Designing and Modeling SolarEnergy Systems

CHAPTER

11The proper sizing of the components of a solar energy system is a complex problem, which includesboth predictable (collector and other components performance characteristics) and unpredictable(weather data) components. In this chapter, various design methods are presented as well as anoverview of the simulation techniques and programs suitable for solar heating and cooling systems. Abrief review of artificial intelligence (AI) methods and their applications in solar energy systems is alsopresented.

The design methods presented include the f-chart, utilizability V, the F; f-chart and the unuti-lizability method. The f-chart is based on the correlation of the results of a large number of simulationsin terms of easily calculated dimensionless variables. The utilizability method is used in cases wherethe collector operating temperature is known or can be estimated and for which critical radiation levelscan be established. The utilizability method is based on the analysis of hourly weather data to obtainthe fraction of the total months radiation that is above a critical level. The F; f-chart method is acombination of the utilizability and f-chart methods, applied in systems where the energy supplied to aload is above a minimum useful temperature and the temperature of this energy supply has no effect onthe performance of the system as long as it is greater than the minimum temperature.

For more detailed results, modeling and simulation are used. In recent years, because of the in-crease of computational speed of personal computers, annual simulations have begun replacing designmethods. Design methods, however, are much faster; therefore, they are still useful in early designstudies. The software programs described briefly in this book include TRNSYS, WATSUN, Polysun,and AI techniques applied in solar-energy systems modeling and prediction.

11.1 f-chart method and programThe f-chart method is used for estimating the annual thermal performance of active building heatingsystems using a working fluid, which is either liquid or air, and where the minimum temperature ofenergy delivery is near 20 C. The system configurations that can be evaluated by the f-chart methodare common in residential applications. With the f-chart method, the fraction of the total heating loadthat can be supplied by the solar energy system can be estimated. Let the purchased energy for a fuel-only system or the energy required to cover the load be L, the purchased auxiliary energy for a solarsystem be LAUX, and the solar energy delivered beQS. For a solar energy system, L LAUXQS. For amonth, i, the fractional reduction of purchased energy when a solar energy system is used, called thesolar fraction, f, is given by the ratio:

f Li LAUX;iLi

QS;iLi

(11.1)

Solar Energy Engineering. http://dx.doi.org/10.1016/B978-0-12-397270-5.00011-X

Copyright 2014 Elsevier Inc. All rights reserved.583

http://dx.doi.org/10.1016/B978-0-12-397270-5.00011-X

584 CHAPTER 11 Designing and Modeling Solar Energy Systems

The f-chart method was developed by Klein et al. (1976, 1977) and Beckman et al. (1977). In themethod, the primary design variable is the collector area, while the secondary variables are storagecapacity, collector type, load and collector heat-exchanger size, and fluid flow rate. The method is acorrelation of the results of many hundreds of thermal performance simulations of solar heating systemsperformed with TRNSYS, in which the simulation conditions were varied over specific ranges ofparameters of practical system designs shown in Table 11.1 (Klein et al., 1976, 1977). The resultingcorrelations give f, i.e., the fraction of the monthly load supplied by solar energy, as a function of twodimensionless parameters. The first is related to the ratio of collector losses to heating load, and thesecond to the ratio of absorbed solar radiation to heating load. The heating load includes both spaceheating and hot water loads. The f-chart method has been developed for three standard system con-figurations: liquid and air systems for space and hot water heating and systems for service hot water only.

Based on the fundamental equation presented in Chapter 6, Section 6.3.3, Klein et al. (1976)analyzed numerically the long-term thermal performance of solar heating systems of the basicconfiguration shown in Figure 6.14. When Eq. (6.60) is integrated over a time period, Dt, such that theinternal energy change in the storage tank is small compared to the other terms (usually one month),we get:

Mcp

s

ZDt

dTsdt

ZDt

QuZDt

Qls ZDt

Qlw ZDt

Qtl (11.2)

The sum of the last three terms of Eq. (11.2) represents the total heating load (including space load andhot water load) supplied by solar energy during the integration period. If this sum is denoted by QS,using the definition of the solar fraction, f, from Eq. (11.1), we get:

f QSL

1L

ZDt

Qu dt (11.3)

where L total heating load during the integration period (MJ).Using Eq. (5.56) for Qu and replacing Gt by Ht, the total (beam and diffuse) insolation over a day,

Eq. (11.3) can be written as:

f AcF0R

L

ZDt

Htsa ULTs Tadt (11.4)

Table 11.1 Range of Design Variables Used in Developing f-Charts for Liquid and Air Systems

Parameter Range

(sa)n 0.6e0.9

F 0RAc5e120 m2

UL 2.1e8.3 W/m2 C

b (collector slope) 30e90

(UA)h 83.3e666.6 W/C

Reprinted from Klein et al. (1976, 1977), with permission from Elsevier.

11.1 f-chart method and program 585

The last term of Eq. (11.4) can be multiplied and divided by the term (Tref Ta), where Tref is areference temperature chosen to be 100 C, so the following equation can be obtained:

f AcF0R

L

ZDt

Htsa UL

Tref Ta

Ts TaTref Ta

dt (11.5)The storage tank temperature, Ts, is a complicated function of Ht, L, and Ta; therefore, the integrationof Eq. (11.5) cannot be explicitly evaluated. This equation, however, suggests that an empirical cor-relation can be found, on a monthly basis, between the f factor and the two dimensionless groupsmentioned above as follows:

X AcF0RULL

ZDt

Tref Ta

dt AcF

0RULL

Tref Ta

Dt (11.6)

AcF0R

ZAcF

0R

Y

LDt

Htsadt L

saHtN (11.7)

where

Lmonthly heating load or demand (MJ);N number of days in a month;Ta monthly average ambient temperature (C);Ht monthly average daily total radiation on the tilted collector surface (MJ/m2); andsa monthly average value of (sa),monthly average value of absorbed over incident solar

radiation S=Ht.For the purpose of calculating the values of the dimensionless parameters X and Y, Eqs (11.6) and(11.7) are usually rearranged to read:

X FRULF0R

FR

Tref Ta

Dt

AcL

(11.8)

F0R sa Ac

Y FRsanFR sanHtN

L(11.9)

The reason for the rearrangement is that the factors FRUL and FR(sa)n are readily available fromstandard collector tests (see Chapter 4, Section 4.1). The ratio F0R=FR corrects the collector perfor-mance because a heat exchanger is used between the collector and the storage tank, which causes thecollector side of the system to operate at higher temperature than a similar system without a heatexchanger and is given by Eq. (5.57) in Chapter 5. For a given collector orientation, the value of thefactor sa=san varies slightly from month to month. For collectors tilted and facing the equator witha slope equal to latitude plus 15, Klein (1976) found that the factor is equal to 0.96 for a one-covercollector and 0.94 for a two-cover collector for the whole heating season (winter months). Using thepreceding definition of sa, we get:

sasan

SHtsan

(11.10)

586 CHAPTER 11 Designing and Modeling Solar Energy Systems

If the isotropic model is used for S and substituted in Eq. (11.10), then:

sasan

HBRBHt

saBsan

HDHt

saDsan

1 cosb

2

HrG

Ht

saGsan

1 cosb

2

(11.11)

In Eq. (11.11), the sa=san ratios can be obtained from Figure 3.27 for the beam component at theeffective angle of incidence, qB, which can be obtained from Figure A3.8 in Appendix 3, and for thediffuse and ground-reflected components at the effective incidence angles at b from Eqs (3.4a) and(3.4b).

The dimensionless parameters, X and Y, have some physical significance. The parameter X rep-resents the ratio of the reference collector total energy loss to total heating load or demand (L) duringthe period Dt, whereas the parameter Y represents the ratio of the total absorbed solar energy to thetotal heating load or demand (L) during the same period.

As was indicated, f-chart is used to estimate the monthly solar fraction, fi, and the energycontribution for the month is the product of fi and monthly load (heating and hot water), Li. To find thefraction of the annual load supplied by the solar energy system, F, the sum of the monthly energycontributions is divided by the annual load, given by:

F P

fiLiPLi

(11.12)

The method can be used to simulate standard solar water and air system configurations and solarenergy systems used only for hot water production. These are examined separately in the followingsections.

EXAMPLE 11.1A standard solar-heating system is installed in an area where the average daily total radiation on thetilted collector surface is 12.5 MJ/m2, average ambient temperature is 10.1 C, and it uses a 35 m2

aperture area collector, which has FR(sa)n 0.78 and FRUL 5.56 W/m2 C, both determinedfrom the standard collector tests. If the space heating and hot water load is 35.2 GJ, the flow rate inthe collector is the same as the flow rate used in testing the collector, F0R=FR 0:98, andsa=san 0:96 for all months, estimate the parameters X and Y.Solution

Using Eqs (11.8) and (11.9) and noting that DT is the number of seconds in a month, equal to31 days 24 h 3600 s/h, we get:

X FRULF0R

FR

Tref Ta

Dt

AcL

5:56 0:98100 10:1 31 24 3600 3535:2 109 1:30

11.1 f-chart method and program 587

Y FRsanF0RFR

sasan

HtN

AcL

0:78 0:98 0:96 12:5 106 31 3535:2 109 0:28

It should be noted that F0RUL and F0R(sa)n can be given instead of FRUL, FR(sa)n and F0R/FR,

given in this problem.

11.1.1 Performance and design of liquid-based solar heating systemsKnowledge of the system thermal performance is required in order to be able to design and optimize asolar heating system. The f-chart for liquid-based systems is developed for a standard solar liquid-based solar energy system, shown in Figure 11.1. This is the same as the system shown inFigure 6.14, drawn without the controls, for clarity. The typical liquid-based system shown inFigure 11.1 uses an antifreeze solution in the collector loop and water as the storage medium. Awater-to-water load-heat exchanger is used to transfer heat from the storage tank to the domestic hot water(DHW) system. Although in Figure 6.14 a one-tank DHW system is shown, a two-tank system couldbe employed, in which the first tank is used for preheating.

The fraction f of the monthly total load supplied by a standard liquid-based solar energy system isgiven as a function of the two dimensionless parameters, X and Y, and can be obtained from the f-chartin Figure 11.2 or the following equation (Klein et al., 1976):

f 1:029Y 0:065X 0:245Y2 0:0018X2 0:0215Y3 (11.13)Application of Eq. (11.13) or Figure 11.2 allows the simple estimation of the solar fraction on amonthly basis as a function of the system design and local weather conditions. The annual value can beobtained by summing up the monthly values using Eq. (11.12). As will be shown in the next chapter, todetermine the economic optimum collector area, the annual load fraction corresponding to differentcollector areas is required. Therefore, the present method can easily be used for these estimations.

