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A parametric study of closed loop solar heating systems—II
Solar Energy Vol. 32, No. 6, pp. 707-723, 1984 Printed in the U.S.A.
0038-092X/84 $3.00 + .00 © 1984 Pergamon Press Ltd.
A PARAMETRIC S T U D Y OF CLOSED LOOP SOLAR HEATING SYSTEMS--II J. P. K ~ N N ^ f
Solar Energy Unit, University College, Cardiff, Wales (Received 5 May 1982; accepted 15 July 1983; revision received 3 October 1983)
Abstract--A nondimensional equation describing the closed loop solar heating system with either parallel or series auxiliary heaters is derived. Using European weather data the monthly solar fraction is calculated for variations in the major non-dimensional groups. A correlation is given relating the solar fraction to three non-dimensional parameters M, Kc and Rc. M is the ratio of energy available on the collector aperture to energy demand. To calculate the energy available it is necessary to know the monthly utilizability. Kc is the ratio of the store temperature required for a 100 per cent solar contribution to the average monthly collector peak stagnation temperature. Both these temperatures are referenced to the demand temperature. Rc is the effective "turn over time" of the store, i.e. the number of days to empty the energy contents of the store. The magnitudes of Kc and Rc depend on the load heat exchanger size. Comparisons between the solar fraction predicted with a dimensional hour by hour computer model and that by the correlation are made for two system types. The agreement is good and it is concluded that the correlation can be used as a reliable method to optimise closed loop solar heating systems. !. INTRODUCTION In the closed loop heating system energy is provided at a constant temperature, usually by means of a heat transfer fluid circulating through heat exchangers. This contrasts with the open loop heating system in which water from a cold storage tank is heated to a higher service temperature and then after serving its purpose is discharged to waste in a contaminated form. For a given demand temperature and the same components, the closed loop will always have a lower efficiency than an open loop system because it is characterised by a minimum operating temperature equal to or greater than the demand temperature, whereas the open loop is characterised by a minimum operating temperature close to ambient. However, the open loop system is energy wasteful and in many cases it may be more economic to convert it into a closed loop system by use of a heat exchanger between the contaminated waste and the inlet. Thus, the closed loop system is a more important market for solar energy. Design and optimisation procedures for solar heating systems are important to ensure that the large initial investment gives a rapid return. The design procedures need to cope with a large number of important variables and at the same time be easy to use. A powerful method that can be used to achieve these objectives is non-dimensional analysis. The open loop solar heating system has already been analysed by this method[I]. This resulted in a simple design procedure valid for a large range of collector types, demand temperatures, demand patterns and locations. The work reported here demonstrates that the
same method can be used to develop a design procedure for closed loop solar heating systems. 2. N O N - D I M E N S I O N A L
SYSTEM MODEL
The system under study is shown in Fig. 1. It can include either a parallel auxiliary heater (Fig. la) or a series auxiliary heater (Fig. lb). Systems with a heat exchanger between collector and store and with pipe losses are accounted for by modifications to the collector performance parameters[l]. Initially the parallel auxiliary heater will be analysed. Later it will be shown that by modifying the demand temperature the series auxiliary heater can be made equivalent to the parallel auxiliary heater. An energy balance on the store, which is assumed to be fully mixed, gives energy stored = energy c o l l e c t e d - store losses -
energy demand
Mscp d T / d t = F"tloA (ibGb + idGd -- U/~lo(T -- To)) + -- U , A , ( T - T,,) - q e C ~ , ( T - Td).(1) Gb, Gd, T, Ta are all time dependent parameters. The energy collected is only accounted for when it is positive (signified by the + sign), ib, ia are incidence angle modifiers for beam and diffuse irradiation respectively and ~/0, U are based on mean collector fluid temperature. is the heat exchanger effectiveness and Cn, is the minimum thermal capacitance flow rate in the heat exchanger, q is a time dependent control parameter and defines the demand pattern.
when q ( t ) = 1.0 energy is required fPresent address: I. T. Power Ltd., Mortimer Hill, Reading, England. 707
when q ( t ) = 0.0 no demand exists.
708
J.P. KE~A
Auxiliary He~er
r
Load Heot -b Exchanger(s)
Thermal Store
(a) Closed Loop Solor Heating with Porallel Auxilx~ry
Auxiliary Heofer
Load __
Exchenger
Thermal Store
~ c for
J
.ipHeot
L
@
(b) Closed Loop Solar Heating System with Ser0es Auxiliary Fig. i. The closed loop solar heating system.
Energy is only extracted from the store when the store temperature is greater than the demand temperature. Heat exchangers with a minimum useful temperature greater than the demand temperature will be considered later. The store output is also subject to a maximum extraction rate Qm,x- When the store output is Qm~ the load has been met. Thus o < Echo(r-
r~) < Q _ .
(2)
If for the purposes of analysis, a load that does not vary with time is considered then ~m,x is constant and is given by p.,,= = ( U A ) , , ( T , -
To).
(3)
Tc is the sink temperature to which the load is referenced. The overall transfer coefficient from the load ( U A ) a is the rate of energy demand per temperature rise. Choice o f non -dimensional parameters
To reduce eqn (1) to non-dimensional terms the non-dimensional variables must be defined. The nondimensional temperature 0 is chosen in the same way as in Ref. [1]. For the open loop system 0 is defined
as
o = (r-
(4)
T~)/(T~- r3.
Temperatures are referenced to the minimum system temperature and expressed as a fraction of the required temperature rise. 0 = 1 specifies that the load has been met. It is unnecessary to achieve temperatures defined by 0 > 1. For the closed loop system the same definition is used (5)
0 = ( T - Td)/(Th -- Ta).
The demand temperature, Td, is now the minimum system temperature and Th is the store temperature required so that the output from the heat exchanger exactly meets the load, i.e. the solar contribution is 100 per cent. For this case the output from the heat exchanger is equal to the load ECmm(T~ -- T~) = (Ua)~(T,~- T3-
(6)
Re-arranging (T~ - T~) = ((UA)J~C~.)(T~--
TJ.
(7)
709
A parametric study of closed loop solar heating systemw--II or re-arranging
Substitute eqn (7) into eqn (5) 0 = ( T - Td)Lx/(T a - T~)
(8)
Gr
where Lx = EC~,/(UA)d and is a dimensionless parameter relating the heat exchanger size to the load. It is noted that this non-dimensional parameter appears also in the f-chart design method[2]. As for the open loop system, 0 = 1 specifies that the load has been met and it is thermodynamically inefficient to store and collect energy at temperatures defined by 0 > 1. The non-dimensional irradiance and time are defined as G' = (G + (UI~o)(To - Ta))IG*
(9)
G = ibGb + idGa,
(I0)
with
(15)
ddays
\Ndays
The r.h.s, of eqn (15) is termed the utilizability ~b, a concept first used by Whillier[3] and studied in detail by Liu and Jordan [4]. For this particular application ~b Is the hourly utilizability at solar noon (this will correspond to the peak in the long term) and at a critical radiation level of (U#lo)(Td -- T°) G* = ~bl~p.
(16)
Non-dimensional equation Substitution of eqns (8), (9) and (11) into eqn (1) and dividing by the daily load L results in the non-dimensional system model Rc dO/dt = X(G' - KcO)+ - LsRc(O - 0~), qO (17)
where
and t" = t/t*.
(11)
t* is taken to be 24hr. Again because Td is the minimum system temperature, the ambient energy input is referenced to Ta in eqn (9). Hence the ambient energy input will generally be a net loss and it is necessary for G to be at least greater than (U#lo)(Td- T~) before the collector can operate. The reference irradiance G* is defined in the same way as in Ref. [1]. For any given day G* = (Gp + (Ul~o)(T, - Ta))
(12)
and
Rc=R/Lx
(18)
Kc=K/Lx
(19)
Ls = U,A,t*/M~cp X = F"~oAG*t*/L
(20)
R = M,%(Ta-
(21)
Tc)/L
(22)
K = U ( T a - Tc)/(~lo~Gp).
In non-dimensional terms the control of the store output, eqn (2), can be expressed as
with
0 < qO < 1.
(13)
G. = ib, G, + i . a . .
Normally monthly or yearly system performance is of interest and G* becomes the average daily peak input. However, for the closed loop system there is a possibility that Gp may be less than (U/~o)(Td-- To) for any given day. In this case there will be no energy collected. Therefore, for long term weather sequences G* must be calculated as
G*=(N~eays(Gp-t-(U#lo)(~ a -
Td))+)/N.
(14)
The + sign indicates that only positive values are included in the summation. N is the number of days in the time period under consideration. For simplicity the monthly average ambient temperature/'o, rather than the individual daily temperature, is used. Equation (14) appears in most studies of solar heating systems. Dividing both sides by Gp gives G*/~p= ( ~ y
Gp + (U/tlo)(l'o- Ta) +)/(N~p)
(23)
Integration of eqn (17) over one day gives the solar fraction E E =
:o
qO dt'
Kc0) d,'
Rc(o
- 0=) d,'
- f°:Rc dO.
