- Email: [email protected]
On generalizing the dynamic performance of solar energy systems
$olarEnergy, 1971,Vol. 13,pp.301-310. Pe~smonP~'ss. PrintedinGreatBritain
ON G E N E R A L I Z I N G THE D Y N A M I C PERFORMANCE OF SOLAR E N E R G Y SYSTEMS* C. L. GUPTAI" (Received 14 October 1969; in revised forra 20 January 1970)
A b s t r a c t - An outline is given of a response factor method for generalizing the dynamic performance of lowtemperature solar energy utilization devices, on the basis of a representative set of experimental test data. The method does not require detailed knowledge of system parameters and is amenable to hand calculation. The time-dependent input function combines solar radiation and ambient temperature in the form of a time series. A natural circulation-type solar-water heater and a basin-type solar still are discussed as examples of the application of the method. R ~ a m ~ - O n donne un aper~u d'une m6thode du facteur r6ponse pour la g6n6ralisation de la performance dynamique des dispositifs d'utilisation d'6n6rige solaire h basse temp6rature, sur la base d'un jeu repr6sentatif de donn6es exp6rimentales. La m6thode ne demande aucune connaissance d6taill6e des param6tres du syst6me et est v6rifiable par des calculs ordinaires. La fonction entr6e d6pendant du temps combine les radiations solaires et la temp6rature ambiante sous la forme d'une s6rie du temps. Un chauffe-eau solaire du type/l circulation naturelle et un distillateur du type h bassin sont pr6sent6s comme exemples de I'application de cette m6thode.
Resumen-Se expone en llneas generales un m6todo de factor de respuesta para la generalizaci6n del comportamiento dimlmico de dispositivos de utilizaci6n de energia solar a baja temperatura, sobre la base de un conjunto representativo de datos procedentes de pruebas experimentales. El m6todo no exige un conocimiento detallado de los parfimetros sistem~.ticos y se presta al cfilculo manual. La funci6n de entrada, dependiente del tiempo, reune la radiac6n solar y la temperatura ambiente en forma de una serie de tiempos. Se comentan como ejemplos de la aplicaci6n del m6todo un heliocalentador de agua de circulaci6n natural y una destiladora solar de tipo cubeta. INTRODUCTION
THE DEVELOPMENT of low-temperature solar energy utilization devices such as solarwater heaters, solar-air heaters and solar stills has reached a stage where it is an accepted practice to formulate analytical prediction models[l,2,3] from a fairly adequate understanding of the relationship between design variables involved in a solar process, and the thermal performance of the corresponding system. These models serve as the basis for the design of experiments [4, 5] which reduce the time and effort involved in purely empirical development in which each successive improvement has to be incorporated into a prototype and tested physically. These can also be coupled to optimization procedures [6] to obtain design controls or economic systems with the help of a computer. Nevertheless, because of the almost inevitable need to make simplifying assumptions in the formulation of such simulation models, it is still good practice to build and test prototypes of the final and near final designs in the laboratory or the field. Further, access to computers is not always available to designers, particularly in underdeveloped countries. The thermal performance data available for a solar design consultant are usually in the form of a figure for utilization efficiency, based on steady-state analysis for the average daily, monthly or annual performance on an integrated basis. In most cases, * 1970 International Solar Energy Society Conference paper. tDivision of Building Research, Commonwealth Scientific and Industrial Research Organization, Melbourne, Australia. 301
302
c.L. GUPTA
the dynamic performance is also given in the form of an experimentally observed outputtime curve, along with climatic variables obtained during the test. A description of the system is also usually available, but it is very brief since most of the commercially available devices are proprietary products. There is a definite need to relate equipment and data developed and tested in some countries to the needs of prospective users in countries with very different climates. The crucial importance of this step with regard to the acceptance of a solar device has been pointed out by L r f e t al. [7]. The purpose of this paper is to outline an approximate but general computational technique amenable to slide rule calculation, which can be used to generalize the dynamic performance of any solar energy system from the data usually available. It can be used in projecting the available dynamic performance data to a different location for technical feasibility studies, in obtaining design information for a specific installation, or for comparative evaluation of different types of any one solar unit, which have been normally tested in different climates. THE APPROACH For a linear and invariable physical system the principle of superposition can be applied to obtain its dynamic response to a given excitation function, provided its unit response function, i.e. the response to a suitably defined unit excitation function, is known. In solar processes, which are highly non-linear, the linear formulation is not a very serious handicap in systems associated with flat-plate collectors at low temperatures, such as water heaters and air heaters. However, in solar stills it is expected to be the limiting consideration because the vapor pressure curves are steep at temperatures near the higher end of the low-temperature range (150-200°F). Another difficulty in applying the response factors approach to solar systems is the absence of a single realistic input, combining the shade air temperature and solar radiation, in a way similar to sol-air temperatures for buildings. It will be shown later how these two problems can be handled, and the magnitude of prediction errors involved in two different solar processes, solar-water heating and solar distillation, will be discussed. THE METHOD Considering the excitation function to be given in the form of a set of numerical values at equally spaced time intervals, the time series concept due to Tustin [8] may be applied. Such an approach has been used in the thermal analysis of buildings for load estimation under very general boundary conditions by Stephenson and Mitalas [9]. In a time series consisting of numbers, each term can be visualized as the magnitude of a triangular pulse centred at a point of time and having a base equal to twice the time interval A. These overlapping triangles centred at successive points of time (Fig. l) are equivalent to a smooth function connecting given points (el, e2...) by straight lines. The usual value of A, in solar processes, is taken to be l hr or 30 min from considerations of availability of data and computational economy. Similarly, the unit response function, i.e. the response to a unit function defined by a triangular pulse as described above and shown in Fig. 2, can be represented by a set of numbers (u0, ul, u2...), which denote the response at equally spaced time intervals apart, with the initial time being defined at the same point as for the excitation function. The system output or response at equally spaced intervals (r0, r~, r2, ra...) can now be very easily calculated on the basis of the principle of superposition, by multiplying
Dynamic performance of solar energy systems
303
40
TIME Fig. 1. Excitation function as time series (e).