Relief valves

Collectorarray

Collectorpump

Storagepump

Collectorheatexchanger

Water supply

Auxiliary

Load heatexchanger House

Warm airducts

Coldairducts

Auxiliary

Fan

Mainstorage

tank

Hotwatertank

Servicehot water

FIGURE 11.1

Schematic diagram of a standard liquid-based solar heating system.

X Reference collector loss/heating load

YA

bsor

bed

sola

r en

ergy

/hea

ting

load

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18

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

Liquid systems

Mw,s 75 l/m2

Z 2

Range: 0 Y 3; 0 X 18

FIGURE 11.2

The f-chart for liquid-based solar heating systems.

588

CHAPTER11

Desig

ningandModelingSolarEnergySyste

ms

11.1 f-chart method and program 589

EXAMPLE 11.2If the solar heating system given in Example 11.1 is liquid-based, estimate the annual solar fractionif the collector is located in an area having the monthly average weather conditions and monthlyheating and hot water loads shown in Table 11.2.

Table 11.2 Average Monthly Weather Conditions and Heating and Hot Water Loads for Example 11.2

Month Ht (MJ/m2) Ta (

C) L (GJ)

January 12.5 10.1 35.2

February 15.6 13.5 31.1

March 17.8 15.8 20.7

April 20.2 19.0 13.2

May 21.5 21.5 5.6

June 22.5 29.8 4.1

July 23.1 32.1 2.9

August 22.4 30.5 3.5

September 21.1 22.5 5.1

October 18.2 19.2 12.7

November 15.2 16.2 23.6

December 13.1 11.1 33.1

Solution

The values of the dimensionless parameters X and Y found from Example 11.1 are equal to 1.30 and0.28, respectively. From the weather and load conditions shown in Table 11.2, these correspond tothe month of January. From Figure 11.2 or Eq. (11.13), f 0.188. The total load in January is35.2 GJ. Therefore, the solar contribution in January is fL 0.188 35.2 6.62 GJ. The samecalculations are repeated from month to month, as shown in Table 11.3.

Table 11.3 Monthly Calculations for Example 11.2

Month Ht (MJ/m2) Ta (

C) L (GJ) X Y f fL

January 12.5 10.1 35.2 1.30 0.28 0.188 6.62

February 15.6 13.5 31.1 1.28 0.36 0.259 8.05

March 17.8 15.8 20.7 2.08 0.68 0.466 9.65

April 20.2 19.0 13.2 3.03 1.18 0.728 9.61

May 21.5 21.5 5.6 7.16 3.06 1 5.60

June 22.5 29.8 4.1 8.46 4.23 1 4.10

July 23.1 32.1 2.9 11.96 6.34 1 2.90

August 22.4 30.5 3.5 10.14 5.10 1 3.50

September 21.1 22.5 5.1 7.51 3.19 1 5.10

October 18.2 19.2 12.7 3.25 1.14 0.694 8.81

November 15.2 16.2 23.6 1.76 0.50 0.347 8.19

December 13.1 11.1 33.1 1.37 0.32 0.219 7.25

Total load 190.8 Total contribution 79.38

590 CHAPTER 11 Designing and Modeling Solar Energy Systems

It should be noted that the values of f marked in bold are outside the range of the f-chart correlationand a fraction of 100% is used, as during these months, the solar energy system covers the load fully.From Eq. (11.12), the annual fraction of load covered by the solar energy system is:

F P

fiLiPLi

79:38190:8

0:416 or 41:6%

It should be noted that the f-chart was developed using fixed nominal values of storage capacity perunit of collector area, collector liquid flow rate per unit of collector area, and load heat-exchanger sizerelative to a space-heating load. Therefore, it is important to apply various corrections for the particularsystem configuration used.

Storage capacity correctionIt can be proven that the annual performance of liquid-based solar energy systems is insensitive to thestorage capacity, as long as this is more than 50 l of water per square meter of collector area. For the f-chart of Figure 11.2, a standard storage capacity of 75 l of stored water per square meter of collectorarea was considered. Other storage capacities can be used by modifying the factor X by a storage sizecorrection factor Xc/X, given by (Beckman et al., 1977):

XcX

Mw;aMw;s

0:25(11.14)

where

Mw,a actual storage capacity per square meter of collector area (l/m2).Mw,s standard storage capacity per square meter of collector area ( 75 l/m2).

Equation (11.14) is applied in the range of 0.5 (Mw,a/Mw,s) 4.0 or 37.5Mw,a 300 l/m2. Thestorage correction factor can also be determined from Figure 11.3 directly, obtained by plottingEq. (11.14) for this range.

EXAMPLE 11.3Estimate the solar fraction for the month of March of Example 11.2 if the storage tank capacity is130 l/m2.

Solution

First the storage correction factor needs to be estimated. By using Eq. (11.14),

XcX

Mw;aMw;s

0:25

130

75

0:25 0:87

For March, the corrected value of X is then Xc 0.87 2.08 1.81. The value of Y remains asestimated before, i.e., Y 0.68. From the f-chart, f 0.481 compared to 0.466 before the correction,an increase of about 2%.

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.7 0.8 0.9 1.0 1.1 1.2

Storage collector factor,Xc/X

Mw

,a/M

w,s

37.5

75

112.5

150

187.5

225

262.5

300

Mw

,a

Liquid systems

Mw,s = 75 l/m2

FIGURE 11.3

Storage correction factor for liquid-based systems.

11.1 f-chart method and program 591

Collector flow rate correction

The f-chart of Figure 11.2 has been generated using a collector antifreeze solution flow rate equal to0.015 l/s m2. A lower flow rate can reduce the energy collection rate significantly, especially if the lowflow rate leads to fluid boiling and relief of pressure through the relief valve. Although the product ofthe mass flow rate and the specific heat of the fluid flowing through the collector strongly affects theperformance of the solar energy system, the value used is seldom lower than the value used for thef-chart development. Additionally, since an increase in the collector flow rate beyond the nominalvalue has a small effect on the system performance, Figure 11.2 is applicable for all practical collectorflow rates.

Load heat exchanger size correctionThe size of the load heat exchanger strongly affects the performance of the solar energy system.This is because the rate of heat transfer across the load heat exchanger directly influences thetemperature of the storage tank, which consequently affects the collector inlet temperature.As the heat exchanger used to heat the building air is reduced in size, the storage tank tem-perature must increase to supply the same amount of heat energy, resulting in highercollector inlet temperatures and therefore reduced collector performance. To account for the

592 CHAPTER 11 Designing and Modeling Solar Energy Systems

load heat exchanger size, a new dimensionless parameter is specified, Z, given by (Beckmanet al., 1977):

Z L_mcp

min

UAL(11.15)

where

L effectiveness of the load heat exchanger. _mcpmin minimum mass flow ratespecific heat product of heat exchanger (W/K).(UA)L building loss coefficient and area product used in degree-day space-heating load

model (W/K).

In Eq. (11.15), the minimum capacitance rate is that of the air side of the heat exchanger. Systemperformance is asymptotically dependent on the value of Z; and for Z> 10, the performance isessentially the same as for an infinitely large value of Z. Actually, the reduction in performance due to asmall-size load heat exchanger is significant for values of Z lower than 1. Practical values of Z arebetween 1 and 3, whereas a value of Z 2 was used for the development of the f-chart of Figure 11.2.The performance of systems having other values of Z can be estimated by multiplying the dimen-sionless parameter Y by the following correction factor:

YcY

0:39 0:65exp0:139

Z

(11.16)

Equation (11.16) is applied in the range of 0.5 Z 50. The load heat-exchanger size correctionfactor can also be determined from Figure 11.4 directly, obtained by plotting Eq. (11.16) for this range.

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

0 5 10 15 20 25 30 35 40 45 50

Dimensionless parameter Z

Load

hea

t exc

hang

er s

ize

corr

ectio

nfa

ctor

, Yc/Y

FIGURE 11.4

Load heat exchanger size correction factor.

11.1 f-chart method and program 593

EXAMPLE 11.4If the liquid flow rate in Example 11.2 is 0.525 l/s, the air flow rate is 470 l/s, the load heat-exchanger effectiveness is 0.65, and the building overall loss coefficientearea product, (UA)L, is422 W/K, find the effect on the solar fraction for the month of November.

Solution

The minimum value of capacitance needs to be found first. Therefore, if we assume that theoperating temperature is 350 K (77 C) and the properties of air and water at that temperature are asfrom Tables A5.1 and A5.2 in Appendix 5, respectively,

Cair 470 0:998 1009=1000 473:3 W=KCwater 0:525 974 4190=1000 2142:6 W=K

Therefore, the minimum capacitance is for the air side of the load heat exchanger.From Eq. (11.15),

Z L_mcp

min

UAL 0:65 473:3

422 0:729

The correction factor is given by Eq. (11.16):

YcY

0:39 0:65exp0:139

Z

0:39 0:65exp

0:1390:729

0:93

From Example 11.2, the value of the dimensionless parameter Y is 0.50. Therefore,Yc 0.50 0.93 0.47. The value of the dimensionless parameter X for this month is 1.76, whichfrom Eq. (11.13) gives a solar fraction f 0.323, a drop of about 2% from the previous value.

Although in the examples in this section only one parameter was different from the standard one, ifboth the storage size and the size of the heat exchanger are different than the standard ones, both Xc andYc need to be calculated for the determination of the solar fraction. Additionally, most of the requiredparameters in this section are given as input data. In the following example, most of the requiredparameters are estimated from the information given in earlier chapters.

EXAMPLE 11.5A liquid-based solar space and domestic water-heating system is located in Nicosia, Cyprus (35Nlatitude). Estimate the monthly and annual solar fraction of the system, which has a total collectorarea of 20 m2 and the following information is given:

1. The collectors face south, installed with 45 inclination. The performance parameters of thecollectors are FR(sa)n 0.82 and FRUL 5.65 W/m2 C, both determined from the standardcollector tests.

2. The flow rate of both the water and antifreeze solution through the collector heat exchanger is0.02 l/s m2 and the factor F0R=FR 0:98:

594 CHAPTER 11 Designing and Modeling Solar Energy Systems

3. The storage tank capacity is equal to 120 l/m2.4. The sa=san 0:96 for October through March and 0.93 for April through September.5. The building UA value is equal to 450 W/K. The water-to-air load heat exchanger has an

effectiveness of 0.75 and air flow rate is 520 l/s.6. The ground reflectivity is 0.2.7. The climatic data and the heating degree days for Nicosia, Cyprus, are taken from Appendix

7 and reproduced in Table 11.4 with the hot-water load.

