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Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
Review for Second MidtermReview for Second Midterm
Larry CarettoMechanical Engineering 483
Alternative Energy Alternative Energy Engineering IIEngineering II
April 19, 2010
2
Outline• Black-body and solar radiation• Emissivity and absorptivity• Path of the sun• Solar collectors
– Basic analysis• Useful gain = Absorbed solar – Heat Loss• Overall loss coefficient, Uc
• Effectiveness terms and factors: F, F’, FR, F’R– f-chart method
3
• Radiation heat transfer by electromagnetic radiation– Part of much larger spectrum– Thermal radiation transfers
heat without contact• Use of fire or electric resistance
heating are best examples• Thermal radiation lies in infrared
and visible part of spectrum (with some in ultraviolet)
Electromagnetic Radiation
Figure 12-3 from Çengel, Heat and Mass Transfer 4
Black-body Radiation• Perfect emitter – no
surface can emit more radiation than a black body
• Diffuse emitter –radiation is uniform in all directions
• Perfect absorber – all radiation striking a black body is absorbed
Figure 12-7 from Çengel, Heat and Mass Transfer
5
Black-Body Radiation II• Basic black body equation: Eb = σT4
– Eb is total black-body radiation energy flux W/m2 or Btu/hr·ft2
– σ is the Stefan-Boltzmann constant• σ = = 2π5k4/(15h3c2) = 5.670x10-8 W/m2·K4 =
0.1714x10-8 Btu/hr·ft2·R4
• k = Boltzmann’s constant = 1.38065x10-23 J/K (molecular gas constant) = Ru/NAvagadro
• h = Planck’s constant = 6.62607x10-34 J·s• c = 299,792,458 m/s = speed of light in a
vacuum
6
Black-body Radiation Spectrum• Energy (W/m2) emitted varies with
wavelength and temperature• Ebλ is spectral radiation
– Units are W/(m2⋅μm)– Ebλdλ is fraction of black body radiation in
range dλ about wavelength λ• Maximum occurs at λT = 2897.8 μm·K
– T increase shifts peak shift to lower λ• Diagram on next chart
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
7
Spectral EbλBlack Body Radiation
1
10
100
1,000
10,000
100,000
1,000,000
10,000,000
100,000,000
0.1 1 10 100Wavelength, λ, μm
Radiation W/m2-μm
T = 6000 KT = 5000 KT = 4000 KT = 3000 KT = 2000 KT = 1000 KT = 800 KT = 600 KT = 400 KT = 300 K
visible infrared
ultra-violet
8
Spectral Black-body Energy• Ebλdλ = black-body emissive power in a
wavelength range dλ about λ– Typical units for Ebλ are W/m2·μm or
Btu/hr·ft2·μm
( ) λ−λ
=λ λλ de
CdE TCb125
1
• C1 = 2πhc2 = 3.74177 W·μm4/m2
• C2 = hc/k = 14387.8 μm/K– h = Planck’s constant, c = speed of light in
vacuum, k = Boltzmann’s constant
9
Partial Black-body Power
∫λ
λλ− λ=1
10
0, dEE bb
∫λ
λλ λσ
=0
4 '1 dET
f b
Black body radiation between λ = 0 and λ = λ1 is Eb,0-λ1
Fraction of total radiation (σT4) between λ = 0 and any given λ is fλ
Figure 12-13 from Çengel, Heat and Mass Transfer10
Radiation Tables• Can show that fλ is function of λT
( ) ( ) ( ) ( )∫∫∫λ
λ
λ
λ
λ
λλ λ−λσ
=λ−λσ
=λσ
=T
TCTCb TdeTCd
eC
TdE
Tf
05
1
05
14
04 1
11
1122
• Radiation tables give fλ versus λT– See table 12-2,
page 118 in Hodge– Extract from similar
table shown at right
11
Δλ
Figure 12-14 from Çengel, Heat and Mass Transfer
• Radiation in finite band, Δλ
( ) ( )TfTf
dET
dET
dET
f
bb
b
120
40
4
4
12
2
1
21
11
1
λ−λ=
λσ
−λσ
=λσ
=
∫∫
∫λ
λ
λ
λ
λ
λλλ−λ
12
Emissivity• Emissivity, ε, is ratio of actual emissive
power to black body emissive power– May be defined on a directional and
wavelength basis, ελ,θ(λ,θ,φ,T) = Iλ,e(λ,θ,φ,T)/Ibλ(λ,T), called spectral, directional emissivity
