Outline. Method for Solar Collectors. Heat Exchangers. Compact Heat Exchangers III. Shell-and-Tube Exchanger. f-chart Method April 12, 2010

The f

The f

-

-

Chart Method for Solar

Chart Method for Solar

Collectors

Collectors

Larry Caretto

Mechanical Engineering 496ALT

Alternative Energy

Alternative Energy

April 12, 2010

2

Outline

• Review heat exchangers

• Solar collector performance equations • Derivation of f-chart method

• Demonstration of f-chart results • Main reference: Duffie and Beckman,

Solar Engineering of Thermal

Processes, Wiley, 2006

• See http://www.fchart.com/ for f-chart software information

3

Heat Exchangers

• Used to transfer energy from one fluid to another

– U = overall heat transfer coefficient, W/m2_·K

• One fluid, the hot fluid, is cooled while the other, the cold fluid, is heated

• May have phase change: temperature of one or both fluids is constant

• Simplest is double pipe heat exchanger

– Parallel flow and counter flow

• More complex designs may be used

4

Figure 11-1 from Çengel, Heat and Mass Transfer

ΔTmax

ΔTmax

5

Compact Heat Exchangers III

Figure 11-3 from Çengel, Heat and Mass Transfer 6

Shell-and-Tube Exchanger

• Counter flow exchanger with larger surface area; baffles promote mixing

7

Shell and Tube Passes II

Tube flow has one complete change of direction giving two tube passes

Figure 11-5(a) from Çengel, Heat and Mass Transfer 8

Shell and Tube Passes III

Tube flow has three complete changes of direction giving four tube passes Shell flow changes direction to give two shell passes Figure 11-5(b) from Çengel, Heat and Mass Transfer

9

Effectiveness-NTU Method

• Analysis approach for heat exchangers when not all temperatures are known

– Based on ratio of actual heat transfer to maximum possible heat transfer

– Maximum possible temperature difference for one fluid, ΔT_maxis T_h,in– T_c,in

– Subscripts cand hfor cold and hot

– Only one fluid, the one with the smaller value of cp, can have ΔTmax

– Define Cc= ( cp)cand Ch= ( cp)h m& m& m& 10

Effectiveness,

ε

• In effectiveness-NTU method we find ε, then find = ε max

– Use C_minΔT_maxto find _max – C1ΔT1= C2ΔT2 or ΔT2= C1ΔT1/C2 – If ΔT1= ΔTmaxand C1/C2> 1, ΔT2> ΔTmax – C_minΔT_maxis maximum heat transfer

without impossible T < Tc,inor T > Th,in

(

)

(

min

)

min , . min max / , min C UA NTU C C C T T C Q Q Q h c in c in h = = − = = _&& & ε

Q& Q& Q&

11

Effectiveness Equations

• Double pipe parallel flow

( )

c

e

NTU c

+

−

=

ε

− +

1

1

1

( )

( )

c NTU c NTU

ce

e

− − − −

−

−

=

ε

1₁

1

1

• Double pipe counter flow

max min C C c= min C UA NTU=

Figures from Figure 11-26 from Çengel, Heat and Mass Transfer 12

More Effectiveness Equations

• Shell and tube _{One shell pass}

and 2, 4, 6, … tube passes 1 1 1 2 2 2 1 1 1 1 2 − + − + − ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − + + + + = ε c NTU c NTU e e c c

• Any geometry with c = 0

NTU

e

−

−

=

ε

1

max min C C c= min C UA NTU =

13

Find

ε

Example chart for finding effectiveness from NTU = UA/Cminand Cmin/Cmaxratio

Figure 11-26 from Çengel, Heat and Mass Transfer

For NTU = 1.5 and Cmin/Cmax = 0.25, ε= ? For NTU = 1.5 and Cmin/Cmax = 0.25, ε= .7 14

Chart

ε

Cmin/Cmax= 0.16 From chart for counter flow: NTU = 2.5 ε= 0.89 15

Basic Collector Performance

• Energy balance on collector

• Useful energy gain = solar energy input adsorbed by collector – losses by heat transfer to ambient