-/ 1,00 -
~-UNIT(&)EXCITATIONFUNCTION ,
0
SE FUNCTION
A
2A TIME
3A
I
4A
Fig. 2. Unit response function as time series (u).
the two time series (e) and (u) to obtain the response series (r). The law of multiplication for time series and its properties with regard to commutativity are defined to be similar to that of multiplace real numbers in algebra, except that instead of digits there are numbers and carrying is not permissible. The general term in the product series may be written as N rn =
~
en_ j . uj
j=0
where N depends upon the number of terms involved and may even tend to infinity. Algebra for time series is more fully discussed in the Appendix. The system response to a unit excitation function can be determined theoretically by using one of the simulation models, which demands a complete knowledge of the system parameters. In this paper, the principle of division of time series has been used to obtain this unit response function in the form of a time series (u) from a known set of test data. The division is carded out in a way similar to polynomial division, the time series terms being considered to be the coefficient of terms in descending order. A graphical representation of this process for the case of terminating series is shown in Fig. 3. This method does not require the detailed knowledge of system parameters and is likely to allow for errors due to linearization in a more realistic way as compared to linearized computational models.
304
C.L. GUPTA
JlNJllllllllllllllllllllllllllll 'lllll'||llll||~l~l 4 "III] 6
O ~ Ill IIIII ..I. ' . 1Z1 ~
'
's" ~
|
I
10
TIME ( & UNITS) (2,5, 7,10,6,4,1 ) : ( 2,1.3, I ) (I,2,1,1)
Fig. 3. Division of time series. DERIVATION
OF EXCITATION
FUNCTIONS
One of the chief requirements associated with deriving the unit response functions from a set of practical test data is the existence of a single excitation function. It is known that incident solar radiation intensity, shade air temperature, wind velocity and sky temperature are some of the climatic parameters which affect the solar processes at low temperatures. Assuming that the effect of wind velocity and sky temperature can be taken into account by choosing a suitable value for the surface coefficient of heat transfer, the two main parameters requiring attention are the incident solar radiation and the shade air temperatures. It will be shown how these two can be combined to form a single excitation function for two types of solar energy systems, those based on fiat-plate solar collectors, and a basin-type solar still. (a) Flat-plate solar collector For a closed-loop system using flat-plate solar collectors, such as natural circulation type solar-water heater, it has been shown by Gupta and Garg [ 1] that the basic governing equation for time varying mean fluid temperatures in the system during sunlit hours is given by
Cwh • ~ffO + Utw = F~,H~-&+ Ut,.
(i)
Equation (1) can be rewritten U
dO F-tu,= ta+
H
(2.1)
Dynamic performance of solar energy systems
dO
LFp~-ffJ
tw
=,-,+r LFp~-~J '-' 1,o.
305 (2.2)
The output is the mean fluid temperature tw, and the excitation function can be considered either as an equivalent air temperature or as an equivalent solar radiation given by the right-hand sides of Eq. (2.1) and Eq. (2.2), respectively. On the basis of existing solar-water heaters and the relevant data reported in the literature[10], the design values for the factor F~(t-~)/U can reasonably be chosen as given in Table I. Table 1. Designvaluesfor Fp(~~)IU G(~)/U
(B.t.u.-tft= hr deg F) Flat black Selective paint black paint
Collectorcover system Single sheet of glass (3 mm)
Two sheetsof glass Outer glass cover with inner plastic sheet
0.5 0.7
0.7 0.9
0.6
1-0
(b) B a s i n - t y p e solar still For a basin-type solar still, it has been shown by Morse and Read[3] that the heat balance equations governing the unsteady-state behaviour are given by Cwo" dtw dO = aoHs + rawHs
-
qua
-
qb
qga = f[Eotr ( To 4 - Ts 4) + ho~ (to - to) ] qb =
Kb(tw -- t , ) .
(3) (3.1) (3.2)
Using the linearization procedure for the radiation term in Eq. (3.1), an effective value of overall surface coefficient of heat transfer at the glassy cover hoo may be obtained such that qga = Jhgo ( to - ta ) .
(3.3)
Although hoo varies with the temperature of the glass cover, sky temperature and air temperature, the dependence is only of the second order as compared with its dependence on wind velocity, which can be taken into account in the calculation of excitation function. Further, there is the heat balance in the still, which gives qga = qr + qc + qe + aoHa .
(4)
As suggested by Baum[11] the heat transfer by evaporation can be calculated by the approximate formula qe= he(tw-tg)
(4.1)
306
C.L. GUPTA
where
h, = ,4 - Bt~ + Ctw 2.
(4.2)
On the basis of exact analysis by Dunkle [12], the quantities ,4, B, C would be nonlinear functions oft, and (tw - to). However. for the purposes of eliminating the unknown (glass cover) temperature, the explicit dependence of he on to has not been considered in the present analysis. This assumption is the main factor in limiting the accuracy of the time series approach in its application to solar stills. The heat transferred by radiation between the water in the still and the glass cover, and by the air in circulating the vapor-air mixture, can also be represented in the linearized form by the equation
qr + qc = hwo(t,,-to).
