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Chapter #2: Solar Geometry Date and day

• Date is represented by month and ‘i’ • Day is represented by ‘n’

Month nth day for ith date January i February 31 + i March 59 + i … … December 334 + i

(See “Days in Year” in Reference Information) 38 Chapter #2: Solar Geometry position from

• Sun rise in the and set in the • “A” sees sun in • “B” sees sun in N A

W E

B S 39 Chapter #2: Solar Geometry Solar

Solar noon is the time when sun is highest above the horizon on that day 40 Chapter #2: Solar Geometry Solar altitude

• Solar altitude angle (αs) is the angle between horizontal and the line passing through sun • It changes every hour and every day

W

αs

S N In northern hemisphere

E 41 Chapter #2: Solar Geometry Solar altitude angle at noon

Solar altitude angle is maximum at “Noon” for a

day, denoted by αs,noon

W

αs,noon S N In northern hemisphere

E 42 Chapter #2: Solar Geometry angle

• Zenith angle (θz) is the angle between vertical and the line passing through sun

• θz = 90 – αs

θz W

S N In northern hemisphere

E 43 Chapter #2: Solar Geometry Zenith angle at noon

• Zenith angle is minimum at “Noon” for a day,

denoted by θz,noon

• ϴz,noon = 90 – αs,noon

θz,noon W

S N In northern hemisphere

E 44 Chapter #2: Solar Geometry Air mass

• Another representation of solar altitude/zenith angle. • Air mass (A.M.) is the ratio of mass of atmosphere through which beam passes, to the mass it would pass through, if the sun were directly overhead. 퐴. 푀. = 1Τcos 휃푧

If A.M.=1 => θz=0° (Sun is directly overhead) If A.M.=2 => θz=60° (Sun is away, a lot of mass of air is present between earth and sun)

45 Chapter #2: Solar Geometry Air mass

퐴. 푀. = 1Τcos 휃푧

46 Chapter #2: Solar Geometry Solar angle

• In any hemisphere, (γs) is the angular displacement of sun from south • It is 0° due south, -ve in east, +ve in west

γs S

E W Morning Noon Evening

(γs = -ve) (γs = 0°) (γs = +ve) 47 Chapter #2: Solar Geometry Solar Important! March Equator faces sun directly June solstice (Spring) Northern hemisphere is towards sun (Summer)

December solstice Northern September equinox hemisphere is away Equator faces sun directly from sun (Autumn) (Winter) 48 Chapter #2: Solar Geometry Solar declination (at solstice) C N N B C A B A (Noon) S S June solstice December solstice

A sees sun in north. A sees sun in south. B sees sun overhead. B sees sun in more south.

C sees sun in south. C sees sun in much more south.49 Chapter #2: Solar Geometry Solar declination (at equinox)

N March equinox C B A sees sun directly overhead A B sees sun in more south C sees sun in much more south

S Same situation happen during September equinox.

(Noon) 50 Chapter #2: Solar Geometry Solar declination from frame of reference of horizontal ground beneath feet ø ø N

ø

S 51 Chapter #2: Solar Geometry Solar declination Declination June solstice Equinox +23.45°

-23.45° W

December solstice 90 - φ φ S N In northern hemisphere E Note: Altitude depends upon latitude but declination is independent.52 Chapter #2: Solar Geometry Solar declination

• For any day in year, solar declination (δ) can be calculated as: 284 + 푛 훿 = 23.45 sin 360 365 Where, n = numberth day of year (See “Days in Year” in Reference Information) • Maximum: 23.45 °, Minimum: -23.45° • Solar declination angle represents “day” • It is independent of time and location!