– Total directional emissivity, average over all wavelengths, εθ(θ,φ,T) = Ie(θ,φ,T)/Ib(T)
– Spectral hemispherical emissivity average over directions, ελ(λ,T) = Iλ(λ,T)/Ibλ(λ,T)
– Total hemispheric emissivity = E(T)/Eb(T)
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
13
Emissivity Assumptions• Diffuse surface – emissivity does not
depend on direction• Gray surface – emissivity does not
depend on wavelength• Gray, diffuse surface – emissivity is the
does not depend on direction or wavelength– Simplest surface to handle and often used
in radiation calculations14
15 16
Properties• When radiation,
G, hits a surface a fraction ρG is reflected; another fraction, αG is absorbed, a third fraction τG is transmitted
• Energy balance: ρ + α + τ = 1
Figure 12-31 from Çengel, Heat and Mass Transfer
17
Properties II• Fractions on
previous chart are properties– Reflectivity, ρ– Absorptivity, α– Transmissivity, τ
• Energy balance: ρ + α + τ = 1Figure 12-31 from
Çengel, Heat and Mass Transfer
18
Properties III• As with emissivity, α, ρ, and τ may be
defined on a spectral and directional basis– Can also take averages over wavelength,
direction or both as with emissivity– Simplest case is no dependence on either
wavelength or direction– Reflectivity may be diffuse or have angle of
reflection equal angle of incidence
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
19
α Data• Solar
radiation has effective source temperature of about 5800 K
Figure 12-33 from Çengel, Heat and Mass Transfer 20
Kirchoff’s Law• Absorptivity equals emissivity (at the
same temperature) αλ = ελ
• True only for values in a given direction and wavelength
• Assuming total hemispherical values of α and ε are the same simplifies radiation heat transfer calculations, but is not always a good assumption
21
Effect of Temperature• Emissivity, ε, depends on surface
temperature• Absorptivity, α, depends on source
temperature (e.g. Tsun ≈ 5800 K)• For surfaces exposed to solar radiation
– high α and low ε will keep surface warm– low α and high ε will keep surface cool– Does not violate Kirchoff’s law since
source and surface temperatures differ 22From Çengel, Heat and Mass Transfer
23
Average Radiation Properties• Integrated average properties over all
wavelengths
∫∫∞∞
==0
40
4
11 λασ
αλεσ
ε λλλλ dET
dET bb
• Look at simple example where ελ = ε1for λ < λ1 and ελ = ε2 for λ > λ1
∫∫∫∞∞
+==1
1
240
140
4
111
λλ
λ
λλλ λεσ
λεσ
λεσ
ε dET
dET
dET bbb
24
Average Radiation Properties II• Rearrange to get fλ, the fraction of black
body radiation between 0 and λ
( )11
1
1
1'' 2142
04
1λλ
∞
λλ
λ
λ −ε+ε=λσε
+λσε
=ε ∫∫ ffdET
dET bb
• Similar equation for absorptivity (αλ = ελ)
( )11
1
1
1'' 2142
04
1λλ
∞
λλ
λ
λ −α+α=λσα
+λσα
=α ∫∫ ffdET
dET bb
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
25
Example• Data: ελ = 0.9 for λ < 3 μm and ελ = 0.2
for λ > 3– Solar T = 5800 K, λT = 17,400 μm·K,
fλ(17,400 μm·K) = 0.980155, find α• Use αλ = ελ
– Earth T = 300 K, λT = 900 μm·K, fλ(17,400 μm·K) = 0.001, find ε
( ) ( ) ( ) 886.0980.012.0980.09.0111 215800 =−+=−α+α=α λλ ffK
( ) ( ) ( ) 201.0001.012.0001.09.0111 21300 =−+=−ε+ε=ε λλ ffK
26
27
Solar Angles III
from the sun center
28
Solar Declination AngleDeclination Angle & Relative Earth-Sun Distance
-30
-20
-10
0
10
20
30
0 40 80 120 160 200 240 280 320 360
Julian Date
Dec
linat
ion
Ang
le (d
egre
es
0.97
0.98
0.99
1.00
1.01
1.02
1.03
Rel
ativ
e Ea
rth-
sun
Dis
tancDeclination angle
Relative distance
29
Tangent plane to earth’s surface at given location 30
