The F-Chart Method for Solar Collectors

Outline The f-Chart Method for Solar Collectors • • • • • 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 Larry Caretto Mechanical Engineering 496ALT Alternative Energy April 12, 2010 2 Heat Exchangers • Used to transfer energy from one fluid to another ΔTmax ΔTmax – 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 Figure 11-1 from Çengel, Heat and Mass Transfer • More complex designs may be used 3 Compact Heat Exchangers III 4 Shell-and-Tube Exchanger • Counter flow exchanger with larger surface area; baffles promote mixing Figure 11-3 from Çengel, Heat and Mass Transfer 5 Figure 11-4 from Çengel, Heat and Mass Transfer 6 1 Shell and Tube Passes II 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 8 Tube flow has one complete change of direction giving two tube passes 7 Figure 11-5(a) from Çengel, Heat and Mass Transfer Figure 11-5(b) from Çengel, Heat and Mass Transfer Effectiveness, ε Effectiveness-NTU Method Q& • Analysis approach for heat exchangers when not all temperatures are known ε= & Q = max – Based on ratio of actual heat transfer to maximum possible heat transfer – Maximum possible temperature difference for one fluid, ΔTmax is Th,in – Tc,in – Subscripts c and h for cold and hot – Only one fluid, the one with the smaller value & cp, can have ΔTmax of m & cp)h & cp)c and Ch = (m – Define Cc = (m Q& Cmin (Th.in − Tc ,in ) Cmin = min (Ch , Cc ) NTU = UA / Cmin • In effectiveness-NTU method we find ε, & = εQ & max then find Q & – Use CminΔTmax to find Q max – C1ΔT1 = C2ΔT2 or ΔT2 = C1ΔT1/C2 – If ΔT1 = ΔTmax and C1/C2 > 1, ΔT2 > ΔTmax – CminΔTmax is maximum heat transfer without impossible T < Tc,in or T > Th,in 9 Effectiveness Equations ε= 1− e • Shell and tube − NTU (1+ c ) 1+ c • Double pipe counter flow ε= More Effectiveness Equations UA NTU = Cmin • Double pipe parallel flow 10 C c = min Cmax 1 − e − NTU (1−c ) 1 − ce − NTU (1−c ) One shell pass and 2, 4, 6, … tube passes ⎧ 1 + e − NTU ⎪ ε = 2⎨1 + c + 1 + c 2 1 − e − NTU ⎩⎪ • Any geometry with c = 0 ε = 1− e − NTU c= 1+ c 2 1+ c 2 ⎫ ⎪ ⎬ ⎪⎭ −1 NTU = 11 Figures from Figure 11-26 from Çengel, Heat and Mass Transfer Cmin Cmax UA Cmin 12 Figure from Figure 11-26 from Çengel, Heat and Mass Transfer 2 Find ε Chart ε Example chart for finding effectiveness from NTU = UA/Cmin and Cmin/Cmax ratio From chart for counter flow: NTU = 2.5 Cmin/Cmax = 0.16 ε = 0.89 For NTU = 1.5 and Cmin/Cmax .7 = 0.25, ε = ? Figure 11-26 from Çengel, Heat and Mass Transfer 13 14 Basic Collector Performance Flat Plate Collector • 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 k = absorber thermal conductivity D = flow tube outer diameter t = absorber plate thickness w = distance between flow tube centerlines = 2L + D 15 Useful Energy Gain • • • • • • • ] Qu = Ac FR H a − U c (T f ,in − Ta ) Hottel-Willier-Bliss equation Di = flow tube inner diameter 16 Heat Removal Factor, FR Qu = rate of useful heat into collector Ac = collector area Ha = solar energy absorbed = Hiτα Uc = collector overall heat-loss coefficient Tf,in = inlet collector fluid temperature Ta = ambient temperature FR = collector heat removal factor [ 2L = distance between outside of flow tubes 17 • Ac and Uc defined on previous chart & c p ) = product of mass flow rate and • (m c heat capacity of collector fluid • F’ = collector efficiency factor • Tf,out = outlet collector fluid temperature F ' AcU c ⎤ • Energy balance ⎡ − m& c p ⎢ (m& c p )c ⎥ on collector fluid c FR = 1− e ⎥ Q = m& c T AcU c ⎢ u p c f ,out − T f ,in ⎥⎦ ⎢⎣ ( ) ( )( ) 18 3 