscow, Idaho)

Hour angle , the time of day, the angular displacement of the sun east or westof the local meridian due to rotation of the earth on its axis at 15 per hour, morning

positive, afternoon negative.The Earths rotation is 15 /hour, 360 /day

Noon: = 0 Each morning hour: - 15 Each afternoon hour: +15 For example, at 11:00 am, = -15 .Declination , time of year, depends on the day of the year, this is the angular

position of the sun at solar noon (i.e., when the sun is on the local meridian) with respect

to the plane of the equator, north positive; -23.45 d 23.45 ; the neutral plane occurs atthe equinoxes. Refer to the solar figures,The sun is higher in the sky in the summer, and lower in the winter. At the

neutral point, = 0 on the vernal and autumnal equinox. The declination tells you howhigh the sun is at noon relative to this neutral plane.

You can also find the declination from the analemma. This graph lists thedeclination along the LHS. Or, use the following equation:

= arcsin(-sin(23.45 ) cos(( 360 (n+10) ) / 365.25)

0 50 100 150 200 250 300 350 4001000

500

0

500

1000

E B n( )( )

n

J F M A M J J A S O N D


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Topic06 Solar2012

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Where n is the day number, n = 1 on January 1 st, n = 31 + 28 = 59 on February 28 th.

Latitude, hour angle, and declination are called fundamental solar angles . In actualapplications, these angles are not the most convenient set to use. It is moreunderstandable if the suns position is expressed in terms of angles relative to a point on

the earths surface. We need two angles for the suns location:

Solar geometry : angles for the suns location

Azimuth s, the suns E-W location, zero due southThe azimuth angle of the sun, s, is measured from due south (where s = 0 ). By

convention, angles to the E are negative, angles to the W are positive. In the summer, s can be greater than 90 .

Altitude , angle of the sun above the horizonThe solar altitude angle, is measured from the horizon where = 0 , positive

upward, 0 90. In the text, Figure 6-5 also shows the zenith angle, s. The zenithangle is measured from the pole (it is the polar angle in spherical coordinates) and itcomplements the altitude.

Use the following equations to relate the altitude and azimuth s to thefundamental angles latitude , hour and declination :

sin( ) = cos( ) cos( ) cos( ) + sin( ) sin( ) = cos( s)

sin( s) = cos( ) sin( ) / sin( s)


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Topic06 Solar2012

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Direct Radiation

The values of a o, a 1, k (and the correction factors r used to find the other three constants)are given for different altitudes and climates in Table 6-2.

The correction terms for the solar constant in the above expression are for a black (andclear or non-participating gas) and a grey (or participating) gas. Participating gasesabsorb and scatter electromagnetic radiation. The degree of interference is stronglywavelength dependent. Monatomic and diatomic gases (Ar, O 2, N 2) are considered non-

participating, whereas H 2O and CO 2 have strong wavelength interference as seen inFigure 6.12 .

Diffuse Radiation

For clear skies, the diffuse radiation can be estimated as:

( ) ( )sdir odif I I I cos2939.0271.0 =

Reflected Radiation

Reflected radiation is a component of radiation on tilted surfaces.

( )[ ] pskyF cos121 +=

F sky is the radiation view factor between the wall (surface) and the sky.

( )[ ] pgrd F cos121 =

F grd is the radiation view factor between the wall (surface) and the ground. The view (orshape) factor is the fraction of the thermal radiation energy leaving one surface that

strikes another. ( ) grd ghor gloskydif idir pglo F I F I I I ,, cos ++=

( ) dif sdir hor glo I I I += cos,

I glo,hor is the rate at which the total radiation (diffuse and direct) strikes the horizontalsurface in front of the wall.

( )

+=

soodir

k aa I I

cosexp1


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g is the reflectivity of the surface dirt, asphalt, concrete, grass, snow, ice

F grd is the radiation view factor between the wall (surface) and the ground. The view (orshape) factor is the fraction of the thermal radiation energy leaving one surface thatstrikes another.

The text describes a method for calculating the long-term average solar gain on a surface.