(4.3)
Combining Eqs. (3), (3.2), (3.3), (4), (4.1), (4.2) and (4.3), two basic equations can be obtained for the mean water temperature in the still and the heat transferred by evaporation, which is directly proportional to the amount of distillate output. These equations are c
dt,,
. -d g + t,e = t. +
[ Ta,e(fhoo + h,,,~+ he) + ao(hwo + he) ] 4-h'
¥
j H.
(5)
where
fh°°(hw"+he) ] C w o = C w g / [ K h + (fhoo+h,,,,,+he)l
(5.1)
and
a-U] U']] D[tw-- [t.+ (mo,
(6)
where
D = (fhoohe)/(fhoo+hu,o+he) -- A1 +Bit,,,,
(6.1)
A I, BI being constants. The right-hand side of Eq. (5) gives the excitation function in terms of equivalent air temperature. As before, it can be modified to give the excitation functions in terms of equivalent solar radiation also and, once the water temperatures are known. Eq. (6) can be used to determine hourly outputs of output on a daily basis. The multiplier of H~ cannot be assumed as a constant in Eq. (5), as was done in the Eq. (2.1) for the case of water heaters, because of strong dependence of he on t,,, as in Eq. (4.2). However, for design purposes, a mean value is taken for the daily cycle. The design value for the multiplier of H, in Eq. (5), corresponding to a constant wind velocity of 15 m.p.h, and for different slopes of the glass cover and values of Kn. may be taken as shown in Table 2. The value of h,,,o is found to be practically constant and equal to 1.5 B.t.u. ft -2 hr -I deg F -1.
Dynamic performance
of solar energy systems
307
EXAMPLES OF APPLICATION To verify the practicability of this technique and to estimate the magnitude of errors arising from simplifying approximations used in deriving excitation functions, three examples of solar energy systems were chosen from the literature such that the dynamic performance data for at least two days under different climatic conditions were available for each. From one set of results, the unit response function has been derived by using the time series division and excitation functions derived in Section 4. This unit response function has been used to predict the performance under different climatic conditions by using the method outlined in Section 3 of the paper. The predicted and Table 2. Design values for multiplier of H. in Eq.
(5)
Multiplier of H, Slope of glass cover 15° 25°
Base insulation and vapor sealing parameter
gb = 1"0 Poorly sealed; no base insulation
0.22 0.26 0'28
Kb = 0-5 Properly sealed or some base insulation Kb = 0"25 Properly sealed and good base insulation
0.22 0.27 0'30
OBSERVED
70
i-2
i~
~
Z
A
~,00
A 0'700
I
I
0900
I
I
I
]
I100 TIME
1300
I
I
1500
l
i
1700
(hr)
Fig. 4. Mean water temperature in a solar-water heater (flat black collector).
observed hourly values of the mean system temperatures for a black-painted solarwater heater reported by Gupta et al. [13], for a selectively black-painted water heater reported by Yellot and Sobotka [ 14] and the mean water temperature and distillate output for a Mark llI-type solar still developed by the Division of Mechanical Engineering, CSIRO [Read, private communication] are shown in Figs, 4, 5 and 6, respectively. For solar-water heaters, the hourly values during daylight hours, for which the basic model is valid, compare to within 3 deg F for both cases. This may be taken as reasonable accuracy for application engineering work, particularly because the test data for the two days were not obtained under identical conditions of draw-off, as indicated by the initial temperature values corresponding to 0800 hr.
308
C.L. GUPTA OOSERVED
170
3OO
,~
H
.
N ~o m
L t.)
5C iO0
0700
ogo0
liO0
1300
1500
1700
TIME (hr) Fig. 5. Mean water temperature in a solar-water heater (selective black collector). r/ J
~ F I
'
5
0500
0
D,ST,LLATERATE
#7'~',,L \
-~ | J
OBSERVED PREDICTED
I
/ I/_\\
1
/ All
ogO(~
/HOURLY
II \'X
"OF
=
"~f-~
\ N'~'t
1300
~
1700
>-
,HOURLYDISTILLATE I
5
2100
OtO0
0500
~LN
.0
TIME (hr)
Fig. 6. Output for Mark I II solar still (plan area = 500 ftz). In the basin-type solar still, the hourly values o f predicted m e a n w a t e r t e m p e r a t u r e s agree within the same range. T h e distillate o u t p u t agrees to within 0.01 lb ft -2 on an hourly basis and within 0.02 Ib ft -2 on the daily total basis. C o n s i d e r i n g that the solar distillation is a highly non-linear process, this o r d e r o f a c c u r a c y should be quite a c c e p t a b l e for application w o r k requiring a k n o w l e d g e o f d y n a m i c p e r f o r m a n c e . Alternatively, a quadratic t e r m in t~ c a n be included in Eq. (6.1) to i m p r o v e the a c c u r a c y . NOMENCLATURE C,~h Thermal capacity of water heater per unit collector area B.t.u. ft-~ deg F-' C~v Thermal capacity of basin-type solar still per unit plan area B.t.u. It-= deg F -' Eg Emissivity of glass; Fg. plate efficiency factor
Dynamic performance of solar energy systems
309