Angles for Tilted Collector
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
31
Equation of Time
Solar Time = Standard Time +
Equation of Time +
(4 min/o) * (Standard Longitude –Local Longitude)Standard Time = DST – 1 hour
32
Computing the Sun Path• Input data: Latitude, L, date, hour h• Find declination from serial date, n
( ) ( ) ( )degreesinδ⎥⎦⎤
⎢⎣⎡ π
+=δ180
284365360sin45.23 no
• Two angles: altitude (α) and azimuth (φ)– sin(α) = sin(L) sin(δ) + cos(L) cos(δ) cos(h)– sin(αs) = sin(φ) = cos(δ) sin(h) / cos(α)– Sun path is plot of α vs. φ = αs for one day– Plot is symmetric about solar noon– Typically plot data for 21st of month
33
Path Calculation Problem• Angles given as sin(angle) = x require
arcsin function calculation• Typical arcsine function returns angle
between –90o and 90o
– Limits correspond to range for sine between –1 and +1
– Special calculation for hour angle limit• hlimit = ±tan(δ) / tan(L)• φ = ±[π – arcsin(sin φ)] for |h| > |hlimit|
34
35
Solar Irradiation by Month in Los Angeles (LAX)Average of Monthly 1961-1990 NREL Data for different collectors
0
1
2
3
4
5
6
7
8
9
10
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
Irrad
iatio
n (k
Wh/
m2 /day
)
Fixed, tilt=0
Fixed, tilt=L-15
Fixed, tilt=LFixed,tilt=L+15
Fixed,tilt=90
1-axis,track,EW horizontal
1-axis,track,NS horizontal
1-axis,track,tilt=L
1axis,tilt=L+15
2-axis,track
Notes:All fixed collectors are facing southThe L in tilt = L means the local latitude (33.93oN)
36
Optimum Fixed Collector Tilt
35o
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
37
Solar
Insolationby collectororientationand monthat 40oNlatitude
38
39 40
41
Active Indirect Solar Water Heating
http://www.dnr.mo.gov/energy/renewables/solar6.htm (accessed March 12, 2007)
42
Passive Direct Solar Water Heatinghttp://southface.org/solar/solar-roadmap/solar_how-to/batch-collector.jpg (accessed March 12, 2007)
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
43
Basic Solar Collector Analysis• Overall heat balance
– Incoming solar radiation– Heat loss from collector to environment– Useful energy gain = Incoming Solar
Radiation – Environmental Heat Loss• Environmental heat loss proportional to
ΔT = Tcollector – Tambient– Applications that require high collector
temperatures will have more heat loss44
Useful Heat Transfer• Heat is added to a collector fluid
– Typically collector fluid is water or water and anti-freeze solution
– Air is also used as collector fluid for home heating
• Energy added from simple first law for open system with consant pressure heat addition
( )infoutfpu TTcmQ ,, −= &&
45
Solar to Useful Energy• Solar transmission through glass covers
provides absorbed radiation, Ha = Hiτα• Consider three losses
– Conduction through bottom of solar collector box
– Conduction through edge of box– Loss through top
• Convection between absorber plate and glass covers with conduction through glass
• Convection from top glass cover to ambient
46
Loss Through Top• In steady state the following heat rates
will be the same– Between absorber plate and bottom glass– From bottom glass to top glass
• Consider two-plate collector– From top glass to ambient– Look at exchange between absorber plate
at temperature TP and bottom glass at temperature Tg2
– Have convection plus radiation
47
Loss from Absorber Plate( ) ( )
1112
42
4
22
−+
−+−= −
gP
gPcgPcgptop
TTATTAhQ
εε
σ
( ) ( )( )( ) ( )22,
2
2222
2
2
42
4
111111 gPgpr
gP
gPgPgPc
gP
gPc TThTTTTTTATTA
−=−+
