Design and optimization of solar industrial hot water systems with storage

Solar Energy Vol. 32, No. I, pp. 121-133. 1984 Printed in Great Britain,

0038-092X/84 $3,00+ .00 © I984 Pergamon Press l.td.

DESIGN AND OPTIMIZATION OF SOLAR INDUSTRIAL HOT WATER SYSTEMS WITH STORAGE

M. COLLARES-PEREIRA,I J. M. GORDON, A. RABL~ and Y. ZARMI§ Applied Solar Calculations Unit. Blaustein Institute for Desert Research, Ben-Gurion University of the Negev. Sede Boqer Campus, Israel

(Revision accepted 10 January 1983) Abstract--This paper presents a new method for the design and optimization of solar industrial process hot water systems with storage. The single-pass open-loop design thermally "decoupies" collectors from storage, hence insuring that collectors a/ways heat the coldest fluid possible and that stored heat can be completely depleted by the nighttime load. So the single-pass open-loop design, in spite of the relatively low flow rates entailed, operates at higher system efficiency than conventional system designs. One solved example for an an industrial hot water application shows that the single-pass open-loop design delivers about 30 per cent more useful energy with roughly 30 per cent less storage than the conventional design. Moreover. storage tanks do not have to stand high pressures and can thus be significantly cheaper than in conventional systems. The effects of collector operating time, heat exchangers, and secondary system losses are also treated. The new method is extended to cover systems that require weekend storage. The introduction of weekend storage may be cost effective because it enables the designer to reduce collector area without reducing the yearly useful energy delivered by the system. NOMENCLATURE ,4 collector area Aa.mv collector area at which the onset of energy dumping must be taken into account ,4 .... maximum collector area as determined by an upper tolerance temperature A,~, economically optimum collector area At,,l~ storage tank surface area B,C parameters in the determination of heat exchanger effectiveness C, fixed cost of solar system C~ collector cost C¢ price of fuel C~ storage cost C, specific heat of water C~,p effective cost of capital F' collector efficiency factor FR collector heat removal factor F~ heat exchanger penalty factor I solar irradiance on the collector aperture I ..... maximum solar irradiance on the collector aperture L life cycle savings M .storage mass (water) &, collector flow rate the, process flow rate

tPermanent address: L.N.E.T.I., Departamento de Energias Renovaveis do Paco Lumiar 22, 1600 Lisboa, Portugal. *Permanent address: Center for Energy and Environmental Studies, Princeton University, Princeton, NJ 08544, U.S.A. §Also at Department of Physics, Ben-Gurion University of the Negev, Beersheva, Israel. ~l'his actually includes many situations where steam is used to heat water baths. It is more efficient for solar equipment to supply hot water than steam. Hence it may be practical to produce solar heated water and use the existing steam distribution system as backup water heater.

N

weekend length = number of days per week plant ,s closed P, power rating of collector pump P, power rating of pump drawing water from storage to process q(X) yearly collectible energy correlation of[1] qo, q~, q2 coefficients of q(X) correlation Q net yearly useful energy delivered by solar system Q,,, yearly collectible energy delivered by collectors Q,~ump yearly dumped or discarded energy Qi.... yearly energy lost from storage tank Qpurnp yearly parasitic energy losses r real discount rate r, real fuel escalation rate T temperature T~.h~e,, ambient temperature Te,v temperature of storage tank environment Tin collector inlet temperature To,,t collector outlet temperature Tj, process temperature T, yearly average storage temperature at the end of daily energy collection U collector heat loss coefficient Ut~.k storage tank heat loss coefficient (UA)tank, effective effective UA value for storage tank (UA).x heat exchanger heat transfer size X,X~o collector operating threshold collector efficiency rto collector optical efficiency r~,~,~,,~, efficiency of backup heater heat exchanger effectiveness r,,~ daily collector operating time rr daily duration of load demand -r~,,~, number of seconds in a day (86400) I. INTRODUCTION

Many industrial processes consume hot water. This is particularly common in the food and textile industries.¶ 121

