outline review for second midterm

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



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


• 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)



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


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



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



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



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



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)



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



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 =−−+=



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



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

=



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



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

outline review for second midterm

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