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A design nomogram for direct thermosyphon solar-energy water heaters
Solar Energy. Vol. 43. No. 2, pp. 85-95, 1989
0038-092X/89 $3.00 + .00 Copyright © 1989 Maxwell Pergamon Macmillan plc
Printed in the U.S.A.
A DESIGN NOMOGRAM FOR DIRECT THERMOSYPHON SOLAR-ENERGY WATER HEATERS* P. A. HOBSON AFRC Institute of Engineering Research, Silsoe, Bedford MK45 4HS, U.K. and B. NORTON Department of Building, University of Ulster, Co. Antrim BT37 0QB, N. Ireland AbstractEA characteristic curve for an individual directly heated thermosyphon solar-energy water heater, when obtained from data of an appropriately chosen test period of 30 days, has been shown to predict the annual solar fraction to within 3% of the corresponding value obtained from a validated numerical simulation model. An extension of this analysis produces two underlying correlations of five dimensionless groups from which the performances of a wide range of thermosyphon solar-energy water heaters with varying thermal loads can be predicted. The estimated uncertainty of this design technique in predicting an annual solar fraction ranges from 8% for multiple-pass systems to 13% for same level. single-pass systems.
I. I N T R O D U C T I O N
X
The identification of generalised dimensionless grouped parameters for both pumped[ 1,2] and buoyancy-driven[3-5] solar-energy water heaters, from which their thermal characteristics can be determined, has provided a practical approach by which their long-term performance can be predicted readily. One of the major inherent drawbacks of correlating parameters derived previously has been that they are based on steady-state analyses and therefore relate to performance over relatively long periods of operation. A method of determining a characteristic correlation curve for an individual natural-circulation solar-energy water heater based on a transient analysis which relates to diurnal performance, has been developed[6]. From a transient heat balance on a generic directly heated thermosyphon solar-energy water heater, the following dimensionless parameters Y, Z and X, designated the Heywood, Yellot and Brooks numbers respectively, have been identified[6]:
1 - exp(-Z)
Heywood number:
Y=
[FAvAcUL + (UA),]t Yellot number:
Z =
Brooks number:
X =
-
T,.)
+
l
(4)
2. THE VERACITYOF THE CORRELATION TECHNIQUE AS A METHOD OF DETERMLNE~G LONG-TERM SYSTEMPERFORMANCE The correlation technique outlined was applied to thermal performance data generated by a validated numerical simulation model[9] using Kew (London, U.K.) weather data[10] for a total of 8 months from March to October. The main features of the simulation model[9] were • A fully transient one-dimensional analysis was applied to the heat and momentum transfer processes occurring within the fluid contained in all the components of the system (i.e., collector, upriser, storage tank, and downcomer). • The collector model was based on a two-dimensional analysis thereby permitting the accurate simulation of its transient performance. • Friction factor correlations relevant to thermally destabilised flows were used in determining the fluid losses within the risers. • All fluid properties and heat transfer coefficients
(I) (2)
(M,C,,,) f Qtot M,C.,(T,.
Z
For experimental data collected via the monitoring of systems operating in England[6,7] and Portugal[8], the linear relationship between X/[ l - e t-z'] and Y/Z, implied by eqn (4) was observed. In this paper, a detailed numerical simulation model[9] is used to investigate the accuracy with which such correlations determine the long-term thermal performances of individual systems. The analysis is extended to produce a simple, but relatively accurate, design method for direct thermosyphon solar-energy water heaters.
FA~Ac(*=)J-/ M, Cw(Ta - 7"=)
Y = -
(3)
They were found (6) to be related by the expression
*Includes material presented to the ASME Winter AnnuM Meeting, Chicago, Illinois, November 1988. Address correspondence to Prof. B. Norton, Dept. of Building, University of Ulster, Co. Antrim, BT37 0QB, N. Ireland. 85
P. A. HOBSONand B. NORTON
86
were calculated from temperature-dependent relationships and updated at each time step in the numerical solution. • The resulting energy and momentum equations were solved using a fully implicit finite-difference scheme employing first-order upstream differences for convective terms and second-order accurate differencing for diffusion terms. The program was run on a VAX 11/750. Input data for both the mains water supply temperatures and the solar water draw-off patterns were based on measurements taken during the long-term monitoring of a group of occupied houses in Wharley End, Bedfordshire, in southeast England[7], in which thermosyphon solar-energy water heaters had been installed. For these eight simulation runs (i.e., one for each month, March through October) Ac = 4 m 2, Ms = 780 kg, ha = 3 m and h~ = 2 m. For the remaining winter months (i.e., November to February), convergence of the solutions to the momentum and energy equations in the simulation model were frequently unobtainable. This was due to the errors introduced by the fluid properties, as predicted by the approximating polynomials, at near-freezing conditions. However, not only were these simulation results realistic but they were also acceptable, as the four months from November to February contribute only 9% of the total annual global radiation received
O3 IJ.I .--I
in the U.K. Furthermore, in practice, an automatically activated solenoid valve, as part of the frost protection system on such directly heated systems, causes the collector to be drained down at 2 (-2)°C[11], thereby rendering it inoperative for periods in the winter months. A plot of daily values of X/[ 1 - e x p ( - Z ) ] against Y/Z is shown in Fig. I for the March through October simulated period of operation. The values of Fay, Us included in Y and Z were calculated on a mean daily basis, based on the temperature-dependent heat transfer coefficients used in the simulation program. Values of ('ro0= were calculated assuming a normal effective angle of incidence of insolation on the collector. The correlation coefficient for these values over the 245 days was 0.97. The gradients of the best straight lines through the data were observed to alter seasonally. The equations of the best straight lines through each of the eight individual monthly graphs of X/[1 - e x p ( - Z ) ] versus Y/Z were each used independently to predict the annual solar fraction. The ensuent errors in the predicted annual solar fractions are shown in Fig. 2. Distinct error minima of - 3 . 0 % and +3.0% were observed to occur in the months of May and June respectively. These results favoured either May or June as being the most appropriate ~reference" months from which a characteristic annual thermal performance curve could be
600.
Z
C) 0 Z I,a,.I
z
¢-t
6000
N
0
i
200
"
X
LLI I
°°
x I
600
-200.0 -1600 -1200 - 0 0 - 6
800
I
1200
I
1600
I
2000
Y / Z (DIMENSIONLESS) 0
-200"
0
-600" 0
0
-600 Fig. 1. Correlation of X/[l - exp(-Z)] against Y/Z using simulated thermal performance data.
A design nomogram for direct thermosyphon solar-energy water heaters
87
40
--
3O
Z
T--.
20
t~
e,,, l.l-
,v
10
_.J
7
-10 I-~J
"'
-2(
a. z
,,-
-30
¢Y
-z,l
t
i
MAR
APR
t
!
MAY
!
JUN
JUL
I
AUG
|
SEP
!