122

M. COLLARES-PEREIRAet al.

For example, in the beverage industry the water used for washing returnable bottles is so contaminated that it must be discarded. One may be able to recover some of the energy with heat exchangers to preheat the incoming fresh water, but the fluid itself is not reused. This feature is of crucial importance for the design of solar heat systems with storage. As shown in this paper, one can design single-pass open-loop systems for this type of application that are significantly more efficient and economical than conventional multi-pass closed-loop systems. A significant fraction of industrial hot water systems have constant loads that operate 24hr/day. Considerations of day-night storage are important in the design of solar heating systems for such applications. Although many plants operate 7 days/week, a significant number of plants close down on weekends. In the latter case considerations of weekend storage are also important. Day-night storage is essential if solar energy is to make a large contribution to our total energy consumption. For example, in order to supply more than about 30 per cent of the energy consumed by plants with 24hr/day loads, one needs large collector fields and storage[I/. The flow diagram for the single-pass open-loop system with storage is shown in Fig. 1. Part (a) shows a direct system, part (b) an indirect system where a heat exchanger separates the collector from storage and process. The heat exchanger may be necessary either to circumvent the freezing of water in the collectors (in cold climates) or in cases where possible corrosion of collectors by water is to be avoided. Our analysis covers both systems, the only difference being an additional heat exchanger penalty factor for the indirect system. For simplicity we consider only the case of greatest practical importance, namely a process that is constant in temperature and flow rate.+ Fluid from the water mains is passed through the collectors at a constant flow rate m,. whenever tho collectors can deliver useful heat. Water enters the collectors at temperature T~,. The outlet temperature To,, varies with insolation. The heated water flows into the storage tank. Whenever hot water is available in the tank, it is withdrawn to the process at constant flow rate rho. The tank should be mixed because that ensures best storage utilization in the single-pass open-loop design. The backup heater is in series with the solar system and operates at variable heat rate to bring the fluid up to the process temperature To. When no solar heat is available, the incoming water bypasses the solar system and goes directly to the backup heater. The storage tank starts out empty at collector turn-on time and is sized to fill up with

¢If the heat rate of the process varies with time, it may be most economical to match the solar system to the constant base component of the load rather than to the total load. If solar equipment is sul~ciently cheap, one may want to choose a larger system with added storage to bridge load variations. If the load varies with season, a monthly calculation based on the monthly utilizability model of[2/ is needed. The basic design philosophy of the single-pass open-loop approach still applies.

r~.c'~

'

~

@ P,

.

.

Lirn.

discard

.

r mains, Tin

mp

mains, TIn Fig. I. Flow diagram for process hot water system with storage:

upper figure--direct; lower figure--with heat exchanger.

solar heated water during the course of an average sunny day. The water tank need not be pressurized because it is not exposed to the pressure of the water mains (typically on the order of five ti~aes atmospheric pressure). Instead, the closed tank can have an internal insulating float. Such tanks are commercially available at much lower cost than pressurized tanks. This system design stands in marked contrast to the conventional multi-pass closed-loop design. In the latter the fluid is returned from storage to the collector for further heating. The flow rate is so high that the storage volume makes on the order of 5-10 passes through the collector during a single sunny day and the temperature rise for each pass is small. In a closed-loop system the tank is exposed to the water mains and must be pressurized. According to the conventional wisdom, the low flow rates of our design imply a low collector efficiency. It is certainly true that for a given inlet temperature the outlet temperature goes up as the flow rate is decreased and hence the instantaneous collector efficiency decreases. However, in a multi-pass system the inlet temperature rises during the course of the day and hence the efficiency decreases anyway. In a single-pass system collection efficiency decreases across the collector from inlet to outlet while in a multi-pass system it decreases during the course of the day. On balance the daily average collector efficiency is essentially the same--if not slightly higher for the one-pass system--as shown by a theorem of[3/and by the computer simulations of[4/. The system e~iciency of the single-pass system actually turns out to be significantly higher than for the multi-pass design, as demonstrated by the examples in this paper. The thermal superiority of the single-pass approach was first established for systems without storage by[l/. The present paper extends the single-pass