f G l a s s cover area per unit plan area of solar still H Total solar radiation incident on inclined plane of collector B.t.u. ft-" hr -~ H, Total solar radiation incident on a horizontal surface B.t.u. ft -2 hr -~ h,a, hoo Surface coefficients of convective and combined heat transfer from glass cover to air B.t.u. ft -2 hr -~ deg F -I h,,~, he Coefficients of convective and radiative, and of evaporative heat transfer from water to glass B.t.u. ft -z hr -I deg F -1 Kb Thermal loss coefficient of still base and vapor leakage B.t.u. ft -2 hr -~ deg F -~ Heat transferred by convection and by radiation between water and glass cover B.t.u. ft -2 hr ' qe, qr Heat transferred by glass cover to surroundings B.t.u. ft z hr-I qua T o, T, Glass cover and sky temperatures (°R) Ta, to Shade air dry bulb, and glass cover temperatures (°F) I u, Mean fluid temperature in system (°F) U Overall heat loss coefficient per unit collector area B.t.u. ft -z hr -1 deg F -~ Solar absorptivity of glass and of water T¢1 Transmittivity of glass Tt~ Effective transmittivity and absorptivity product O" Stefan's Constant = 17.2 × I0 -~° B.t.u. ft z hr -~ o Time REFERENCES [ 1] C. k. Gupta and H. P. Garg, System design in solar water heaters with natural circulation. Solar Energy 12, 163 (1968). [2] A. Whillier, Performance of black-painted solar air heaters of conventional design. Solar Energy 8, 31 (1964). [3] R. N. Morse and W. R. W. Read, A rational basis for the engineering development of a solar still. Solar Energy 12, 5 (1968). [4] D . J . Close, A design approach for solar processes. Solar Energy 11, 112 (1967). [ 5] P. I. Cooper. Digital simulation of transient solar still processes, Solar Energy 12, 313 (1969). [6] H. Buchberg and J. R. Roulet, Simulation and optimization of solar collection and storage for house heating. Solar Energy 12.31 (1968). [7] G. O. G. Ltif, D. J. Close and J. A. Duflie, A philosophy for solar energy development. Solar Energy 12.243 (1968). [8] A. Tustin, A method of analysing the behaviour of linear systems in terms of time series. J. Inst. elect. Engrs 94 {1I-A), 130 (1947). [9] D. G. Stephenson and G. P. Mitalas, Cooling load calculations by thermal response factor method. Trans. Am. Soc. Heat. Ref. A ir-Condit. Engrs 73, paper I I I. I (1967). [ 10] R. C. Jordan (Ed.), Report o f A S H R A E Technical Committee on Solar Energy Utilization (1967). [ I 1] V. A. Baum, Solar distiller, U.N. Conference on N e w Sources of Energy, Rome, paper S/i 19, ( 1961 ). [12] R. V. Dunkle, Solar water distillation: the roof-type still and in multiple-offset diffusion still. International Developments in Heat Transfer. pp. 895-912. The American Society of Mechanical Engineers (1961). [ 13] C. L. Gupta, H. P. Garg and R. Gangnly, Solar cure electric water heater. Indian east. Engr 111(1), 21 (1969). [ 14] J. I. Yellot and R. Sobotka, An investigation of solar water heater performance, Trans. Am. Soc. Heat. Re.[. A ir-Condit. Engrs 70, 425 (1964). APPENDIX
Algebra o f time series The time series are meant to represent time functions, which are zero up to a point of time and then are given in form of numbers at equally spaced time intervals. The algebraic operations o f addition, subtraction, multiplication and division correspond to physical operations and are best understood by the following examples: Addition If a parameter is subject to two or more excitations in parallel, e.g. solar energy systems are subject to shade air temperature and solar radiation, the resulting excitation is the sum of these two. (Col, ezl, ezt, eaj .... ) + (eo2, etz, ez2, e.~z.... ) -= (e01 + e0z, eH + el2, e2t -k e2z, e3t -4-ez2.... )
Subtraction is the corresponding procedure when one of the branches of a system, in parallel, is considered. Multiplication If two or more linear systems act in series, e.g. an excitation function and a physical system represented by its unit response function or two physical systems in series such as insulated panels, then the
S E V o L 13 No. 3 - B
310
C.L. GUPTA
overall response function or transfer function is obtained by multiplication as below: Time
A
Time series for input or first system Time series for unit response function or second system Output due to eo Output due to e~ Output due to e~
Response function or overall transfer function = 2 columns
2A
3A
4A
e0
el
e~
e3
e4. •.
u0
ul
us
u3
u4 • • •
uoeo
u l eo
u:eo
useo
u4eo • • •
uoe~
llles
u~el
I~sel . . .
uoe~
ule2
u~e2...
V~
V3
V,...
V0
V~
5A...
The resulting time series terminates only if the constituting series terminate and the expression for any N
general term of the resulting series is V. = ~ e~_j,. Uj where N depends upon the number of terms involved J-0
in the constituting series. Division The process of division corresponds to the determination of a unit response function when the response of a linear system to a specified excitation is known. The method is similar to long division except that digits can be nun:lbers and carrying is not permissible. The example shown graphically in Fig. 3 is the original illustration given by Tustin in [8], and is worked out below: 1, 2, 1, 1 J 2, 5,7, 10,6,4, I~ 2.1,3, 1 2,4,2, 2 1,5, 8,6 1,2, 1,1 3, 7 , 5 , 4 3, 6 , 3 , 3 1,2,1,1 1,2,1,1 The quotient is (2, 1, 3, I). The division is exactly an inverse process to multiplication. As can be seen, multiplication and addition are both associative and commutative.