OCT
MONTH FROM WHICH CHARACTERISTIC CURVE IS TAKEN Fig. 2. Error incurred in predicting the annual solar fraction using the characteristic curves for the months of March to October.
obtained; for June this is given by
X/[ 1 -
exp(-Z)] =
mj(glZ)
(5)
In general it was observed that the gradient mj, (calculated from eqns 8 and 9: see later) gave values of less than unity. This was due to the assumption of equal mean store and mean collector temperatures which was implicit in the simplified analysis from which eqn 5 was derived[6]; an assumption that would lead to an over-estimate of the system performance. A value of ms less than unity, in effect represented a "thermal penalty" coefficient which brings the overoptimistic thermal performance predicted by the simplified analysis, in line with the more realistic data generated by the high-level simulation. Plots of eqn (5) were found to pass through the origin and the constant (of unity) in eqn (4) was therefore discarded. The cumulative solar fraction and associated error over the year calculated by using eqn (5), is shown in Fig. 3. Using dally average monthly data, a single calculation giving a representative predicted dally solar contribution for each particular month was determined using eqn (5). To determine the total monthly solar contribution this daily value was assumed to remain invariant over the month and was therefore multiplied by the number of days in the month. Using
this procedure, the predicted annual solar fraction was within +6.3% of that calculated using the high-level simulation model. The number of basic calculations was therefore reduced from 245 (i.e., the total number of operational days) to 8 (i.e., one average day for each operational month) with a relatively small decrease in accuracy (shown in Fig. 3).
3. DEVELOPMENTOF A DESIGN ALGORITHM An investigation was undertaken into the effects on the gradient, mj, of changes in the primary dimensionless system parameter, the Yellot number, Z, evaluated, for reference purposes, for the month of June (Zj). This was carried out initially by varying the collector area of a reference system with h3 = 2.48 m and hi = 1.5 m for a total of six systems (designated in row three of Table 1). An asymtopic maximum gradient of approximately 0.3 was observed (see Fig. 4) for values of Zj greater than 0.54. When the collector area increases relative to the store capacity, the temperatures are such that the buoyancy-driven flow rates around the system increase. With this progressive increase, the conditions are therefore approached in which the assumption (pertinent to multiple-pass systems made in the original analysis[6]) of equivalent
88
P. A. HOBSONand B. NORTON
g
ts
,:
':
-5 z
.~-. . . . . .
,/
\
-10 -~s
-10
OPERATIONAL PERIOD
•~ 5 0
--I "~
.400
Z
.350 p j/
.300 eY LL
•250
.,.J
.200
,~
H~H LEVEL SIMULATION MODEL
LU
~'/
.150
CORRELATION BASED ON DALLY DATA
I"-,.,.I
/~//'
.100
',r
CORRELATION BASED ON DALLY AVERAGE MONTHLY DATA
I
EQUATION
.050 JAN FEB MAR APR HAY
JUN JUL
AUG SEP
OCT
NOV DEC
TIME
Fig. 3. Comparison of solar fraction calculated from the numerical simulation model with corresponding values determined from: (1) the correlation technique based on daily data, and (2) the correlation technique based on daily mean monthly data. mean store and collector temperatures[12] becomes valid• For the relative position of the collector and store, used in the reference system, the degree of multiple-pass behaviour is sufficient for the above assumption to be valid at and above values of Zs greater than 0.54. A further series of nine simulations were run in which the Yellot number, Zs, was varied by changing the thermal mass of the store for two values of the collector area (see columns 3 and 5 of Table I). A plot of the resulting gradients against corresponding mean monthly values of the parameter Zs indicated a fundamental change in the relationship between ms and Zs when compared with simulations in which Table 1. Corresponding system data for the first parametric analysis series of simulations Area of Collector A~ (m~) 0.5 Mass of water in StOl~
M, (kg)
50 too 200 300 500 800 l,OOO
/
1.0
2.0
3.0
/
/ / / / /
/
4.0
/ / / /
5.0
variations in Zj were due to area changes alone. However, when Zj was plotted against corresponding values of the mean daily circulation number (Ne) (Fig. 5) for all the simulations indicated in Table 1, a linear relationship emerges, regardless of whether Zj is varied due to changes either in the mass of water in the store or collector area. Because the linear relationship observed in Fig. 5 also contains data from the six simulations between which the collector area was changed, then changes in the gradient, ms which conform to the same relationship as that shown in Fig. 4 must also be occurring for the simulations between which the mass of the store was changed. These changes were therefore being overwhelmed by an additional change in ms due to another factor which, although dependent on the mass of the store, is independent of the circulation number. It also follows that because the effect of Zj on system performance is dependent essentially on the corresponding circulation number, it is independent of this additional storecapacity-dependent factor. By changing the mass of the store alone in the reference system, not only does this change the value of Zj, but also, since the total daily draw-off remains constant, affects the specific load, W[13], where
/ /
Specific load:
ML W = -Ms
(6)
A design nomogram for direct thermosyphon solar-energy water heaters
89
3.50
3.25
u~ LL=
Z 0
0
3.00
Z
0
X ¢:3
--
2.75
0
2.50 er
2.2S L~ 2.00
0 -r-
1.75
1.5( 0.10
I
I
0.20
0.30
I
I
0,40
0,50
I
0,60
I
I
0.70
0.80
Zj (DIHENSIONLESS) Fig. 4. Variation of m~ with Zj due to changes in collector area alone.
V oq .-J Z m (/'1 Z X ,,-t
V
IJ=l
MASS OF STORE (kgs):
:D Z Z O I,-,,-I 4"
)-
X
+x
V +
107 195
x
302
O
553
0
830
o
1107
13
C~
13+
Z lad 'I"
o0 0.00
I
I
0.10
0.20
I
0.30
I
I
I
I
0,40 0.50 0.60 0.70 Zj (OlMENSIONLESS}
I
0.80
I
0.90
I
I
1.00
1.10
Fig. 5. Variation of Np with Zj due to changes in collector area and store capacity.
90
P.A. HOBSONand B. NORTON
The gradients of the characteristic curves obtained from all the first series of simulations (i.e., for the ranges of systems indicated in Table 1) are shown plotted against W in Fig. 6. The variations in the gradients due to W are seen to be large relative to those changes in which W remained constant while Zj (and therefore the mean daily circulation number) alone, was varied. The variation of gradient with W is due to it being assumed in the original analysis[6] that the withdrawal of heated water occurs as a single event at the end of the insolation period. Changing the relative vertical distance of the store from the collector from that of a reference system, changes the prevalent flow rate, and therefore the mean daily circulation number of the system; this will alter the gradient of the reference characteristic curve. A family of curves is therefore required (for a fixed arbitrary value of W), each of which relates the gradient to the Yellot number, Zj, for different relative collector/store heights. All the curves will converge to the same maximum gradient, this value being dependent on the magnitude of W chosen. The relationship between the displacement of the gradient from the maximum value represented by these family of curves, will be valid for all values of W because the circulation number is independent of W. From this family of curves, a system can then be chosen for which the gradient of the characteristic curve is a maximum. Using this system, a plot similar to that in Fig. 6 of mj.ma~ against W, can be determined and
which will then be representative of the performance of all systems in which the critical circulation number (beyond which no further change in the gradient occurs), has been exceeded. The Bailey number, K, which represents the system parameters effecting flow within the system[13] is defined as;
pf3gAT~t.{h3 h:/2] vm~f[L,./N(D~) + L,/O~)] -
Bailey number: K =
(7)
where ATref and mref are given the values 10°C and 10 -t kgs -t respectively. The basic parameters of a reference system for which Ms = 280 kg, were varied in a second series of simulations according to Table 2. Values of the resulting gradient displacements Amj from the observed maximum value of 0.429 are shown in Fig. 7. A correlating function (with a correlation co-efficient of 0.94), relating the displacements of the characteristic gradients of this set of simulated systems to their respective system parameters, is given by Amj = 2.541 x 10 -3 + 0.780(m*) + 1.967(m*) 2
(8)
over the range 0_< m*__O.15 where m* = 0.195 exp[(0.402 - 0.387K)Zj].