54 Chapter #2: Solar Geometry Solar declination

Days to Remember δ March, 21 0° June, 21 +23.45° September, 21 0° δ December, 21 -23.45°

Can you prove this?

n 55 Chapter #2: Solar Geometry Solar altitude and zenith at noon

• As solar declination (δ) is the function of day (n) in year, therefore, solar altitude at noon can be calculated as:

αs,noon = 90 – ø + δ • Similarly zenith angle at noon can be calculated as:

ϴz,noon = 90 – αs,noon= 90 – (90 – ø + δ)= ø - δ

56 Chapter #2: Solar Geometry

• The time in your clock (local time) is not same as “solar time” • It is always “Noon” at 12:00pm solar time

Solar time “Noon” Local time (in your clock) 58 Chapter #2: Solar Geometry Solar time

The difference between solar time (ST) and local time (LT) can be calculated as: 4 × 푆퐿 − 퐿퐿 푆푇 − 퐿푇 = 퐸 − 60 Where, ST: Solar time (in 24 hours format) LT: Local time (in 24 hours format) SL: Standard longitude (depends upon GMT) LL: Local longitude (+ve for east, -ve for west) E: (in hours)

Try: http://www.powerfromthesun.net/soltimecalc.html59 Chapter #2: Solar Geometry Solar time

• Standard longitude (SL) can be calculated as: SL = (퐺푀푇 × 15) • Where GMT is Greenwich Mean Time, roughly: If LL > 0 (Eastward): 퐺푀푇 = 푐푒푖푙 퐿퐿Τ15 If LL < 0 (Westward): 퐺푀푇 = −푓푙표표푟 퐿퐿 Τ15 • GMT for Karachi is 5, GMT for Tehran is 3.5. • It is recommended to find GMT from standard database e.g. http://wwp.greenwichmeantime.com/ 60 Chapter #2: Solar Geometry Solar time

• The term Equation of time (E) is because of earth’s tilt and orbit eccentricity. • It can be calculated as: 0.000075 +0.001868 cos 퐵 229.2 퐸 = × −0.032077 sin 퐵 60 −0.014615 cos 2퐵 −0.04089 sin 2퐵 Where,

퐵 = 푛 − 1 360Τ365 61 Chapter #2: Solar Geometry

• Hour angle (ω) is another representation of solar time • It can be calculated as: 휔 = (푆푇 − 12) × 15 (-ve before solar noon, +ve after solar noon)

11:00am 12:00pm 01:00pm ω = -15° ω = 0° ω = +15°

62 Chapter #2: Solar Geometry A plane at earth’s surface

• Tilt, pitch or slope angle: β (in degrees) • Surface azimuth or orientation: γ (in degrees, 0° due south, -ve in east, +ve in west)

W (γ = +ve) β (γ = 0) S N γ (γ = -ve) E 65 Chapter #2: Solar Geometry Summary of solar angles

Can you write symbols of different solar angles shown in this diagram? 66 Chapter #2: Solar Geometry Interpretation of solar angles Angle Interpretation Set#

Latitude φ Site location 1 Declination δ Day (Sun position) 2 Hour angle ω Time (Sun position)

Solar altitude αs Sun direction (Sun position)

Zenith angle θz Sun direction (Sun position) 3

Solar azimuth γs Sun direction (Sun position) Tilt angle β Plane direction 4 Surface azimuth γ Plane direction 67 Chapter #2: Solar Geometry Angle of incidence

Angle of incidence (θ) is the angle between normal of plane and line which is meeting plane and passing through the sun

θ W

S N

E 68 Chapter #2: Solar Geometry Angle of incidence

• Angle of incidence (θ) depends upon: – Site location (1): θ changes place to place – Sun position (2/3): θ changes in every instant of time and day – Plane direction (4): θ changes if plane is moved • It is 0° for a plane directly facing sun and at this angle, maximum solar radiations are collected by plane.