• Look at variation throughout year to get overall performance

– Detailed hour-by-hour computer analysis for large installations

– Simplified f-chart method for residences

16

Flat Plate Collector

2L = 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 17

Useful Energy Gain

• Q_u= rate of useful heat into collector • A_c= collector area

• Ha= solar energy absorbed = Hiτα

• Uc= collector overall heat-loss coefficient

• T_f,in= inlet collector fluid temperature • T_a= ambient temperature

• FR= collector heat removal factor

(

)

[

a c f in a

]

R c u

A

F

H

U

T

T

Q

=

−

_,

−

Hottel-Willier-Bliss equation 18

Heat Removal Factor, F

_R

• A_cand U_cdefined on previous chart • = product of mass flow rate and

heat capacity of collector fluid • F’ = collector efficiency factor

• Tf,out= outlet collector fluid temperature

( )

( )

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = − c p c c c m U A F c c c p R e U A c m F & & ' 1

( ) (

p _c f out f in

)

u mc T T Q = & , − , • Energy balance on collector fluid

( )

m&cp_c

19

Collector Efficiency Factor

• F = tanh(mL)/mL m2_{= U}

c/tk, t, k = absorber plate thickness, thermal conductivity

• CB= bond conductance = kBwB/kB • hc,i= flow tube internal heat transfer

coefficient ambient and fluid collector between resistance Thermal ambient and plate absorber between resistance Thermal = ' F ( ) ⎥_⎦⎤ ⎢ ⎣ ⎡ + + + = i i c B c c D h C D LF U w U F π , 1 1 2 1 1 ' 20 FR/F' Chart 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1 10 100 FR /F' ( ) c c c p U F A c m ' & 21

F-chart Water Heating Only

22

System with Solar Air Heating

23 F-Chart Water and Space Heating

24 T_f,in Tf,out T_w,in Tw,out http://starfiresolar.com/db2/00144/starfiresolar.com/_uimages/solardiagramallred.jpg (C)ollector fluid loop (S)torage fluid loop

( ) (

p fout win

) ( )

p _s

(

wout win

)

c u mc T T mc T T

Q =ε & _min , − , = & , − ,

εc= heat-exchanger effectiveness

( )

( ) ( )

[

p _s p_c

]

p c m c m c m & & & , min min=

25

Combine Equations

• Eliminate Tf,inin favor of Tf,out

(

)

[

a c f in a

]

R c u

A

F

H

U

T

T

Q

=

−

_,

−

( ) (

)

_{( )}

c p u out f in f in f out f c p u c m Q T T T T c m Q & & − ⇒ = − = , , , ,

( )

_⎥⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − = _a c p u out f c R c a R c u T c m Q T U F A H F A Q & ,

( )

c R a c R c

(

f out a

)

c p c R c u AF H AFU T T c m U F A Q = − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − _, 1 & 26

Heat Exchanger

• Substitute into previous Quequation,

rearrange, and define F’_R

( )

(

)

( )

(

win u c p a

)

c R c a R c a out f c R c a R c c p c R c u T c m Q T U F A H F A T T U F A H F A c m U F A Q − + − = − − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − min , , 1 & & ε

( )

( )

c R a c R c

(

win a

)

p c c R c c p c R c u AF H AFU T T c m U F A c m U F A Q = − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − _, min 1 & & ε

(

)

[

a c win a

]

R c u AF H U T T Q = ' − _, −

( ) (

p min f,out w,in

)

f,out w,in u c

( )

pmin

c

u mc T T T T Q mc

Q =ε & − ⇒ = + ε &

27

Heat Exchanger II

• F’_R/F_Rabout 0.97 (Hodge – for water) • If Tw,out> Tmax= 100oC, water is vented

• Define Qdas solar heat delivered

( )

( )

( )

( )

( )

1 min 1 min ' 1 1 1 − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ε + = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ε + − = p c c p c p c R c R p c c R c c p c R c R R c m c m c m U F A F c m U F A c m U F A F F & & & & &

( )

(

)

[

]

( )

(

)