ON G E N E R A L I Z I N G THE D Y N A M I C PERFORMANCE OF SOLAR E N E R G Y SYSTEMS* C. L. GUPTAI" (Received 14 October 1969; in revised forra 20 January 1970)
A b s t r a c t - An outline is given of a response factor method for generalizing the dynamic performance of lowtemperature solar energy utilization devices, on the basis of a representative set of experimental test data. The method does not require detailed knowledge of system parameters and is amenable to hand calculation. The time-dependent input function combines solar radiation and ambient temperature in the form of a time series. A natural circulation-type solar-water heater and a basin-type solar still are discussed as examples of the application of the method. R ~ a m ~ - O n donne un aper~u d'une m6thode du facteur r6ponse pour la g6n6ralisation de la performance dynamique des dispositifs d'utilisation d'6n6rige solaire h basse temp6rature, sur la base d'un jeu repr6sentatif de donn6es exp6rimentales. La m6thode ne demande aucune connaissance d6taill6e des param6tres du syst6me et est v6rifiable par des calculs ordinaires. La fonction entr6e d6pendant du temps combine les radiations solaires et la temp6rature ambiante sous la forme d'une s6rie du temps. Un chauffe-eau solaire du type/l circulation naturelle et un distillateur du type h bassin sont pr6sent6s comme exemples de I'application de cette m6thode.
Resumen-Se expone en llneas generales un m6todo de factor de respuesta para la generalizaci6n del comportamiento dimlmico de dispositivos de utilizaci6n de energia solar a baja temperatura, sobre la base de un conjunto representativo de datos procedentes de pruebas experimentales. El m6todo no exige un conocimiento detallado de los parfimetros sistem~.ticos y se presta al cfilculo manual. La funci6n de entrada, dependiente del tiempo, reune la radiac6n solar y la temperatura ambiente en forma de una serie de tiempos. Se comentan como ejemplos de la aplicaci6n del m6todo un heliocalentador de agua de circulaci6n natural y una destiladora solar de tipo cubeta. INTRODUCTION
THE DEVELOPMENT of low-temperature solar energy utilization devices such as solarwater heaters, solar-air heaters and solar stills has reached a stage where it is an accepted practice to formulate analytical prediction models[l,2,3] from a fairly adequate understanding of the relationship between design variables involved in a solar process, and the thermal performance of the corresponding system. These models serve as the basis for the design of experiments [4, 5] which reduce the time and effort involved in purely empirical development in which each successive improvement has to be incorporated into a prototype and tested physically. These can also be coupled to optimization procedures [6] to obtain design controls or economic systems with the help of a computer. Nevertheless, because of the almost inevitable need to make simplifying assumptions in the formulation of such simulation models, it is still good practice to build and test prototypes of the final and near final designs in the laboratory or the field. Further, access to computers is not always available to designers, particularly in underdeveloped countries. The thermal performance data available for a solar design consultant are usually in the form of a figure for utilization efficiency, based on steady-state analysis for the average daily, monthly or annual performance on an integrated basis. In most cases, * 1970 International Solar Energy Society Conference paper. tDivision of Building Research, Commonwealth Scientific and Industrial Research Organization, Melbourne, Australia. 301
302
c.L. GUPTA
the dynamic performance is also given in the form of an experimentally observed outputtime curve, along with climatic variables obtained during the test. A description of the system is also usually available, but it is very brief since most of the commercially available devices are proprietary products. There is a definite need to relate equipment and data developed and tested in some countries to the needs of prospective users in countries with very different climates. The crucial importance of this step with regard to the acceptance of a solar device has been pointed out by L r f e t al. [7]. The purpose of this paper is to outline an approximate but general computational technique amenable to slide rule calculation, which can be used to generalize the dynamic performance of any solar energy system from the data usually available. It can be used in projecting the available dynamic performance data to a different location for technical feasibility studies, in obtaining design information for a specific installation, or for comparative evaluation of different types of any one solar unit, which have been normally tested in different climates. THE APPROACH For a linear and invariable physical system the principle of superposition can be applied to obtain its dynamic response to a given excitation function, provided its unit response function, i.e. the response to a suitably defined unit excitation function, is known. In solar processes, which are highly non-linear, the linear formulation is not a very serious handicap in systems associated with flat-plate collectors at low temperatures, such as water heaters and air heaters. However, in solar stills it is expected to be the limiting consideration because the vapor pressure curves are steep at temperatures near the higher end of the low-temperature range (150-200°F). Another difficulty in applying the response factors approach to solar systems is the absence of a single realistic input, combining the shade air temperature and solar radiation, in a way similar to sol-air temperatures for buildings. It will be shown later how these two problems can be handled, and the magnitude of prediction errors involved in two different solar processes, solar-water heating and solar distillation, will be discussed. THE METHOD Considering the excitation function to be given in the form of a set of numerical values at equally spaced time intervals, the time series concept due to Tustin [8] may be applied. Such an approach has been used in the thermal analysis of buildings for load estimation under very general boundary conditions by Stephenson and Mitalas [9]. In a time series consisting of numbers, each term can be visualized as the magnitude of a triangular pulse centred at a point of time and having a base equal to twice the time interval A. These overlapping triangles centred at successive points of time (Fig. l) are equivalent to a smooth function connecting given points (el, e2...) by straight lines. The usual value of A, in solar processes, is taken to be l hr or 30 min from considerations of availability of data and computational economy. Similarly, the unit response function, i.e. the response to a unit function defined by a triangular pulse as described above and shown in Fig. 2, can be represented by a set of numbers (u0, ul, u2...), which denote the response at equally spaced time intervals apart, with the initial time being defined at the same point as for the excitation function. The system output or response at equally spaced intervals (r0, r~, r2, ra...) can now be very easily calculated on the basis of the principle of superposition, by multiplying
Dynamic performance of solar energy systems
303
40
TIME Fig. 1. Excitation function as time series (e).