69 Chapter #2: Solar Geometry Angle of incidence

If the sun position is known in terms of declination (day) and hour angle, angle of incidence (θ) can be calculated as:

cos 휃 = sin 훿 sin ∅ cos 훽 − sin 훿 cos ∅ sin 훽 cos 훾 + cos 훿 cos ∅ cos 훽 cos 휔 + cos 훿 sin ∅ sin 훽 cos 훾 cos 휔 + cos 훿 sin 훽 sin 훾 sin 휔

(Set 1+2+4) 70 Chapter #2: Solar Geometry Angle of incidence

If the sun position is known in terms of sun direction (i.e. solar altitude/zenith and solar azimuth angles), angle of incidence (θ) can be calculated as:

cos 휃 = cos 휃푧 cos 훽 + sin 휃푧 sin 훽 cos 훾푠 − 훾

Remember, θz = 90 – αs Note: Solar altitude/zenith angle and solar azimuth angle depends upon location.

(Set 1+3+4) 71 Chapter #2: Solar Geometry Special cases for angle of incidence

• If the plane is laid horizontal (β=0°) – Equation is independent of γ (rotate!)

– θ becomes θz because normal to the plane becomes vertical, hence:

cos 휃푧 = cos ∅ cos 훿 cos 휔 + sin ∅ sin 훿

Remember, θz = 90 – αs Note: Solar altitude/zenith angle depends upon location, day and hour.

73 Chapter #2: Solar Geometry Solar altitude and azimuth angle

Solar altitude angle (αs) can be calculated as:

sin훼푠 = cos ∅ cos 훿 cos 휔 + sin ∅ sin 훿

Solar azimuth angle (γs) can be calculated as:

−1 cos 휃푧 sin ∅ − sin 훿 훾푠 = sign 휔 cos sin 휃푧 cos ∅

75 Chapter #2: Solar Geometry diagram or sun charts

αs

-150° -120° -90° - 60° -30° γ0° 30° 60° 90° 120° 150° s 76 Note: These diagrams are different for different . Chapter #2: Solar Geometry Shadow analysis (objects at distance)

• Shadow analysis for objects at distance (e.g. trees, buildings, poles etc.) is done to find: – Those moments (hours and days) in year when plane will not see sun. – Loss in total energy collection due to above. • Mainly, following things are required: – Sun charts for site location – Inclinometer – Compass and information of M.D.

78 Chapter #2: Solar Geometry Inclinometer A simple tool for finding and altitudes of objects

http://rimstar.org/renewnrg/solar_site_survey_shading_location.htm 79 Chapter #2: Solar Geometry Shadow analysis using sun charts

αs

-150° -120° -90° - 60° -30° 0° 30° 60° 90° 120° 150°

γs 80 Chapter #2: Solar Geometry hour angle and daylight hours

• Sunset occurs when θ z = 90° (or αs = 0°). Sunset hour angle (ωs) can be calculated as:

cos 휔푠 = − tan ∅ tan 훿

• Number of daylight hours (N) can be calculated as: 2 푁 = 휔 15 푠 For half-day ( to noon or noon to sunrise), number of daylight hours will be half of above.

81 Chapter #2: Solar Geometry Profile angle It is the angle through which a plane that is initially horizontal must be rotated about an axis in the plane of the given surface in order to include the sun.

83 Chapter #2: Solar Geometry Profile angle

• It is denoted by αp and can be calculated as follow:

tan 훼푠 tan 훼푝 = cos 훾푠 − 훾

• It is used in calculating shade of one collector (row) on to the next collector (row). • In this way, profile angle can also be used in calculating the minimum distance between collector (rows).

84 Chapter #2: Solar Geometry Profile angle

• Collector-B will be in shade of collector-A, only when: ҧ 훼푝 < 훽

85 Chapter #2: Solar Geometry Angles for tracking surfaces

• Some solar collectors "track" the sun by moving in prescribed ways to minimize the angle of incidence of beam radiation on their surfaces and thus maximize the incident beam radiation. • Tracking the sun is much more essential in concentrating systems e.g. parabolic troughs and dishes.