(24)
d0~ As with the study of open loop systems (I) the dependence of sunrise/sunset time is removed by changing the reference time for collectedenergy. This is defined as t" = t/t** with t** = H / G * and H given by
H = J(G + U/no(/',--T~))+ dt,
(25)
i.e.H is the energy available to the collector. Equation (24) must only be integrated over positive values
J.P. ~ ^
710
of the integrand. For monthly weather sequences with hourly radiation data
H--
~
(G + U/tlo(J'o- Td))+dt
tion is exactly the same as eqn (28) except that L x is replaced by Ld. K c and R c then become Kc = K/Ld;
(26)
(34)
Rc = R/Ld.
all hours
or in terms of the monthly average utilizability (27)
H=~[=H,
where i-. is the monthly average incidence angle modifier. The monthly average utilizability ff is the fraction of the monthly solar radiation that is available to the collector. A correlation relating ff to the monthly clearness index ~ r has been given by Klein[5]. Substituting t" into eqn (24) gives E = M
(G' - KcO)dt" -
L s R c ( O - O~,)dt'
Note that the utilizability functions and are now determined at a critical radiation level of (U/~o)(T, - :r,,) instead of ( U / ~ o ) / ( T d - J',). The store return temperature 7", can be expressed in terms of the demand temperature Td by the energy balance on the heat exchanger (eqn (31)). Substituting Th from eqn (6) into eqn (31) and re-arranging T, = T d + ( T , t - T ~ ) ( I / L x -- 1/Ld). Space heating s y s t e m s
For space heating systems or other systems with time varying loads the maximum extraction rate from the store will not be constant but is given by O,m~ = ( U A ) d ( T d - 7",)
-
Rc~°:dO
(35)
(36)
(28) where To is time varying. In these cases Th can be defined as the average temperature required to meet the load in which case eqn (6) is replaced by
dot
with M = F%loAH/L.
Thus, the dosed loop non-dimensional equation is identical to the open loop non-dimensional equation. Series auxiliary
With a series auxiliary heater the feed temperature to the load heat exchanger is always Th and hence for a constant load the return temperature is constant. The energy extracted from the store is given by =C(T-
(29)
T,).
C, is the thermal capacitance flow rate in the store loop of the heat exchanger. The minimum temperature in the system is now the return temperature from the heat exchanger T,, so the non-dimensional temperature 0 must be defined as 0 = (T - T,)/(Th -
1",).
(30)
0 can be expressed in terms of Td and T¢ by an energy balance on the heat exchanger. The energy input to the heat exchanger always meets the load such that C(Th-- T,)ffi (UA)~(T~- To).
(31)
ECmi,,(Th-- Ta) =
(37)
L/AtN
where N is the average number of operating hours per day (i.e. the number of hours when a load is required. The average sink temperature T~ becomes Tc = T a - L x ( T s -
(38)
Ta).
Space heating systems with a parallel auxiliary heater may also be characterised by a load heat exchanger with a minimum operating temperature greater than the demand temperature. For example if the demand (room) temperature is 20°C then an energy supply at temperatures as low as this will be perceived as a draught (cool air current) and it may be necessary to operate a fan coil heat exchanger at temperatures greater than say 30°C. This type of system can be accounted for by taking a reference temperature of T=i.. The non-dimensional temperature 0 becomes 0 = (T - Tmi.)Lx/(Ta-
(39)
To)
and the non-dimensional store equation involves one extra parameter
Substituting for ( T h - T,) in eqn (30) RcdO/dt = X(G'-
0 = (T - T,)Ld/(Tz - Tc)
(32)
L d = C,/(UA)d.
(33)
KcO) - L s R c ( O - Oa~) (40)
- q(O + SmiJ
where where s.,° =Ec=i.(r=i~ - r J / ( U A ) ~ ( T ~ By choosing 0 in this way the series auxiliary heater can be analysed in exactly the same way as the parallel auxiliary heater. The non-dimensional equa-
I'o).
(41)
The only difference between the non-dimensional parameters for this case and the parallel auxiliary
A parametric study of closed loop solar heating systems---II case analysed above is that the utilizability functions are determined at a critical radiation level of (U/~lo)(Tmi,- T°). Note that for the series auxiliary heater or the parallel auxiliary heater with T~, = Td, S,,~, becomes zero and eqn (40) reduces to eqn (17). 3. PHYSICAL SIGNIFICANCE O F PARAMETERS
The parameters M, K, Ls, R are the same as those for the open loop system. M is the ratio of energy available to energy demand; K is the ratio of rate of heat loss when the collector is operating at the demand temperature (now the lowest temperature in the system) to peak rate of energy input; Ls defines the store loss and R is the number of days storage available. However, for the closed loop system the parameters K and R have to be replaced by Kc = K / L x and Rc = R / L x (replace L x by Ld for the series auxiliary). In fact this is not really the change that it appears to be because Kc and Rc hold the same physical significance as K and R for the open loop system. Kc can be re-arranged as Kc = (Th -- Ta)/(T, -
(42)
Td)
where T, is the maximum temperature obtainable from the collector. Thus Kc is the ratio of desired temperature to maximum possible temperature with both temperatures referenced to Ta.
Rc can be re-arranged as Rc = M,cp/~Cm~,t* (parallel aux)
(43)
Rc = M,cp/C,t* (series aux).
(44)
Hence Rc is the "turn over time" of the store. For the open loop system with R = 1 the contents of the store are emptied every day, with R = 2 every 2 days, etc. Similarly for the closed loop system with Rc = 1 the energy content of the store is withdrawn every day. 4. P E R F O R M A N C E OF CLOSED L O O P S O L A R HEATING SYSTEMS
The solution can be obtained to the nondimensional store eqn (28) in the same way as for the open loop system. The solar fraction E depends on the parameters M, Kc, Re, Ls, T d (or 7", for a series auxiliary), Tc and Lx. The latter three are included because they determine the non-dimensional ambient temperature. For parallel mode heat exchangers with minimum operating temperatures greater than the demand temperature the dependence of the solar fraction E on Smi, must also be included. The monthly solar fraction has been computed for the base system of Table 1 using meteorological data from 6 locations (Table 2). The results are shown in Fig. 2. The similarity between these results and those for the open loop system is expected because the
Table 1. Specification of the ba~ system Collector
tilt:
45°
Collector
azimuth:
due South
Collector incidence angle modifier:
Store slze:
Rc:I
Store loss :
Ls=O. 2
Ambient around
store:
1.0
-
0.1(1/cos(8)
-1)
0as :0.O
Demand temperature :
60°C
Demand pattern :
Constant
Load Heat Exchanger
Lx=I.O Smln=
0.0
Table 2. Locations used for simulations Location
Year
Latitude
Longitude
Kew, England
Nov 63 - Oct 64
51029 ' N
00°19 ' W
1973
48048 ' N
02°02 ' E
Valentia, Ireland
CornIX)S l t e
51°57 ' N
10o13 ' W
Copenhagen, Denmark
Cornpo si te
55°41 ,
12°34 ' E
Hamburg, W.Germany
Composite
53°30 ' N
10°00 ' E
Hohenpeizenberg, W. Germany
Compo slte
4~48'
11001 ' E
Trappes, France
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M Fig. 2. Performance of the base system. non-dimensional system equations are similar. A least squares regression to the data of Fig. 2 gives a similar performance correlation E = 0.61M/(0.73 + MK*)
(45)
K* = K c / ( l . O + O . l l K c ) for Kc >0.65
(46)
with
and
carried out for the base system with a range of Rc between 0.1 and 2.0. The universal performance curves for this range are shown in Fig. 4. Data points are omitted for clarity. The effect of Rc is dependent on the parameter MK*/Rc. These curves can be represented by the equation E l M = a/(b + MK*/Rc)
(47)
K* =Kc/(l +0.11Kc) for Kc >0.65
(48)
with
K* = (Kc + 0.061M)/(0.69 + 0.87Kc) for Kc < 0.65.
and This correlation is shown by the solid lines in Fig. 2. As for the open loop system the family of curves can be collapsed to a single curve. This is shown in Fig. 3.
K* = (Kc + cM)/(0.69 + 0.87Kc) for Kc < 0.65 (49)
a = 1.67/(1 + 1.74Rc)
(50)
Effect o f Rc
b = 1.13/0 + 0.55Rc)
(51)
The performance correlation above has a limited range because it is only valid for Rc = 1 and, therefore, does not include the effect of different heat exchangers or store sizes. Simulations have been
c = 0.136Rc/(1 + 1.23Rc).
(52)
Equations (50) to (52) are valid for 0.1 < Rc < 2.0.
713
A parametric study of closed loop solar heating systemsIII
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. v
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Kc= vl~= ,. K c = I × Kc= I r~ K c = 3 . 0
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.40.
.30.
.20.
.10
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~0.2
~0.5
'I .0
~2.0
I5
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~I0 .0
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The validity of ¢qn (45) in hotter locations than northern Europe has been examined. Hourly meteoroiogical data for Madison, U.S.A. Oat. 43 °) and Albuquerque, U.S.A. Oat. 37°) were synthesized from the monthly average dearness index ~r. A sequence of days with daily clearness index distribution comparable to the long term average distribution was generated using data given by Liu and Jordan[5]. The hourly irradiance was obtained from the daily irradiation using the correlation of Collares-Perc~ra and Rabl[6]. The results are shown in Fig. 5. Comparison with ¢qn (45) (shown by the solid line in Fig. 5) indicates the general application of the correlation to other locations. $. SENSITIVITYTO OTHER PARAMETERS The sensitivity of system performance to changes in the base system has been examined. Variations in Td, T~, L x only effect the value of the non-dimensional ambient temperature, so their effect should be small. This has been found to be the case
as shown in Figs. 6 and 7. In Fig. 6 predictions of monthly solar fraction for a demand temperature of 20°C are shown compared with the correlation given in cqn (45). Because this demand temperature corresponds to a space heating system, only the winter months (October-March inclusive) were used for the 6 locations given in Table 2. It is clear that although the scatter is somewhat increased for the lower demand temperature, the average solar fraction for a given M and Kc does not change significantly. Figure 7 shows data for the base system but with a heat exchanger defined by L x = 2.0. When compared with Fig. 3 it can be seen that the scatter is also increased but again the average value is still the same. This confirms that the effect of T~, T~ and L x is adequately accounted for in the parameters Kc and Rc. Sw is the minimum non-dimensional energy input to the load heat exchanger below which it cannot be used (usually because of thermal comfort limitations in space heating systems). It represents the proportion of the instantaneous load that is required before the heat exchanger operates. Simulations reported above are for Smi, ----0.0. This corresponds to either a
714
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.eo_
.50
.40
Rc
= 0.i = 0.25 -Rc = 0.50
.30
~ R c
= 1.0
=2.0
.20.