Design and optimizationof solar industrial hot water systems with storage approach to systems with day-night and/or weekend storage. But while,the no-storage design of/l] can be used for all process heat systems, whether the fluid is discarded or not. the design of the present paper is appropriate only if the fluid is discarded or if orre can afford to build an additional storage tank to hold one day's worth of process fluid. The lower system efficiency of multi-pass closed-loop systems is primarily caused by mixing in storage, during charging and/or during discharging, an effect that prevents complete utilization of the heat in the storage tank. This explanation is actually not required for the present paper, because all performance equations are calculated from first principles with simple analytical formulae. Thus the reader can readily verify the superior performance of the single-pass open-loop design by comparing it with results for multi-pass systems as obtained by computer simulations or curve fits. e.g. the a]-f chart of[5]. The single-pass open-loop design has an additional advantage that does not show up in a comparison with 4~-f chart results because transients are neglected. In a single-pass design penalties due to collector warm-up and cool-down are much smaller than in a multi-pass design because the collector operates at lower inlet temperature and any heat above T~, is useful. The low flow rate has another benefit beyond the comparison of thermal efficiency. In multi-pass systems the parasitic power consumption for pumping has been found to represent typically 3-10 per cent of the collector energy[6,7]. Since pumping energy is supplied as electricity, its cost can easily eat up a sizable portion of the energy savings from solar heat. The one-pass design prescribed herein represents significant reductions in parasitic energy requirements. The analysis presented herein also applies to plants with batch processing. That is, loads that periodically, but not continuously, require fixed quantities of hot water; for example, washing baths that must be refilled with hot water once every hour. For batch processing, the solar system should include additional "buffer" storage that is large enough to hold the water heated in between batches. The one-pass nature of the storage design remains unaltered, as will calculations of collector area and yearly useful energy delivered by the system. In the following sections the single-pass open-loop system is analyzed in detail for applications that require day-night and/or weekend storage. Formulae and graphs are presented for choosing the system parameters (flow rates, collector area, storage mass) and for calculating annual delivered energy. The parameter values are approximately optimal in the sense of maximizing the life cycle savings. As usual in such situations, the optimum is very broad, and the deviation from the optimum is small compared to typical uncertainties in the input, especially future fuel prices. For a hot water application with day-night storage, the method is illustrated by an example where flat plate collectors are used to heat mains water for a process requiring 70°C water. For the same collector area, the single-pass design turns out to be 28 per cent more SE Vol 32. No b - I

123

efficient than the multi-pass design while requiring about 30 per cent less storage. A similar exatflple for a hot water application with weekend "storage illustrates the interesting result that at current (Israeli) costs, weekend storage may be one of the more cost effective inclusions in system design. 2. SYSTEMEQUATIONS

2.1 Instantaneous efficiency and annual collectible energy The calculation of system performance requires as input the instantaneous efficiency curve of the solar collector field. For the purpose of this paper, the efficiency r) should be parameterized in linear form as ,O = F,[r/o

U(Tnua~T~,,,b~,t)]

(2.1)

where ~()= optical efficiency; U = heat loss coefficient (W/m2K); F ' = f a c t o r to account for temperature difference between absorber surface and fluid (collector efficiency factor); T =temperature. with appropriate subscripts (in K); and I = solar irradiance in W/m'- of aperture area (hemispherical irradiance for flat plate, beam irradiance for focusing collectors). In practice only the products F'~o and F'U are measured and needed for our calculations. Sometimes the efficiency is reported in terms of a fit with an additional parameter to allow for temperature dependent U. In that case one first needs to linearize "0 by approximating the efficiency curve by a straight line which gives the best agreement at anticipated operating temperatures. Because of heat losses from the lines between collectors and from collector field to storage, the parameters F'~/o and F'U for an individual collector should be modified according to the correction procedure of Section 10.3

of/5]. An expression equivalent to eqn (2.1) but based on fldid inlet temperature T~. instead of mean fuid temperature is [5]

"~=FR['I~O U(Tin --Zambient)]

(2.2)

_ (rhcCp) [1 / F'UA\] F= - - - - 0 T - e x p , - m----~)j

(2.3)

with

where rn,./A is the total flow rate divided by the total area A of the collector field, and Cp is the heat capacity of the collector fluid. Equation (2.3) is relevant for our design because the collector inlet temperature is specified to be constant and.equal to T~.. The flow rate is also constant. The collector is turned on whenever insolation exceeds the threshold X~., Xin =

(U/T~o)(Zin-Tambient),

(2.4)

Xj. is to be based on the yearly daytime average ambient temperature. Seasonal or diurnal variations of the instantaneous threshold about its average value have a