Table 11.4 Climate Data and Heating Degree Days for Example 11.5

Month H (MJ/m2) Ta (C)

ClearnessIndex KT

Heating CDegree Days

Hot Water Load,Dw (GJ)

January 8.96 12.1 0.49 175 3.5

February 12.38 11.9 0.53 171 3.1

March 17.39 13.8 0.58 131 2.8

April 21.53 17.5 0.59 42 2.5

May 26.06 21.5 0.65 3 2.1

June 29.20 29.8 0.70 0 1.9

July 28.55 29.2 0.70 0 1.8

August 25.49 29.4 0.68 0 1.9

September 21.17 26.8 0.66 0 2.0

October 15.34 22.7 0.60 1 2.7

November 10.33 17.7 0.53 36 3.0

December 7.92 13.7 0.47 128 3.3

Solution

The loads need to be estimated first. For the month of January, from Eq. (6.24):

Dh UA

DDh 450W=K 24h=day 3600J=Wh 175C days 6:80 GJThe monthly heating load (including hot-water load) 6.80 3.5 10.30 GJ. The results for allthe months are shown in Table 11.5.

Next, we need to estimate the monthly average daily total radiation on the tilted collectorsurface from the daily total horizontal solar radiation, H: For this estimation, the average day ofeach month is used, shown in Table 2.1, together with the declination for each day. For each ofthose days, the sunset hour angle, hss, is required, given by Eq. (2.15), and the sunset hour angle onthe tilted surface, h0ss, given by Eq. (2.109). The calculations for the month of January are asfollows.From Eq. (2.15),

hss cos1tanLtand cos1tan35tan20:92 74:5

Table 11.5 Heating Load for All the Months in Example 11.5

Month Heating C Degree Days Dh (GJ) Dw (GJ) L (GJ)

January 175 6.80 3.5 10.30

February 171 6.65 3.1 9.75

March 131 5.09 2.8 7.89

April 42 1.63 2.5 4.13

May 3 0.12 2.1 2.22

June 0 0 1.9 1.90

July 0 0 1.8 1.80

August 0 0 1.9 1.90

September 0 0 2.0 2.00

October 1 0.04 2.7 2.74

November 36 1.40 3.0 4.40

December 128 4.98 3.3 8.28

Total 57.31

11.1 f-chart method and program 595

From Eq. (2.109),

h0ss minhss; cos

1tanL btand min74:5; cos1tan35 45tan20:92 minf74:5; 93:9g 74:5

From Eq. (2.105b),

HD

H 0:775 0:00653hss 90 0:505 0:00455hss 90cos

115KT 103

0:775 0:0065374:5 90 0:505 0:0045574:5 90cos115 0:49 103 0:38

From Eq. (2.108),

RB cos

L bcosdsinh0ss p=180h0sssinL bsindcosLcosdsinhss p=180hsssinLsind

cos35 45cos20:92sin74:5 p=18074:5sin35 45sin20:92cos35cos20:92sin74:5 p=18074:5sin35sin20:92 2:05

From Eq. (2.107),

R HtH

1 HD

H

RB HD

H

1 cosb

2

rG

1 cosb

2

1 0:382:05 0:381 cos452

0:2

1 cos45

2

1:62

Table 11.6 Monthly Average Calculations for Example 11.5

Month N d () hss () h0ss () HD=H RB R Ht (MJ/m2)

January 17 20.92 74.5 74.5 0.38 2.05 1.62 14.52February 47 12.95 80.7 80.7 0.37 1.65 1.38 17.08March 75 2.42 88.3 88.3 0.36 1.27 1.15 20.00April 105 9.41 96.7 88.3 0.38 0.97 0.96 20.67

May 135 18.79 103.8 86.6 0.36 0.78 0.84 21.89

June 162 23.09 107.4 85.7 0.35 0.70 0.78 22.78

July 198 21.18 105.7 86.1 0.34 0.74 0.81 23.13

August 228 13.45 99.6 87.6 0.34 0.88 0.90 22.94

September 258 2.22 91.6 89.6 0.33 1.14 1.07 22.65

October 288 9.6 83.2 83.2 0.34 1.52 1.32 20.25November 318 18.91 76.1 76.1 0.36 1.94 1.58 16.32December 344 23.05 72.7 72.7 0.38 2.19 1.71 13.54

596 CHAPTER 11 Designing and Modeling Solar Energy Systems

And finally,Ht RH 1:62 8:96 14:52 MJ=m2: The calculations for all months are shown inTable 11.6.We can now move along in the f-chart estimation. The dimensionless parameters X and Y areestimated from Eqs (11.8) and (11.9):

X FRULF0R

FR

Tref Ta

Dt

AcL

5:65 0:98100 12:1 31 24 3600 2010:30 109 2:53

Y FRsanF0RFR

sasan

HtN

AcL

0:82 0:98 0:96 14:52 106 31 2010:30 109 0:67

The storage tank correction is obtained from Eq. (11.14):

XcX

Mw;aMw;s

0:25

120

75

0:25 0:89

Then, the minimum capacitance value needs to be found (at an assumed temperature of 77 C):

Cair 520 0:998 1009=1000 523:6 W=KCwater 0:02 20 974 4190=1000 1632 W=K

Therefore, the minimum capacitance is for the air side of the load heat exchanger.

11.1 f-chart method and program 597

From Eq. (11.15),

Z L_mcp

min

UAL 0:75 523:6

450 0:87

The correction factor is given by Eq. (11.16):

YcY

0:39 0:65exp0:139

Z

0:39 0:65exp

0:139

0:87

0:94

Therefore,

Xc 2:53 0:89 2:25and

Yc 0:67 0:94 0:63When these values are used in Eq. (11.13), they give f 0.419. The complete calculations for allmonths of the year are shown in Table 11.7.

Table 11.7 Complete Monthly Calculations for the f-Chart for Example 11.5

Month X Y Xc Yc f fL

January 2.53 0.67 2.25 0.63 0.419 4.32

February 2.42 0.76 2.15 0.71 0.483 4.71

March 3.24 1.21 2.88 1.14 0.714 5.63

April 5.73 2.24 5.10 2.11 1 4.13

May 10.49 4.57 9.34 4.30 1 2.22

June 11.21 5.38 9.98 5.06 1 1.90

July 11.67 5.95 10.39 5.59 1 1.80

August 11.02 5.59 9.81 5.25 1 1.90

September 10.51 5.08 9.35 4.78 1 2.00

October 8.37 3.53 7.45 3.32 1 2.74

November 5.37 1.72 4.78 1.62 0.846 3.72

December 3.09 0.78 2.75 0.73 0.464 3.84

Total 38.91

From Eq. (11.12), the annual fraction of load covered by the solar energy system is:

F P

fiLiPLi

38:9157:31

0:679 or 67:9%

11.1.2 Performance and design of air-based solar heating systemsKlein et al. (1977) developed for air-based systems a design procedure similar to that for liquid-basedsystems. The f-chart for air-based systems is developed for the standard solar air-based solar energy

Auxiliary

BuildingPebblebed

storagePre heat

tankDHWAuxiliary

Hot watersupply

Cold watersupply

Fan Warmairsupply

Coldairreturn

Collectorarray

Air-to-waterheat exchanger

Bypassline

FIGURE 11.5

Schematic diagram of the standard air-based solar heating system.

598 CHAPTER 11 Designing and Modeling Solar Energy Systems

t

,

,

system, shown in Figure 11.5. This is the same as the system shown in Figure 6.12, drawn without thecontrols, for clarity. As can be seen, the standard configuration of air-based solar heating system uses apebble-bed storage unit. The energy required for the DHW is provided through the air-to-water heaexchanger, as shown. During summertime, when heating is not required, it is preferable not to storeheat in the pebble bed, so a bypass is usually used, as shown in Figure 11.5 (not shown in Figure 6.12)which allows the use of the collectors for water heating only.

The fraction f of the monthly total load supplied by a standard solar air-based solar energy systemshown in Figure 11.5, is also given as a function of the two parameters, X and Y, and can be obtainedfrom the f-chart given in Figure 11.6 or from the following equation (Klein et al., 1977):

f 1:040Y 0:065X 0:159Y2 0:00187X2 0:0095Y3 (11.17)

EXAMPLE 11.6A solar air-heating system of the standard configuration is installed in the same area as the one inExample 11.2 and the building has the same load. The air collectors have the same area as inExample 11.2 and are double glazed, with FRUL 2.92 W/m2 C, FR(sa)n 0.52, andsa=san 0:93: Estimate the annual solar fraction.Solution

A general condition of air systems is that no correction factor is required for the collector heatexchanger and ducts are well insulated; therefore, heat losses are assumed to be small, soF0R=FR 1. For the month of January and from Eqs (11.8) and (11.9).

X FRULF0R

FR

Tref Ta

Dt

AcL

2:92100 10:1 31 24 3600 3535:2 109 0:70

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18

X Reference collector loss/heating load

Y

Abs

orbe

d so

lar

ener

gy/h

eatin

g lo

ad

f 0.6

f 0.9

f 0.8

f 0.4

f 0.3

f 0.2

f 0.1

f 0.5

f 0.7

Air systemsRange: 0 Y 3; 0 X 18

Mb,s 0.25 m3/m2

m s 10 l/s m2

FIGURE 11.6

The f-chart for air-based solar heating systems.

11.1

f-chartmethodandprogram

599

600 CHAPTER 11 Designing and Modeling Solar Energy Systems

Y FRsanF0RFR

sasan

HtN

AcL

0:52 0:93 12:5 106 31 3535:2 109 0:19

From Eq. (11.17) or Figure 11.6, f 0.147. The solar contribution is fL 0.147 35.2 5.17 GJ.The same calculations are repeated for the other months and tabulated in Table 11.8.