.10.
.00
3.1
'0.5
'1.0
r5~T10.e
1100
M. K ~/R~. Fig. 4. Effect of Rc.
heat exchanger that can work down to inlet temperatures equal to the demand temperature or to a series auxiliary heater. Simulations have been carried out for the base system but with S~, in the range 0.1-2.0. Figure 8 shows the results for S~, = 1.0 together with the performance correlation for S ~ = 0.0. It is implicit in these results that the utilizability functions have been calculated at a critical radiation level of ( U / ~ I o ) ( T ~ - f ' o ) . Also the ambient temperature around the store is assumed to be equal to the demand temperature (corresponding to 0~, = - S~,). It is seen from Fig. 8 that for S~, = 1.0 there is a region at low M K * at which the normalised system efficiency departs from the correlation found for Sm~, = 0. In the case of S ~ = 1.0 the correlation may be used for M K * > 0.5. The influence of demand pattern on system performance has been examined by Klein and Beckman[7]. Differences were noted between a 6 a.m. and 6p.m. load, a 6p.m. and 6a.m. load and a constant 24 hr load. These studies were for an infinite
load heat exchanger. However, if the heat exchanger size is finite its size must influence the effect of demand pattern. Taken to the extreme if the draw off is only 1 hr per day an extremely large heat exchanger will be required, whereas if the draw off is 24 hr per day a smaller heat exchanger can be used. Small variations in demand pattern such as those of a space heating system have no effect on the system performance. This has been demonstrated by the data in Fig. 6 which were generated assuming the limitation in store output was given by eqn (36). Larger variations such as a mid-day demand (constant between 10.30 and 3.30p.m.) can have a significant effect. With R c = 0.1 (Fig. 9) the solar fraction is not significantly degraded but with Rc = 1.0 (Fig. 10) there is a large decrease in performance. F o r the 5 hr demand the non-dimensional temperature must be greater than 4.8 (24/5) for a 100 per cent solar contribution. When Rc = 1.0 the non-dimensional temperature does not reach this as often as with R c - - 0 . 1 . Hence the solar fraction is reduced. This means that when the demand occupies only 5 hr per
A parametric study of closed loop solar heating systems---II
715
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I'.00
I'.50
2'.00
2'.50
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M Fig. 5. System performance using synthesised U.S. weather data. day instead of 24 hr, a heat exchanger defined by high Lx is required to ensure that Rc is low. It can be concluded from this sensitivity study tllat both S~, and demand pattern can influence the solar fraction significantly. Until further studies are made on the effect of these parameters the application of eqn (47) must be restricted to 24 hr or space heating demand patterns and for non zero values of S u , to the range MK* > 0.5.
at temperatures close to the demand temperature. This in turn means that the average store temperature can be lower and the collector operates with lower heat losses. Conversely a poor heat exchanger (Lx~O) will require high inlet temperatures, high store temperatures and hence high collector losses. At first sight this suggests that poor collectors can be compensated by good heat exchangers. However, this is not the case because poor collectors will lead to low utilizability (hence reducing M and increasing
Kc). 6. I~;~-~-KCTOF HEAT EXCHANGERSIZE The load heat exchanger size can have a significant effect of the performance of the closed loop system. This has been partially demonstrated in the discussion on demand patterns. It has been shown that the effect of heat exchanger size is adequately accounted for in the parameters Kc and Rc which in turn influence the solar fraction E. The physical link between heat exchanger size and collector loss (as shown mathematically in Kc) is because a perfect heat exchanger (Lx ~ oo) will be able to meet the load
The physical link between heat exchanger size and store size (shown mathematically by Rc) is because good heat exchangers deplete the energy contents of the store in a shorter time period, thereby requiring a higher storage mass to obtain the same daily energy storage. In practice the store output is limited so that it does not over supply the load. Consequently it is possible to have a heat exchanger that is oversized, because its output is always limited. To determine the optimum heat exchanger size for a given application it will be necessary to offset the
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increased solar fraction (due to a reduction in Kc) against the increased cost of a good heat exchanger. 7. SERIES V. PARALLEL AUXILIARY HEATERS
The parallel auxiliary heater is generally thought to be a better choice for solar heating systems because the system has a lower minimum operating temperature. The choice of a series or parallel heater can be examined using the non-dimensional parameters. The criteria governing which mode of operation is better depend solely on the properties of the heat exchanger. The important characteristics are the minimum operating temperature (parallel auxiliary), the heat exchanger return temperature (series auxiliary), and the heat exchanger sizing parameters L x (parallel) and Ld (series). Generally the series auxiliary heater will be such that 7", (series) > Ta (parallel) and Ld (series) > L x (parallel). The former condition acts to decrease system performance and the latter favours an increase in system performance. For a heat exchanger with a high effectiveness it is possible that the return temperature when operating in a series
mode is lower than the minimum useful temperature when operating in a parallel mode, in which case the series auxiliary heater would be the better choice. However, to obtain an optimum performance from a given collector/store combination it can be expected that in practice for many cases a heat exchanger operating with a low minimum useful temperature and in a parallel mode is the better choice. 8. THE DESIGN METHOD
This section will illustrate how the correlation given in eqn (47) will serve as a design method for closed loop solar heating systems. As with th e open loop design method the closed loop method can be split into three stages (1) Determine the three meteorological parameters H s, Gp and it=. Gp can either be obtained from hourly weather data or by using the correlation of Collares-Periera and Rabl relating monthly average daily peak irradiance to monthly average daily total irradiation [6]. (2) Parameters Kc, M and Rc are calculated. To
717
A parametric study of closed loop solar heating systems---II I .80
o x
o
x bt~K X .X~. ~ 0 +x
v VvW~ •
%
.70_
I:1 O
~, K c = v Kc : + Kc=I × Kc: I o Kc=3.0
II
E/M
.G8
÷
X
v v l ~ • xO
0
.58
v~
.2 .5 .8 .5
•
~o
÷ .40_
.30
V V~
.20
.I@.
.00 .I
TO .2
'0,5~--T~o
12. ~ . . . . . . . % . 0
L10.~[
V1 oKm Fig. 7. Effect of Lx.
determine Kc and M two utilizability functions are required, one to calculate the monthly average utilizability and the other to calculate the hourly utilizability of the daily peak irradiation. The latter can be obtained from charts given by Liu and Jordan [4]. is given as a function of the critical radiation level Xc, the monthly clearness index £ r and the monthly average ratio of beam radiation on a tilted surface to beam radiation on a horizontal surface Rb. In terms of the notation used here the critical radiation level Xc is xc = v ( r . , , . -
L)/(nod,).
An alternative method of obtaining 0 and ~ is to generate them from hour by hour weather data. When comparing the design method with an hour by hour computer model this is a necessity because the correlations for 0 and ~ relate to the long term average and the year under consideration may not be representative of the long term average. For systems with heat exchangers between the collector and store and with pipe losses the parameters M and Kc are modified as in Ref. [1]. (3) Providing M < 3 and Rc < 2 the monthly solar fraction is given by
(53) E = aM/(b + MK*/Rc)
(54)
(Replace Tmin by T, for the series auxiliary heater). The monthly average utilizability ~ can be obtained from a correlation given by Klein[8] in terms of Xc, £ r and Rr/R,. Rr is the ratio of monthly total radiation on a tilted surface to that on a horizontal surface and R, is the ratio of radiation at noon on the tilted surface to that on the horizontal surface for an average day of the month.
where a, b a n d K * are given by eqns (48)-(52). To compare the design method with a full simulation two systems have been simulated with a dimensional hour by hour computer model. These are specified in Tables 3 and 4. Also shown is a comparison between the solar fractions predicted by the design method and those predicted with the hour by
718
J. P. ~qCNA Table 3. Performance predictions: System A System specification Collector :
Type Flat plate Performance parameters no = 0.67; U = 3.OWm-2K ~ Tilt 45 ° Azimuth due South Area 50 m2
Store :
Mass 2500 kg. Loss coefflclent I W m-2K-1 Surface area 10 m 2
Load Heat Exchanger: Lx 1.0 Minimum useful temperature: 25°C
Demand :
Temperature 20°C -1 Space heating (UA~ : 100 W K
System Performance Monthly and Seasonal solar f r a c t i o n s Location
Trappes
Kew
Month
Full Slm.
Design Method
Full Slm.