M. COt,LARES-PEREIRAet ~ll.

124

negligible effect on the calculations in this paper, as shown in Section 2.4 of [8]. As shown in/I] and/8], the yearly collectible energy for the principal collector types can be calculated with excellent accuracy by evaluating a simple polynomial, if the yearly average threshold X is known. For this purpose one interprets T,,~b~.... in eqn (2.4) as the yearly average daytime ambient temperature. In terms of X, the yearly energy collected Qc,,~ by a collector field of area A, is given by an expression of the form

days have the same operating period. As shown later on, the resulting values of M and rh,. are good even if the operating period varies from day to day. The length of time per day when the collector does not operate is rd~ly Top with Tday= 24 hr = 86400 sec. During this time, hot water (if available) is withdrawn from storage at the process flow rate vfi,,. Clearly one should not make storage any larger than M .... given by

Q~o, = AFRrloq(X)

(2.5)

q(X) = q o - q i X + q2X2

(2.6)

because any storage in excess of this value would never be depleted, i.e. it would be useless. To determine the flow rate rh,. through the collector we note that the collector is to supply hot water both to the process, at rate rG, and to storage. If the tank is to just fill up during the operating period, then the net inflow r h , - mr, into the tank must be equal to M/zoo. This fixes the collector flow rate in terms of storage mass M as

with

(assuming that no energy is discarded), The coefficients qo, q~ and q2 are positive, and they depend on collector type and/or tracking mode as well as on yearly average (daytime) direct normal irradiance I- and, for some types, on latitude. Explicit expressions for the correlations q(X) are given in/l] and/8] for --flat plates at tilt equal latitude; --CPCs of concentration ratio 1.5 and fixed tilt equal latitude; --focusing collectors which track about one axis (horizontal east-west axis. horizontal north-south axis, or polar axis) and focusing collectors which track about two axes. 2.2 Collector operating period An important parameter is the average daily operating period Too of the collector. It depends on collector type, threshold and location. As shown in/l], the operating period top (in sol/day) is equal to minus the derivative of q(X) (the annual energy (in GJ/m 2) collectible per unit area by a collector with FR-0o= l) with respect to threshold X (in KW/m2):

Zoo-

106 dq(X) 3~ ~ x=x~,

(2.7)

The number 365 represents the number of days in a year, and the factor 106 is for consistency of units, 106= (GJ/J)/(KW/W). Even though eqn (2.7) is exact if q(X) is known exactly and even though the quadratic polynomial eqn (2.6) reproduces the data for annual energy within 2-4 per cent, ~he quadratic fit is less reliable for taking derivatives. Hence the formula 106 ,

too = - 365 tq, - 2qzX~,)

(2.8)

can have errors on the order of 10 per cent; fortunately this is acceptable for the present purpose since the consequent errors in Q are of the order of 1 per cent. 2.3 Storage mass and flow rate To explain the selection of storage mass M and collector flow rate rh,. let us for the moment assume that all

-

-

M -< M ...... = th~(rd~,~- zoo),

m, = rh, + (M/zoo).

(2,9)

(2.10)

If the flow rate were larger, hot water would be discarded at the end of the day, while smaller flow rates would leave the top of the tank unutilized. Hence we choose the flow rate according to eqn (2.10). The choice of storage mass will be addressed in Section 4. 2.4 Annual energy • If the collector area is A then the annual collected solar energy is, by eqns (2.3) and (2.5).

Qc~,, = AFRrloq( Xi,)

_ re, c,,,,, [ U

r

F'UA]I

1 - exp[ - m,----~-jj q (Xin).

(2.11)

The net useful energy Q delivered by the system during the year is obtained by subtracting losses from storage Q,o~ and any excess energy Qd,,mp(if any) that cannot be utilized, Q = Qc,,, - Q,o~ - Qd..... •

(2,12)

In well designed systems Q,o~ is a few percent of Q, hence it need not be calculated very accurately. For example, if Q~o~ is 5 per cent of Q, then a 20 per cent error in Q,o~ causes only a I per cent error in Q; such a small error is insignificant. Hence we estimate Q~o~ by considering the steady state heat loss from the storage tank. The average tank temperature at the end of an average day's energy collection T, can be obtained from the daily energy balance of the tank. Since the daily water flow through the collectors is #brop, the water mass M collected in the tank by the end of the day represents a fraction Ml(m,'voo) of the total water flow. All the water from the collectors passes through the tank where it is mixed. Therefore the fraction M/(m, Zoo) of the water flow approximately equals the fraction of the daily collected energy Q~o,/365 that is in the storage tank at the end of the day. Thus one obtains the energy

Design and optimization of solar industrial hot water systems with storage balance of the storage tank as 109 M - - -

365 th, r,,p

Q~,,n = M C , , ( T , - T~.)