Table 11.8 Solar Contribution and f-Values for all Months for Example 11.6

Month Ht (MJ/m2) Ta (

C) L (GJ) X Y f fL

January 12.5 10.1 35.2 0.70 0.19 0.147 5.17

February 15.6 13.5 31.1 0.69 0.24 0.197 6.13

March 17.8 15.8 20.7 1.11 0.45 0.367 7.60

April 20.2 19.0 13.2 1.63 0.78 0.618 8.16

May 21.5 21.5 5.6 3.84 2.01 1 5.60

June 22.5 29.8 4.1 4.54 2.79 1 4.10

July 23.1 32.1 2.9 6.41 4.18 1 2.90

August 22.4 30.5 3.5 5.44 3.36 1 3.50

September 21.1 22.5 5.1 4.03 2.10 1 5.10

October 18.2 19.2 12.7 1.74 0.75 0.587 7.45

November 15.2 16.2 23.6 0.94 0.33 0.267 6.30

December 13.1 11.1 33.1 0.74 0.21 0.164 5.43

Total load 190.8 Total contribution 67.44

It should be noted here that, again, the values of f marked in bold are outside the range of thef-chart correlation and a fraction of 100% is used because during these months, the solar energysystem covers the load fully. From Eq. (11.12), the annual fraction of load covered by the solarenergy system is:

F P

fiLiPLi

67:44190:8

0:353 or 35:3%

Therefore, compared to the results from Example 11.2, it can be concluded that, due to the lower

collector optical characteristics, F is lower.

Air systems require two correction factors: one for the pebble-bed storage size and one for the airflow rate, which affects stratification in the pebble bed. There are no load heat exchangers in airsystems, and care must be taken to use the collector performance parameters FRUL and FR(sa)n,determined at the same air flow rate as used in the installation; otherwise, the correction outlined inChapter 4, Section 4.1.1, needs to be used.

Pebble-bed storage size correctionFor the development of the f-chart of Figure 11.6, a standard storage capacity of 0.25 cubic metersof pebbles per square meter of collector area was considered, which corresponds to 350 kJ/m2 Cfor typical void fractions and rock properties. Although the performance of air-based systems is not

0.5

1

1.5

2

2.5

3

3.5

4

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3Storage size correction factor, Xc/X

Mb,

a/Mb,

s

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

Mb,

a

Air systems

Mb,s = 0.25 m3/m2

FIGURE 11.7

Storage size correction factor for air-based systems.

11.1 f-chart method and program 601

as sensitive to the storage capacity as in liquid-based systems, other storage capacities can beused by modifying the factor X by a storage size correction factor, Xc/X, as given by (Klein et al.,1977):

XcX

Mb;aMb;s

0:30(11.18)

where

Mb,a actual pebble storage capacity per square meter of collector area (m3/m2);Mb,s standard storage capacity per square meter of collector area 0.25 m3/m2.

Equation (11.18) is applied in the range of 0.5 (Mb,a/Mb,s) 4.0 or 0.125Mb,a 1.0 m3/m2.The storage correction factor can also be determined from Figure 11.7 directly, obtained by plottingEq. (11.18) for this range.

Air flow rate correctionAir-based heating systems must also be corrected for the flow rate. An increased air flow rate tendsto increase system performance by improving FR, but it tends to decrease performance byreducing the pebble bed thermal stratification. The standard collector flow rate is 10 l/s of air persquare meter of collector area. The performance of systems having other collector flow rates can beestimated by using appropriate values of FR and Y, then modifying the value of X by a collector air

0.5

0.75

1

1.25

1.5

1.75

2

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 Collector air flow rate correction factor, Xc/X

5

7.5

10

12.5

15

17.5

20

Air systems

sm = 10 l/s m2

am / sm ma

FIGURE 11.8

Correction factor for air flow rate to account for stratification in pebble bed.

602 CHAPTER 11 Designing and Modeling Solar Energy Systems

flow rate correction factor, Xc/X, to account for the degree of stratification in the pebble bed(Klein et al., 1977):

XcX

_ma_ms

0:28(11.19)

where

_ma actual collector flow rate per square meter of collector area (l/s m2);_ms standard collector flow rate per square meter of collector area 10 l/s m2.

Equation (11.19) is applied in the range of 0:5 _ma= _ms 2:0 or 5 _ms 20 l/s m2. The air flow ratecorrection factor can also be determined from Figure 11.8 directly, obtained by plotting Eq. (11.19) forthis range.

EXAMPLE 11.7If the air system of Example 11.6 uses a flow rate equal to 17 l/s m2, estimate the solar fraction forthe month of January. At the new flow rate, the performance parameters of the collector areFRUL 3.03 W/m2 C, FR(sa) 0.54.

11.1 f-chart method and program 603

Solution

From Eq. (11.19),

XcX

_ma_ms

0:28

17

10

0:28 1:16

The increased air flow rate also affects the FR and the performance parameters, as shown in theproblem definition. Therefore, the value of X to use is the value from Example 11.6 corrected for thenew air flow rate through the collector and the pebble bed. Hence,

Xc XFRUL

new

FRULtest

XcX

0:703:03

2:92

1:16 0:84

The dimensionless parameter Y is affected only by the FR. So,

Yc YFR

sa

new

FRsa

test

0:19

0:54

0:52

0:20

Finally, from the f-chart of Figure 11.6 or Eq. (11.17), f 0.148 or 14.8%. Compared to the previousresult of 14.7%, there is no significant reduction for the increased flow rate, but there will be anincrease in fan power.

If, in a solar energy system, both air flow rate and storage size are different from the standard ones,two corrections must be done on dimensionless parameter X. In this case, the final X value to use is theuncorrected value multiplied by the two correction factors.

EXAMPLE 11.8If the air system of Example 11.6 uses a pebble storage tank equal to 0.35 m3/m2 and the flow rate isequal to 17 l/s m2, estimate the solar fraction for the month of January. At the new flow rate, theperformance parameters of the collector are as shown in Example 11.7.

Solution

The two correction factors need to be estimated first. The correction factors for X and Y for theincreased flow rate are as shown in Example 11.7. For the increased pebble bed storage, fromEq. (11.18),

XcX

Mb;aMb;s

0:30

0:35

0:25

0:30 0:90

The correction for the air flow rate is given in Example 11.7 and is equal to 1.16. By consideringalso the correction for the flow rate on FR and the original value of X,

Xc XFRUL

new

FRULtest

XcX

flow

XcX

storage

0:703:03

2:92

1:16 0:90 0:76

The dimensionless parameter Y is affected only by the FR. So, the value of Example 11.7 is used

here ( 0.20). Therefore, from the f-chart of Figure 11.6 or Eq. (11.17), f 0.153 or 15.3%.

604 CHAPTER 11 Designing and Modeling Solar Energy Systems

11.1.3 Performance and design of solar service water systemsThe f-chart of Figure 11.2 or Eq. (11.13) can also be used to estimate the performance of solar servicewater-heating systems with a configuration like that shown in Figure 11.9. Although a liquid-basedsystem is shown in Figure 11.9, air or water collectors can be used in the system with the appro-priate heat exchanger to transfer heat to the preheat storage tank. Hot water from the preheat storagetank is then fed to a water heater where its temperature can be increased, if required. A tempering valvemay also be used to maintain the supply temperature below a maximum temperature, but this mixingcan also be done at the point of use by the user.

The performance of solar water heating systems is affected by the public mains water temperature,Tm, and the minimum acceptable hot water temperature, Tw; both affect the average system operatingtemperature and thus the collector energy losses. Therefore, the parameter X, which accounts for thecollector energy losses, needs to be corrected. The additional correction factor for the parameter X isgiven by (Beckman et al., 1977):

XcX

11:6 1:18Tw 3:86Tm 2:32Ta100 Ta

(11.20)

where

Tmmains water temperature (C);Twminimum acceptable hot water temperature (C); andTa monthly average ambient temperature (C).

The correction factor Xc/X is based on the assumption that the solar preheat tank is well insulated. Tanklosses from the auxiliary tank are not included in the f-chart correlations. Therefore, for systemssupplying hot water only, the load should also include the losses from the auxiliary tank. Tank lossescan be estimated from the heat loss coefficient and tank area (UA) based on the assumption that theentire tank is at the minimum acceptable hot water temperature, Tw.

Relief valves

Collectorarray

Collectorpump

Storagepump

Collector-storageheat exchanger

Water supply

Auxiliary

Pre-heatstorage

tank

Waterheater

Hot watersupply

Tempering valve

FIGURE 11.9

Schematic diagram of the standard of water-heating system configuration.

11.1 f-chart method and program 605

The solar water heater performance is based on storage capacity of 75 l/m2 of collector aperturearea and a typical hot water load profile, with little effect by other distributions on the system per-formance. For different storage capacities, the correction given by Eq. (11.14) applies.

EXAMPLE 11.9A solar water heating system is installed in an area where, for the 31-day month under investigation,the average daily total radiation on the tilted collector surface is 19.3 MJ/m2, average ambienttemperature is 18.1 C, and it uses a 5 m2 aperture area collector that has FR(sa)n 0.79 andFRUL 6.56 W/m2 C, both determined from the standard collector tests. The water heating load is200 l/day, the public mains water temperature, Tm, is 12.5

C, and the minimum acceptable hotwater temperature, Tw, is 60

C. The storage capacity of the preheat tank is 75 l/m2 and auxiliarytank has a capacity of 150 l, a loss coefficient of 0.59 W/m2 C, diameter 0.4 m, and height of 1.1 m;it is located indoors, where the environment temperature is 20 C. The flow rate in thecollector is the same as the flow rate used in testing the collector, the F0R=FR 0:98, and thesa=san 0:94: Estimate the solar fraction.Solution

The monthly water-heating load is the energy required to heat the water from Tm to Tw plus theauxiliary tank losses. For the month investigated, the water heating load is:

200 31 419060 12:5 1:234 GJThe auxiliary tank loss rate is given by UA(Tw Ta). The area of the tank is:

pd2=2 pdl p0:42=2 p 0:4 1:1 1:63 m2

Thus, auxiliary tank loss 0.59 1.63(60 20) 38.5 W. The energy required to cover this lossin a month is:

38:5 31 24 3600 0:103 GJ:Therefore,

Total heating load 1:234 0:103 1:337 GJUsing Eqs (11.8) and (11.9), we get:

X FRULF0R

FR

Tref Ta

Dt

AcL

6:56 0:98100 18:1 31 24 3600 51:337 109 5:27

Y FRsanF0RFR

sasan

HtN

AcL

0:79 0:98 0:94 19:3 106 31 51:337 109 1:63

606 CHAPTER 11 Designing and Modeling Solar Energy Systems

From Eq. (11.20), the correction for X is:

XcX

11:6 1:18Tw 3:86Tm 2:32Ta100 Ta

11:6 1:18 60 3:86 12:5 2:32 18:1100 18:1 1:08

Therefore, the corrected value of X is:

Xc 5:27 1:08 5:69

From Figure 11.2 or Eq. (11.13), for Xc and Y, we get f 0.808 or 80.8%.