Design Method
Oct Nov Dec Jan Feb Mar Apr Heating Season
0.97 0.63 0.18 0.19 0.50 0.58 0.75 0.50
0.96 O.60 0.18 0.21 0.46 0.57 0.97 0.51
1.00 0.86 0.37 O. 2o 0.44 I. O0
1.00 0.73 0.35 O. 22 O. 45 O. 99 O. 96 0.63
I . O0
0.65
Table 4. Performance predictions: System B System specification Collector:
Type Flat Plate -I Performance parameters no : 0.67; U : 2.0 Wm -2 K Tilt 45° Azimuth due South Area 500 m2
Store:
Mass 2500 kg. Loss coefficient I Wm-2 K- i Surface area 10 m 2
Load Heat Exchanger: Lx : 4.0 Minimum u s e f u l temperature: 60°C Demand:
Temperature 60°C (UA~ : 400 W Kq
System Performance Monthly and Yearly solar fractions Location
Kew
Trappes
Month
Full Sim.
Design Method
Ful 1 Sim.
Design Method
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year
O. 08 0.18 O. 22 O. 47 0.70 O.59 O.78 0.73 O. 82 0.45 0.18 O. 07 0.41
0.10 0.20 0.25 O. 47 0.68 0.59 0.73 O. 70 O. 76 O. 46 0.20 O. 09 0.41
O. 09 0.17 0.73 O. 52 d.64 0.81 0.71 O.86 O. 76 0.57 O. 35 0,16 0.51
0.12 0.23 O. 67 O. 50 O. 65 O. 78 0.69 0.81 O. 76 O. 58 0.37 0.19 0.51
A parametric study of closed loop solar heating systems--II
719
I .00.
.90.
.80_
.70_ o
• , " ¢ " , . :v Eli'1
.00_ •• .
.-" _
•
.50_
• _" •
+,
•
I
• ~" .'sb_ • i,
;,
;
.
:
x
+
_wo N . . + , ~ ,,-.,x,~.'._~," ,,, - +~.-x
.',
d
,j.,",
_~o-
a .''°o
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n
~ Kc=
.2
v Kc=
.5
+ K=:I.~
"K
[]
"o
c:
3
.~
"t,_'o+O~ ~'
,+-
o
.
o
/ + x o °°
=+,-~..~,~"
°
Ill
.10.
.00
•I
~0.2
'0.5
~I .0
~2.0
~5.0
n10.g
H .K~ Fig. 8. Effect of S~a~.
hour model. These results are shown graphically in Fig. 11.
9. CONCLUSIONS
The main parametric groups influencing the performance of the closed loop system are M, Kc and Rc. A correlation relating the monthly solar fraction E to M, Kc and Rc has been obtained for a wide range of weather conditions. The parameters M and Kc are dependent on the monthly and hourly utilizability, respectively. The critical radiation level for the utilizability calculation depends on the mode of operation. For the parallel auxiliary heater with a load heat exchanger capable of operating down to the demand temperature the critical radiation level is U(Td--;P~)/r/0; for the scrim auxiliary heater the
critical radiation level is U(T, - T~)/rl0; for the parallel auxiliary heater with a minimum operating temperature greater than the demand temperature the critical radiation level is U(Tmin- I'o)/%. For the latter case the correlation is only valid for MK* > 0.5. Other areas where this correlation is not applicable have been examined and it has been found that a further parameter not yet identified, will be required to take account of load patterns which depart significantly from a uniform demand over 24 hr. Comparison of the results from a full hour by hour computer model with predictions made by the design method shows that the correlation gives satisfactory. results. It can be used to optimisc the choice of collector type, collector area, store size and heat exchanger size for a given demand.
720
J.P. I ¢ ~ A
1 .gg.
.90.
.80.
• W~
.70.
Kc=
.2
Kc= .5 K c = I .0 + x K c = I .5 o Kc=3.0
,i, T
E/M _,,~. v@ .50
.40_
.30
.20.
.10.
.00
'0.2
'8.5
'1 .0
'2.0
M ,K~ Fig. 9. Effect of demand pattern,
R c = 0.1.
15.0
Ilg.g
A parametric study of closed loop solar heating systems--lI
721
1 .00.
.90.
.80.
.70.
Z/~
v + x rl
.60.
Kc: .2 Kc: .5 Kc=I .0 Kc:I .5 Kc=3.0
.50_ I ilm
m
.30
•" ~ = ~
•
.20L
.10.
.00
.1
f0.2
~0.5
~1 .0
'2.0
~5.0
K1 . K ~ Fig. I0. Effect of demand pattern,
Rc =
1.0.
I10.0
722
J.P. K~,rN^ 1 .00. O C3 "r FW r"
.90.
Z .80. CD W 0
Nc
SYST~ t
V
3Y3TI~ B
,70.
W
.60.
.50.
V
.410. .30. / .20.
/
.10. .00 .00
'
.20
~. 4- 0- '
. 6" 0 E FULL
'
. 8' 0
'
1 .00
SIMULATION
Fig. 11. Comparison of design method and full simulation.
Acknowledgements--The author wishes to thank Dr. R.H. Marshall for his contribution to the work reported here and to Prof. B. J. Brinkworth, Dr. W. B. Gillett and colleagues at the Solar Energy Unit, Cardiff for many helpful cornments and discussions, NOMENCLATURE
(Note that To, etc. indicates a monthly average.) A collector area, m 2 A, store surface area, m 2 C, store thermal capacitance flow rate, J K - i Cram minimum thermal capacitance flow rate in heat exchanger, J K - I cp specific heat of heat transfer fluid, J kg- t K - m E solar fraction F" collector flow factor Gb direct solar radiation, W m -2 Gd diffuse solar irradiance, W m-2 Gbp direct solar irradiance at daily peak, W m-2 G~ diffuse solar irradiance at daily peak, W m-2 Gp peak irradiance (including incidence angle moditier), W m - 2 G* reference solar + ambient power input, W m-2 G' non-dimensional power input H, solar irradiation on collector tilt, J m-2 H total solar + ambient energy available on collector aperture, J m-2 F,~ mean incidence angle modifier id diffuse incidence angle modifier ibe direct incidence angle modifier at daily peak K ratio of reference collector loss to reference collector gain Kc ratio of demand stagnation temperature difference to desired store temperature difference Ls store loss parameter L daily load, J
Ld Lx M, M q
Q
Qm~
Rr
Rb R. R
Rc S~o t
t,, t, t' t" T
Tc Td T~ T~ 7", Tm~ Th 7", U Uj
(UA)d
series auxiliary heat exchanger sizing parameter parallel auxiliary heat exchanger sizing parameter store mass, kg ratio of energy available to energy demand draw off control parameter energy output from solar system, J maximum output allowed from store, J ratio of monthly radiation on tilted surface to that on a horizontal surface ratio of beam radiation on the tilted surface to that on the horizontal surface at noon ratio of radiation at noon on a tilted surface to that on a horizontal surface number of days storage available "turn over time" of store heat exchanger minimum input parameter time, s sunrise time, (superscript indicates non-dimensional), s sunset time, (superscript indicates non-dimensional), s non dimensional time (referenced to 1 day) non dimensional time (referenced to H/G*) store temperature, K reference temperature, K demand temperature, K ambient temperature, K ambient temperature around store, K maximum daily collector temperature possible, K minimum useful temperature for heat exchanger, K inlet temperature to heat exchanger for I00 per cent solar, K store return temperature with series auxiliary, K collector loss coefficient, W/m2K store loss coefficient, W/m2K energy demand rate of heat transfer, W K - '
A parametric study of dosed loop solar heating systems--II Xc r/0 0 0~ E
critical irradiance collector zero loss efficiency non-dimensional temperature non-dimensional temperature around store heat exchanger effectiveness hourly utilizability monthly utilizability fl angle of incidence of beam radiation on collector aperture At 3600 sec (1 hr)
4.
5. 6.
REFERENCES
I. J. P. Kenna, A parametric study of open loop solar heating systems--I. Solar Energy 32, 687-705 (1984). 2. S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar heating systems. Solar Energy lg, 113-127 (1976). 3. A. Whillier, Sc.D. Thesis, Mechanical Engineering, Mas-
7. 8.
723
sachusetts Institute of Technology. Solar Energy Collection and its Utilizationfor House Heating (1953). B. Y. H. Liu and R. C. Jordan, A rational procedure for predicting the long-term average performance of flat-plate solar energy collectors. Solar Energy 7, 53-74 (1963). B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). M. Collares-Pereira and A. Rabl, The average distribution of solar radiation. Correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22, 155-164 (1979). S. A. Klein and W. A. Beckman, A general design method for closed loop solar heating systems. Solar Energy 22, 269-282 (1979). S. A. Klein, Calculation of flat plate collector utilizability. Solar Energy 21, 393--402 (1978).
0038-092X/84 $3.00 + .00 © 1984 Pergamon Press Ltd.