1.z~O
1.30 1.20 1.10 trl ELI ,--J Z 0 Z W =E
1.0C
0.90 0.80 0.'/0
t--" Z
0.60 0.50 O.~C
I--" t~
0.30 0.20 0.10
"r
0.0
I
0.00
0.50
I
1.O0
I
I
1.50 2.00
I
I
I
2.50 3.00 3.50 W (DIHENSIONLESS)
I
4.00
I
~.50
Fig. 6. Variation of m~ with W due to changes in store capacity.
I
5.00
I
5.50
!
6.00
A design nomogram for direct thermosyphon solar-energy water heaters ficient of 0.999) through this data is given by
Table 2. Corresponding system data for the second parametric analysis series of simulations
mj..,,~
=
Area of Collector Ac (m 2) 1.0 Vertical height of the store upriser port above the collector inlet, h3 (m)
0.788 1.288 1.788 2.288 4.788
1.5
,/ ,/ ,/ ,/ ,/
2.0
,/ ,/ ,/ ,/ ,/
,/ ,/ ,/ ,/ ,/
3.0
6.0
,/ ,/ ,/ ,/ ,/
,/ ,/ ,/ ,/
91
0.4817(W) -0"937 (9)
Within the stated range, eqn (9) represents the second of two "universal" curves. 4. A C C U R A C Y O F P R E D I C T I O N O F " U N I V E R S A L " CORRELATIONS
Equation 8 is applicable ~universally" to all systems within the stated range. The relationship between the absolute value of the maximum gradient mj.=~, and W, was investigated by varying the value of the mass of water in the store in six further simulations. For the chosen reference systemAc = 2 m 2 and ha = 4.788 m. The high Bailey number, K, of this system (i.e., K = 31.2), ensured that the gradient deviated (due to changes in Zfl by a negligible amount from the observed maximum of 0.429, as the mass stored varied. A plot of the maximum gradients, mj.r,~ as a function of W, is shown in Fig. 8. The best fit line (with a correlation coef-
Errors in determining the gradient of the characteristic curve were dominated by errors in calculating the gradient displacement Amj. For the five sets of simulations (as indicated by each row of Table 2) each for a particular vertical height of the store upriser port above the collector inlet, the mean error, e, incurred when eqns 5, 8, and 9 are used to determine the solar fraction, were found to be a function of the Bailey number, K, given (with a correlation coefficient of 0.904) by, = -+[5.8 + 14.52 K -°~2]
(10)
Equation (10) indicates expected mean errors which vary from 13% for a Bailey number, K, of 4 (typical of a same-level, single-pass system), to 8% for a Bai-
t/3 O'1 Z
o
2.00
t/1 Z w
1.80
& ~
1.60
K-VALUES
<3
z~
1.40
T
X ,< ~.-
1.20
~
k
o
\
~.C~ or"
IA.
1oo
o\
÷
&
4.3
v
10.2
*
15.2
x
20.1
o
30.5
I,-Z m
Lt O t--" Z tl.I
0.80
'~
0,
x ,i
~
0.L
L
=3
t.~ t,_/ '< ,--J
,7
tn ,
0
0.10
0.20
•
0.30
0.~0
0.50
,
0.60
,
0.70
,
0.80
I
0.90
!
!
1.00
1.10
REFERENCE YELLOT NUHBER, Zj, (DIMENSIONLESS) Fig. 7. DisplacementAm: of the gradientsof the characteristiccurvesdue to variationsin the reference Ycllot number,Z: for differentBailey numbers,K.
P. A. HOBSONand B. NORTON
92
6"60 I /,.~,0
&,.20 l,.O0
3.80 3.60 3.40 "-'m xg
3.20
E--'~
- " ,,,.. i--i
3.00 2.80 2.60 2.40 2.20 2.00 3.80
I
/,.20
I
4.60
I
5.00
I
I
I
I
S.40
5.80
6.20
6.60
In (WxIO0) Fig. 8. Variations of values of ms.m,~ as a function of the specific load, W.
ley number, K, of 30 (typical of a distributed, multiple-pass system). 5. A DESIGN NOMOGRAM The relationships between the Yellot, Z, Bailey, K, Heywood, Y, and Brooks, X, numbers and the specific load, W, may be summarised as a nomogram as shown in Fig. 9. The quadrants in the nomogram correspond (reading anticlockwise from the top righthand side) to eqns 8, 9, and 5 (both lower quadrants) respectively. The Heywood, Y, and Yellot, Z, numbers and the specific load, W, are functions of the applied conditions, whereas the Bailey number, K, is a function essentially of the system design. However, all these dimensionless groups include information available readily to a designer who, using the nomogram in Fig. 9, can thus determine the Brooks number, X, and thus the solar fractions. A worked example is given in the Appendix.
6. CONCLUSIONS Five dimensionless groups pertaining to the dimensions, thermal characteristics and operating conditions of a thermosyphon solar-energy water heater, have been identified. A characteristic thermal performance curve for an individual system can be determined in terms of three of these dimensionless groups. Using such a curve, the calculated annual solar fraction agreed well with the corresponding value computed from the numerical simulation. As the basis for a test method for characterising the long-term performance of individual systems from short-term data, this technique requires a much-reduced test period when compared with previous methods. Using the numerical simulation model, values representing the characteristic gradients of a wide range of systems were investigated from which two universal curves were developed. These two CUlWesformed the basis of a technique by which the thermal perfor*
A design nomogram for direct thermosyphon solar-energy water heaters
o
0~--~-~_~
~
93
~
=
~- ~
Z
0'7 - =~
~'l~
~,
~~i~ t~
XVI~ WOU:I J.N:IIO'v~ ~O 1N:~I~33V~SIO ~u9 ~ O
00000~<~
~.
o "N
,',."
~.
~,,c::,~ o.e-NNN
Z
P. A. HOBSON and B. NORTON
94
mances of directly heated systems (within a stated range) could be predicted. The veracity o f these design correlations for climatic conditions other than those experienced in the U.K. remains to be determined.
ref Refers to a reference value s Pertaining to the storage tank
Acknowledgment--The authors would like to thank the Science and Engineering Research Council, Swindon, U.K., for their financial support.