(See “Tracking surfaces” in Reference Information) 87 Chapter #3: Solar Radiations Types of solar radiations

1. Types by components: Total = Beam + Diffuse or Direct or Sky

89 Chapter #3: Solar Radiations Types of solar radiations

2. Types by terrestre: Extraterrestrial Terrestrial • Solar radiations • Solar radiations received on earth received on earth in without the presence the presence of of atmosphere OR solar atmosphere. radiations received • We can measure or outside earth estimate these atmosphere. radiations. Ready • We always calculate databases are also these radiations. available e.g. TMY. 90 Chapter #3: Solar Radiations Measurement of solar radiations

1. Magnitude of solar radiations: Irradiance Irradiation/Insolation • Rate of Energy received per unit area in a energy given time (power) received per unit area Hourly: I Daily: H Monthly avg. daily: H • Symbol: G Unit: J/m2 Unit: J/m2 Unit: J/m2 • Unit: W/m2 91 Chapter #3: Solar Radiations Measurement of solar radiations

2. Tilt (β) and orientation (γ) of measuring instrument: – Horizontal (β=0°, irrespective of γ)

– Normal to sun (β=θz, γ= γs) – Tilt (any β, γ is usually 0°) – Latitude (β=ø, γ is usually 0°)

92 Chapter #3: Solar Radiations Representation of solar radiations

• Symbols: – Irradiance: G – Irradiations: I (hourly), H (daily), H (monthly average daily) • Subscripts: – Ex.terr.: o Terrestrial: - – Beam: b Diffuse: d Total - – Normal: n Tilt: T Horizontal -

93 Chapter #3: Solar Radiations Extraterrestrial solar radiations

Solar Irradiance Irradiance at constant at normal horizontal

(Gsc) (Gon) (Go)

Mathematical integration…

Hourly Daily Monthly avg. irradiations on irradiations on daily irrad. on horizontal horizontal horizontal

(Io) (Ho) (Ho)

95 Chapter #3: Solar Radiations

Solar constant (Gsc)

Extraterrestrial solar radiations received at normal, when earth is at an average distance (1 au) away from sun.

2 퐺푠푐 = 1367 푊Τ푚

Adopted by World Radiation Center (WRC)

G96 sc Chapter #3: Solar Radiations Ex.terr. irradiance at normal

Extraterrestrial solar radiations received at

normal. It deviates from GSC as the earth move near or away from the sun. 360푛 퐺 = 퐺 1 + 0.033 cos 표푛 푠푐 365

G99on Chapter #3: Solar Radiations Ex.terr. irradiance on horizontal

Extraterrestrial solar radiations received at

horizontal. It is derived from Gon and therefore, it deviates from GSC as the earth move near or away from the sun.

퐺표 = 퐺표푛 × cos ∅ cos 훿 cos 휔 + sin ∅ sin 훿

101Go Chapter #3: Solar Radiations Ex.terr. hourly irradiation on horizontal

퐼표 12 × 3600 360푛 = 퐺 × 1 + 0.033 cos 휋 푠푐 365

× ቈcos ∅ cos 훿 sin 휔2 − sin 휔1

103Io Chapter #3: Solar Radiations Ex.terr. daily irradiation on horizontal

퐻표 24 × 3600 360푛 = 퐺 × 1 + 0.033 cos 휋 푠푐 365 휋휔 × cos ∅ cos 훿 sin 휔 + 푠 sin ∅ sin 훿 푠 180

105Ho Chapter #3: Solar Radiations Ex.terr. monthly average daily irradiation on horizontal

퐻ഥ표 24 × 3600 360푛 = 퐺 × 1 + 0.033 cos 휋 푠푐 365 휋휔 × cos ∅ cos 훿 sin 휔 + 푠 sin ∅ sin 훿 푠 180

Where day and time dependent parameters are calculated on average day of a particular month i.e. 푛 = 푛ത 107Ho Chapter #3: Solar Radiations Terrestrial radiations

Can be… • measured by instruments • obtained from databases e.g. TMY, NASA SSE etc. • estimated by different correlations

109 Chapter #3: Solar Radiations Terrestrial radiations measurement

• Total irradiance can • Diffuse irradiance can be measured using be measured using Pyranometer Pyranometer with shading ring