A PARAMETRIC S T U D Y OF CLOSED LOOP SOLAR HEATING SYSTEMS--II J. P. K ~ N N ^ f
Solar Energy Unit, University College, Cardiff, Wales (Received 5 May 1982; accepted 15 July 1983; revision received 3 October 1983)
Abstract--A nondimensional equation describing the closed loop solar heating system with either parallel or series auxiliary heaters is derived. Using European weather data the monthly solar fraction is calculated for variations in the major non-dimensional groups. A correlation is given relating the solar fraction to three non-dimensional parameters M, Kc and Rc. M is the ratio of energy available on the collector aperture to energy demand. To calculate the energy available it is necessary to know the monthly utilizability. Kc is the ratio of the store temperature required for a 100 per cent solar contribution to the average monthly collector peak stagnation temperature. Both these temperatures are referenced to the demand temperature. Rc is the effective "turn over time" of the store, i.e. the number of days to empty the energy contents of the store. The magnitudes of Kc and Rc depend on the load heat exchanger size. Comparisons between the solar fraction predicted with a dimensional hour by hour computer model and that by the correlation are made for two system types. The agreement is good and it is concluded that the correlation can be used as a reliable method to optimise closed loop solar heating systems. !. INTRODUCTION In the closed loop heating system energy is provided at a constant temperature, usually by means of a heat transfer fluid circulating through heat exchangers. This contrasts with the open loop heating system in which water from a cold storage tank is heated to a higher service temperature and then after serving its purpose is discharged to waste in a contaminated form. For a given demand temperature and the same components, the closed loop will always have a lower efficiency than an open loop system because it is characterised by a minimum operating temperature equal to or greater than the demand temperature, whereas the open loop is characterised by a minimum operating temperature close to ambient. However, the open loop system is energy wasteful and in many cases it may be more economic to convert it into a closed loop system by use of a heat exchanger between the contaminated waste and the inlet. Thus, the closed loop system is a more important market for solar energy. Design and optimisation procedures for solar heating systems are important to ensure that the large initial investment gives a rapid return. The design procedures need to cope with a large number of important variables and at the same time be easy to use. A powerful method that can be used to achieve these objectives is non-dimensional analysis. The open loop solar heating system has already been analysed by this method[I]. This resulted in a simple design procedure valid for a large range of collector types, demand temperatures, demand patterns and locations. The work reported here demonstrates that the
same method can be used to develop a design procedure for closed loop solar heating systems. 2. N O N - D I M E N S I O N A L
SYSTEM MODEL
The system under study is shown in Fig. 1. It can include either a parallel auxiliary heater (Fig. la) or a series auxiliary heater (Fig. lb). Systems with a heat exchanger between collector and store and with pipe losses are accounted for by modifications to the collector performance parameters[l]. Initially the parallel auxiliary heater will be analysed. Later it will be shown that by modifying the demand temperature the series auxiliary heater can be made equivalent to the parallel auxiliary heater. An energy balance on the store, which is assumed to be fully mixed, gives energy stored = energy c o l l e c t e d - store losses -
energy demand
Mscp d T / d t = F"tloA (ibGb + idGd -- U/~lo(T -- To)) + -- U , A , ( T - T,,) - q e C ~ , ( T - Td).(1) Gb, Gd, T, Ta are all time dependent parameters. The energy collected is only accounted for when it is positive (signified by the + sign), ib, ia are incidence angle modifiers for beam and diffuse irradiation respectively and ~/0, U are based on mean collector fluid temperature. is the heat exchanger effectiveness and Cn, is the minimum thermal capacitance flow rate in the heat exchanger, q is a time dependent control parameter and defines the demand pattern.
when q ( t ) = 1.0 energy is required fPresent address: I. T. Power Ltd., Mortimer Hill, Reading, England. 707
when q ( t ) = 0.0 no demand exists.
708
J.P. KE~A
Auxiliary He~er
r
Load Heot -b Exchanger(s)
Thermal Store
(a) Closed Loop Solor Heating with Porallel Auxilx~ry
Auxiliary Heofer
Load __
Exchenger
Thermal Store
~ c for
J
.ipHeot
L
@
(b) Closed Loop Solar Heating System with Ser0es Auxiliary Fig. i. The closed loop solar heating system.
Energy is only extracted from the store when the store temperature is greater than the demand temperature. Heat exchangers with a minimum useful temperature greater than the demand temperature will be considered later. The store output is also subject to a maximum extraction rate Qm,x- When the store output is Qm~ the load has been met. Thus o < Echo(r-
r~) < Q _ .
(2)
If for the purposes of analysis, a load that does not vary with time is considered then ~m,x is constant and is given by p.,,= = ( U A ) , , ( T , -
To).
(3)
Tc is the sink temperature to which the load is referenced. The overall transfer coefficient from the load ( U A ) a is the rate of energy demand per temperature rise. Choice o f non -dimensional parameters
To reduce eqn (1) to non-dimensional terms the non-dimensional variables must be defined. The nondimensional temperature 0 is chosen in the same way as in Ref. [1]. For the open loop system 0 is defined
as
o = (r-
(4)
T~)/(T~- r3.
Temperatures are referenced to the minimum system temperature and expressed as a fraction of the required temperature rise. 0 = 1 specifies that the load has been met. It is unnecessary to achieve temperatures defined by 0 > 1. For the closed loop system the same definition is used (5)
0 = ( T - Td)/(Th -- Ta).
The demand temperature, Td, is now the minimum system temperature and Th is the store temperature required so that the output from the heat exchanger exactly meets the load, i.e. the solar contribution is 100 per cent. For this case the output from the heat exchanger is equal to the load ECmm(T~ -- T~) = (Ua)~(T,~- T3-
(6)
Re-arranging (T~ - T~) = ((UA)J~C~.)(T~--
TJ.
(7)
709
A parametric study of closed loop solar heating systemw--II or re-arranging
Substitute eqn (7) into eqn (5) 0 = ( T - Td)Lx/(T a - T~)
(8)
Gr
where Lx = EC~,/(UA)d and is a dimensionless parameter relating the heat exchanger size to the load. It is noted that this non-dimensional parameter appears also in the f-chart design method[2]. As for the open loop system, 0 = 1 specifies that the load has been met and it is thermodynamically inefficient to store and collect energy at temperatures defined by 0 > 1. The non-dimensional irradiance and time are defined as G' = (G + (UI~o)(To - Ta))IG*
(9)
G = ibGb + idGa,
(I0)
with
(15)
ddays
\Ndays
The r.h.s, of eqn (15) is termed the utilizability ~b, a concept first used by Whillier[3] and studied in detail by Liu and Jordan [4]. For this particular application ~b Is the hourly utilizability at solar noon (this will correspond to the peak in the long term) and at a critical radiation level of (U#lo)(Td -- T°) G* = ~bl~p.
(16)
Non-dimensional equation Substitution of eqns (8), (9) and (11) into eqn (1) and dividing by the daily load L results in the non-dimensional system model Rc dO/dt = X(G' - KcO)+ - LsRc(O - 0~), qO (17)
where
and t" = t/t*.
(11)
t* is taken to be 24hr. Again because Td is the minimum system temperature, the ambient energy input is referenced to Ta in eqn (9). Hence the ambient energy input will generally be a net loss and it is necessary for G to be at least greater than (U#lo)(Td- T~) before the collector can operate. The reference irradiance G* is defined in the same way as in Ref. [1]. For any given day G* = (Gp + (Ul~o)(T, - Ta))
(12)
and
Rc=R/Lx
(18)
Kc=K/Lx
(19)
Ls = U,A,t*/M~cp X = F"~oAG*t*/L
(20)
R = M,%(Ta-
(21)
Tc)/L
(22)
K = U ( T a - Tc)/(~lo~Gp).
In non-dimensional terms the control of the store output, eqn (2), can be expressed as
with
0 < qO < 1.
(13)
G. = ib, G, + i . a . .
Normally monthly or yearly system performance is of interest and G* becomes the average daily peak input. However, for the closed loop system there is a possibility that Gp may be less than (U/~o)(Td-- To) for any given day. In this case there will be no energy collected. Therefore, for long term weather sequences G* must be calculated as
G*=(N~eays(Gp-t-(U#lo)(~ a -
Td))+)/N.
(14)
The + sign indicates that only positive values are included in the summation. N is the number of days in the time period under consideration. For simplicity the monthly average ambient temperature/'o, rather than the individual daily temperature, is used. Equation (14) appears in most studies of solar heating systems. Dividing both sides by Gp gives G*/~p= ( ~ y
Gp + (U/tlo)(l'o- Ta) +)/(N~p)
(23)
Integration of eqn (17) over one day gives the solar fraction E E =
:o
qO dt'
Kc0) d,'
Rc(o
- 0=) d,'
- f°:Rc dO.
(24)
d0~ As with the study of open loop systems (I) the dependence of sunrise/sunset time is removed by changing the reference time for collectedenergy. This is defined as t" = t/t** with t** = H / G * and H given by
H = J(G + U/no(/',--T~))+ dt,
(25)
i.e.H is the energy available to the collector. Equation (24) must only be integrated over positive values
J.P. ~ ^
710
of the integrand. For monthly weather sequences with hourly radiation data
H--
~
(G + U/tlo(J'o- Td))+dt
tion is exactly the same as eqn (28) except that L x is replaced by Ld. K c and R c then become Kc = K/Ld;
(26)
(34)
Rc = R/Ld.
all hours
or in terms of the monthly average utilizability (27)
H=~[=H,
where i-. is the monthly average incidence angle modifier. The monthly average utilizability ff is the fraction of the monthly solar radiation that is available to the collector. A correlation relating ff to the monthly clearness index ~ r has been given by Klein[5]. Substituting t" into eqn (24) gives E = M
(G' - KcO)dt" -
L s R c ( O - O~,)dt'
Note that the utilizability functions and are now determined at a critical radiation level of (U/~o)(T, - :r,,) instead of ( U / ~ o ) / ( T d - J',). The store return temperature 7", can be expressed in terms of the demand temperature Td by the energy balance on the heat exchanger (eqn (31)). Substituting Th from eqn (6) into eqn (31) and re-arranging T, = T d + ( T , t - T ~ ) ( I / L x -- 1/Ld). Space heating s y s t e m s
For space heating systems or other systems with time varying loads the maximum extraction rate from the store will not be constant but is given by O,m~ = ( U A ) d ( T d - 7",)
-
Rc~°:dO
(35)
(36)
(28) where To is time varying. In these cases Th can be defined as the average temperature required to meet the load in which case eqn (6) is replaced by
dot
with M = F%loAH/L.