NOMENCLATURE
REFERENCES
Where units are not specified, quantities are dimensionless. At Area of collector, m 2 Cp Specific heat capacity of water, J kg -~ K -J D Internal diameter of considered pipe, m FAy Collector efficiency factor based on the mean fluid temperature within the collector f Daily solar fraction g Acceleration due to gravity, m s -~ H Total daily global radiation incident on the plane of the collector, J m -2 h, Vertical height of store downcomer port above collector inlet, m h2 Vertical distance between the inlet and outlet of the collector, m h3 Vertical height of store upriser port above collector inlet, m K Bailey number defined by eqn (7) L Component length, m M Mass of water, kg mj Gradient of characteristic curve determined using data from the month of June and defined by eqn (5) m* Correlating function defined by eqn (8) m,f Reference thermosyphonic mass flow rate kg s -t N Number of riser pipes in collector Np Mean dally circulation number, dayQ,,~ Total daily hot water energy requirements (J) T Temperature, *C t Time, duration of insolation period, s U General effective heat transfer coefficient (Wm -2 K-') W Specific load ratio defined by eqn (6) X Brooks number defined by eqn (3) Y Heywood number defined by eqn (1) Z Yellot number defined by eqn (2) ct Absorptanee of collector absorber plate surface 13 Cubic expansivity of water (K -~) A( ) Increment in the quantity in parenthesis Error value (%) v Kinematic viscosity of water (m z s -J) p Density of water (kg/m -3) ~r Transmittance of glass collector cover
1. S. A. Klein, W. A. Beckman, and J. A. Duffle, A design procedure for solar heating systems, Solar Energy 18, 113-127 (1976). 2. S. T. Liu and J. E. Hill, A proposed technique for correlating the performance of solar domestic water heating systems, ASHRAE Transactions, 85, part (i), 96-109 (1979). 3. M. P. Malkin, S. A. Klein, and J. A. Duffle, A design method for thermosyphon solar domestic hot water systems, Solar Engineering 1986, Proc. of the ASME Solar Energy Conf., Aneheim, CA (1986). 4. G. L. Morrison and N. H. Tran, Correlation of solar water heater test data, Solar Energy 39, 135-142 (1987). 5. Z. P. Song and H. J. Zhang, Prediction of system performance of solar water heaters for a specified locality, Solar Energy 28, 433-441 (1982). 6. P. A. Hobson, S. N. G. Lo, B. Norton, and S. D. Probert, Correlating the daily performance of indirect thermosyphon solar energy water heaters, Advances in Solar Energy Technology Proc. ISES Solar World Congress, Hamburg, F.R. Germany, pp. 1102-1114 (1988). 7. B. Norton, P. D. Fleming, S. N. G. Lo, H. G. W. Wilson, and S. D. Probert, Data acquisition from thermosyphon solar-energy water heaters for a terrace of three dwellings, Proc. UK-ISES Workshop on Solar Energy and Building Design, Birmingham, U.K. pp. 17-21 (1985). 8. B. Norton, A. Reis, S. N. G. Lo, P. A. Hohson, and A. C. Inverno, Correlation of the performance of passive solar-energy water-heaters for the Portuguese climate, Proc. P.L.E.A. 88 Passive and Low Energy Architecture, Porto, Portugal, pp. 687-692 (1988). 9. P. A. Hobson and B. Norton, Verified accurate performance simulation model for direct thermosyphon solar water heaters, ASME Journal of Solar Energy Engineering 110, 282-292 (1988). 10. A. Smith, J. K. Page and J. L. Thomson, A meterological data base system for architectural and building engineering designers, 1, Science and Engineering Research Council, 2nd edition, Swindon, U.K. (1983). 11. S. N. G. Lo, B. Norton and P. A. Hobson, Demonstration of thermosyphon solar-energy water heating in a group of three dwellings, Final Report to the Commission of the European Communities, Project No. SE 03483, Brussels, Belgium (1987). 12. D. J. Close, The performance of solar water heaters with natural circulation, Solar Energy 6, 33-40 (1962). 13. P. A. Hobson, B. Norton, and E. Kovolos, Verified high level simulation of thermosyphon solar water heaters, Final Report on Grant GR/D/66593 for Science and Engineering Research Council, Swindon, U.K. (1988).
Subscripts a c d e L m max p
Refers to ambient conditions Pertaining to the collector Pertaining to the downcomer pipe Representing an effective overall value Refers to daily thermal load on solar water heater Refers to mains water supply Refers to a maximum value Pertaining to characteristics common to both the upriser and downcomer pipes r Pertaining to the collector riser pipes
APPENDIX
A worked example of using the correlations to predict the daily solar fraction The component specifications of a thermosyphon solarenergy water heater and climatic conditions (for a typical day in June) used in the following example are given in Table 3. Evaluating the parameters K, W, Y and Z from equations 7, 6, 1 and 2 gives K = 12, W --- 0.7, Y = 20, and Z =
0.3 respectively. Also, since the thermal performance is being determined for the reference month of June, Zj ffi Z = 0.3. A three-stage algorithm is used to determine the dimensionless Brooks number, X, from which the solar fraction can be calculated. Stage I. Determine the deviation, Amj (due to circulation number effects) of the characteristic gradient from the maximum value, rnj.,~.
A design nomogram for direct thermosyphon solar-energy water heaters Table 3. Details of the configuration, operating conditions and thermal properties o f the thermosyphon solar-energy water-heater used in the sample calculation a~ = 2 . 0 FAy = 0.9
System Parameters
( , a ) . = 0.72 UL = 3.5 (UA), = 3 N=8 M, = 297 h3 = 1.8 he = 0.7 L, = 1 D, = 0.015 L, = 8.72 D . = 0.025
(m 2)
(W m -z K -I) (W K -~)
(m)
Weather Conditions
(MJ m -2) (°C) (°C) (s)
Hot-water Demand
Mc = 208 T, = 46
(kg) (°C)
p. = p~ = v. = C~ = [3. =
(kg m -3) (Ns m -z) (m: s) (J kg -1 K -I) (K -l)
Fluid Properties
From eqn (8), m* = 0.055, giving Ares = 0.051. This stage corresponds to the fhst quadrant of the nomogram shown in Fig. 9. S t a g e l l . Determine the m a x i m u m gradient, m s . ~ for the system and subtract the gradient displacement, Ares, to give the actual gradient, ms, of the characteristic curve. E i t h e r using eqn 9, rns.M~ = 0.673, the actual gradient, ms is then given by m j = ms,m~ - (Ams) = 0.622
(kg) (m) (m) (m) (m) (m)
H,a = 19.2 To = 16 T. = 15 t = 59,220
998 10 -3 1.00 x 10 -6 4190 2.1 x 10 -~
95
o r using the nomogram, the value of ms can be read off from the second quadrant (moving anticlockwise). Stage 111. Using the characteristic curve, calculate X from which the solar fraction can be determined. From equation 5, X --- 10.71. This calculation corresponds to the third and fourth quadrants of the nomogram. The total thermal load, Q,,~ when M, kg of water are heated from the mains cold water supply temperature, T . to the required temperature of TL is Q.
= MLCw (TL - T . ) = 27.02 x 106./
From the definition of the Brooks number, X, given in eqn 3, the daily solar fraction can be determined; f =
XMLCw (Ta - Tin)
Q~
= 0.345
The estimated mean error hand, e, associated with this value for the daily solar fraction is, from eqn (10), • = ---7%.