11.1.4 Thermosiphon solar water-heating systemsIt is a fact that globally, the majority of solar water heaters installed are of the thermosiphontype. Therefore, it is important to develop a simple method to predict their performancesimilar to the forced circulation or active systems. This method may be a modification of theoriginal f-chart method presented in the previous sections to account for the natural circulationoccurring in thermosiphon systems, which exhibit also a strong stratification of the hot water in thestorage tank.

In fact, the original f-chart cannot be used as it is to predict the thermal performance of thermo-siphon solar water heaters with good accuracy, for two reasons (Fanney and Klein, 1983; Malkin et al.,1987):

1. The f-chart design tool was developed for pumped systems and the assumption that the fluid flowrate through the collector loop is known and it is fixed. The varying flow rates encountered inthermosiphon solar water-heating systems, driven by the strength of solar radiation, lead todifferent FR and FRUL values from those encountered in active systems. These parameters aredetermined experimentally with the procedures outlined in Chapter 4 at fixed flow rate.

2. Additionally, a major assumption made for the development of the f-chart design method is that thehot-water storage tank is fully mixed. Therefore, the thermal performance of thermosiphon solar-water heaters would be greatly underestimated due to the enhanced thermal stratification producedbecause of the lower mass flow rate from the collector to storage tank. If this is ignored, f-chartwould give wrong results leading to system oversizing and would predict reduced costeffectiveness of the thermosiphon solar water-heating system.

The modification to the f-chart design method presented here is suggested by Malkin et al. (1987). Infact they used a correction factor to take into account on the system performance the effect of theenhanced thermal stratification within the hot-water storage tank. This was obtained by using, simi-larly as in the original f-chart method, a number of TRNSYS simulations for various thermosiphonsolar water-heating systems operating in three different locations in the USA (Albuquerque, Madisonand Seattle). The characteristics of these three systems are shown in Table 11.9. In addition to thesecharacteristics, in the simulations the overall loss coefficient for the hot-water storage tank wasassumed to be constant and equal to 1.46 W/K.

Table 11.9 Range of Design Variables Used in Developingthe Modified f-Chart Method for Thermosiphon Units

Parameter Range

Load draw 150e500 l

Hot-water storage tank size 100e500 l

Collector slope 30e90

FRUL 3.6e8.6 W/m2 C

FR(sa) 0.7e0.8

Malkin (1985).

11.1 f-chart method and program 607

According to Malkin et al. (1987) it is possible to modify the f-chart method to enable prediction ofthe improved performance of systems exhibiting stratified storage tanks. This can be done byconsidering the varying flow rate through the thermosiphon system to be approximated by anequivalent average fixed flow rate for each month in a corresponding active system of the same size.In this case, the active system operating at this fixed flow rate may yield similar results for monthlyfractional energy savings by solar as the thermosiphon system. Therefore, in this way the long-termperformance of a thermosiphon system may be easily predicted, using the modified form of thef-chart method but the same chart shown in Figure 11.2 for liquid systems.

Once the equivalent monthly average fluid flow rate has been calculated from the density differ-ences in the thermosiphon fluid flow loop then the modified values of FR(sa)n and FRUL need to becalculated using the corrective factor (r) given for different test and use flow rates in Chapter 4, Section4.1.1 by Eq. (4.17) or for the thermosiphon flow in Chapter 5, Section 5.1.1 by Eq. (5.4b) against thenormal (or test) flow rate.

It should be noted that thermally stratified storage tanks return the fluid to the collector at atemperature below that of the average temperature of the storage tank. This results in improvedcollector efficiency as the temperature entering the collector is lower, closer to ambient and thus thethermal losses from the collector are lower. Thermosiphon systems usually circulate domestic waterthrough the collectors, so they do not include a heat exchanger, therefore by combining Eqs (11.8) and(11.20) the X parameter, called Xmix, is given by:

Xmix AcFRUL

11:6 1:18Tw 3:86Tm 2:32Ta

Dt

L(11.21)

Similarly the Y parameter without the presence of the heat exchanger term can be given by modifyingEq. (11.9) as:

Ymix AcFRsaHtNL

(11.22)

It should be pointed again that the Xmix and Ymix parameters shown in Eqs (11.21) and (11.22) assumethat the hot-water storage tank is fully mixed. Copsey (1984) was the first to develop a modification ofthe f-chart to account for stratified storage tanks. He has shown that the solar fraction of a stratified tanksystem (f) can be obtained by analyzing an otherwise identical fully mixed tank system with a reduced

608 CHAPTER 11 Designing and Modeling Solar Energy Systems

collector loss coefficient (UL). As shown by Eq. (3.58) the collector heat-removal factor FR is afunction of the collector heat-loss coefficient and the flow rate through the collector, therefore amodification of the f-chart method that is based on the collector loss coefficient will also requiremodification of the heat removal factor (FR).

The predicted value of the solar fraction of a thermosiphon solar water heater exhibiting stratifiedstorage (fstr) would be between the solar fraction that could be met with a fully mixed hot-storage tank(fmix) using a loss coefficient equal to that reported at test conditions and the solar fraction that could besupplied from a solar water-heating system with a fully mixed hot-water storage tank with UL equal tozero. If the collector had no thermal losses then the X parameter calculated using Eq. (11.21) would bezero and the maximum value of Y, denoted as Ymax, for which FR 1, can be estimated from:

Ymax AcsaHtNL

(11.23)

As reported from Malkin et al. (1987) the relationship between the mixed and stratified coordinatesused to predict the solar fraction of the heating load could be related using an X-parameter correctivefactor, denoted as (DX/DXmax), for stratified storage. This factor is a function of the monthly averagecollector to load flow ratio Mc=ML and mixed tank solar fraction (fmix) and is given by (Malkin,1985):

DX

DXmax Xmix Xstr

Xmix Ystr Ymix

Ymax Ymix 1:040

Mc=ML

h0:726

Mc=ML

1:564fmix 2:760f 2mix

i2 1 (11.24)where the coefficients used are as reported by Malkin (1985) which minimize the root mean squareerror between the predicted thermal performance of a solar water heater with a stratified hot-waterstorage tank simulated using TRNSYS and that predicted using the f-chart procedure modified forstratified hot-water storage tanks. Equation (11.24) is valid when the value of Mc=ML is greater than0.3, which is usual.

The daily average flow through the collector could be estimated from a correlation equivalent to thenumber of hours (Np) that a solar collector would have a useful energy output. According to Mitchellet al. (1981), for a zero degree differential temperature controller:

Np Ht dfdIc

(11.25)

Where

fmonthly average daily utilizability;Ic critical radiation level (W/m2); andHt solar radiation incident of the collector (kJ/m2 day).

It should be noted that by dividing Ht by 3.6 the units of Np is hours.Utilizability is defined as the fraction of the incident solar radiation that can be converted into

useful heat by a collector having FR(sa) 1 and operating at a fixed collector inlet to ambient tem-perature difference. It should be noted that although the collector has no optical losses and has aheat removal factor of 1, the utilizability is always less than 1 since the collector has thermal losses

Table 11.10 Recommended Monthly Optimal Tilt Angle

Month bm (deg.)

January L 29February L 18March L 3April L 10May L 22June L 25July L 24August L 10September L 2October L 10November L 23December L 30L local latitude (deg.).Evans et al. (1982).

11.1 f-chart method and program 609

(see Section 11.2.3). Evans et al. (1982) developed an empirical relationship to calculate the monthlyaverage daily utilizability as a function of the critical radiation level, the monthly average clearnessindex KT; given by Eq. (2.82a), the collector tilt angle (b) and the latitude (L):

f 0:97 AIc BIc2 (11.26a)where:

A 4:86 103 7:56 103K 0T 3:81 103K0T

2(11.26b)

B 5:43 106 1:23 105K 0 7:62 106K0 2

(11.26c)

T T

For

K0T KTcos0:8bm b (11.27)

where bm is the monthly optimal collector tilt shown in Table 11.10.Differentiating Eq. (11.26a) and substituting into Eq. (11.25), yields an expression for the monthly

average daily collector operating time as:

Np HtA 2BIc

(11.28)

The monthly average critical radiation level is defined as the level above which useful energy may becollected and is given by:

Ic FRUL

T i Ta

FRsa (11.29)

610 CHAPTER 11 Designing and Modeling Solar Energy Systems

As can be seen from Eq. (11.29) in order to find the monthly average critical radiation level, a valuefor the monthly average collector inlet temperature T i must be known. This however cannot bedetermined analytically and it is a function of the thermal stratification in the storage tank. It may beapproximated by the Phillips stratification coefficient, which is described subsequently. By assumingthat the storage tank remains stratified, the monthly average collector inlet temperature may initially beestimated to be the mains water temperature.

To apply the method an initial estimated value for a flow rate is used and the solar fraction ofan active system stratified tank is estimated using the f-chart method by applying the Copseysmodification for stratified storage. Subsequently, the collector parameters FRUL and FR(sa) arecorrected for the estimated flow using Eq. (4.17a). An iterative method is required to determine theequivalent average flow rate of a thermosiphon solar water-heating system using the initial estimatedvalue of flow rate (Malkin et al., 1987) as explained subsequently. The average temperature in thestorage tank is calculated from a correlation developed between the solar fraction of a thermosiphonsystem and a non-dimensional form of the monthly average tank temperature obtained from thesimulations with TRNSYS indicated above. The least square regression equation obtained is (Malkinet al., 1987):

T tank TmTset Tm 0:117fstr 0:356f

2str 0:424f 3str (11.30)

The average temperature distribution in the system can be predicted from the fraction calculated usingthe initial estimate of the flow rate allowing an approximate calculation to be made of the thermo-siphon head for the system geometry shown in Figure 11.10 (Malkin et al., 1987). This value is then

FIGURE 11.10

Thermosiphon system configuration and geometry.

11.1 f-chart method and program 611

compared to the frictional losses in the collector loop. Iterations need to be made until the agreementbetween the thermosiphon head and frictional losses are within one percent.