(2.13)

125

dumping the following day. We opt instead for a choice of collector area that avoids temperature dumping. As a simple criterion for avoiding temperature dumping, we take the condition (to be explained in the Appendix)

where Cp = 4186J/(kg-K) is the specific heat of water; Q~o. is in GJ/yr; M is in kg: top is in sec/day; and rn,. is in kg/sec. Using eqn (2.11), eqn (2.13) is readily solved for T~. If the tank is located in an environment at average temperature T~,~ then the annual loss from storage is approximated by

where Adumpis the collector area that corresponds to the onset of temperature dumping, and a is a number between zero and unity. As shown in the Appendix, proper solution of the problem of temperature dumping shows that selecting

Q,o~ =(3.1536x 10 2)(]r~ - T,.~)(UA),,,k

a =0.8

(2.14)

where (UA),,,k is the effective tank surface/heat transfer coefficient of the tank in W/K, and Q~o~ is in G J/yr. 3. TEMPERATURE LIMITS AND ENERGY DUMPING

To complete the calculation of Q we need to know the excess energy Qd.mp in eqn (2.12). If the collector area is sufficiently small in relation to the load then no energy dumping will ever occur and the calculation is complete. Before worrying about energy dumping it is advisable to check how large the collector area can be without exceeding the upper tolerance temperature of collector or heat transfer fluid during periods of peak insolation. Typically one wants to keep the collector outlet temperature below 90-95°C to prevent boiling in the collector. Suppose the peak outlet temperature is specified as T . . . . corresponding to peak insolation I,, .... Then the largest permissible collector area Amax is given by the instantaneous energy banance rh, C,,(T .... - T~,) = Ama×FR'oo(Im,×- X~,)

FR'rh,(l .... - X,.)A,, .... a : rh, Cp(Tp - T~,)

(3.4)

affords a conservative estimate of Aaump (i.e. a lower limit for Adump). Since FR depends on area, eqn (3.3) is not yet the explicit solution for Ad,,mo. After inserting eqn (2.3) for FR with A = Ad.,,r. one obtains the explicit result tfi"Cpln[1-[ Aj.mp = - F'U

U(T"zT~")

v(rm.x-r,o)]

(35) -

If A is less than this value then we can safely assume that energy dumping is negligible. 4. OPTIMIZATION OF SYSTEM SIZE

The optimal system is the one that maximizes life cycle savings or, equivalently, minimizes life cycle costs. In the present design the flow rates are fixed by the process and by the storage mass, and thus only the collector area A and the storage mass M are to be optimized: The capital cost of the system is assumed to increase linearly with A and M as Capital Cost = C,, + CAA + C~jM

F'V

1]

Lb.8(-~.~- x,.)n,,JJ

(3.1)

where FR is evaluated at A = A . . . . Inserting eqn (2.3) for FR and solving for Am.x one finds

A .... : -

(3.3)

(3.?)

In the single-pass open-loop design with fixed flow rate there are two distinct mechanisms that can cause energy dumping: mass dumping and temperature dumping. Mass dumping occurs if a larger mass of heated water is collected than can be consumed by the process. Our choice of storage mass, eqn (2.9), and flow rate, eqn (2.10), guarantees that mass dumping does not occur, at least if all days have the same operating period. As shown in the Appendix, the effect of mass dumping is negligible even when variations in operating period are taken into account. Temperature dumping occurs if the storage tank temperature exceeds the process temperature. One could minimize temperature dumping by adding a tempering valve and varying the flow from storage to process; this complicates the controls. Furthermore the use of a tempering valve will not necessarily solve the problem due to conservation of mass, wherein the heated water not consumed on one day by the load will lead to mass

(4.1)

where C.=fixed cost; CA = collector cost ($/m2); and CM - storage cost ($/kg). In a given application the range of A and M is usually small enough that the cost can indeed be described in this form by making a linear fit to cost quotes by equipment suppliers. The life cycle savings L are the difference between fuel savings and the cost of capital and operation and maintenance, + CMM)

(4.2)

C, tl + r,) [1 (1 + r, lN 1 Cf., "r/backup(r--rf) [ - [T-~TJ J

(4.3)

L --- Q G , I - Ccap(Co "}"C A A

where

is the levelized effective fuel cost with CI = price of fuel ($/GJ): rtback.p=efficiency of backup; r1=real fuel escalation rate; r = real discount rate; N = system lifetime (yr). The second term is the effective cost of capital including expenses for operation and maintenance Cc,p= 1 + - ~ [ 1 - ( 1

+r) N]

(4,4)

M COLLARES-PEREIRAet al.