−++=
−+
−−
εε
σ
εε
σ
( ) ( )2
222,2
gp
gPgPcgprgptop R
TTTTAhhQ
−−−
−=−+=
48
Remaining Top Loss Path
• Between glass plates
( ) ( )12
121212,12
gg
ggggcggrggtop R
TTTTAhhQ
−−−
−=−+=
( )( )111
21
2122
21
12,
−+
++=−
gg
ggggcggr
TTTTAh
εε
σ
( ) ( )12
111,1
gg
agagcagragtop R
TTTTAhhQ
−−−
−=−+=
( )( )ag
skyg
gg
skygskygcagr TT
TTTTTTAh
−
−
−+
++=−
1
1
21
122
11,
111εε
σ
• Top plate to ambient
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
49
Loss Through Top/Bottom• Combine three resistances in series to
get Rtop = RP-g2 + Rg2-g1 + Rg1-a– Qtop = (TP – Ta)/Rtop = UtopAc(TP – Ta)
• Loss through bottom is conduction through insulation (kins, Δxins) in series with convection to ambient with hb-a
( )aPcbottom
cabcins
ins
aP
convins
aPbottom TTAU
AhAxk
TTRRTTQ −=
+Δ
−=
+−
=
−
1
50
Total Loss• Qsides = U’sideAside(TP – Ta)
– Can estimate U’side = 0.5 W/m2·K– Use UsideAc = U’sideAside for common area– Qsides = UsideAc(TP – Ta) = (TP – Ta)/Aside
• Total is sum of individual losses• Qloss = UcAc(TP – Ta)= (TP – Ta)/Rc
• Overall conductance and resistance• Uc = Utop + Ubottom + Usides
sidebottomtopc RRRR1111
++=
51
Approximate Utop Equation
( )( )
( ) NBNN
TTTT
hBNTT
TA
U
gpp
apap
w
ap
p
top
−⎟⎟⎠
⎞⎜⎜⎝
⎛
ε−+
+ε−+ε
++σ+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
12105.0
1
1'
1
22
33.0
N = number of glass coversA’ = 250[1 – 0.0044(s – 90)]s = tilt angle (degrees)B = (1 – 0.04hw +
0.0005hw2)(1 + 0.091N)
hw = heat transfer coeffi-cient from top to ambient
Other symbols have previous definitions
Equation uses SI units: Uc and h in W/m2·K, T in K, σ= 5.670x10-8 W/m2·K4, εg is same for all glass covers
52
Absorber Plate Analysis• Three analysis steps for solar energy to
heat fluid (Hottel-Whillier-Bliss equation)– Solar energy into plate flows across plate
to location of tubes at some line on plate– At same line heat flow into collector fluid
from plate is determined– Integrate heat flow into fluid from inlet to
exit to get total useful heat transfer to fluid
( )[ ]ainfcacRu TTUHAFQ −−= .&
53
Flat Plate Collector2L = distance between outside of flow tubes
D = flow tube outer diameter
t = absorber plate thickness
w = distance between flow tube centerlines = 2L + D
Di = flow tube inner diameter
k = absorber thermal conductivity
54
Absorber Plate Analysis
• Define m2 = Uc/(tkplate)• Effectiveness factor, F = tanh(mL) / (mL)• Total (useful) heat transfer per unit length of
tube ( ) ( )[ ] '2 uabcatotal qTTUHDLFq =−−+=
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
55
Absorber Plate Analysis II
• Heat flow into fluid at any point
( )[ ]
( )
( )[ ]afca
iicBc
afcac
u TTUHwF
DhCUDLF
TTUHUq −−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++
+
−−= '
112
1
1
,
'
π
56
Factors, F’ and FR
AmbientandFluidBetweenResistanceThermalAmbientandPlateBetweenResistanceThermal
='F
( )[ ] ( )( )[ ]iicBc
cDhCUDLFw
UFπ+++
=,1121
1'
• Collector efficiency factor, F’
( )p
ccacmFAU
cc
pRcm
FAUaea
eFAU
cmFF p
cc
&
& & '111''
'
=−=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−= −
−
• Heat removal factor, FR
[ ] [ ]∞→→ ,0/1,1' aasaFFR
57
FR/F' Chart
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1 10 100
F R/F
'
( )cc
cp
UFA
cm
'
&58
Summary of Results• Qu = useful heat transfer to working fluid
( ) ⎥⎦
⎤⎢⎣
⎡++
+
=
iiBcc hDCDLFUUw
F
π11
21
1'
⎥⎥⎦
⎤
⎢⎢⎣
⎡−=
−p
ccmAFU
c
pR e
AUcm
F &&'
1
( )[ ]p
uinfoutfainfcaRu cm
QTTTTUHAFQ&
+=−−= ,,,
( )τα= ia HH
59
Collector Efficiency, ηc = Qu/AcHi
• Replace Ha by Hiτα
( )[ ]ainfcacRu TTUHAFQ −−= .