-/ 1,00 -
~-UNIT(&)EXCITATIONFUNCTION ,
0
SE FUNCTION
A
2A TIME
3A
I
4A
Fig. 2. Unit response function as time series (u).
the two time series (e) and (u) to obtain the response series (r). The law of multiplication for time series and its properties with regard to commutativity are defined to be similar to that of multiplace real numbers in algebra, except that instead of digits there are numbers and carrying is not permissible. The general term in the product series may be written as N rn =
~
en_ j . uj
j=0
where N depends upon the number of terms involved and may even tend to infinity. Algebra for time series is more fully discussed in the Appendix. The system response to a unit excitation function can be determined theoretically by using one of the simulation models, which demands a complete knowledge of the system parameters. In this paper, the principle of division of time series has been used to obtain this unit response function in the form of a time series (u) from a known set of test data. The division is carded out in a way similar to polynomial division, the time series terms being considered to be the coefficient of terms in descending order. A graphical representation of this process for the case of terminating series is shown in Fig. 3. This method does not require the detailed knowledge of system parameters and is likely to allow for errors due to linearization in a more realistic way as compared to linearized computational models.
304
C.L. GUPTA
JlNJllllllllllllllllllllllllllll 'lllll'||llll||~l~l 4 "III] 6
O ~ Ill IIIII ..I. ' . 1Z1 ~
'
's" ~
|
I
10
TIME ( & UNITS) (2,5, 7,10,6,4,1 ) : ( 2,1.3, I ) (I,2,1,1)
Fig. 3. Division of time series. DERIVATION
OF EXCITATION
FUNCTIONS
One of the chief requirements associated with deriving the unit response functions from a set of practical test data is the existence of a single excitation function. It is known that incident solar radiation intensity, shade air temperature, wind velocity and sky temperature are some of the climatic parameters which affect the solar processes at low temperatures. Assuming that the effect of wind velocity and sky temperature can be taken into account by choosing a suitable value for the surface coefficient of heat transfer, the two main parameters requiring attention are the incident solar radiation and the shade air temperatures. It will be shown how these two can be combined to form a single excitation function for two types of solar energy systems, those based on fiat-plate solar collectors, and a basin-type solar still. (a) Flat-plate solar collector For a closed-loop system using flat-plate solar collectors, such as natural circulation type solar-water heater, it has been shown by Gupta and Garg [ 1] that the basic governing equation for time varying mean fluid temperatures in the system during sunlit hours is given by
Cwh • ~ffO + Utw = F~,H~-&+ Ut,.
(i)
Equation (1) can be rewritten U
dO F-tu,= ta+
H
(2.1)
Dynamic performance of solar energy systems
dO
LFp~-ffJ
tw
=,-,+r LFp~-~J '-' 1,o.
305 (2.2)
The output is the mean fluid temperature tw, and the excitation function can be considered either as an equivalent air temperature or as an equivalent solar radiation given by the right-hand sides of Eq. (2.1) and Eq. (2.2), respectively. On the basis of existing solar-water heaters and the relevant data reported in the literature[10], the design values for the factor F~(t-~)/U can reasonably be chosen as given in Table I. Table 1. Designvaluesfor Fp(~~)IU G(~)/U
(B.t.u.-tft= hr deg F) Flat black Selective paint black paint
Collectorcover system Single sheet of glass (3 mm)
Two sheetsof glass Outer glass cover with inner plastic sheet
0.5 0.7
0.7 0.9
0.6
1-0
(b) B a s i n - t y p e solar still For a basin-type solar still, it has been shown by Morse and Read[3] that the heat balance equations governing the unsteady-state behaviour are given by Cwo" dtw dO = aoHs + rawHs
-
qua
-
qb
qga = f[Eotr ( To 4 - Ts 4) + ho~ (to - to) ] qb =
Kb(tw -- t , ) .
(3) (3.1) (3.2)
Using the linearization procedure for the radiation term in Eq. (3.1), an effective value of overall surface coefficient of heat transfer at the glassy cover hoo may be obtained such that qga = Jhgo ( to - ta ) .
(3.3)
Although hoo varies with the temperature of the glass cover, sky temperature and air temperature, the dependence is only of the second order as compared with its dependence on wind velocity, which can be taken into account in the calculation of excitation function. Further, there is the heat balance in the still, which gives qga = qr + qc + qe + aoHa .
(4)
As suggested by Baum[11] the heat transfer by evaporation can be calculated by the approximate formula qe= he(tw-tg)
(4.1)
306
C.L. GUPTA
where
h, = ,4 - Bt~ + Ctw 2.
(4.2)
On the basis of exact analysis by Dunkle [12], the quantities ,4, B, C would be nonlinear functions oft, and (tw - to). However. for the purposes of eliminating the unknown (glass cover) temperature, the explicit dependence of he on to has not been considered in the present analysis. This assumption is the main factor in limiting the accuracy of the time series approach in its application to solar stills. The heat transferred by radiation between the water in the still and the glass cover, and by the air in circulating the vapor-air mixture, can also be represented in the linearized form by the equation
qr + qc = hwo(t,,-to).