The temperature at the bottom of the storage tank is between the mains temperature (Tm) and theaverage tank temperature T tank: The proximity of the two temperatures depends on the degree ofstratification present. This may be measured approximately using the stratification coefficient, Kstr,defined by Phillips and Dave (1982):

Kstr AcItFRsa FRULTi TaAc

ItFR

sa

FRULT tank Ta (11.31)The stratification coefficient is also a function of two dimensionless variables; the mixing number, M,and the collector effectiveness, E, given by:

M Acylk_mcpH

(11.32)

E FRUL (11.33)

_mcp

where k is the thermal conductivity of water and H is the storage tank height. Physically, themixing number is the ratio of conduction to convection heat transfer in the storage tank and inthe limit as conduction becomes negligible M approaches zero. Phillips and Dave (1982) showedthat:

Kstr ln1=1 E

E1Mln

1=1 E

(11.34)The temperature of the return fluid from the tank to the collector can be found by solving Eq. (11.31)for Ti. Thus:

Ti KstrTtank 1 KstrFRsaFRUL

It Ta

(11.35)

Therefore, by using the pump-operating time, estimated from Eq. (11.28), the monthly averagetemperature of return fluid is approximated as:

T i KstrT tank 1 Kstr

FRsaFRUL

Np

Ht Ta

(11.36)

It should be noted that if a value is obtained from Eq. (11.36) which is lower than Tm, then Tm is usedinstead as it is considered impossible to have a storage tank temperature which is lower than the mainstemperature.

The collector outlet temperature is found by equating Eq. (3.60), by using It instead of Gt, and anenergy balance across the collector:

_mcpTo Ti AcFRItsa ULTi Ta (11.37)

612 CHAPTER 11 Designing and Modeling Solar Energy Systems

By integrating Eq. (11.37) on a monthly basis, the monthly average collector fluid outlet tem-perature can be obtained:

To T i Ac_mcpNp

HtFRsa FRULNp

T i Ta

T i Ac_mcpNp

fHtFRsa

(11.38)

This procedure is undertaken on a monthly basis allowing the equivalent average flow rate to bedetermined and the solar fraction calculated from this value.

Once the monthly average collector fluid inlet and outlet temperatures are known, an estimate ofthe thermosiphon head may be found based on the relative positions of the tank and the flat-platecollectors as shown in Figure 11.10. Close (1962) has shown that the thermosiphon head generatedby the differences in density of fluid in the system may be approximated by making the followingassumptions:

1. There are no thermal losses in the connecting pipes.2. Water from the collector rises to the top of the tank.3. The temperature distribution in the tank is linear.

Therefore, according to the dimensions indicated in Figure 11.10, the thermosiphon head generated isgiven by (Close, 1962):

hT 12Si So

"2H3 H1 H2 H1 H3 H5

2

H4 H5

#(11.39)

where Si and So are the specific gravities of the fluid at the collector inlet and outlet respectively. Hereonly direct circulation thermosiphon systems are considered in which water is the collection fluid. Thespecific gravity according to the temperature (in C) of water is given by:

S 1:0026 3:906 105T 4:05 106T2 (11.40)The equivalent average flow rate is that which balances the thermosiphon buoyancy force with thefrictional resistance in the flow circuit on a monthly average basis. As indicated in Chapter 5 (Section5.1.1) the flow circuit comprises the collector headers and risers, connecting pipes and storage tank.For each component of the flow circuit, the DarcyWeisbach equation for friction head loss needs to beemployed, given by Eq. (5.6).

The Reynolds number, used to find the type of flow as indicated in Section 5.1.1, at the estimatedflow rate is calculated using the following correlation for viscosity as a function of temperature in C:

m 0:12:1482

T 8:435

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8087:4 T 8:4352

q 120

(11.41)

The last term in Eq. (5.6) is included to account for minor friction losses associated with bends, tees,and other restrictions in the piping circuit. It should be noted that although the majority of the pressuredrop in the flow circuit occurs across the relatively small diameter collector risers, the minor frictionallosses are included to enhance the accuracy of the flow rate estimation. Details of these losses are givenin Section 5.1.1 in Chapter 5 and Table 5.2.

11.1 f-chart method and program 613

l

.

All the components of the friction head loss in the flow circuit, at the estimated flow rate, are combinedand comparison is made with the previously calculated thermosiphon head. If the thermosiphon headdoes not balance the frictional losses to within 1% a new guess of the flow rate through the connectingpipes is made by successive substitution. The procedure is repeated with the new estimate of flow rateuntil the convergence is within 1%.

A new guess of the thermosiphon flow rate can be obtained from (Malkin, 1985):

_mnew rAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ghTfld k

pfld k

rfld k

h

vuut (11.42)where subscript p stands for connecting pipes, r for riser, and h for headers.

Convergence is usually obtained by applying no more than three iterations. The resulting singlevalue for monthly average flow rate is that which balances the thermosiphon head with the frictionalosses in the flow circuit. As was pointed out before, the solar fraction is calculated, assuming a fixedflow rate operating in an equivalent active system. As in the standard f-chart method, this procedure iscarried out for each month of the year, with a previous months equivalent average flow rate as theinitial guess of flow rate for the new month. The fraction of the annual load supplied by solar energy isthe sum of the monthly solar energy contributions divided by the annual load as given by Eq. (11.12)

Thedifferencebetween theTRNSYSsimulations and themodified f-chart design method had a reportedroot mean square error of 2.6% (Malkin, 1985). The procedure is clarified by the following example.

EXAMPLE 11.10Calculate the solar contribution of a thermosiphon solar water-heating system located in Nicosia,Cyprus for the month of January. The system has the following characteristics:

1. Slope of collector 452. Monthly average solar radiation 8960 kJ/m2 day (From Appendix 7, Table A7.12)3. Monthly average ambient temperature 12.1 C (From Appendix 7, Table A7.12)4. Monthly average clearness index 0.49 (From Appendix 7, Table A7.12)5. Number of collector panels 26. Collector area per panel 1.35 m27. Collector test FRUL 21.0 kJ/h m2 C8. Collector FR(sa) 0.799. Collector test flow rate 71.5 kg/h m2

10. Number of risers per panel 1011. Riser diameter 0.015 m12. Combined header length per panel 1.9 m13. Header diameter 0.028 m14. Tank-collector connecting pipe length 2.5 m15. Collector-tank connecting pipe length 0.9 m16. Connecting pipe diameter 0.022 m17. Number of bends in connecting pipe 2

614 CHAPTER 11 Designing and Modeling Solar Energy Systems

18. Connecting pipe heat-loss coefficient 10.0 kJ/h m2 C19. Storage tank volume 160 l20. Storage tank height 1 m21. Storage tank diameter 0.45 m22. Daily load draw off 150 l23. Mains water temperature 18 C24. Auxiliary set temperature 60 C25. Height H1 0.05 m26. Height H2 1.12 m27. Height H3 2.1 m28. Height H5 1.27 mSolution

Initially the radiation on the collector surface is required and to save some space the results ofExample 11.5 are used. So from Table 11.6 the required value is equal to 14,520 kJ/m2.

An initial estimate of the equivalent average flow rate is 15 kg/h m2 or 40.5 kg/h for a 2.7 m2

collector area. The collector performance parameters FRUL and FR(sa) are corrected for theassumed flow rate, which is different than the test flow rate by using Eq. (5.4b) using the parameterF0UL estimated from Eq. (5.3). Thus:

From Eq. (5.3):

F0UL _mTcpAc

ln

1 FRULAc

_mTcp

71:5 2:7 4:19

2:7ln

1 21 2:7

71:5 2:7 4:19

21:77 kJ=h m2 CFrom Eq. (5.4b):

r _mt

1 exp

F0ULAc_mtcp

_mT

1 exp

F0ULAc_mTcp

40:51 exp 21:772:740:54:19

2:7 71:51 exp 21:772:771:52:74:19 0:876and

FRUL 0:876 21 18:40 kJ=h m2 C

FRsa 0:876 0:79 0:692Thermal losses from the connecting pipes are estimated from Eq. (5.64b) and Eq. (5.64c):

sa0sa

1

1 UpAp;o _mcpc 1

1 10p0:0220:940:54:19 0:996

U0LUL

1 UpAp;i _mcpc

UpAp;iAp;oAcFRUL

1 UpAp;o _mcpc 1

10p0:0220:940:54:19 10p0:0220:9p0:0222:52:718:4

1 10p0:0222:540:54:19 1:033

11.1 f-chart method and program 615

Therefore:

FRU0L 1:033 18:4 19:01 kJ=h m2 C

FRsa0 0:996 0:692 0:689For simplification it is assumed that FRsa FRsa 0:689From Eq. (11.21):

Xmix ACFRU

0L

11:6 1:18Tw 3:86Tm 2:32Ta

Dt

L

2:7 19:0111:6 1:18 60 3:86 18 2:32 12:1 24150 4:1960 18

0:047123:808 5:82From Eq. (11.22):

Ymix AcFRsaHtNL

2:7 0:692 14520150 4:1960 18 1:03

It should be noted that in both the above equations we do not multiply by the number of days in amonth as the load is estimated on a daily basis.

Storage capacity 160/2.7 59.3 l/m2, which is different from the standard value of 75 l/m2, soa correction is required estimated from Eq. (11.14): Xmix,c 5.82 (59.3/75)0.25 6.17.From Eq. (11.13): fmix 1.029Ymix 0.065Xmix 0.245Ymix2 0.0018Xmix2 0.0215Ymix3 1.029 1.03 0.065 6.17 0.245 (1.03)2 0.0018 (6.17)2 0.0215 (1.03)3 0.49.

To estimate the collector pump operation time a number of parameters are required. FromTable 11.10 bm L 29 35.15 29 64.15.

From Eq. (11.27): K0T KTcos0:8bm b 0:49 cos0:8 64:15 45 0:47:

From Eq. (11.26b):

A 4:86 103 7:56 103K 0T 3:81 103K0T

2 4:86 103 7:56 103 0:47 3:81 1030:472 0:00215

From Eq. (11.26c):

B 5:43 106 1:23 105K 0T 7:62 106K0T

2 5:43 106 1:23 105 0:47 7:62 1060:472 1:332 106

From Eq. (11.29):

Ic FRUL

T i Ta

FRsa

19:0118 12:13:6 0:689 45:22 W=m

2

From Eq. (11.28):

Np HtA 2BIc

14;5200:00215 2 1:332 106 45:223:6 8:2 h

616 CHAPTER 11 Designing and Modeling Solar Energy Systems

The ratioMc=ML

is equal to (8.2 40.5)/150 2.21.