• Start with Hottel-Whillier-Bliss Equation
( )[ ]ainfcicRu TTUHAFQ −−= .τα
• Substitute into efficiency equation
( )[ ] ( )i
ainfcRR
ic
ainfcicR
ic
uc H
TTUFF
HATTUHAF
HAQ −
−=−−
== .. τατα
η
60
Solar Collector Efficiency Tests
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.05 0.1 0.15
Effic
ienc
y
1-cover, black1-cover, selective2-cover, black2-cover, selective
Slope = –FRUc
Intercept = FR(τα)n
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
61
Sample Rating Sheet
http://www.builditsolar.com/References/Ratings/SRCCRating.htm 62
Sample Rating Sheet II
slope = –FRUcintercept = FR(τα)n
63
F-chart Water Heating Only
64
System with Solar Air Heating
65
F-Chart Water and Space Heating
66
Tf,in
Tf,out
Tw,in
Tw,out
http://starfiresolar.com/db2/00144/starfiresolar.com/_uimages/solardiagramallred.jpg
(C)ollectorfluid loop(S)torage
fluid loop
( ) ( ) ( ) ( )inwoutwspinwoutfpcu TTcmTTcmQ ,,,,min−=−= &&ε
εc = heat-exchanger effectiveness
( )( ) ( )[ ]
cpsp
p
cmcmcm
&&
&
,minmin
=
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
67
Heat Exchanger• Start with Hottel-Whillier-Bliss equation
– Replace Tf,in by Tw,in
( ) ( ) ( )ainwcRcaRcpc
cRc
cp
cRcu TTUFAHFA
cmUFA
cmUFAQ −−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+− ,
min
1&& ε
( )[ ]ainwcaRcu TTUHFAQ −−= ,'
( ) ( ) ( )( )( )
1
min
1
min
' 111
−−
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
ε+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
ε+−=
pc
cp
cp
cRcR
pc
cRc
cp
cRcRR cm
cm
cmUFAF
cmUFA
cmUFAFF
&
&
&&&
68
F'R/FR Chart
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100
F'R/F
R
ratio = 0.9ratio = 0.8ratio = 0.7ratio = 0.6ratio = 0.5ratio = 0.4ratio = 0.3ratio = 0.2ratio = 0.1
( )( )
cp
pc
cm
cm&
&min
ε=ratio
( )cRc
cp
UFA
cm&
69
f-chart Method• Predicts fraction of demand over a time
period (usually monthly) than can be supplied by solar
• Two empirical pamameters, X and Y– X is ratio of reference collector loss to total
heating load– Y is ratio of absorbed solar energy to total
heating load
( )arefcRc TT
DUFAX −='
totaliRc H
DFAY ,
' τα=
70
Computing X (dimensionless)( )⎥
⎦
⎤⎢⎣
⎡−
Δ= aref
R
RcRc TT
Dt
FFUFAX
'
• FRUc (W/m2·K) from slope of collector test data
• F’R/FR computed or assumed = 0.97• Usual averaging period, Δt = 1 month,
converted to seconds• D = heating demand for averaging
period (J)• Tref = 100oC; from NREL dataaT
• Ac = collector area (m2)
71
Computing Y (dimensionless)
• FR(τα)n from intercept of collector test• F’R/FR computed or assumed = 0.97• Ratio = 0.94 (October – March),
= 0.90 (April – September) or computed• Hi,total is available from NREL data for Δt
= 1 month (convert to J/m2)• D is heating demand J
( ) ( ) ⎥⎦
⎤⎢⎣
⎡=
DH
FFFAY totali
nR
RnRc
,'
τατατα
( )nτατα /
• Ac = collector area (m2)
72
f Equations• For water heating: f = 1.029Y – 0.065X