0038-092X/89 $3.00 + .00 Copyright © 1989 Maxwell Pergamon Macmillan plc
Printed in the U.S.A.
A DESIGN NOMOGRAM FOR DIRECT THERMOSYPHON SOLAR-ENERGY WATER HEATERS* P. A. HOBSON AFRC Institute of Engineering Research, Silsoe, Bedford MK45 4HS, U.K. and B. NORTON Department of Building, University of Ulster, Co. Antrim BT37 0QB, N. Ireland AbstractEA characteristic curve for an individual directly heated thermosyphon solar-energy water heater, when obtained from data of an appropriately chosen test period of 30 days, has been shown to predict the annual solar fraction to within 3% of the corresponding value obtained from a validated numerical simulation model. An extension of this analysis produces two underlying correlations of five dimensionless groups from which the performances of a wide range of thermosyphon solar-energy water heaters with varying thermal loads can be predicted. The estimated uncertainty of this design technique in predicting an annual solar fraction ranges from 8% for multiple-pass systems to 13% for same level. single-pass systems.
I. I N T R O D U C T I O N
X
The identification of generalised dimensionless grouped parameters for both pumped[ 1,2] and buoyancy-driven[3-5] solar-energy water heaters, from which their thermal characteristics can be determined, has provided a practical approach by which their long-term performance can be predicted readily. One of the major inherent drawbacks of correlating parameters derived previously has been that they are based on steady-state analyses and therefore relate to performance over relatively long periods of operation. A method of determining a characteristic correlation curve for an individual natural-circulation solar-energy water heater based on a transient analysis which relates to diurnal performance, has been developed[6]. From a transient heat balance on a generic directly heated thermosyphon solar-energy water heater, the following dimensionless parameters Y, Z and X, designated the Heywood, Yellot and Brooks numbers respectively, have been identified[6]:
1 - exp(-Z)
Heywood number:
Y=
[FAvAcUL + (UA),]t Yellot number:
Z =
Brooks number:
X =
-
T,.)
+
l
(4)
2. THE VERACITYOF THE CORRELATION TECHNIQUE AS A METHOD OF DETERMLNE~G LONG-TERM SYSTEMPERFORMANCE The correlation technique outlined was applied to thermal performance data generated by a validated numerical simulation model[9] using Kew (London, U.K.) weather data[10] for a total of 8 months from March to October. The main features of the simulation model[9] were • A fully transient one-dimensional analysis was applied to the heat and momentum transfer processes occurring within the fluid contained in all the components of the system (i.e., collector, upriser, storage tank, and downcomer). • The collector model was based on a two-dimensional analysis thereby permitting the accurate simulation of its transient performance. • Friction factor correlations relevant to thermally destabilised flows were used in determining the fluid losses within the risers. • All fluid properties and heat transfer coefficients
(I) (2)
(M,C,,,) f Qtot M,C.,(T,.
Z
For experimental data collected via the monitoring of systems operating in England[6,7] and Portugal[8], the linear relationship between X/[ l - e t-z'] and Y/Z, implied by eqn (4) was observed. In this paper, a detailed numerical simulation model[9] is used to investigate the accuracy with which such correlations determine the long-term thermal performances of individual systems. The analysis is extended to produce a simple, but relatively accurate, design method for direct thermosyphon solar-energy water heaters.
FA~Ac(*=)J-/ M, Cw(Ta - 7"=)
Y = -
(3)
They were found (6) to be related by the expression
*Includes material presented to the ASME Winter AnnuM Meeting, Chicago, Illinois, November 1988. Address correspondence to Prof. B. Norton, Dept. of Building, University of Ulster, Co. Antrim, BT37 0QB, N. Ireland. 85
P. A. HOBSONand B. NORTON
86
were calculated from temperature-dependent relationships and updated at each time step in the numerical solution. • The resulting energy and momentum equations were solved using a fully implicit finite-difference scheme employing first-order upstream differences for convective terms and second-order accurate differencing for diffusion terms. The program was run on a VAX 11/750. Input data for both the mains water supply temperatures and the solar water draw-off patterns were based on measurements taken during the long-term monitoring of a group of occupied houses in Wharley End, Bedfordshire, in southeast England[7], in which thermosyphon solar-energy water heaters had been installed. For these eight simulation runs (i.e., one for each month, March through October) Ac = 4 m 2, Ms = 780 kg, ha = 3 m and h~ = 2 m. For the remaining winter months (i.e., November to February), convergence of the solutions to the momentum and energy equations in the simulation model were frequently unobtainable. This was due to the errors introduced by the fluid properties, as predicted by the approximating polynomials, at near-freezing conditions. However, not only were these simulation results realistic but they were also acceptable, as the four months from November to February contribute only 9% of the total annual global radiation received
O3 IJ.I .--I
in the U.K. Furthermore, in practice, an automatically activated solenoid valve, as part of the frost protection system on such directly heated systems, causes the collector to be drained down at 2 (-2)°C[11], thereby rendering it inoperative for periods in the winter months. A plot of daily values of X/[ 1 - e x p ( - Z ) ] against Y/Z is shown in Fig. I for the March through October simulated period of operation. The values of Fay, Us included in Y and Z were calculated on a mean daily basis, based on the temperature-dependent heat transfer coefficients used in the simulation program. Values of ('ro0= were calculated assuming a normal effective angle of incidence of insolation on the collector. The correlation coefficient for these values over the 245 days was 0.97. The gradients of the best straight lines through the data were observed to alter seasonally. The equations of the best straight lines through each of the eight individual monthly graphs of X/[1 - e x p ( - Z ) ] versus Y/Z were each used independently to predict the annual solar fraction. The ensuent errors in the predicted annual solar fractions are shown in Fig. 2. Distinct error minima of - 3 . 0 % and +3.0% were observed to occur in the months of May and June respectively. These results favoured either May or June as being the most appropriate ~reference" months from which a characteristic annual thermal performance curve could be
600.
Z
C) 0 Z I,a,.I
z
¢-t
6000
N
0
i
200
"
X
LLI I
°°
x I
600
-200.0 -1600 -1200 - 0 0 - 6
800
I
1200
I
1600
I
2000
Y / Z (DIMENSIONLESS) 0
-200"
0
-600" 0
0
-600 Fig. 1. Correlation of X/[l - exp(-Z)] against Y/Z using simulated thermal performance data.
A design nomogram for direct thermosyphon solar-energy water heaters
87
40
--
3O
Z
T--.
20
t~
e,,, l.l-
,v
10
_.J
7
-10 I-~J
"'
-2(
a. z
,,-
-30
¢Y
-z,l
t
i
MAR
APR
t
!
MAY
!
JUN
JUL
I
AUG
|
SEP
!