From Eq. (11.24):

DX

DXmax

1:040Mc=ML

h0:726

Mc=ML

1:564fmix 2:760f 2mix

i2 1 1:040 2:21h

0:726 2:21 1:564 0:49 2:760 0:492i2 1 0:59

So

Xstr Xmix1 DX

DXmax

6:171 0:59 2:5297

To find sa when FR 1 we solve for FR,high/FR,use at a very high flow rate, say 10,000 kg/h.This gives r 1.04 and sa 0:79 1:04 0:8216:

Therefore, from Eq. (11.23):

Ymax AcsaHtNL

2:7 0:8216 14520150 4:1960 18 1:22

So from Eq. (11.24):

Ystr Ymix Ymax Ymix DXDXmax

1:03 1:22 1:03 0:59 1:1421From Eq. (11.13): fstr 1.029Ystr 0.065Xstr 0.245Ystr2 0.0018Xstr2 0.0215Ystr3 1.029

1.1421 0.065 2.5297 0.245 (1.1421)2 0.0018 (2.5297)2 0.0215 (1.1421)3 0.73.Then the Phillips stratification coefficient is evaluated:From Eq. (11.32):

M Acylk_mcpH

hp0:452=4

i 0:6

40:5 4:19 1 0:00056

From Eq. (11.33):

E FRUL_mcp

2:7 19:0140:5 4:19 0:30

From Eq. (11.34):

Kstr ln1=1 E

E1M ln

1=1 E

ln1=1 0:3

0:3

1 0:00056 ln

1=1 0:3

1:19

11.1 f-chart method and program 617

From Eq. (11.30):

T tank TmTset Tm 0:117fstr 0:356f

2str 0:424f 3str

0:117 0:73 0:356 0:732 0:424 0:733 0:440Therefore: T tank Tm 0:440Tset Tm 18 0:44060 18 36:5 CFrom Eq. (11.36):

T i KstrT tank 1 Kstr

FRsaFRUL

Np

Ht Ta

1:19 36:5 1 1:19 0:69219:01 8:2 14;520 12:1

28:9 C

Similarly from Eq. (11.38):

To T i Ac_mcpNp

HtFRsa FRULNp

T i Ta

28:9 2:7

40:5 4:19 8:2 14;520 0:689 19:01 8:228:9 12:1 43:2C

Then the specific gravity is estimated from these two temperatures using Eq. (11.40) as:Si 0.995749 and So 0.991014. Using the various values of heights and noting that H4 is equal toH5 plus the tank height [1.27 1] in Eq. (11.39):

hT 12Si So

"2H3 H1 H2 H1 H3 H5

2

H4 H5

#

120:995749 0:991014

"22:1 0:05 1:12 0:05 2:1 1:27

2

2:27 1:27

#

0:005543 mFrom Eq. (11.41) using the mean tank temperature: m 0.000701 kg/m2 s.

Hydraulic ConsiderationsThe specific gravity of the water in the storage tank, obtained using the mean storage tank tem-perature in Eq. (11.40) is equal to 0.993439. The mean tank temperature is assumed to flow throughthe connecting pipes. Therefore, in connecting pipes the velocity is equal to:

vc _mt3600StrAc

40:53600 0:993439 1000 p0:0222=4 0:0298 m=s

which gives: Re StrvcDcm

0:993439 1000 0:0298 0:0220:000701

929

618 CHAPTER 11 Designing and Modeling Solar Energy Systems

Using Eq. (5.7a): f 64/929 0.069. Correcting for developing flow in pipes using Eq. (5.7c):

f 1 0:038L

dRe

0:964 1 0:0382:50:9

0:0229290:964 1:2141

Therefore f 0.069 1.2141 0.084.Finally, the equivalent pipe length is equal to the actual lengthof connectingpipes plus the numberof

bends multiplied by 30 (Table 5.2) multiplied by the pipe diameter (2.5 0.9) 2 300.022 4.72 m. The pipes are of equal diameter so there is no contraction or expansion loss. There isonly a loss due to tank entry which is equal to 1 (Table 5.2), so k 1. Using Eq. (5.6):

Hf fLv2

2dg kv

2

2g 0:084 4:72 0:0298

2

2 0:022 9:81 1 0:02982

2 9:81 0:000861 m

Similar calculations for risers give:

Flow rate in risers 2.025 kg/hVelocity in risers 0.0032 m/sRe 68f 0.965 mHf 5.147 105 mFor headers the flow is given by:

P20i1

_mr=20 21:263 kg=hFlow rate in risers 0.0097 kg/hVelocity in headers 0.0096 m/sRe 384.9f 0.0012 mHf 0.00106 mTherefore the total friction head Hf is equal to: Hf 0.000861 5.147 105

0.00012 0.00103 m.By comparing this value with the thermosiphon head estimated before we get a percentage

difference of:

% difference hT HfhT

100 0:005543 0:001030:005543

100 81:4%

As the percentage difference is more than 1% the new flow rate is estimated by Eq. (11.42) whichgives 93.8 kg/h.

By repeating this procedure for the new flow rate we get a percentage difference of 29.5%. Anew guess of Eq. (11.42) gives a flow rate of 82.4 kg/h which gives a percentage difference of 5.8%.So a third guessing is required which gives a new flow rate equal to 83.7 kg/h which gives:

fstr 0.66hT 0.002671Hf 0.002670% difference 0.04%, which is below 1% so this solution is considered as final. Therefore, the

solar contribution at this flow rate is 0.66. This flow rate can also be used as the initial guess for theestimation of the next months solar contribution if the estimation is done on an annual basis.

11.1 f-chart method and program 619

If should be noted that this flow rate is equal to 31 kg/h/m2 [ 83.7 kg/h/2.7 m2] which is about43% of the test flow rate. Additionally, the number of iterations required depends on how far is theinitial guessing of thermosiphonic flow rate from the actual one.

It should be noted that as in earlier examples and due to the large number of calculations required,

the use of a spreadsheet program greatly facilitates the calculations, especially at the iteration stage.

11.1.5 General remarksThe f-chart design method is used to quickly estimate the long-term performance of solarenergy systems of standard configurations. The input data needed are the monthly average radiationand temperature, the monthly load required to heat a building and its service water, and thecollector performance parameters obtained from standard collector tests. A number of assumptions aremade for the development of the f-chart method. The main ones include assumptions that the systemsare well built, system configuration and control are close to the ones considered in the development ofthe method, and the flow rate in the collectors is uniform. If a system under investigation differsconsiderably from these conditions, then the f-chart method cannot give reliable results.

It should be emphasized that the f-chart is intended to be used as a design tool for residentialspace and domestic water-heating systems of standard configuration. In these systems, theminimum temperature at the load is near 20 C; therefore, energy above this value of temperature isuseful. The f-chart method cannot be used for the design of systems that require minimum tem-peratures substantially different from this minimum value. Therefore, it cannot be used for solar airconditioning systems using absorption chillers, for which the minimum load temperature is around80 C.

It should also be understood that, because of the nature of the input data used in the f-chart method,there are a number of uncertainties in the results obtained. The first uncertainty is related to themeteorological data used, especially when horizontal radiation data are converted into radiation fallingon the inclined collector surface, because average data are used, which may differ considerably fromthe real values of a particular year, and that all days were considered symmetrical about a solar noon. Asecond uncertainty is related to the fact that solar energy systems are assumed to be well built withwell-insulated storage tanks and no leaks in the system, which is not very correct for air systems, all ofwhich leak to some extent, leading to a degraded performance. Additionally, all liquid storage tanksare assumed to be fully mixed, which leads to conservative long-term performance predictions becauseit gives overestimation of collector inlet temperature. The final uncertainty is related to the buildingand hot water loads, which strongly depend on variable weather conditions and the habits of theoccupants.

Despite these limitations, the f-chart method is a handy method that can easily and quickly be usedfor the design of residential-type solar heating systems. When the main assumptions are fulfilled, quiteaccurate results are obtained.

11.1.6 f-chart programAlthough the f-chart method is simple in concept, the required calculations are tedious, particularly forthe manipulation of radiation data. The use of computers greatly reduces the effort required. Program

620 CHAPTER 11 Designing and Modeling Solar Energy Systems

f-chart (Klein and Beckman, 2005), developed by the originators of TRNSYS, is very easy to use andgives predictions very quickly. Again, in this case, the model is accurate only for solar heating systemsof a type comparable to that which was assumed in the development of the f-chart.

The f-chart program is written in the BASIC programming language and can be used to estimate thelong-term performance of solar energy systems that have flat-plate, evacuated tube collectors, com-pound parabolic collectors, and one- or two-axis tracking concentrating collectors. Additionally, theprogram includes an activepassive storage system and analyzes the performance of a solar energysystem in which energy is stored in the building structure rather than a storage unit (treated withmethods presented in the following section) and a swimming pool-heating system that providesestimates of the energy losses from the swimming pool. The complete list of solar energy systems thatcan be handled by the program is as follows:

Pebble-bed storage space and domestic water heating systems. Water-storage space and/or domestic water heating systems. Active collection with building storage space-heating systems. Direct gain passive systems. Collector-storage wall passive systems. Pool heating systems. General heating systems, such as process heating systems. Integral collector-storage domestic water-heating systems.

The program can also perform economic analysis of the systems. The program, however, does notprovide the flexibility of detailed simulations and performance investigations, as TRNSYS does.

11.2 Utilizability methodIn the previous section, the f-chart method is presented. Due to the limitations outlined in Section11.1.5, the f-chart method cannot be used for systems in which the minimum temperature supplied to aload is not near 20 C. Most of the systems that cannot be simulated with f-chart can be modeled withthe utilizability method or its enhancements.

The utilizability method is a design technique used for the calculation of the long-term thermalcollector performance for certain types of systems. Initially originated by Whillier (1953), the method,referred to as the V-curve method, is based on the solar radiation statistic, and the necessary calcu-lations have to be done at hourly intervals about solar noon each month. Subsequently, the method wasgeneralized for the time of year and geographic location by Liu and Jordan (1963). Their generalizedF-curves, generated from daily data, gave the ability to calculate utilizability curves for any locationand tilt by knowing only the clearness index, KT. Afterward, the work by Klein (1978) and Collares-Pereira and Rabl (1979a) eliminated the necessity of hourly calculations. The monthly average dailyutilizability, F; reduced much of the complexity and improved the utility of the method.

11.2.1 Hourly utilizabilityThe utilizability method is based on the concept that only radiation that is above a critical or thresholdintensity is useful. Utilizability, F, is defined as the fraction of insolation incident on a collectorssurface that is above a given threshold or critical value.