126

where OM is the annual charge for operation and maintenance expressed as a fraction of the capital cost. This analysis is done in constant currency. The inflation rate need not be specified because the discount rate and fuel escalation rate are real rates, i.e. above inflation" (real rate = market rate minus inflation ratet. If Q is known as a function Q(A, M) of A and M, then the maximum of the life cycle savings L can be found by trial and error or by setting the derivatives of L with respect to A and M equal to zero. We have an explicit analytic expression for Q(A, M) only if Qd,~r = 0. (Calculating Qa,mp is more difficult and will not be attempted in this paper.) Strictly speaking, we can therefore find the optimum only if it occurs within the parameter domain where dumping does not occur. Fortunately this is no serious drawback in practice for two reasons. First. it will often turn out that the upper limit for collector area due to an upper tolerance temperature Am~x(eqn (3.2)) is less than the collector area at which the onset of energy dumping occurs A,,,mo (eqn 3.5). Second, it will often be the case that the economically optimum collector area is less than Ad,,~. In this case the economic optimum is rigorously correct. In the rarer instance where the optimum collector area may exceed Ad,mo, we note that the incremental value of collectors beyond the dumping threshold decreases rapidly and therefore the optimum is close to the dumping threshold. Furthermore, the optimum is so broad that the life cycle savings are very close to the maximum if the area is chosen as the dumping threshold instead of the true optimum, as shown by[l].* In fact the error in L is negligible compared to uncertainties in the input parameters. Furthermore by choosing a no-dump design one errs on the side of greater system efficiency and hence higher rate of return from the solar investment. To find the optimum, one can even omit the storage losses because they are typically small. Thus one can simply take Q as Q~o, of eqn (2.11). To proceed with the optimization we evaluate the derivatives of L. The derivative with respect to A at fixed M is ,

F' UA

(OL/OA)M =C,-,q(X,.)F r l o e x p [ - m - - ~ ] - C¢,pCA (4.5) and the one with respect to M at fixed A is

(OLIOM)A [

~

rh,.C,, JJ

(F'UA) exp[ V'tJa ]l m,.C,

-~lJ

- c~"°c~

(4.6)

where the constraint of eqn (2.10) between rh,. and M is understood. The standard criterion of requiring both derivatives to vanish yields two equations, but both ~In [1] an incorrect statement was made to the effect that Q (and with it the rate of return) increases linearly with A. Because of the area dependence of FR. Q decreases slightlyless than linearly with A. However, all the equations in[l] are correct.

depend only on the combination (Alto,.), not on A and 61, separately. Since one of these equatidns is transcendental while the other one is no't, these two equations cannot be satisfied simultaneously, except for very special cons.tellations of input parameters. In general there is no simultaneous solution. This means that there is no optimum unless the space of A and M values is constrained to a finite region. But this is indeed the case. M must not exceed M .... of eqn (2.9), and A is bounded by eqns (3.2) and (3.5). Therefore L attains its maximum when the system is as large as permitted by one of the constraints, unless of course the system is not cost effective at all. This is to be expected because we have not included energy dumping in the equation for L. The optimization procedure then reduces to the following. Start with M = Mm~x and solve eqn (4.5) for the area A = Aop, which maximizes L for this value of M, the result being A

_ mcC,, j_[C~.,q(X,.)F'~lo] opt

--

~

vv

,II

[

"

-

co°oc -

J

(4.7)

If Aop, > A .... (eqn 3.2) or Aop, > Aa,,,p (eqn 3.5), reduce A,,p, to the smaller of the two. Now calculate (OL/OM)A (eqn 4.6) at this value of A (with M still at Mm~x). If this derivative is not negative, the optimal solution has been found. If this derivative is negative, then the optimal solution is the no-storage solution (M = 0) for which the optimal, area Aopt shquld be recalculated with eqn (4.7) with rn, replaced by trip. (In the latter case (M = 0), accurate treatment of energy dumping (when necessary) is presented in[l].) If with the final values of A and M the life cycle savings L turn out to be negative, then the system is not cost effective. 5. SOLVEDEXAMPLE