110 Chapter #3: Solar Radiations Terrestrial radiations measurement

• Beam irradiance can • Beam irradiance can be measured using also be measured Pyrheliometer by taking difference in readings of pyranometer with and without shadow band:

beam = total - diffuse

111 Chapter #3: Solar Radiations Terrestrial radiations databases

1. NASA SSE: Monthly average daily total irradiation on horizontal surface (퐻ഥ) can be obtained from NASA Surface meteorology and (SSE) Database, accessible from:

http://eosweb.larc.nasa.gov/sse/RETScreen/

(See “NASA SSE” in Reference Information)

112 Chapter #3: Solar Radiations Terrestrial radiations databases

2. TMY files: Information about hourly solar radiations can be obtained from Typical Meteorological Year files.

(See “TMY” section in Reference Information)

113 Chapter #3: Solar Radiations Terrestrial irradiation estimation

• Angstrom-type regression equations are generally used: 퐻ഥ 푛ത = 푎 + 푏 퐻ഥ표 푁ഥ

(See “Terrestrial Radiations Estimations” section in Reference Information)

114 Chapter #3: Solar Radiations Terrestrial irradiation estimation

For Karachi:

퐻ഥ 푛ത = 0.324 + 0.405 퐻ഥ표 푁ഥ

Where, 푛ത is the representation of cloud cover and 푁ഥ is the day length of average day of month.

115 Chapter #3: Solar Radiations Clearness index

• A ratio which mathematically represents sky clearness. =1 (clear day) <1 (not clear day) • Used for finding: – frequency distribution of various radiation levels – diffuse components from total irradiations

117 Chapter #3: Solar Radiations Clearness index

1. Hourly clearness index: 퐼 푘푇 = 퐼표 2. Daily clearness index: 퐻 퐾푇 = 퐻표 3. Monthly average daily clearness index: 퐻ഥ 퐾ഥ푇 = 퐻ഥ표

118 Chapter #3: Solar Radiations Diffuse component of hourly irradiation (on horizontal) Orgill and Holland correlation: 1 − 0.249푘 , 푘 ≤ 0.35 퐼 푇 푇 푑 = ቐ1.557 − 1.84푘 , 0.35 < 푘 < 0.75 퐼 푇 푇 0.177, 푘푇 ≥ 0.75

Erbs et al. (1982) 121 Chapter #3: Solar Radiations Diffuse component of daily irradiation (on horizontal) Collares-Pereira and Rabl correlation:

0.99, 퐾푇 ≤ 0.17 1.188 − 2.272퐾푇 2 +9.473퐾푇 퐻푑 3 , 0.17 < 퐾푇 < 0.75 = −21.865퐾푇 퐻 4 +14.648퐾푇 −0.54퐾푇 + 0.632, 0.75 < 퐾푇 < 0.8 0.2, 퐾푇 ≥ 0.8

123 Chapter #3: Solar Radiations Diffuse component of monthly average daily irradiation (on horizontal) Collares-Pereira and Rabl correlation:

퐻ഥ푑 퐻ഥ = 0.775 + 0.00606 휔푠 − 90 − ሾ0.505

124 Chapter #3: Solar Radiations Hourly total irradiation from daily irradiation (on horizontal) For any mid-point (ω) of an hour, 퐼 = 푟푡퐻 According to Collares-Pereira and Rabl: 휋 cos 휔 − cos 휔푠 푟푡 = 푎 + 푏 cos 휔 휋휔 24 sin 휔 − 푠 cos 휔 푠 180 푠 Where, 푎 = 0.409 + 0.5016 sin 휔푠 − 60 푏 = 0.6609 − 0.4767 sin 휔푠 − 60

125 Chapter #3: Solar Radiations Hourly diffuse irradiations from daily diffuse irradiation (on horizontal) For any mid-point (ω) of an hour, 퐼푑 = 푟푑퐻푑

From Liu and Jordan: 휋 cos 휔 − cos 휔푠 푟푑 = 휋휔 24 sin 휔 − 푠 cos 휔 푠 180 푠

126 Chapter #3: Solar Radiations Air mass and radiations

• Terrestrial radiations depends upon the path length travelled through atmosphere. Hence, these radiations can be characterized by air mass (AM). • Extraterrestrial solar radiations are symbolized as AM0. • For different air masses, spectral distribution of solar radiations is different.