FR/F' Chart 1 Collector Efficiency Factor Thermal resistance between absorber plate and ambient Thermal resistance between collector fluid and ambient 1 Uc F'= ⎡ 1 1 1 ⎤ + + w⎢ ⎥ ⎣U c (2 LF + D ) C B hc ,iπDi ⎦ • F = tanh(mL)/mL m2 = Uc/tk, t, k = absorber plate thickness, thermal conductivity • CB = bond conductance = kBwB/kB • hc,i = flow tube internal heat transfer coefficient 19 0.95 0.9 0.85 FR/F' F'= 0.8 0.75 0.7 0.65 (m& c p )c 0.6 1 F-chart Water Heating Only 10 20 Ac F 'U c 100 System with Solar Air Heating 21 22 Qu = ε c (m& c p )min (T f ,out − Tw,in ) = (m& c p )s (Tw,out − Tw,in ) F-Chart Water and Space Heating εc = heat-exchanger effectiveness Tf,out Tf,in Tw,out (S)torage fluid loop (C)ollector fluid loop ( Tw,in ) m& c p min = min m& c p s , m& c p [( ) ( ) ] c 23 4 Combine Equations [ ] Qu = Ac FR H a − U c (T f ,in − Ta ) Qu = (m& c p )c (T f ,out − T f ,in ) ⇒ T f ,in = T f ,out − Qu = ε c (m& c p Qu p c ( ) f ,out w,in f ,out w,in + Qu ε c (m& c p )min ⎡ AFU ⎤ Qu ⎢1 − c R c ⎥ = Ac FR H a − Ac FRU c (T f ,out − Ta ) (m& c p )c ⎦⎥ ⎣⎢ = Ac FR H a − Ac FRU c Tw,in + Qu ε c (m& c p )min − Ta ( ⎡ AFU ⎤ Qu ⎢1 − c R c ⎥ = Ac FR H a − Ac FRU c T f ,out − Ta m& c p ⎥ ⎢⎣ c ⎦ ( min • Substitute into previous Qu equation, rearrange, and define F’R (m& c ) • Eliminate Tf,in in favor of Tf,out ⎡ ⎤ Qu Qu = Ac FR H a − Ac FRU c ⎢T f ,out − − Ta ⎥ (m& c p )c ⎦⎥ ⎣⎢ Heat Exchanger ) (T − T ) ⇒ T = T ) ) ⎡ AFU AFU ⎤ Qu ⎢1 − c R c + c R c ⎥ = Ac FR H a − Ac FRU c (Tw,in − Ta ) (m& c p )c ε c (m& c p )min ⎦⎥ ⎣⎢ Qu = Ac FR' [H a − U c (Tw,in − Ta )] 25 26 F'R/FR Chart 1 Heat Exchanger II ⎡ AFU Ac FRU c ⎤ = FR ⎢1 − c R c + ⎥ m& c p ε c m& c p ⎢⎣ c min ⎥ ⎦ ( ) ( ) −1 ( ( ⎡ A F U ⎛ m& c p = FR ⎢1 + c R c ⎜ m& c p ⎜ ε c m& c p ⎢⎣ c ⎝ ( ) • F’R/FR about 0.97 (Hodge – for water) • If Tw,out > Tmax = 100oC, water is vented • Define Qd as solar heat delivered [ ] 0.8 0.7 0.6 0.5 ⎧max (m& c p ) (Tw,out − Tw,in ),0 Tw,out ≤ Tmax s ⎪ Qd = ⎨ ⎪ (m& c ) (T − T ) Tw,out > Tmax p s max w,in ⎩ ratio = ε c m& c p min ( ) (m& c p )c 0.4 0.3 27 Storage Tank Energy Change dT Mc p s = Qd − QL,s − Qw,s − QTL dt • • • • • • 0.9 )c ⎞⎟⎤ −1 ⎥ )min − 1⎟⎠⎥⎦ F'R/FR FR' 1 10 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 (m& c p )c 100 28 Ac FRU c f-chart Method Development dTs dt = Qd dt − QL, s dt − Qw, s dt − QTL dt dt Δt Δt Δt Δt Δt 0 • For a long time period, the initial = –QS ∫ Mc p Mcp = mass * cp for storage tank liquid QL,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) ∫ ∫ ∫ ∫ 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 29 30 5 f-chart Method Development II ∫ ∫ 0 = Qd dt − Qs ⇒ Qs = Qd dt ⇒ Δt f = Δt f-chart Method Development III Qs 1 = Qd dt D D ∫ • The absorbed solar radiation, Ha, is the incident radiation, Hi, times the factor τα Δt – τ is the fraction transmitted through the glass cover(s) – α is the fraction absorbed • D = total energy demand • f = fraction of total supplied by solar • Use the definition of Qs, total energy supplied by solar from previous slide Q 1 1 f = s = ∫ Qd dt = ∫ Ac FR' [H a − U c (Tw,in − Ta )]dt D D Δt D Δt • Set Ha = Hiτα and multiply the last term, top and bottom, by Tref - Ta T −T ⎤ Q A F' ⎡ f = s = c R ∫ ⎢ H iτα − U c (Tref − Ta ) w,in a ⎥dt D D Δt ⎢⎣ Tref − Ta ⎥⎦ 31 32 f-chart Method Development IV What are X and Y? • Cannot evaluate integral because of the dependence of Ts on other variables X= Ac FR' D Y= ∫ H iταdt − Δt Ac FR' D ∫ H i ταdt Y= T −T Ac FR' U c (Tref − Ta ) w,in a dt ∫ D Δt Tref − Ta X= Δt ∫ ( ) • Integrate X and Y to get averages Ac FR' D X= ∫ H i ταdt = Δt ∫ ( 34 ∫ Δt ( Ac FR' A F'U U c Tref − Ta dt = c R c Tref − Ta D D f-chart Method Development VI Y= Ac FR' τα A F ' τα H i dt = c R H i ,total D D ) ∫ H i ταdt Δt 33 f-chart Method Development V Y= Ac FR' D • X is ratio of reference collector loss to total heating load • Y is ratio of absorbed solar energy to total heating load Ac FR' U c Tref − Ta dt D Δt ) Δt – Use engineering judgment to identify important variables and form empirical parameters based on f equation f = ∫ ( Ac FR' U c Tref − Ta dt D ) Δt • Rearrange equations to introduce factors FRUc and FR(τα)n available from standard collector test results 35 FR (τα )n Ac FR' τα H i ,total FR (τα )n D X= ( ) ⇒ FR Ac FR' U c Tref − Ta Δt ⇒ FR D Y F ' τα H i ,total = FR (τα )n R Ac FR (τα )n D ( X F ' Δt = FRU c R Tref − Ta Ac FR D ) • (τα)n = value of the τα product for normal radiation measured in tests • Hi,total is available from NREL data for Δt = 1 month 36 6 Collector Efficiency Equations 0.9 • From March 22-24 lecture [ 1-cover, black 1-cover, selective 2-cover, black 2-cover, selective intercept = FR (τα )n 0.8 0.7 Efficiency Qu = (m& c p )c (T f ,out − T f ,in ) Solar Collector Efficiency Tests 1 ] Qu = Ac FR H i (τα )n − FRU c (T f ,in − Ta ) Qu = FR (τα )n Ac H i ⎛T −T ⎞ − FRU c ⎜⎜ i a ⎟⎟ ⎝ Hi ⎠ 0.6 η= 0.5 0.4 0.3 ⎛T −T ⎞ Q η = u = FR (τα )n − FRU c ⎜⎜ f ,in a ⎟⎟ Ac H i ⎝ Hi ⎠ slope = − FRU c 0.2 0.1 0 • Plot of η versus (Tf,in – Ta)/Hi is straight line with slope = –FRUc and intercept FR(τα)n 0 0.05 0.1 0.15 37 38 Sample Rating Sheet Sample Rating Sheet II slope = –FRUc intercept = FR(τα)n 9 40 f-Chart 3.0 Back to f Equation 41 2.5 2.0 f = 0.9 f = 0.8 Y • 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 + 0.0018X2 + 0.0215Y3 • Klein, Beckman and Duffie developed method and software f = 0.7 f = 0.6 1.5 f = 0.5 f = 0.4 f = 0.3 f = 0.2 1.0 f = 0.1 0.5 0.0 0 2 4 6 8 10 12 14 16 42 X 7 Computing X (dimensionless) ( ) ⎡ ⎤ F ' Δt X = Ac ⎢ FRU c R Tref − Ta ⎥ FR D ⎣ ⎦ • Ac = collector area (m2) • 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; Ta from NREL data Computing Y (dimensionless) ⎡ F ' τα H i ,total ⎤ Y = Ac ⎢ FR (τα )n R ⎥ FR (τα )n D ⎦ ⎣ • Ac = collector area (m2) • FR(τα)n from intercept of collector test • F’R/FR computed or assumed = 0.97 • Ratio τα / (τα )n = 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 43 NREL Data 44 NREL Solar 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 • 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 – Hourly data on radiation components – Compute resultant for given collector geometry • http://www.nrel.gov/rredc/solar_data.html 45 NREL Collector Types ’61-’90 46 NREL Collector Types ’61-’90 II • Data available at different tilt levels for flat-plate collectors facing south – Horizontal (0o) – Latitude – 15o – Latitude – Latitude + 15o – Vertical (90o) 47 48 8 NREL Collector Types ’61-’90 III 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 49 50 LAX Solar Radiation 1961-1990 8 o Flat-plate titted at latitude = 34 7 Fchart Input data screen 5 2 Radiation (kWh/m /day) 6 4 30-year average Individual years 3 2 http://rredc.nrel.gov/solar/old_data/nsrdb/redbook/sum2/23174.txt 1 http://rredc.nrel.gov/solar/old_data/nsrdb/redbook/mon2/23174.txt 0 Jan Feb Mar Apr May Jun Jul Month Aug Sep Oct Nov Dec 51 52 Sample Fchart Output 53 54 9 Adjustments Another Adjustment • 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 ( Z = ε L m& c p • 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 CF = )min (UA)L 55 11.6 + 1.18Tw + 3.86Tm − 2.32Ta 100 − Ta 56 10