11.2 Utilizability method 621

We saw in Chapter 3, Section 3.3.4, Eq. (3.61), that a solar collector can give useful heat only ifsolar radiation is above a critical level. When radiation is incident on the tilted surface of a collector,the utilizable energy for any hour is (It Itc), where the plus sign indicates that this energy can beonly positive or zero. The fraction of the total energy for the hour that is above the critical level iscalled the utilizability for that hour, given by:

Fh It Itc

It(11.43)

Utilizability can also be defined in terms of rates, using Gt and Gtc, but because radiation data areusually available on an hourly basis, the hourly values are preferred and are also in agreement with thebasis of the method.

Utilizability for a single hour is not very useful, whereas utilizability for a particular hour of amonth having N days, in which the average radiation for the hour is It; is very useful, given by:

F 1N

XN1

It ItcIt

(11.44)

In this case, the average utilizable energy for the month is given by NItF: Such calculations can bedone for all hours of the month, and the results can be added up to get the utilizable energy of themonth. Another required parameter is the dimensionless critical radiation level, defined as:

Xc ItcIt

(11.45)

For each hour or hour pair, the monthly average hourly radiation incident on the collector is given by:

It Hr HDrd

RB HDrd

1 cosb

2

HrrG

1 cosb

2

(11.46)

Dividing by H and using Eq. (2.82a),

It KTHo

r HDH

rd

RB HD

Hrd

1 cosb

2

rrG

1 cosb

2

(11.47)

The ratios r and rd can be estimated from Eqs. (2.83) and (2.84), respectively.Liu and Jordan (1963) constructed a set ofV curves for various values of KT:With these curves, it

is possible to predict the utilizable energy at a constant critical level by knowing only the long-termaverage radiation. Later on Clark et al. (1983) developed a simple procedure to estimate the gener-alized V functions, given by:

F

8>>>>>>>>>:

0 if Xc Xm1 Xc

Xm

2if Xm 2

otherwise;jgj hg2 1 2g1 XcXm

2i1=2(11.48a)

622 CHAPTER 11 Designing and Modeling Solar Energy Systems

where

g Xm 12 Xm (11.48b)

Rh cosb kT

Xm 1:85 0:169

k2T

0:0696k2T

0:981cos2d (11.48c)

The monthly average hourly clearness index, kT; based on Eq. (2.82c), is given by:

kT IIo

(11.49)

and can be estimated using Eqs (2.83) and (2.84) as:

kT IIo

rrdKT r

rd

H

Ho a bcoshKT (11.50)

where a and b can be estimated from Eqs (2.84b) and (2.84c), respectively. If necessary, Ho can beestimated from Eq. (2.79) or obtained directly from Table 2.5.

The ratio of monthly average hourly radiation on a tilted surface to that on a horizontal surface, Rh;is given by:

Rh ItI It

rH(11.51)

TheV curves are used hourly, which means that three to six hourly calculations are required per monthif hour pairs are used. For surfaces facing the equator, where hour pairs can be used, the monthlyaverage daily utilizability, F; presented in the following section can be used and is a more simple wayof calculating the useful energy. For surfaces that do not face the equator or for processes that havecritical radiation levels that vary consistently during the days of a month, however, the hourlyV curvesneed to be used for each hour.

11.2.2 Daily utilizabilityAs can be understood from the preceding description, a large number of calculations are required touse the V curves. For this reason, Klein (1978) developed the monthly average daily utilizability, F;concept. Daily utilizability is defined as the sum over all hours and days of a month of the radiationfalling on a titled surface that is above a given threshold or critical value, which is similar to the oneused in the V concept, divided by the monthly radiation, given by:

F Xdays

Xhours

It ItcNHt

(11.52)

The monthly utilizable energy is then given by the product NHtF. The value of F for a monthdepends on the distribution of hourly values of radiation in that month. Klein (1978) assumed that all

days are symmetrical about solar noon, and this means that F depends on the distribution of daily totalradiation, i.e., the relative frequency of occurrence of below-average, average, and above-average

11.2 Utilizability method 623

daily radiation values. In fact, because of this assumption, any departure from this symmetry within

days leads to increased values of F. This means that the F calculated gives conservative results.

Klein developed the correlations ofF as a function of KT, a dimensionless critical radiation level,

Xc; and a geometric factor R=Rn. The parameter R is the monthly ratio of radiation on a tilted surface

to that on a horizontal surface,Ht=H; given by Eq. (2.107), and Rn is the ratio for the hour centered atnoon of radiation on the tilted surface to that on a horizontal surface for an average day of the month,which is similar to Eq. (2.99) but rewritten for the noon hour in terms of rdHD and rH as:

Rn ItI

n

1 rd;nHD

rnH

RB;n

rd;nHDrnH

1 cosb

2

rG

1 cosb

2

(11.53)

where rd,n and rn are obtained from Eqs (2.83) and (2.84), respectively, at solar noon (h 0). It shouldbe noted that Rn is calculated for a day that has a total radiation equal to the monthly average daily totalradiation, i.e., a day for which H H and Rn is not the monthly average value of R at noon. The termHD/H is given from Erbs et al. (1982) as follows.

For hss 81.4,HDH

1:0 0:2727KT 2:4495K2T 11:9514K3T 9:3879K4T for KT < 0:7150:143 for KT 0:715 (11.54a)

For hss> 81.4,

HDH

1:0 0:2832KT 2:5557K2T 0:8448K3T for KT < 0:7220:175 for KT 0:722 (11.54b)

The monthly average critical radiation level, Xc; is defined as the ratio of the critical radiation level tothe noon radiation level on a day of the month in which the radiation is the same as the monthlyaverage, given by:

Xc ItcrnRnH

(11.55)

The procedure followed by Klein (1978) was that, for a given KT a set of days was established that hadthe correct long-term average distribution of KT values. The radiation in each of the days in a sequencewas divided into hours, and these hourly values of radiation were used to find the total hourly radiationon a tilted surface, It. Subsequently, critical radiation levels were subtracted from the It values andsummed as shown in Eq. (11.52) to get the F values. The F curves calculated in this manner can beobtained from graphs or the following relation:

F exp

A BRn

R

hXc CX2c

i(11.56a)

where

A 2:943 9:271KT 4:031K2T (11.56b)B 4:345 8:853KT 3:602K2T (11.56c)

C 0:170 0:306KT 2:936K2T (11.56d)

624 CHAPTER 11 Designing and Modeling Solar Energy Systems

EXAMPLE 11.11A north-facing surface located in an area that is at 35S latitude is tilted at 40. For the month ofApril, when H 17.56 MJ/m2, critical radiation is 117 W/m2, and rG 0.25, calculate F and theutilizable energy.

Solution

For April, the mean day from Table 2.1 is N 105 and d 9.41. From Eq. (2.15), the sunset timehss 83.3. From Eqs (2.84b), (2.84c), and (2.84a), we have:

a 0:409 0:5016 sinhss 60 0:409 0:5016 sin83:3 60 0:607

b 0:6609 0:4767 sinhss 60 0:6609 0:4767 sin83:3 60 0:472

rn p24

a bcosh cosh coshss

sinhss 2phss360

coshss

p24

0:607 0:472cos0 cos0 cos83:3sin83:3

2p83:3

360

cos83:3

0:152

From Eq. (2.83), we have:

rd;n p

24

cosh coshss

sinhss 2phss360

coshss

p

24

cos0 cos83:3

sin83:3 h2p83:3

360

icos83:3

0:140

From Eq. (2.90a), for the Southern Hemisphere (plus sign instead of minus),

RB;n sinL bsind cosL bcosdcoshsinLsind cosLcosdcosh

sin35 40sin9:41 cos35 40cos9:41cos0sin35sin9:41 cos35cos9:41cos0 1:396

From Eq. (2.79) or Table 2.5, Ho 24.84 kJ/m2, and from Eq. (2.82a),

KT 17:5624:84

0:707:

For a day in which H H; KT 0.707, and from Eq. (11.54b),HDH

1:0 0:2832KT 2:5557K2T 0:8448K3T 1:0 0:2832 0:707 2:55570:7072 0:84480:7073 0:221

11.2 Utilizability method 625

Then, from Eq. (11.53),

Rn 1 rd;nHD

rnH

RB;n

rd;nHDrnH

1 cosb

2

rG

1 cosb

2

1 0:140 0:221

0:152

1:396

0:140 0:221

0:152

1 cos40

2

0:251 cos40

2

1:321

From Eq. (2.109), for the Southern Hemisphere (plus sign instead of minus),

h0ss minhss; cos

1tanL btand min83:3; cos1tan35 40tan9:41 minf83:3; 90:8g 83:3

From Eq. (2.108), for the Southern Hemisphere (plus sign instead of minus),

RB cos

L bcosdsinh0ss p=180h0sssinL bsindcosLcosdsinhss p=180hsssinLsind

cos35 40cos9:41sin83:3 p=18083:3 sin35 40sin9:41cos35cos9:41sin83:3 p=18083:3 sin35sin9:41 1:496

From Eq. (2.105d),

HD

H 1:311 3:022KT 3:427K2T 1:821K3T 1:311 3:022 0:707 3:4270:7072 1:8210:7073 0:244

From Eq. (2.107),

R HtH

1 HD

H

RB HD

H

1 cosb

2

rG

1 cosb

2

1 0:244 1:496 0:2441 cos40

2

0:25

1 cos40

2

1:376

Now,

Rn

R 1:321

1:376 0:96

From Eq. (11.55), the dimensionless average critical radiation level is:

Xc ItcrnRnH

117 36000:152 1:321 17:56 106 0:119

626 CHAPTER 11 Designing and Modeling Solar Energy Systems

From Eq. (11.56):

A 2:943 9:271KT 4:031K2T 2:943 9:271 0:707 4:0310:7072 1:597

B 4:345 8:853KT 3:602K2T 4:345 8:853 0:707 3:6020:7072 0:114

C 0:170 0:306KT 2:936K2T 0:170 0:306 0:707 2:9360:7072 1:081

F exp

A BRn

R

hXc CX2c

i

expnh

1:597 0:1140:96ih0:119 1:0810:1192io 0:819Finally, the month utilizable energy is:

NHtF NH RF 30 17:56 1:376 0:819 593:7 MJ=m2

Both theV and theF concepts can be applied in a variety of design problems, such as heating systemsand passively heated buildings, where the unutilizable energy (excess energy) that cannot be stored in thebuilding mass can be estimated. Examples of these application