(4.3)
Combining Eqs. (3), (3.2), (3.3), (4), (4.1), (4.2) and (4.3), two basic equations can be obtained for the mean water temperature in the still and the heat transferred by evaporation, which is directly proportional to the amount of distillate output. These equations are c
dt,,
. -d g + t,e = t. +
[ Ta,e(fhoo + h,,,~+ he) + ao(hwo + he) ] 4-h'
¥
j H.
(5)
where
fh°°(hw"+he) ] C w o = C w g / [ K h + (fhoo+h,,,,,+he)l
(5.1)
and
a-U] U']] D[tw-- [t.+ (mo,
(6)
where
D = (fhoohe)/(fhoo+hu,o+he) -- A1 +Bit,,,,
(6.1)
A I, BI being constants. The right-hand side of Eq. (5) gives the excitation function in terms of equivalent air temperature. As before, it can be modified to give the excitation functions in terms of equivalent solar radiation also and, once the water temperatures are known. Eq. (6) can be used to determine hourly outputs of output on a daily basis. The multiplier of H~ cannot be assumed as a constant in Eq. (5), as was done in the Eq. (2.1) for the case of water heaters, because of strong dependence of he on t,,, as in Eq. (4.2). However, for design purposes, a mean value is taken for the daily cycle. The design value for the multiplier of H, in Eq. (5), corresponding to a constant wind velocity of 15 m.p.h, and for different slopes of the glass cover and values of Kn. may be taken as shown in Table 2. The value of h,,,o is found to be practically constant and equal to 1.5 B.t.u. ft -2 hr -I deg F -1.
Dynamic performance
of solar energy systems
307
EXAMPLES OF APPLICATION To verify the practicability of this technique and to estimate the magnitude of errors arising from simplifying approximations used in deriving excitation functions, three examples of solar energy systems were chosen from the literature such that the dynamic performance data for at least two days under different climatic conditions were available for each. From one set of results, the unit response function has been derived by using the time series division and excitation functions derived in Section 4. This unit response function has been used to predict the performance under different climatic conditions by using the method outlined in Section 3 of the paper. The predicted and Table 2. Design values for multiplier of H. in Eq.
(5)
Multiplier of H, Slope of glass cover 15° 25°
Base insulation and vapor sealing parameter
gb = 1"0 Poorly sealed; no base insulation
0.22 0.26 0'28
Kb = 0-5 Properly sealed or some base insulation Kb = 0"25 Properly sealed and good base insulation
0.22 0.27 0'30
OBSERVED
70
i-2
i~
~
Z
A
~,00
A 0'700
I
I
0900
I
I
I
]
I100 TIME
1300
I
I
1500
l
i
1700
(hr)
Fig. 4. Mean water temperature in a solar-water heater (flat black collector).
observed hourly values of the mean system temperatures for a black-painted solarwater heater reported by Gupta et al. [13], for a selectively black-painted water heater reported by Yellot and Sobotka [ 14] and the mean water temperature and distillate output for a Mark llI-type solar still developed by the Division of Mechanical Engineering, CSIRO [Read, private communication] are shown in Figs, 4, 5 and 6, respectively. For solar-water heaters, the hourly values during daylight hours, for which the basic model is valid, compare to within 3 deg F for both cases. This may be taken as reasonable accuracy for application engineering work, particularly because the test data for the two days were not obtained under identical conditions of draw-off, as indicated by the initial temperature values corresponding to 0800 hr.
308
C.L. GUPTA OOSERVED
170
3OO
,~
H
.
N ~o m
L t.)
5C iO0
0700
ogo0
liO0
1300
1500
1700
TIME (hr) Fig. 5. Mean water temperature in a solar-water heater (selective black collector). r/ J
~ F I
'
5
0500
0
D,ST,LLATERATE
#7'~',,L \
-~ | J
OBSERVED PREDICTED
I
/ I/_\\
1
/ All
ogO(~
/HOURLY
II \'X
"OF
=
"~f-~
\ N'~'t
1300
~
1700
>-
,HOURLYDISTILLATE I
5
2100
OtO0
0500
~LN
.0
TIME (hr)
Fig. 6. Output for Mark I II solar still (plan area = 500 ftz). In the basin-type solar still, the hourly values o f predicted m e a n w a t e r t e m p e r a t u r e s agree within the same range. T h e distillate o u t p u t agrees to within 0.01 lb ft -2 on an hourly basis and within 0.02 Ib ft -2 on the daily total basis. C o n s i d e r i n g that the solar distillation is a highly non-linear process, this o r d e r o f a c c u r a c y should be quite a c c e p t a b l e for application w o r k requiring a k n o w l e d g e o f d y n a m i c p e r f o r m a n c e . Alternatively, a quadratic t e r m in t~ c a n be included in Eq. (6.1) to i m p r o v e the a c c u r a c y . NOMENCLATURE C,~h Thermal capacity of water heater per unit collector area B.t.u. ft-~ deg F-' C~v Thermal capacity of basin-type solar still per unit plan area B.t.u. It-= deg F -' Eg Emissivity of glass; Fg. plate efficiency factor
Dynamic performance of solar energy systems
309