127 Chapter #3: Solar Radiations Air mass and radiations

128 Chapter #3: Solar Radiations Air mass and radiations

• The standard spectrum at the Earth's surface generally used are: – AM1.5G, (G = global) – AM1.5D (D = direct radiation only) • AM1.5D = 28% of AM0 18% (absorption) + 10% (scattering). • AM1.5G = 110% AM1.5D = 970 W/m2.

129 Chapter #3: Solar Radiations Air mass and radiations

130 Chapter #3: Solar Radiations Radiations on a tilted plane

To calculate radiations on a tilted plane, following information are required: • tilt angle • total, beam and diffused components of radiations on horizontal (at least two of these) • diffuse sky assumptions (isotropic or anisotropic) • calculation model 131 Chapter #3: Solar Radiations Diffuse sky assumptions

132 Chapter #3: Solar Radiations Diffuse sky assumptions

Diffuse radiations consist of three parts: 1. Isotropic (represented by: iso) 2. Circumsolar brightening (represented by : cs) 3. Horizon brightening (represented by : hz)

There are two types of diffuse sky assumptions: 1. Isotropic sky (iso) 2. Anisotropic sky (iso + cs, iso + cs + hz)

133 Chapter #3: Solar Radiations General calculation model

푋푇 = 푋푏푅푏 + 푋푑,𝑖푠표퐹푐−푠 + 푋푑,푐푠푅푏 + 푋푑,ℎ푧퐹푐−ℎ푧 + 푋휌𝑔퐹푐−𝑔 Where,

• X, Xb, Xd: total, beam and diffuse radiations (irradiance or irradiation) on horizontal • iso, cs and hz: isotropic, circumsolar and horizon brightening parts of diffuse radiations

• Rb: beam radiations on tilt to horizontal ratio

• Fc-s, Fc-hz and Fc-g: shape factors from collector to sky, horizon and ground respectively

• ρg: albedo 134 Chapter #3: Solar Radiations Calculation models

1. Liu and Jordan (LJ) model (iso, 훾=0°, 퐼) 2. Liu and Jordan (LJ) model (iso, 훾=0°, 퐻ഥ) 3. Hay and Davies (HD) model (iso+cs, 훾=0°, 퐼) 4. Hay, Davies, Klucher and Reindl (HDKR) model (iso+cs+hz, 훾=0°, 퐼) 5. Perez model (iso+cs+hz, 훾=0°,퐼) 6. Klein and Theilacker (K-T) model (iso+cs, 훾=0°, 퐻ഥ) 7. Klein and Theilacker (K-T) model (iso+cs, 퐻ഥ)

(See “Sky models” in Reference Information) 135 Chapter #3: Solar Radiations Optimum tilt angle

136 Chapter #4:Flat-Plate Collectors Introduction 1. Flat-plate collectors are special type of heat-exchangers 2. Energy is transferred to fluid from a distant source of radiant energy 3. Incident solar radiations is not more than 1100 W/m2 and is also variable 4. Designed for applications requiring energy delivery up to 100°C above ambient temperature.

138 Chapter #4:Flat-Plate Collectors Introduction

1. Use both beam and diffuse solar radiation 2. Do not require sun tracking and thus require low maintenance 3. Major applications: solar water heating, building heating, air conditioning and industrial process heat.