Thus, the dosed loop non-dimensional equation is identical to the open loop non-dimensional equation. Series auxiliary
With a series auxiliary heater the feed temperature to the load heat exchanger is always Th and hence for a constant load the return temperature is constant. The energy extracted from the store is given by =C(T-
(29)
T,).
C, is the thermal capacitance flow rate in the store loop of the heat exchanger. The minimum temperature in the system is now the return temperature from the heat exchanger T,, so the non-dimensional temperature 0 must be defined as 0 = (T - T,)/(Th -
1",).
(30)
0 can be expressed in terms of Td and T¢ by an energy balance on the heat exchanger. The energy input to the heat exchanger always meets the load such that C(Th-- T,)ffi (UA)~(T~- To).
(31)
ECmi,,(Th-- Ta) =
(37)
L/AtN
where N is the average number of operating hours per day (i.e. the number of hours when a load is required. The average sink temperature T~ becomes Tc = T a - L x ( T s -
(38)
Ta).
Space heating systems with a parallel auxiliary heater may also be characterised by a load heat exchanger with a minimum operating temperature greater than the demand temperature. For example if the demand (room) temperature is 20°C then an energy supply at temperatures as low as this will be perceived as a draught (cool air current) and it may be necessary to operate a fan coil heat exchanger at temperatures greater than say 30°C. This type of system can be accounted for by taking a reference temperature of T=i.. The non-dimensional temperature 0 becomes 0 = (T - Tmi.)Lx/(Ta-
(39)
To)
and the non-dimensional store equation involves one extra parameter
Substituting for ( T h - T,) in eqn (30) RcdO/dt = X(G'-
0 = (T - T,)Ld/(Tz - Tc)
(32)
L d = C,/(UA)d.
(33)
KcO) - L s R c ( O - Oa~) (40)
- q(O + SmiJ
where where s.,° =Ec=i.(r=i~ - r J / ( U A ) ~ ( T ~ By choosing 0 in this way the series auxiliary heater can be analysed in exactly the same way as the parallel auxiliary heater. The non-dimensional equa-
I'o).
(41)
The only difference between the non-dimensional parameters for this case and the parallel auxiliary
A parametric study of closed loop solar heating systems---II case analysed above is that the utilizability functions are determined at a critical radiation level of (U/~lo)(Tmi,- T°). Note that for the series auxiliary heater or the parallel auxiliary heater with T~, = Td, S,,~, becomes zero and eqn (40) reduces to eqn (17). 3. PHYSICAL SIGNIFICANCE O F PARAMETERS
The parameters M, K, Ls, R are the same as those for the open loop system. M is the ratio of energy available to energy demand; K is the ratio of rate of heat loss when the collector is operating at the demand temperature (now the lowest temperature in the system) to peak rate of energy input; Ls defines the store loss and R is the number of days storage available. However, for the closed loop system the parameters K and R have to be replaced by Kc = K / L x and Rc = R / L x (replace L x by Ld for the series auxiliary). In fact this is not really the change that it appears to be because Kc and Rc hold the same physical significance as K and R for the open loop system. Kc can be re-arranged as Kc = (Th -- Ta)/(T, -
(42)
Td)
where T, is the maximum temperature obtainable from the collector. Thus Kc is the ratio of desired temperature to maximum possible temperature with both temperatures referenced to Ta.
Rc can be re-arranged as Rc = M,cp/~Cm~,t* (parallel aux)
(43)
Rc = M,cp/C,t* (series aux).
(44)
Hence Rc is the "turn over time" of the store. For the open loop system with R = 1 the contents of the store are emptied every day, with R = 2 every 2 days, etc. Similarly for the closed loop system with Rc = 1 the energy content of the store is withdrawn every day. 4. P E R F O R M A N C E OF CLOSED L O O P S O L A R HEATING SYSTEMS
The solution can be obtained to the nondimensional store eqn (28) in the same way as for the open loop system. The solar fraction E depends on the parameters M, Kc, Re, Ls, T d (or 7", for a series auxiliary), Tc and Lx. The latter three are included because they determine the non-dimensional ambient temperature. For parallel mode heat exchangers with minimum operating temperatures greater than the demand temperature the dependence of the solar fraction E on Smi, must also be included. The monthly solar fraction has been computed for the base system of Table 1 using meteorological data from 6 locations (Table 2). The results are shown in Fig. 2. The similarity between these results and those for the open loop system is expected because the
Table 1. Specification of the ba~ system Collector
tilt:
45°
Collector
azimuth:
due South
Collector incidence angle modifier:
Store slze:
Rc:I
Store loss :
Ls=O. 2
Ambient around
store:
1.0
-
0.1(1/cos(8)
-1)
0as :0.O
Demand temperature :
60°C
Demand pattern :
Constant
Load Heat Exchanger
Lx=I.O Smln=
0.0
Table 2. Locations used for simulations Location
Year
Latitude
Longitude
Kew, England
Nov 63 - Oct 64
51029 ' N
00°19 ' W
1973
48048 ' N
02°02 ' E
Valentia, Ireland
CornIX)S l t e
51°57 ' N
10o13 ' W
Copenhagen, Denmark
Cornpo si te
55°41 ,
12°34 ' E
Hamburg, W.Germany
Composite
53°30 ' N
10°00 ' E
Hohenpeizenberg, W. Germany
Compo slte
4~48'
11001 ' E
Trappes, France
711
N
N
712
J. P. KENt^ i
.00 ~, K c = 0 . 2 v KC : ~ . 5 Kc=T
.9oJ
w
'l~ l
0
× Kc: I.~ m Kc=3.0 0
~
• 8o_1
Z
,,'" 7:,
•- , ' ; . , V ; L---
v
w
,
w
v ~
i v
~
vv ~
v
v~
v
v
.70 v
•v~
~
v
~v v v
G~
~v
.v
~:vv~
~v
v
O~v v v
v
v vv
v
1'.50
2'.00
~ v v
%
<~ __J .50 C3 Lr~ .'t0
.30 ~x
.20. I3 ira., ~ iQ
o
.10.
.00 [ .00
• 50
1'.00
2'.50
3'.00
M Fig. 2. Performance of the base system. non-dimensional system equations are similar. A least squares regression to the data of Fig. 2 gives a similar performance correlation E = 0.61M/(0.73 + MK*)
(45)
K* = K c / ( l . O + O . l l K c ) for Kc >0.65
(46)
with
and
carried out for the base system with a range of Rc between 0.1 and 2.0. The universal performance curves for this range are shown in Fig. 4. Data points are omitted for clarity. The effect of Rc is dependent on the parameter MK*/Rc. These curves can be represented by the equation E l M = a/(b + MK*/Rc)
(47)
K* =Kc/(l +0.11Kc) for Kc >0.65
(48)
with
K* = (Kc + 0.061M)/(0.69 + 0.87Kc) for Kc < 0.65.
and This correlation is shown by the solid lines in Fig. 2. As for the open loop system the family of curves can be collapsed to a single curve. This is shown in Fig. 3.
K* = (Kc + cM)/(0.69 + 0.87Kc) for Kc < 0.65 (49)
a = 1.67/(1 + 1.74Rc)
(50)
Effect o f Rc
b = 1.13/0 + 0.55Rc)
(51)
The performance correlation above has a limited range because it is only valid for Rc = 1 and, therefore, does not include the effect of different heat exchangers or store sizes. Simulations have been
c = 0.136Rc/(1 + 1.23Rc).
(52)
Equations (50) to (52) are valid for 0.1 < Rc < 2.0.
713
A parametric study of closed loop solar heating systemsIII
.00
I
"90 t
m
vV
. v
tE/I'I
%
v
Kc= vl~= ,. K c = I × Kc= I r~ K c = 3 . 0
.6~
.2 .5 .0 .5
nJ
.50.
DII
1 v
.40.
.30.
.20.
.10
.00
~0.2
~0.5
'I .0
~2.0
I5
.0
~I0 .0
M oK~* Fig. 3. Universal performance curve for Rc = 1.0. System performance in other locations
The validity of ¢qn (45) in hotter locations than northern Europe has been examined. Hourly meteoroiogical data for Madison, U.S.A. Oat. 43 °) and Albuquerque, U.S.A. Oat. 37°) were synthesized from the monthly average dearness index ~r. A sequence of days with daily clearness index distribution comparable to the long term average distribution was generated using data given by Liu and Jordan[5]. The hourly irradiance was obtained from the daily irradiation using the correlation of Collares-Perc~ra and Rabl[6]. The results are shown in Fig. 5. Comparison with ¢qn (45) (shown by the solid line in Fig. 5) indicates the general application of the correlation to other locations. $. SENSITIVITYTO OTHER PARAMETERS The sensitivity of system performance to changes in the base system has been examined. Variations in Td, T~, L x only effect the value of the non-dimensional ambient temperature, so their effect should be small. This has been found to be the case
as shown in Figs. 6 and 7. In Fig. 6 predictions of monthly solar fraction for a demand temperature of 20°C are shown compared with the correlation given in cqn (45). Because this demand temperature corresponds to a space heating system, only the winter months (October-March inclusive) were used for the 6 locations given in Table 2. It is clear that although the scatter is somewhat increased for the lower demand temperature, the average solar fraction for a given M and Kc does not change significantly. Figure 7 shows data for the base system but with a heat exchanger defined by L x = 2.0. When compared with Fig. 3 it can be seen that the scatter is also increased but again the average value is still the same. This confirms that the effect of T~, T~ and L x is adequately accounted for in the parameters Kc and Rc. Sw is the minimum non-dimensional energy input to the load heat exchanger below which it cannot be used (usually because of thermal comfort limitations in space heating systems). It represents the proportion of the instantaneous load that is required before the heat exchanger operates. Simulations reported above are for Smi, ----0.0. This corresponds to either a
714
J.P.