⎪ ⎩ ⎪ ⎨ ⎧ > − ≤ − = max , , max max , , , ,0 max T T T T c m T T T T c m Q out w in w s p out w in w out w s p d & & 28 F'R/FR Chart 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 10 100 F'R /FR ratio = 0.9 ratio = 0.8 ratio = 0.7 ratio = 0.6 ratio = 0.5 ratio = 0.4 ratio = 0.3 ratio = 0.2 ratio = 0.1 ( ) ( )pc p c c m c m & & min εratio= ( ) c R c c p U F A c m& 29

Storage Tank Energy Change

• Mcp= mass * cpfor storage tank liquid

• Q_L,s= space heating supplied by solar • Qw,s= water heating supplied by solar

• QTL= storage tank heat loss

• t = time

• f-chart method integrates this equation over some time period (1 month)

TL s w s L d s p Q Q Q Q dt dT Mc = − , − , − 30

f-chart Method Development

• For a long time period, the initial transient term is negligible

• The tank loss warms the house and contributes to the home heating load • The last three integrals are the total

energy supplied by solar, Qs

t d Q dt Q dt Q dt Q dt dt dT Mc t TL t s w t s L t d t s p

∫

∫

∫

∫

∫

Δ Δ Δ Δ Δ − − − = , , = –Q_S 0

31

f-chart Method Development II

• D = total energy demand

• f = fraction of total supplied by solar • Use the definition of Qs, total energy

supplied by solar from previous slide

dt Q D D Q f dt Q Q Q dt Q t d s t d s s t d

∫

∫

∫

Δ Δ Δ = = ⇒ = ⇒ − = 1 0

(

)

[

H U T T

]

dt F A D dt Q D D Q f t a in w c a R c t d s

∫

∫

Δ Δ − − = = = , ' 1 1 32

f-chart Method Development III

• The absorbed solar radiation, H_a, is the incident radiation, H_i, times the factor τα

– τis the fraction transmitted through the glass cover(s)

– αis the fraction absorbed

• Set H_a= H_iταand multiply the last term, top and bottom, by Tref- Ta

(

)

dt T T T T T T U H D F A D Q f t ref a a in w a ref c i R c s

∫

Δ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − − = = , ' τα 33

f-chart Method Development IV

• Cannot evaluate integral because of the dependence of T_son other variables

– Use engineering judgment to identify important variables and form empirical parameters based on f equation

(

)

∫

∫

Δ Δ − − − − = t ref a a in w a ref c R c t i R c _dt T T T T T T U D F A dt H D F A f , ' ' τα

(

)

∫

∫

Δ Δ − = τα = t a ref c R c t i R c _{U T T dt} D F A X dt H D F A Y ' ' 34

What are 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

(

)

∫

Δ − = t a ref c R c _{U T T dt} D F A X '

∫

Δ τα = t i R c _{H dt} D F A Y ' 35

f-chart Method Development V

• Integrate X and Y to get averages

total i R c t t i R c i R c _H D F A dt H D F A dt H D F A Y _, ' ' ' _τα = τα = τα =

∫

∫

Δ Δ

(

)

c R c

(

ref a

)

t a ref c R c _{T T} D U F A dt T T U D F A X=

∫

− = − Δ ' '

• Rearrange equations to introduce factors F_RU_cand F_R(τα)_navailable from standard collector test results

36

f-chart Method Development VI

(

)

(

ref a

)

R R c R c a ref c R c R R _{T T} D t F F U F A X t T T D U F A F F X= − Δ ⇒ = Δ − ' '

• (τα)n= value of the ταproduct for

normal radiation measured in tests

• Hi,totalis available from NREL data for Δt

= 1 month ( ) ( ) ( ) ( ) D H F F F A Y H D F A F F Y itotal n R R n R c total i R c n R n R ' , , ' τα τα τα = ⇒ τα τα τα =

37

Collector Efficiency Equations

( ) (

p _c fout fin

)

u

m

c

T

T

Q

=

&

_,

−

_,

( )

_⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−

=

=

i a in f c R n R i c u

H

T

T

U

F

F

H

A

Q

τα

,

η

• From March 22-24 lecture

• Plot of ηversus(Tf,in– Ta)/Hiis straight line

with slope = –FRUcand intercept FR(τα)n

( )

(

)