139 Chapter #4:Flat-Plate Collectors

140 Chapter #4:Flat-Plate Collectors

141 Chapter #4:Flat-Plate Collectors

142 Chapter #4:Flat-Plate Collectors

Out (hot)

To tap

In (cold)

143 Chapter #4:Flat-Plate Collectors

Installation of flat-plate collectors at Mechanical Engineering Department, NED University of Engg. & Tech., Pakistan 144 Chapter #4:Flat-Plate Collectors Heat transfer: Fundamental

Heat transfer, in general: 2 푞 = 푄/퐴 = 푇1 − 푇2 Τ푅 = ∆푇Τ푅 = 푈∆푇[W/m ] Where,

푇1 > 푇2: Heat is transferred from higher to lower temperature ∆푇 is the temperature difference [K] R is the thermal resistance [m2K/W] A is the heat transfer area [m2] U is overall H.T. coeff. U=1/R [W/m2K] 145 Chapter #4:Flat-Plate Collectors Heat transfer: Circuits

Resistances in series: q q 1 2 푅 = 푅1 + 푅2 T 1 T1 R1 R2 2 푈 = 푅 + 푅 q=q1=q2 1 2 Resistances in parallel: q 1 1 R 푅 = 1 1ൗ + 1ൗ T1 T2 푅1 푅2 R2 1 1 q2 푈 = ൗ + ൗ 푅1 푅2 q=q1+q2 146 Chapter #4:Flat-Plate Collectors Example-1 Heat transfer: Circuits Determine the heat transfer per unit area(q) and overall heat transfer coefficient (U) for the following circuit:

R3 T1 R1 R2 T2 R4

q

147 Chapter #4:Flat-Plate Collectors Heat transfer: Radiation Radiation heat transfer between two infinite parallel plates: 푹풓 = ퟏΤ풉풓 and, 흈 푻 ퟐ + 푻 ퟐ 푻 + 푻 풉 = ퟏ ퟐ ퟏ ퟐ 풓 ퟏ ퟏ + − ퟏ 휺ퟏ 휺ퟐ Where, 휎 = 5.67 × 10−8 W/m2K4 휖 is the emissivity of a plate 148 Chapter #4:Flat-Plate Collectors Heat transfer: Radiation Radiation heat transfer between a small object surrounded by a large enclosure:

푹풓 = ퟏΤ풉풓 and, 흈 푻 ퟐ + 푻 ퟐ 푻 + 푻 풉 = ퟏ ퟐ ퟏ ퟐ 풓 ퟏΤ휺 ퟐ ퟐ = 휺흈 푻ퟏ + 푻ퟐ 푻ퟏ + 푻ퟐ [W/m2K]

149 Chapter #4:Flat-Plate Collectors Heat transfer: Sky Temperature

1. Sky temperature is denoted by Ts

2. Generally, Ts = Ta may be assumed because sky temperature does not make much difference in evaluating collector performance. 3. For a bit more accuracy:

In hot climates: Ts = Ta+ 5°C

In cold climates: Ts = Ta+ 10°C 150 Chapter #4:Flat-Plate Collectors Heat transfer: Convection Convection heat transfer between parallel plates:

푹풄 = ퟏΤ풉풄 and 풉풄 = 푵풖풌Τ푳 Where, 푵풖 = ퟏ + ퟏퟕퟎퟖ 퐬퐢퐧 ퟏ. ퟖ휷 ퟏ.ퟔ ퟏퟕퟎퟖ + ퟏ. ퟒퟒ ퟏ − ퟏ − 푹풂 풄풐풔 휷 푹풂 풄풐풔 휷 ퟏΤퟑ + 푹 퐜퐨퐬 휷 + 풂 − ퟏ ퟓퟖퟑퟎ Note: Above is valid for tilt angles between 0° to 75°. ‘+’ indicates that only positive values are to be considered. Negative values should be discarded. 151 Chapter #4:Flat-Plate Collectors Heat transfer: Convection

𝑔훽′∆푇퐿3 푅 = also 푃 = 휗Τ훼 푎 휗훼 푟 Where, Fluid properties are evaluated at mean temperature Ra Rayleigh number Pr Prandtl number L plate spacing k thermal conductivity g gravitational constant β‘ volumetric coefficient of expansion for ideal gas, β‘ = 1/T [K-1] 휗, α kinematic viscosity and thermal diffusivity 152 Chapter #4:Flat-Plate Collectors Heat transfer: Conduction Conduction heat transfer through a material:

푹풌 = 푳Τ풌

Where, L material thickness [m] k thermal conductivity [W/mK]

154 Chapter #4:Flat-Plate Collectors General energy balance equation In steady-state: Useful Energy = Incoming Energy – Energy Loss [W] 푸풖 = 푨풄 푺 − 푼푳 푻풑풎 − 푻풂 Incoming Energy A = Collector area [m2] c Energy T = Absorber plate temp. [K] pm Loss Ta = Ambient temp. [K] 2 UL = Overall heat loss coeff. [W/m K] Qu = Useful Energy [W] Useful Energy SAc = Incoming (Solar) Energy [W] AcUL(Tpm-Ta) = Energy Loss [W]

155 Chapter #4:Flat-Plate Collectors Thermal network diagram Ta q a loss (top) Ambient (a) Rr(c-a) Rc(c-a) c Top losses Cover (c) Rr(p-c) Rc(p-c) p Plate (p) S Qu Rk(p-b)

b Structure (b) Bottom losses Rr(b-a) Rc(b-a) a qloss (bottom) Ta 156 Chapter #4:Flat-Plate Collectors Thermal network diagram T a T a a R R r(c-a) c(c-a) R(c-a) =1/(1/Rr(c-a) + 1/Rc(c-a)) c Tc

Rr(p-c) Rc(p-c) R(p-c) =1/(1/Rr(p-c) + 1/Rc(p-c)) p Tp S Qu S Qu R k(p-b) Rk(p-b)

b Tb Rr(b-a) Rc(b-a) R(b-a) =1/(1/Rr(b-a) + 1/Rc(b-a)) a Ta Ta 157 Chapter #4:Flat-Plate Collectors Cover temperature T 1. Ambient and plate temperatures are a generally known. R(c-a) Tc 2. Utop can be calculated as:

R(p-c) Utop= 1/(R(c-a) +R(p-c)) Tp S Qu Rk(p-b) 3. From energy balance: qp-c = qp-a Tb (T -T )/R = U (T -T ) R(b-a) p c (p-c) top p a

T =>Tc = Tp- Utop (Tp-Ta)x R(p-c) a 158 Chapter #4:Flat-Plate Collectors Thermal resistances

2 2 Rr(c-a) =1/hr(c-a) = 1/εcσ(Ta +Tc ) (Ta+Tc)

Rc(c-a) = 1/hc(c-a) = 1/hw

2 2 Rr(p-c) = 1/hr(p-c) = 1/[σ(Tc +Tp ) (Tc+Tp)/(1/εc+ 1/εp-1)]

Rc(p-c) =1/hc(p-c) = 1/hc

Rk(p-b) = L/k

2 2 Rr(b-a) =1/hr(b-a) = 1/εbσ(Ta +Tb ) (Ta+Tb)

Rc(b-a) =1/hc(b-a) = 1/hw

159 Chapter #4:Flat-Plate Collectors Solution methodology

Utop= 1/(R(c-a) + R(p-c)) Cover to ambient Plate to cover

R(c-a) =1/(1/Rr(c-a) + 1/Rc(c-a)) R(p-c) =1/(1/Rr(p-c) + 1/Rc(p-c))

Radiation Radiation

Rr(c-a) =1/hr(c-a) Rr(p-c) = 1/hr(p-c) 2 2 2 2 = 1/εcσ(Ta +Tc ) (Ta+Tc) = 1/[σ(Tc +Tp ) (Tc+Tp)/(1/εc+ 1/εp-1)]

Convection Convection

Rc(c-a) = 1/hc(c-a) = 1/hw Rc(p-c) =1/hc(p-c) = 1/hc

Assumption validation Assume Tc between ambient and absorber ! plate temperature

Tc = Tp- Utop (Tp-Ta)x R(p-c)

161