KENNA
I .00
.90
.80.
.70.
E/M
.eo_
.50
.40
Rc
= 0.i = 0.25 -Rc = 0.50
.30
~ R c
= 1.0
=2.0
.20.
.10.
.00
3.1
'0.5
'1.0
r5~T10.e
1100
M. K ~/R~. Fig. 4. Effect of Rc.
heat exchanger that can work down to inlet temperatures equal to the demand temperature or to a series auxiliary heater. Simulations have been carried out for the base system but with S~, in the range 0.1-2.0. Figure 8 shows the results for S~, = 1.0 together with the performance correlation for S ~ = 0.0. It is implicit in these results that the utilizability functions have been calculated at a critical radiation level of ( U / ~ I o ) ( T ~ - f ' o ) . Also the ambient temperature around the store is assumed to be equal to the demand temperature (corresponding to 0~, = - S~,). It is seen from Fig. 8 that for S~, = 1.0 there is a region at low M K * at which the normalised system efficiency departs from the correlation found for Sm~, = 0. In the case of S ~ = 1.0 the correlation may be used for M K * > 0.5. The influence of demand pattern on system performance has been examined by Klein and Beckman[7]. Differences were noted between a 6 a.m. and 6p.m. load, a 6p.m. and 6a.m. load and a constant 24 hr load. These studies were for an infinite
load heat exchanger. However, if the heat exchanger size is finite its size must influence the effect of demand pattern. Taken to the extreme if the draw off is only 1 hr per day an extremely large heat exchanger will be required, whereas if the draw off is 24 hr per day a smaller heat exchanger can be used. Small variations in demand pattern such as those of a space heating system have no effect on the system performance. This has been demonstrated by the data in Fig. 6 which were generated assuming the limitation in store output was given by eqn (36). Larger variations such as a mid-day demand (constant between 10.30 and 3.30p.m.) can have a significant effect. With R c = 0.1 (Fig. 9) the solar fraction is not significantly degraded but with Rc = 1.0 (Fig. 10) there is a large decrease in performance. F o r the 5 hr demand the non-dimensional temperature must be greater than 4.8 (24/5) for a 100 per cent solar contribution. When Rc = 1.0 the non-dimensional temperature does not reach this as often as with R c - - 0 . 1 . Hence the solar fraction is reduced. This means that when the demand occupies only 5 hr per
A parametric study of closed loop solar heating systems---II
715
1 .00. ~Kc=0.2 ~, K c = 0 . 5 +Kc=l.0 × K c = I .5 D Kc=3.0
.90
,w
" "'".., v
,,, " " '~ ~
LU
"
¢
.80-
•
~
?v ,
Z 0 .70_ 0 rY LL
.60_
n/ _] C] CO
/
.50_
.40.
.30.
.20.
.10.
.00 .00
.50
I'.00
I'.50
2'.00
2'.50
3'.00
M Fig. 5. System performance using synthesised U.S. weather data. day instead of 24 hr, a heat exchanger defined by high Lx is required to ensure that Rc is low. It can be concluded from this sensitivity study tllat both S~, and demand pattern can influence the solar fraction significantly. Until further studies are made on the effect of these parameters the application of eqn (47) must be restricted to 24 hr or space heating demand patterns and for non zero values of S u , to the range MK* > 0.5.
at temperatures close to the demand temperature. This in turn means that the average store temperature can be lower and the collector operates with lower heat losses. Conversely a poor heat exchanger (Lx~O) will require high inlet temperatures, high store temperatures and hence high collector losses. At first sight this suggests that poor collectors can be compensated by good heat exchangers. However, this is not the case because poor collectors will lead to low utilizability (hence reducing M and increasing
Kc). 6. I~;~-~-KCTOF HEAT EXCHANGERSIZE The load heat exchanger size can have a significant effect of the performance of the closed loop system. This has been partially demonstrated in the discussion on demand patterns. It has been shown that the effect of heat exchanger size is adequately accounted for in the parameters Kc and Rc which in turn influence the solar fraction E. The physical link between heat exchanger size and collector loss (as shown mathematically in Kc) is because a perfect heat exchanger (Lx ~ oo) will be able to meet the load
The physical link between heat exchanger size and store size (shown mathematically by Rc) is because good heat exchangers deplete the energy contents of the store in a shorter time period, thereby requiring a higher storage mass to obtain the same daily energy storage. In practice the store output is limited so that it does not over supply the load. Consequently it is possible to have a heat exchanger that is oversized, because its output is always limited. To determine the optimum heat exchanger size for a given application it will be necessary to offset the
716
1
.00 901
J . P . KENNA
00
•
X
¢)
+
O
x +
,~
x
~
o X
•
•
VIW" •
It
v
.70
%
x O_
0 0
0 0
E/M
Kc=
0
.2
v Kc: .5 + Kc=I .0 x Kc=I .5 [] K c = 3 . 0
.c0 IIv o O
.50.
O+
o
'b i•
I O
g
0 30
.40.
.30. v a t•
.20.
.10_
.00
a0.2
10.5
I1 .0
12.0
'5.0 I]0.0
M .K~ Fig. 6. Effect of a demand temperature of 20°C.
increased solar fraction (due to a reduction in Kc) against the increased cost of a good heat exchanger. 7. SERIES V. PARALLEL AUXILIARY HEATERS
The parallel auxiliary heater is generally thought to be a better choice for solar heating systems because the system has a lower minimum operating temperature. The choice of a series or parallel heater can be examined using the non-dimensional parameters. The criteria governing which mode of operation is better depend solely on the properties of the heat exchanger. The important characteristics are the minimum operating temperature (parallel auxiliary), the heat exchanger return temperature (series auxiliary), and the heat exchanger sizing parameters L x (parallel) and Ld (series). Generally the series auxiliary heater will be such that 7", (series) > Ta (parallel) and Ld (series) > L x (parallel). The former condition acts to decrease system performance and the latter favours an increase in system performance. For a heat exchanger with a high effectiveness it is possible that the return temperature when operating in a series
mode is lower than the minimum useful temperature when operating in a parallel mode, in which case the series auxiliary heater would be the better choice. However, to obtain an optimum performance from a given collector/store combination it can be expected that in practice for many cases a heat exchanger operating with a low minimum useful temperature and in a parallel mode is the better choice. 8. THE DESIGN METHOD
This section will illustrate how the correlation given in eqn (47) will serve as a design method for closed loop solar heating systems. As with th e open loop design method the closed loop method can be split into three stages (1) Determine the three meteorological parameters H s, Gp and it=. Gp can either be obtained from hourly weather data or by using the correlation of Collares-Periera and Rabl relating monthly average daily peak irradiance to monthly average daily total irradiation [6]. (2) Parameters Kc, M and Rc are calculated. To
717
A parametric study of closed loop solar heating systems---II I .80
o x
o
x bt~K X .X~. ~ 0 +x
v VvW~ •
%
.70_
I:1 O
~, K c = v Kc : + Kc=I × Kc: I o Kc=3.0
II
E/M
.G8
÷
X
v v l ~ • xO
0
.58
v~
.2 .5 .8 .5
•
~o
÷ .40_
.30
V V~
.20
.I@.
.00 .I
TO .2
'0,5~--T~o
12. ~ . . . . . . . % . 0
L10.~[
V1 oKm Fig. 7. Effect of Lx.
determine Kc and M two utilizability functions are required, one to calculate the monthly average utilizability and the other to calculate the hourly utilizability of the daily peak irradiation. The latter can be obtained from charts given by Liu and Jordan [4]. is given as a function of the critical radiation level Xc, the monthly clearness index £ r and the monthly average ratio of beam radiation on a tilted surface to beam radiation on a horizontal surface Rb. In terms of the notation used here the critical radiation level Xc is xc = v ( r . , , . -
L)/(nod,).
An alternative method of obtaining 0 and ~ is to generate them from hour by hour weather data. When comparing the design method with an hour by hour computer model this is a necessity because the correlations for 0 and ~ relate to the long term average and the year under consideration may not be representative of the long term average. For systems with heat exchangers between the collector and store and with pipe losses the parameters M and Kc are modified as in Ref. [1]. (3) Providing M < 3 and Rc < 2 the monthly solar fraction is given by
(53) E = aM/(b + MK*/Rc)
(54)
(Replace Tmin by T, for the series auxiliary heater). The monthly average utilizability ~ can be obtained from a correlation given by Klein[8] in terms of Xc, £ r and Rr/R,. Rr is the ratio of monthly total radiation on a tilted surface to that on a horizontal surface and R, is the ratio of radiation at noon on the tilted surface to that on the horizontal surface for an average day of the month.
where a, b a n d K * are given by eqns (48)-(52). To compare the design method with a full simulation two systems have been simulated with a dimensional hour by hour computer model. These are specified in Tables 3 and 4. Also shown is a comparison between the solar fractions predicted by the design method and those predicted with the hour by
718
J. P. ~qCNA Table 3. Performance predictions: System A System specification Collector :
Type Flat plate Performance parameters no = 0.67; U = 3.OWm-2K ~ Tilt 45 ° Azimuth due South Area 50 m2
Store :
Mass 2500 kg. Loss coefflclent I W m-2K-1 Surface area 10 m 2
Load Heat Exchanger: Lx 1.0 Minimum useful temperature: 25°C
Demand :
Temperature 20°C -1 Space heating (UA~ : 100 W K
System Performance Monthly and Seasonal solar f r a c t i o n s Location
Trappes
Kew
Month
Full Slm.