f G l a s s cover area per unit plan area of solar still H Total solar radiation incident on inclined plane of collector B.t.u. ft-" hr -~ H, Total solar radiation incident on a horizontal surface B.t.u. ft -2 hr -~ h,a, hoo Surface coefficients of convective and combined heat transfer from glass cover to air B.t.u. ft -2 hr -~ deg F -I h,,~, he Coefficients of convective and radiative, and of evaporative heat transfer from water to glass B.t.u. ft -z hr -I deg F -1 Kb Thermal loss coefficient of still base and vapor leakage B.t.u. ft -2 hr -~ deg F -~ Heat transferred by convection and by radiation between water and glass cover B.t.u. ft -2 hr ' qe, qr Heat transferred by glass cover to surroundings B.t.u. ft z hr-I qua T o, T, Glass cover and sky temperatures (°R) Ta, to Shade air dry bulb, and glass cover temperatures (°F) I u, Mean fluid temperature in system (°F) U Overall heat loss coefficient per unit collector area B.t.u. ft -z hr -1 deg F -~ Solar absorptivity of glass and of water T¢1 Transmittivity of glass Tt~ Effective transmittivity and absorptivity product O" Stefan's Constant = 17.2 × I0 -~° B.t.u. ft z hr -~ o Time REFERENCES [ 1] C. k. Gupta and H. P. Garg, System design in solar water heaters with natural circulation. Solar Energy 12, 163 (1968). [2] A. Whillier, Performance of black-painted solar air heaters of conventional design. Solar Energy 8, 31 (1964). [3] R. N. Morse and W. R. W. Read, A rational basis for the engineering development of a solar still. Solar Energy 12, 5 (1968). [4] D . J . Close, A design approach for solar processes. Solar Energy 11, 112 (1967). [ 5] P. I. Cooper. Digital simulation of transient solar still processes, Solar Energy 12, 313 (1969). [6] H. Buchberg and J. R. Roulet, Simulation and optimization of solar collection and storage for house heating. Solar Energy 12.31 (1968). [7] G. O. G. Ltif, D. J. Close and J. A. Duflie, A philosophy for solar energy development. Solar Energy 12.243 (1968). [8] A. Tustin, A method of analysing the behaviour of linear systems in terms of time series. J. Inst. elect. Engrs 94 {1I-A), 130 (1947). [9] D. G. Stephenson and G. P. Mitalas, Cooling load calculations by thermal response factor method. Trans. Am. Soc. Heat. Ref. A ir-Condit. Engrs 73, paper I I I. I (1967). [ 10] R. C. Jordan (Ed.), Report o f A S H R A E Technical Committee on Solar Energy Utilization (1967). [ I 1] V. A. Baum, Solar distiller, U.N. Conference on N e w Sources of Energy, Rome, paper S/i 19, ( 1961 ). [12] R. V. Dunkle, Solar water distillation: the roof-type still and in multiple-offset diffusion still. International Developments in Heat Transfer. pp. 895-912. The American Society of Mechanical Engineers (1961). [ 13] C. L. Gupta, H. P. Garg and R. Gangnly, Solar cure electric water heater. Indian east. Engr 111(1), 21 (1969). [ 14] J. I. Yellot and R. Sobotka, An investigation of solar water heater performance, Trans. Am. Soc. Heat. Re.[. A ir-Condit. Engrs 70, 425 (1964). APPENDIX
Algebra o f time series The time series are meant to represent time functions, which are zero up to a point of time and then are given in form of numbers at equally spaced time intervals. The algebraic operations o f addition, subtraction, multiplication and division correspond to physical operations and are best understood by the following examples: Addition If a parameter is subject to two or more excitations in parallel, e.g. solar energy systems are subject to shade air temperature and solar radiation, the resulting excitation is the sum of these two. (Col, ezl, ezt, eaj .... ) + (eo2, etz, ez2, e.~z.... ) -= (e01 + e0z, eH + el2, e2t -k e2z, e3t -4-ez2.... )
Subtraction is the corresponding procedure when one of the branches of a system, in parallel, is considered. Multiplication If two or more linear systems act in series, e.g. an excitation function and a physical system represented by its unit response function or two physical systems in series such as insulated panels, then the
S E V o L 13 No. 3 - B
310
C.L. GUPTA
overall response function or transfer function is obtained by multiplication as below: Time
A
Time series for input or first system Time series for unit response function or second system Output due to eo Output due to e~ Output due to e~
Response function or overall transfer function = 2 columns
2A
3A
4A
e0
el
e~
e3
e4. •.
u0
ul
us
u3
u4 • • •
uoeo
u l eo
u:eo
useo
u4eo • • •
uoe~
llles
u~el
I~sel . . .
uoe~
ule2
u~e2...
V~
V3
V,...
V0
V~
5A...
The resulting time series terminates only if the constituting series terminate and the expression for any N
general term of the resulting series is V. = ~ e~_j,. Uj where N depends upon the number of terms involved J-0
in the constituting series. Division The process of division corresponds to the determination of a unit response function when the response of a linear system to a specified excitation is known. The method is similar to long division except that digits can be nun:lbers and carrying is not permissible. The example shown graphically in Fig. 3 is the original illustration given by Tustin in [8], and is worked out below: 1, 2, 1, 1 J 2, 5,7, 10,6,4, I~ 2.1,3, 1 2,4,2, 2 1,5, 8,6 1,2, 1,1 3, 7 , 5 , 4 3, 6 , 3 , 3 1,2,1,1 1,2,1,1 The quotient is (2, 1, 3, I). The division is exactly an inverse process to multiplication. As can be seen, multiplication and addition are both associative and commutative.