[

i n R c fin a

]

R c u

A

F

H

F

U

T

T

Q

=

τα

−

,

−

38 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 E ff iciency 1-cover, black 1-cover, selective 2-cover, black 2-cover, selective ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = = i a i c R n R i c u H T T U F F H A Q _τα η c RU F slope=− ( )n R F τα = intercept 39

Sample Rating Sheet

http://www.builditsolar.com/References/Ratings/SRCCRating.htm 40

Sample Rating Sheet II

slope = –FRUc

intercept = FR(τα)n

41

Back to f Equation

• Had equation for f depending on factors like X and Y

• Form empirical relationship between f, X, and Y

• This is basis for f-chart method

• For water heating: f = 1.029Y – 0.065X – 0.245Y2_{+ 0.0018X}2_{+ 0.0215Y}3

• Klein, Beckman and Duffie developed method and software

42 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 = 0.9 f = 0.8 f = 0.7 f = 0.6 f = 0.5 f = 0.4 f = 0.3 f = 0.2 f = 0.1

43

Computing X (dimensionless)

(

)

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ ₋ = ref a R R c R c T T D t F F U F A X '

• F_RU_c(W/m2_{·K) from slope of collector}

test data

• F’R/FRcomputed or assumed = 0.97

• Usual averaging period, Δt = 1 month, converted to seconds

• D = heating demand for averaging period (J)

• Tref = 100o_{C; from NREL data} a T • Ac= collector area (m2₎ 44

Computing Y (dimensionless)

• FR(τα)nfrom intercept of collector test

• F’R/FRcomputed or assumed = 0.97

• Ratio = 0.94 (October – March), = 0.90 (April – September) or computed

• Hi,totalis available from NREL data for Δt

= 1 month (convert to J/m2₎ • D is heating demand J ( ) _{( )} ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = D H F F F A Y itotal n R R n R c , ' τα τα τα

( )

τα n τα/ • Ac= collector area (m2₎ 45

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

46

NREL Solar Data

• 1961-1990 measured data for 239 sites

– 56 sites measured; other had some modeled data

• 1991-2005 update for 1,454 locations

– 99% of sites contain modeled data

– Van Nuys Airport is in this update, but LAX is only LA site in original data set

– Data format different from 1961-1990 data

• TMY3 data: Typical Meteorological Year

– Hourly data on radiation components – Compute resultant for given collector geometry

• http://www.nrel.gov/rredc/solar_data.html

47

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₎ 48

NREL Collector Types ’61-’90 II

49

NREL Collector Types ’61-’90 III

50

NREL 1961-1990 LAX Average

SOLAR RADIATION FOR FLAT-PLATE COLLECTORS FACING SOUTH AT A FIXED-TILT (kWh/m2/day) Percentage Uncertainty = 9

Tilt(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 5.5 Minimum 2.9 3.6 4.5 5.8 5.7 5.4 6.3 6.3 4.7 4.4 3.4 2.7 5.2 Maximum 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 5.6 Minimum 3.3 3.8 4.7 5.6 5.4 5.0 5.9 6.1 4.8 4.7 3.7 3.0 5.3 Maximum 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 5.4 Minimum 3.4 3.8 4.5 5.2 4.8 4.4 5.2 5.5 4.5 4.7 3.9 3.1 5.1 Maximum 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 51 LAX Solar Radiation 1961-1990

Flat-plate titted at latitude = 34o

0 1 2 3 4 5 6 7 8

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month R a d ia ti on (k Wh/ m 2/d ay) 30-year average Individual years http://rredc.nrel.gov/solar/old_data/nsrdb/redbook/sum2/23174.txt http://rredc.nrel.gov/solar/old_data/nsrdb/redbook/mon2/23174.txt 52 Fchart Input data screen 53 54

55

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

( )

p

( )

L L mc UA Z min & ε = 56

Another Adjustment

• For systems with only water heating

– Tw= water temperature to household

– T_m= cold water supply temperature

– Ta= monthly average ambient temperature

• Multiply X by correction factor, CF, below a a m w T T T T CF − − + + = 100 32 . 2 86 . 3 18 . 1 6 . 11