OCT
MONTH FROM WHICH CHARACTERISTIC CURVE IS TAKEN Fig. 2. Error incurred in predicting the annual solar fraction using the characteristic curves for the months of March to October.
obtained; for June this is given by
X/[ 1 -
exp(-Z)] =
mj(glZ)
(5)
In general it was observed that the gradient mj, (calculated from eqns 8 and 9: see later) gave values of less than unity. This was due to the assumption of equal mean store and mean collector temperatures which was implicit in the simplified analysis from which eqn 5 was derived[6]; an assumption that would lead to an over-estimate of the system performance. A value of ms less than unity, in effect represented a "thermal penalty" coefficient which brings the overoptimistic thermal performance predicted by the simplified analysis, in line with the more realistic data generated by the high-level simulation. Plots of eqn (5) were found to pass through the origin and the constant (of unity) in eqn (4) was therefore discarded. The cumulative solar fraction and associated error over the year calculated by using eqn (5), is shown in Fig. 3. Using dally average monthly data, a single calculation giving a representative predicted dally solar contribution for each particular month was determined using eqn (5). To determine the total monthly solar contribution this daily value was assumed to remain invariant over the month and was therefore multiplied by the number of days in the month. Using
this procedure, the predicted annual solar fraction was within +6.3% of that calculated using the high-level simulation model. The number of basic calculations was therefore reduced from 245 (i.e., the total number of operational days) to 8 (i.e., one average day for each operational month) with a relatively small decrease in accuracy (shown in Fig. 3).
3. DEVELOPMENTOF A DESIGN ALGORITHM An investigation was undertaken into the effects on the gradient, mj, of changes in the primary dimensionless system parameter, the Yellot number, Z, evaluated, for reference purposes, for the month of June (Zj). This was carried out initially by varying the collector area of a reference system with h3 = 2.48 m and hi = 1.5 m for a total of six systems (designated in row three of Table 1). An asymtopic maximum gradient of approximately 0.3 was observed (see Fig. 4) for values of Zj greater than 0.54. When the collector area increases relative to the store capacity, the temperatures are such that the buoyancy-driven flow rates around the system increase. With this progressive increase, the conditions are therefore approached in which the assumption (pertinent to multiple-pass systems made in the original analysis[6]) of equivalent
88
P. A. HOBSONand B. NORTON
g
ts
,:
':
-5 z
.~-. . . . . .
,/
\
-10 -~s
-10
OPERATIONAL PERIOD
•~ 5 0
--I "~
.400
Z
.350 p j/
.300 eY LL
•250
.,.J
.200
,~
H~H LEVEL SIMULATION MODEL
LU
~'/
.150
CORRELATION BASED ON DALLY DATA
I"-,.,.I
/~//'
.100
',r
CORRELATION BASED ON DALLY AVERAGE MONTHLY DATA
I
EQUATION
.050 JAN FEB MAR APR HAY
JUN JUL
AUG SEP
OCT
NOV DEC
TIME
Fig. 3. Comparison of solar fraction calculated from the numerical simulation model with corresponding values determined from: (1) the correlation technique based on daily data, and (2) the correlation technique based on daily mean monthly data. mean store and collector temperatures[12] becomes valid• For the relative position of the collector and store, used in the reference system, the degree of multiple-pass behaviour is sufficient for the above assumption to be valid at and above values of Zs greater than 0.54. A further series of nine simulations were run in which the Yellot number, Zs, was varied by changing the thermal mass of the store for two values of the collector area (see columns 3 and 5 of Table I). A plot of the resulting gradients against corresponding mean monthly values of the parameter Zs indicated a fundamental change in the relationship between ms and Zs when compared with simulations in which Table 1. Corresponding system data for the first parametric analysis series of simulations Area of Collector A~ (m~) 0.5 Mass of water in StOl~
M, (kg)
50 too 200 300 500 800 l,OOO
/
1.0
2.0
3.0
/
/ / / / /
/
4.0
/ / / /
5.0
variations in Zj were due to area changes alone. However, when Zj was plotted against corresponding values of the mean daily circulation number (Ne) (Fig. 5) for all the simulations indicated in Table 1, a linear relationship emerges, regardless of whether Zj is varied due to changes either in the mass of water in the store or collector area. Because the linear relationship observed in Fig. 5 also contains data from the six simulations between which the collector area was changed, then changes in the gradient, ms which conform to the same relationship as that shown in Fig. 4 must also be occurring for the simulations between which the mass of the store was changed. These changes were therefore being overwhelmed by an additional change in ms due to another factor which, although dependent on the mass of the store, is independent of the circulation number. It also follows that because the effect of Zj on system performance is dependent essentially on the corresponding circulation number, it is independent of this additional storecapacity-dependent factor. By changing the mass of the store alone in the reference system, not only does this change the value of Zj, but also, since the total daily draw-off remains constant, affects the specific load, W[13], where
/ /
Specific load:
ML W = -Ms
(6)
A design nomogram for direct thermosyphon solar-energy water heaters
89
3.50
3.25
u~ LL=
Z 0
0
3.00
Z
0
X ¢:3
--
2.75
0
2.50 er
2.2S L~ 2.00
0 -r-
1.75
1.5( 0.10
I
I
0.20
0.30
I
I
0,40
0,50
I
0,60
I
I
0.70
0.80
Zj (DIHENSIONLESS) Fig. 4. Variation of m~ with Zj due to changes in collector area alone.
V oq .-J Z m (/'1 Z X ,,-t
V
IJ=l
MASS OF STORE (kgs):
:D Z Z O I,-,,-I 4"
)-
X
+x
V +
107 195
x
302
O
553
0
830
o
1107
13
C~
13+
Z lad 'I"
o0 0.00
I
I
0.10
0.20
I
0.30
I
I
I
I
0,40 0.50 0.60 0.70 Zj (OlMENSIONLESS}
I
0.80
I
0.90
I
I
1.00
1.10
Fig. 5. Variation of Np with Zj due to changes in collector area and store capacity.
90
P.A. HOBSONand B. NORTON
The gradients of the characteristic curves obtained from all the first series of simulations (i.e., for the ranges of systems indicated in Table 1) are shown plotted against W in Fig. 6. The variations in the gradients due to W are seen to be large relative to those changes in which W remained constant while Zj (and therefore the mean daily circulation number) alone, was varied. The variation of gradient with W is due to it being assumed in the original analysis[6] that the withdrawal of heated water occurs as a single event at the end of the insolation period. Changing the relative vertical distance of the store from the collector from that of a reference system, changes the prevalent flow rate, and therefore the mean daily circulation number of the system; this will alter the gradient of the reference characteristic curve. A family of curves is therefore required (for a fixed arbitrary value of W), each of which relates the gradient to the Yellot number, Zj, for different relative collector/store heights. All the curves will converge to the same maximum gradient, this value being dependent on the magnitude of W chosen. The relationship between the displacement of the gradient from the maximum value represented by these family of curves, will be valid for all values of W because the circulation number is independent of W. From this family of curves, a system can then be chosen for which the gradient of the characteristic curve is a maximum. Using this system, a plot similar to that in Fig. 6 of mj.ma~ against W, can be determined and
which will then be representative of the performance of all systems in which the critical circulation number (beyond which no further change in the gradient occurs), has been exceeded. The Bailey number, K, which represents the system parameters effecting flow within the system[13] is defined as;
pf3gAT~t.{h3 h:/2] vm~f[L,./N(D~) + L,/O~)] -
Bailey number: K =
(7)
where ATref and mref are given the values 10°C and 10 -t kgs -t respectively. The basic parameters of a reference system for which Ms = 280 kg, were varied in a second series of simulations according to Table 2. Values of the resulting gradient displacements Amj from the observed maximum value of 0.429 are shown in Fig. 7. A correlating function (with a correlation co-efficient of 0.94), relating the displacements of the characteristic gradients of this set of simulated systems to their respective system parameters, is given by Amj = 2.541 x 10 -3 + 0.780(m*) + 1.967(m*) 2
(8)
over the range 0_< m*__O.15 where m* = 0.195 exp[(0.402 - 0.387K)Zj].