– 0.245Y2 + 0.0018X2 + 0.0215Y3
– Adjustments required• Adjust X for hot water supply only and storage
capacity different from standard• Adjust Y for load heat exchanger capacity
• For air heating: f = 1.040Y – 0.065X –0.159Y2 +0.00187X2 – 0.0095Y3
– Solar collectors heating air have no heat exchanger so F’R = FR
Review for Second Midterm April 19, 2010
ME 483 Alternative Energy Engineering II
73
f-Chart
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 2 4 6 8 10 12 14 16
X
Y
f = 0.9f = 0.8f = 0.7f = 0.6f = 0.5f = 0.4f = 0.3f = 0.2f = 0.1
F-Chart for Collectors Using Water
74
Adjustments• Adjust X for storage capacity, M, in L/m2
X’ = X(75/M)1/4
• Adjust Y for load heat exchanger factor, Z: Y’ = Y(0.39 + 0.65e-0.139/Z)– εL = heat exchanger effectiveness– mass flow times heat capacity and UA
factors defined previously
( ) ( )LpL UAcmZmin
&ε=
75
Another Adjustment• For systems with only water heating
– Tw = water temperature to household– Tm = cold water supply temperature– Ta = monthly average ambient temperature
• Multiply X by correction factor, CF, below
a
amwT
TTTCF−
−++=
10032.286.318.16.11
76
NREL Data• National Renewable Energy Laboratory• Collector data for 1961-1990 for 360
individual months and monthly averages– Available for variety of collectors
• Flat plate collector data for several angles
• TMY3 data: Typical Meteorological Year– Hourly data on radiation components– Compute resultant for given collector
geometry
77
NREL Collector Types ’61-’90• Data available at
different tilt levels for flat-plate collectors facing south– Horizontal (0o)– Latitude – 15o
– Latitude– Latitude + 15o
– Vertical (90o)78
NREL 1961-1990 LAX AverageSOLAR RADIATION FOR FLAT-PLATE COLLECTORS FACING SOUTH AT A FIXED-TILT (kWh/m2/day) Percentage Uncertainty = 9Tilt(deg) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year 0 Average 2.8 3.6 4.8 6.1 6.4 6.6 7.1 6.5 5.3 4.2 3.2 2.6 4.9
Minimum 2.3 3.0 4.0 5.5 5.7 5.6 6.4 6.1 4.4 3.8 2.7 2.1 4.7 Maximum 3.3 4.4 5.6 6.8 7.2 7.7 8.0 7.0 5.8 4.5 3.6 3.0 5.1
Lat - 15 Average 3.8 4.5 5.5 6.4 6.4 6.4 7.1 6.8 5.9 5.0 4.2 3.6 5.5Minimum 2.9 3.6 4.5 5.8 5.7 5.4 6.3 6.3 4.7 4.4 3.4 2.7 5.2Maximum 4.6 5.7 6.4 7.3 7.3 7.3 7.9 7.2 6.6 5.6 4.9 4.3 5.7
Lat Average 4.4 5.0 5.7 6.3 6.1 6.0 6.6 6.6 6.0 5.4 4.7 4.2 5.6Minimum 3.3 3.8 4.7 5.6 5.4 5.0 5.9 6.1 4.8 4.7 3.7 3.0 5.3Maximum 5.4 6.4 6.7 7.2 6.8 6.7 7.3 7.0 6.7 6.0 5.6 5.0 5.9
Lat + 15 Average 4.7 5.1 5.6 5.9 5.4 5.2 5.8 6.0 5.7 5.5 5.0 4.5 5.4Minimum 3.4 3.8 4.5 5.2 4.8 4.4 5.2 5.5 4.5 4.7 3.9 3.1 5.1Maximum 5.9 6.6 6.6 6.7 6.1 5.8 6.3 6.4 6.5 6.1 6.0 5.4 5.7
90 Average 4.1 4.1 3.8 3.3 2.5 2.2 2.4 3.0 3.6 4.2 4.3 4.1 3.5 Minimum 2.9 3.0 3.1 2.9 2.3 2.1 2.3 2.8 2.9 3.5 3.2 2.7 3.3 Maximum 5.2 5.4 4.5 3.6 2.7 2.3 2.5 3.2 4.1 4.7 5.2 5.0 3.7