Fenestration Heat Gain

The absorbed solar gain warms glass up. This energy is convected from the glass by airmovement. Consequently, the

Total Solar Heat Gain = transmitted solar heat gain + absorbed solar heat gain

Note that the solar effect is not included in the Topic 5 conduction/resistance tables.Consequently:

Total Heat Gain = total solar heat gain + transmission heat gain + infiltration heat gain

Transmission Heat Gain and Infiltration Heat Gain: Topics 5 and 7

Solar Heat Gain

SHGF SC AQsol

=

Procedure: calculate the SHGF for a reference (DSA Double-strength Sheet Glass)glass.

SHGF : for standard glass

FI SHGF =


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Topic06 Solar2012

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87.0=F F value of this constant for DSA glass

+++=

osi

oo hh

U hU

F 11

F solar heat gain coefficient

osi hhhU

111 ++=

h i , h s , h o are the combined convection and radiation heat transfer coefficients forthe inner surface, space surfaces (for double-pane glass), and outside surface of the glass;

is the transmissivity of the glass; o and i are the absorptivities of the outer and inner panes, respectively.

Shading coefficient (SC)

SC = (actual SHG) / (DSA glass SHG) = F act / 0.87 = 1.15 F act

Table Description6-5 SC and for different types of glass and seasons6-6 SC and for high performance glass

We will do an example in class of shading (see also, example 6-6). The shaded area of aglass window or door changes as the sun moves across the sky. Because the shaded areareceives diffuse solar radiation, the total solar heat gain due to the glass which alreadychanges in time because of the suns movement has additional changes imposed by theinstantaneous fraction of shaded/unshaded area.

Heres a step-by-step procedure for calculating fenestration solar heat gain (assuming that glass reflection can be neglected):

1. Convert local time to solar time2. Get , , (latitude, declination, hour angle)3. Calculate the suns location, , s (solar altitude and solar azimuth)4. Find the wall-solar azimuth - make a sketch to get this correct!5. Calculate the solar incidence angle i 6. Find the normal direct irradiance

(See Table 6-2 for constants.)( )

+=

soodir

k aa I I

cosexp1


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7. Find ( ) ( )sdir odif I I I cos2939.0271.0 = , diffuse irradiation8. Find the total irradiance striking the window, I glo,p

( )[ ] pskyF cos121 +=

( )[ ] pgrd F cos121 =

( ) dif sdir hor glo I I I += cos, ( ) grd ghor gloskydif idir pglo F I F I I I ,, cos ++=

9. Find SHGF for DSA (standard) glass

FI SHGF = here, I is I glo,p , the global irradiance striking the glass87.0=F

10. Find the actual solar heat gain,

SHGF SC AQ sol = SC is the shading coefficient

Stay sane and set the calculations up on a spreadsheet or mathematics program!

We will solve example problems in class.

Solar Data for the 21 st Day of EachMonth

Equationof Time,

min

Declination,degrees

Jan -11.2 -20.2Feb -13.9 -10.8Mar -7.5 0.0Apr 1.1 11.6

May 3.3 20.0Jun -1.4 23.45Jul -6.2 20.6

Aug -2.4 12.3Sep 7.5 0.0Oct 15.4 -10.5

Nov 13.8 -19.8Dec 1.6 -23.45


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ME 414/514 HVAC Systems - Solar Angles

Solar Geometry - Angle Definitions

Type Symbol Name Description Sign Location 2 Latitude Angular location N or S of the equator North positive

Location 1,2 Declination Angular position of the sun at solar noonwith respect to the equatorial plane

North positive

Location 2 Hour angle Displacement of the sun E or W of due S East negativeWest positive

Solar position

(2) Solar altitude Angle of the sun above the horizon Positive

Solar position

s Solar zenith Angle of the sun from the normal of theearths surface, 90 -

Positive

Solar position

s(2)

Solar azimuth E or W position of the sun from due S East negativeWest positive

Wallorientation

p Wall azimuth E or W position of the wall from due S East negativeWest positive

Wallorientation

p Surface tilt Angle of surface relative to the horizontal Positive9

Wall-sunorientation

Wall-solarazimuth

Angle between the solar azimuth and thewall azimuth

Positive

Wall-sunorientation

i(3) Solar incident Angle between the surface normal and thesun

Positive

i


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solar-angles2010.docx

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1. You need to either use Table 6.1 or know the day of the year n to find the declination:

2. Relate and to , , and :

sin( ) = cos( ) cos( ) cos( ) + sin( ) sin( ) = cos( s)

sin( s) = cos( ) sin( ) / sin( s)

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Solar Calculations