Design Method
Full Slm.
Design Method
Oct Nov Dec Jan Feb Mar Apr Heating Season
0.97 0.63 0.18 0.19 0.50 0.58 0.75 0.50
0.96 O.60 0.18 0.21 0.46 0.57 0.97 0.51
1.00 0.86 0.37 O. 2o 0.44 I. O0
1.00 0.73 0.35 O. 22 O. 45 O. 99 O. 96 0.63
I . O0
0.65
Table 4. Performance predictions: System B System specification Collector:
Type Flat Plate -I Performance parameters no : 0.67; U : 2.0 Wm -2 K Tilt 45° Azimuth due South Area 500 m2
Store:
Mass 2500 kg. Loss coefficient I Wm-2 K- i Surface area 10 m 2
Load Heat Exchanger: Lx : 4.0 Minimum u s e f u l temperature: 60°C Demand:
Temperature 60°C (UA~ : 400 W Kq
System Performance Monthly and Yearly solar fractions Location
Kew
Trappes
Month
Full Sim.
Design Method
Ful 1 Sim.
Design Method
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year
O. 08 0.18 O. 22 O. 47 0.70 O.59 O.78 0.73 O. 82 0.45 0.18 O. 07 0.41
0.10 0.20 0.25 O. 47 0.68 0.59 0.73 O. 70 O. 76 O. 46 0.20 O. 09 0.41
O. 09 0.17 0.73 O. 52 d.64 0.81 0.71 O.86 O. 76 0.57 O. 35 0,16 0.51
0.12 0.23 O. 67 O. 50 O. 65 O. 78 0.69 0.81 O. 76 O. 58 0.37 0.19 0.51
A parametric study of closed loop solar heating systems--II
719
I .00.
.90.
.80_
.70_ o
• , " ¢ " , . :v Eli'1
.00_ •• .
.-" _
•
.50_
• _" •
+,
•
I
• ~" .'sb_ • i,
;,
;
.
:
x
+
_wo N . . + , ~ ,,-.,x,~.'._~," ,,, - +~.-x
.',
d
,j.,",
_~o-
a .''°o
4t
n
~ Kc=
.2
v Kc=
.5
+ K=:I.~
"K
[]
"o
c:
3
.~
"t,_'o+O~ ~'
,+-
o
.
o
/ + x o °°
=+,-~..~,~"
°
Ill
.10.
.00
•I
~0.2
'0.5
~I .0
~2.0
~5.0
n10.g
H .K~ Fig. 8. Effect of S~a~.
hour model. These results are shown graphically in Fig. 11.
9. CONCLUSIONS
The main parametric groups influencing the performance of the closed loop system are M, Kc and Rc. A correlation relating the monthly solar fraction E to M, Kc and Rc has been obtained for a wide range of weather conditions. The parameters M and Kc are dependent on the monthly and hourly utilizability, respectively. The critical radiation level for the utilizability calculation depends on the mode of operation. For the parallel auxiliary heater with a load heat exchanger capable of operating down to the demand temperature the critical radiation level is U(Td--;P~)/r/0; for the scrim auxiliary heater the
critical radiation level is U(T, - T~)/rl0; for the parallel auxiliary heater with a minimum operating temperature greater than the demand temperature the critical radiation level is U(Tmin- I'o)/%. For the latter case the correlation is only valid for MK* > 0.5. Other areas where this correlation is not applicable have been examined and it has been found that a further parameter not yet identified, will be required to take account of load patterns which depart significantly from a uniform demand over 24 hr. Comparison of the results from a full hour by hour computer model with predictions made by the design method shows that the correlation gives satisfactory. results. It can be used to optimisc the choice of collector type, collector area, store size and heat exchanger size for a given demand.
720
J.P. I ¢ ~ A
1 .gg.
.90.
.80.
• W~
.70.
Kc=
.2
Kc= .5 K c = I .0 + x K c = I .5 o Kc=3.0
,i, T
E/M _,,~. v@ .50
.40_
.30
.20.
.10.
.00
'0.2
'8.5
'1 .0
'2.0
M ,K~ Fig. 9. Effect of demand pattern,
R c = 0.1.
15.0
Ilg.g
A parametric study of closed loop solar heating systems--lI
721
1 .00.
.90.
.80.
.70.
Z/~
v + x rl
.60.
Kc: .2 Kc: .5 Kc=I .0 Kc:I .5 Kc=3.0
.50_ I ilm
m
.30
•" ~ = ~
•
.20L
.10.
.00
.1
f0.2
~0.5
~1 .0
'2.0
~5.0
K1 . K ~ Fig. I0. Effect of demand pattern,
Rc =
1.0.
I10.0
722
J.P. K~,rN^ 1 .00. O C3 "r FW r"
.90.
Z .80. CD W 0
Nc
SYST~ t
V
3Y3TI~ B
,70.
W
.60.
.50.
V
.410. .30. / .20.
/
.10. .00 .00
'
.20
~. 4- 0- '
. 6" 0 E FULL
'
. 8' 0
'
1 .00
SIMULATION
Fig. 11. Comparison of design method and full simulation.
Acknowledgements--The author wishes to thank Dr. R.H. Marshall for his contribution to the work reported here and to Prof. B. J. Brinkworth, Dr. W. B. Gillett and colleagues at the Solar Energy Unit, Cardiff for many helpful cornments and discussions, NOMENCLATURE
(Note that To, etc. indicates a monthly average.) A collector area, m 2 A, store surface area, m 2 C, store thermal capacitance flow rate, J K - i Cram minimum thermal capacitance flow rate in heat exchanger, J K - I cp specific heat of heat transfer fluid, J kg- t K - m E solar fraction F" collector flow factor Gb direct solar radiation, W m -2 Gd diffuse solar irradiance, W m-2 Gbp direct solar irradiance at daily peak, W m-2 G~ diffuse solar irradiance at daily peak, W m-2 Gp peak irradiance (including incidence angle moditier), W m - 2 G* reference solar + ambient power input, W m-2 G' non-dimensional power input H, solar irradiation on collector tilt, J m-2 H total solar + ambient energy available on collector aperture, J m-2 F,~ mean incidence angle modifier id diffuse incidence angle modifier ibe direct incidence angle modifier at daily peak K ratio of reference collector loss to reference collector gain Kc ratio of demand stagnation temperature difference to desired store temperature difference Ls store loss parameter L daily load, J
Ld Lx M, M q
Q
Qm~
Rr
Rb R. R
Rc S~o t
t,, t, t' t" T
Tc Td T~ T~ 7", Tm~ Th 7", U Uj
(UA)d
series auxiliary heat exchanger sizing parameter parallel auxiliary heat exchanger sizing parameter store mass, kg ratio of energy available to energy demand draw off control parameter energy output from solar system, J maximum output allowed from store, J ratio of monthly radiation on tilted surface to that on a horizontal surface ratio of beam radiation on the tilted surface to that on the horizontal surface at noon ratio of radiation at noon on a tilted surface to that on a horizontal surface number of days storage available "turn over time" of store heat exchanger minimum input parameter time, s sunrise time, (superscript indicates non-dimensional), s sunset time, (superscript indicates non-dimensional), s non dimensional time (referenced to 1 day) non dimensional time (referenced to H/G*) store temperature, K reference temperature, K demand temperature, K ambient temperature, K ambient temperature around store, K maximum daily collector temperature possible, K minimum useful temperature for heat exchanger, K inlet temperature to heat exchanger for I00 per cent solar, K store return temperature with series auxiliary, K collector loss coefficient, W/m2K store loss coefficient, W/m2K energy demand rate of heat transfer, W K - '
A parametric study of dosed loop solar heating systems--II Xc r/0 0 0~ E
critical irradiance collector zero loss efficiency non-dimensional temperature non-dimensional temperature around store heat exchanger effectiveness hourly utilizability monthly utilizability fl angle of incidence of beam radiation on collector aperture At 3600 sec (1 hr)
4.
5. 6.
REFERENCES
I. J. P. Kenna, A parametric study of open loop solar heating systems--I. Solar Energy 32, 687-705 (1984). 2. S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar heating systems. Solar Energy lg, 113-127 (1976). 3. A. Whillier, Sc.D. Thesis, Mechanical Engineering, Mas-
7. 8.
723
sachusetts Institute of Technology. Solar Energy Collection and its Utilizationfor House Heating (1953). B. Y. H. Liu and R. C. Jordan, A rational procedure for predicting the long-term average performance of flat-plate solar energy collectors. Solar Energy 7, 53-74 (1963). B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). M. Collares-Pereira and A. Rabl, The average distribution of solar radiation. Correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22, 155-164 (1979). S. A. Klein and W. A. Beckman, A general design method for closed loop solar heating systems. Solar Energy 22, 269-282 (1979). S. A. Klein, Calculation of flat plate collector utilizability. Solar Energy 21, 393--402 (1978).