1.z~O
1.30 1.20 1.10 trl ELI ,--J Z 0 Z W =E
1.0C
0.90 0.80 0.'/0
t--" Z
0.60 0.50 O.~C
I--" t~
0.30 0.20 0.10
"r
0.0
I
0.00
0.50
I
1.O0
I
I
1.50 2.00
I
I
I
2.50 3.00 3.50 W (DIHENSIONLESS)
I
4.00
I
~.50
Fig. 6. Variation of m~ with W due to changes in store capacity.
I
5.00
I
5.50
!
6.00
A design nomogram for direct thermosyphon solar-energy water heaters ficient of 0.999) through this data is given by
Table 2. Corresponding system data for the second parametric analysis series of simulations
mj..,,~
=
Area of Collector Ac (m 2) 1.0 Vertical height of the store upriser port above the collector inlet, h3 (m)
0.788 1.288 1.788 2.288 4.788
1.5
,/ ,/ ,/ ,/ ,/
2.0
,/ ,/ ,/ ,/ ,/
,/ ,/ ,/ ,/ ,/
3.0
6.0
,/ ,/ ,/ ,/ ,/
,/ ,/ ,/ ,/
91
0.4817(W) -0"937 (9)
Within the stated range, eqn (9) represents the second of two "universal" curves. 4. A C C U R A C Y O F P R E D I C T I O N O F " U N I V E R S A L " CORRELATIONS
Equation 8 is applicable ~universally" to all systems within the stated range. The relationship between the absolute value of the maximum gradient mj.=~, and W, was investigated by varying the value of the mass of water in the store in six further simulations. For the chosen reference systemAc = 2 m 2 and ha = 4.788 m. The high Bailey number, K, of this system (i.e., K = 31.2), ensured that the gradient deviated (due to changes in Zfl by a negligible amount from the observed maximum of 0.429, as the mass stored varied. A plot of the maximum gradients, mj.r,~ as a function of W, is shown in Fig. 8. The best fit line (with a correlation coef-
Errors in determining the gradient of the characteristic curve were dominated by errors in calculating the gradient displacement Amj. For the five sets of simulations (as indicated by each row of Table 2) each for a particular vertical height of the store upriser port above the collector inlet, the mean error, e, incurred when eqns 5, 8, and 9 are used to determine the solar fraction, were found to be a function of the Bailey number, K, given (with a correlation coefficient of 0.904) by, = -+[5.8 + 14.52 K -°~2]
(10)
Equation (10) indicates expected mean errors which vary from 13% for a Bailey number, K, of 4 (typical of a same-level, single-pass system), to 8% for a Bai-
t/3 O'1 Z
o
2.00
t/1 Z w
1.80
& ~
1.60
K-VALUES
<3
z~
1.40
T
X ,< ~.-
1.20
~
k
o
\
~.C~ or"
IA.
1oo
o\
÷
&
4.3
v
10.2
*
15.2
x
20.1
o
30.5
I,-Z m
Lt O t--" Z tl.I
0.80
'~
0,
x ,i
~
0.L
L
=3
t.~ t,_/ '< ,--J
,7
tn ,
0
0.10
0.20
•
0.30
0.~0
0.50
,
0.60
,
0.70
,
0.80
I
0.90
!
!
1.00
1.10
REFERENCE YELLOT NUHBER, Zj, (DIMENSIONLESS) Fig. 7. DisplacementAm: of the gradientsof the characteristiccurvesdue to variationsin the reference Ycllot number,Z: for differentBailey numbers,K.
P. A. HOBSONand B. NORTON
92
6"60 I /,.~,0
&,.20 l,.O0
3.80 3.60 3.40 "-'m xg
3.20
E--'~
- " ,,,.. i--i
3.00 2.80 2.60 2.40 2.20 2.00 3.80
I
/,.20
I
4.60
I
5.00
I
I
I
I
S.40
5.80
6.20
6.60
In (WxIO0) Fig. 8. Variations of values of ms.m,~ as a function of the specific load, W.
ley number, K, of 30 (typical of a distributed, multiple-pass system). 5. A DESIGN NOMOGRAM The relationships between the Yellot, Z, Bailey, K, Heywood, Y, and Brooks, X, numbers and the specific load, W, may be summarised as a nomogram as shown in Fig. 9. The quadrants in the nomogram correspond (reading anticlockwise from the top righthand side) to eqns 8, 9, and 5 (both lower quadrants) respectively. The Heywood, Y, and Yellot, Z, numbers and the specific load, W, are functions of the applied conditions, whereas the Bailey number, K, is a function essentially of the system design. However, all these dimensionless groups include information available readily to a designer who, using the nomogram in Fig. 9, can thus determine the Brooks number, X, and thus the solar fractions. A worked example is given in the Appendix.
6. CONCLUSIONS Five dimensionless groups pertaining to the dimensions, thermal characteristics and operating conditions of a thermosyphon solar-energy water heater, have been identified. A characteristic thermal performance curve for an individual system can be determined in terms of three of these dimensionless groups. Using such a curve, the calculated annual solar fraction agreed well with the corresponding value computed from the numerical simulation. As the basis for a test method for characterising the long-term performance of individual systems from short-term data, this technique requires a much-reduced test period when compared with previous methods. Using the numerical simulation model, values representing the characteristic gradients of a wide range of systems were investigated from which two universal curves were developed. These two CUlWesformed the basis of a technique by which the thermal perfor*
A design nomogram for direct thermosyphon solar-energy water heaters
o
0~--~-~_~
~
93
~
=
~- ~
Z
0'7 - =~
~'l~
~,
~~i~ t~
XVI~ WOU:I J.N:IIO'v~ ~O 1N:~I~33V~SIO ~u9 ~ O
00000~<~
~.
o "N
,',."
~.
~,,c::,~ o.e-NNN
Z
P. A. HOBSON and B. NORTON
94
mances of directly heated systems (within a stated range) could be predicted. The veracity o f these design correlations for climatic conditions other than those experienced in the U.K. remains to be determined.
ref Refers to a reference value s Pertaining to the storage tank
Acknowledgment--The authors would like to thank the Science and Engineering Research Council, Swindon, U.K., for their financial support.
NOMENCLATURE
REFERENCES
Where units are not specified, quantities are dimensionless. At Area of collector, m 2 Cp Specific heat capacity of water, J kg -~ K -J D Internal diameter of considered pipe, m FAy Collector efficiency factor based on the mean fluid temperature within the collector f Daily solar fraction g Acceleration due to gravity, m s -~ H Total daily global radiation incident on the plane of the collector, J m -2 h, Vertical height of store downcomer port above collector inlet, m h2 Vertical distance between the inlet and outlet of the collector, m h3 Vertical height of store upriser port above collector inlet, m K Bailey number defined by eqn (7) L Component length, m M Mass of water, kg mj Gradient of characteristic curve determined using data from the month of June and defined by eqn (5) m* Correlating function defined by eqn (8) m,f Reference thermosyphonic mass flow rate kg s -t N Number of riser pipes in collector Np Mean dally circulation number, dayQ,,~ Total daily hot water energy requirements (J) T Temperature, *C t Time, duration of insolation period, s U General effective heat transfer coefficient (Wm -2 K-') W Specific load ratio defined by eqn (6) X Brooks number defined by eqn (3) Y Heywood number defined by eqn (1) Z Yellot number defined by eqn (2) ct Absorptanee of collector absorber plate surface 13 Cubic expansivity of water (K -~) A( ) Increment in the quantity in parenthesis Error value (%) v Kinematic viscosity of water (m z s -J) p Density of water (kg/m -3) ~r Transmittance of glass collector cover
1. S. A. Klein, W. A. Beckman, and J. A. Duffle, A design procedure for solar heating systems, Solar Energy 18, 113-127 (1976). 2. S. T. Liu and J. E. Hill, A proposed technique for correlating the performance of solar domestic water heating systems, ASHRAE Transactions, 85, part (i), 96-109 (1979). 3. M. P. Malkin, S. A. Klein, and J. A. Duffle, A design method for thermosyphon solar domestic hot water systems, Solar Engineering 1986, Proc. of the ASME Solar Energy Conf., Aneheim, CA (1986). 4. G. L. Morrison and N. H. Tran, Correlation of solar water heater test data, Solar Energy 39, 135-142 (1987). 5. Z. P. Song and H. J. Zhang, Prediction of system performance of solar water heaters for a specified locality, Solar Energy 28, 433-441 (1982). 6. P. A. Hobson, S. N. G. Lo, B. Norton, and S. D. Probert, Correlating the daily performance of indirect thermosyphon solar energy water heaters, Advances in Solar Energy Technology Proc. ISES Solar World Congress, Hamburg, F.R. Germany, pp. 1102-1114 (1988). 7. B. Norton, P. D. Fleming, S. N. G. Lo, H. G. W. Wilson, and S. D. Probert, Data acquisition from thermosyphon solar-energy water heaters for a terrace of three dwellings, Proc. UK-ISES Workshop on Solar Energy and Building Design, Birmingham, U.K. pp. 17-21 (1985). 8. B. Norton, A. Reis, S. N. G. Lo, P. A. Hohson, and A. C. Inverno, Correlation of the performance of passive solar-energy water-heaters for the Portuguese climate, Proc. P.L.E.A. 88 Passive and Low Energy Architecture, Porto, Portugal, pp. 687-692 (1988). 9. P. A. Hobson and B. Norton, Verified accurate performance simulation model for direct thermosyphon solar water heaters, ASME Journal of Solar Energy Engineering 110, 282-292 (1988). 10. A. Smith, J. K. Page and J. L. Thomson, A meterological data base system for architectural and building engineering designers, 1, Science and Engineering Research Council, 2nd edition, Swindon, U.K. (1983). 11. S. N. G. Lo, B. Norton and P. A. Hobson, Demonstration of thermosyphon solar-energy water heating in a group of three dwellings, Final Report to the Commission of the European Communities, Project No. SE 03483, Brussels, Belgium (1987). 12. D. J. Close, The performance of solar water heaters with natural circulation, Solar Energy 6, 33-40 (1962). 13. P. A. Hobson, B. Norton, and E. Kovolos, Verified high level simulation of thermosyphon solar water heaters, Final Report on Grant GR/D/66593 for Science and Engineering Research Council, Swindon, U.K. (1988).
Subscripts a c d e L m max p
Refers to ambient conditions Pertaining to the collector Pertaining to the downcomer pipe Representing an effective overall value Refers to daily thermal load on solar water heater Refers to mains water supply Refers to a maximum value Pertaining to characteristics common to both the upriser and downcomer pipes r Pertaining to the collector riser pipes
APPENDIX
A worked example of using the correlations to predict the daily solar fraction The component specifications of a thermosyphon solarenergy water heater and climatic conditions (for a typical day in June) used in the following example are given in Table 3. Evaluating the parameters K, W, Y and Z from equations 7, 6, 1 and 2 gives K = 12, W --- 0.7, Y = 20, and Z =
0.3 respectively. Also, since the thermal performance is being determined for the reference month of June, Zj ffi Z = 0.3. A three-stage algorithm is used to determine the dimensionless Brooks number, X, from which the solar fraction can be calculated. Stage I. Determine the deviation, Amj (due to circulation number effects) of the characteristic gradient from the maximum value, rnj.,~.
A design nomogram for direct thermosyphon solar-energy water heaters Table 3. Details of the configuration, operating conditions and thermal properties o f the thermosyphon solar-energy water-heater used in the sample calculation a~ = 2 . 0 FAy = 0.9
System Parameters
( , a ) . = 0.72 UL = 3.5 (UA), = 3 N=8 M, = 297 h3 = 1.8 he = 0.7 L, = 1 D, = 0.015 L, = 8.72 D . = 0.025
(m 2)
(W m -z K -I) (W K -~)
(m)
Weather Conditions
(MJ m -2) (°C) (°C) (s)
Hot-water Demand
Mc = 208 T, = 46
(kg) (°C)
p. = p~ = v. = C~ = [3. =
(kg m -3) (Ns m -z) (m: s) (J kg -1 K -I) (K -l)
Fluid Properties
From eqn (8), m* = 0.055, giving Ares = 0.051. This stage corresponds to the fhst quadrant of the nomogram shown in Fig. 9. S t a g e l l . Determine the m a x i m u m gradient, m s . ~ for the system and subtract the gradient displacement, Ares, to give the actual gradient, ms, of the characteristic curve. E i t h e r using eqn 9, rns.M~ = 0.673, the actual gradient, ms is then given by m j = ms,m~ - (Ams) = 0.622
(kg) (m) (m) (m) (m) (m)
H,a = 19.2 To = 16 T. = 15 t = 59,220
998 10 -3 1.00 x 10 -6 4190 2.1 x 10 -~
95
o r using the nomogram, the value of ms can be read off from the second quadrant (moving anticlockwise). Stage 111. Using the characteristic curve, calculate X from which the solar fraction can be determined. From equation 5, X --- 10.71. This calculation corresponds to the third and fourth quadrants of the nomogram. The total thermal load, Q,,~ when M, kg of water are heated from the mains cold water supply temperature, T . to the required temperature of TL is Q.
= MLCw (TL - T . ) = 27.02 x 106./
From the definition of the Brooks number, X, given in eqn 3, the daily solar fraction can be determined; f =
XMLCw (Ta - Tin)
Q~
= 0.345
The estimated mean error hand, e, associated with this value for the daily solar fraction is, from eqn (10), • = ---7%.