AS an illustration of the design method delineated above, we consider a solar retrofit for an industrial hot water application in Sede Boqer, Israel (latitude = 30.9°N; yearly average normal beam irradiance = 502W/m2). The process runs 365 days/yr and heats mains water (mains water being, to a good approximation, equal to the daily average ambient temperature) to 70°C at the rate of l kg/sec. Flat plate collectors (at tilt=latitude) of efficiency parameters F'~o-- 0.75 and F'U = 5.0 W/m2K (including estimated piping losses) and at a price of $200]m 2 installed, and unpressurized hot water storage at a price of $200/m 3 installed, are to be considered. All relevant system and economic parameters are summarized in Table 1. Determination of the optimal system size proceeds as follows: (l) From[l], determine the coefficients qo, qt and q2 of eqn (2.6), yielding qo=7.236; q~ = 13.362; q2 =6.158. (2) Calculate the yearly average operating time Top from eqn (2.8), 1-o~= 10,17 hr/day = 3.661 × 104 sec/day. (3) Determine the maximumstorage size Mmax to be considered from eqn (2.9): M,,a× = 4.98 × 104kg water = 49.8 m 3 water. (4) Calculate the collector flow rate rn,, from eqn (2.10) m, = 2.360 kg/sec.

Design and optimization of solar industrial hot water systems with storage

127

Table I. Solar industrial process hot water system in Sede Boqer, Israel (latitude = 30.9°N yearly average normal beam irradiance = 502 W/m2) Plant operation 24 hr/day, 365 days/yr Load:

~p : 1 kg/s

Collectors:

Tin = Tenv = T-ambient = 18.25°C

Flat Plates

F'n o = 0.75

q(X) = 7.236 z 13.362 X + 6.158 X2 Imax = 1040 W/m2

F'U = 5.0 W/m2K Xin = 0

Tmax = 95°C

Storage Tank: height/diameter = 1.75 Economic Parameters: N = 10 years Co = 0

r = 0.05

Cf = $ 7.20/GJ

Results of calculations:

Utank = 0.42 W/m2K

r f = 0.00

0M = 0.02

nbackup = 0.80

CA = $ 200/m2 installed

CM = $ 200/m3 installed

Top = 10.17 hr/day = 3.661 x 104 s/day

Mmax = 4.98 x 104 kg water = 49.8 m3 water Amax = 1338 m2

Adump = 1058 m2

Qcoll = 4160 GJ/year

Q = 4128 GJ/year

Atank = 77.4 m2

~c : 2~36 kg/s

Aopt = 970 m2

(aL/BM)A > O, so Mopt =d~max

Storage Losses: (UA)tank = 32.51W/K Net:

Tp = 70°C

Ts-Tin = 31.5°C

Qloss = 32 Gd/year

L = $ 5.14 x 104

annual solar fraction = 0.604 of 24 hr/day load

(5) Assuming that in order to prevent boiling in the collectors one sets an upper tolerance temperature in the collector at Tmax= 95°C, we calculate with eqn (3.2) the maximum collector area permitted: Amax = 1338 m2. (6) From eqn (3.5), calculate that value of A at which the onset of temperature (energy) dumping must be taken into account, Adumo= 1058 m 2. (7) From eqn (4.7), calculate the optimal collector area Aop, = 970 m2 and the corresponding yearly collectible energy (eqn (2.11)) Qcon = 4160 GJ/yr. We check that indeed mopt < A . . . . and that since Adump> Aopt, energy dumping will not enter into our considerations. Hence we can safely set Qdump= 0. (8) To check if M "--Mmax is indeed the optimal storage size, check the sign of (aL/aM),~ from eqn (4.6). Since in this case it is positive, Mm~x and Aop, are economically optimal. (9) Assuming a storage tank with a height-to-diameter ratio of 1.75 and Utank=0.42W/m2K, we estimate storage heat losses from eqn (2.14), Q~o~=32GJ/yr. Now evaluate net system performance: Q = Q¢o,- Qd,mp- Q~o~ = 4160-- o - 32 = 4128 GJ/yr. This corresponds to a solar fraction of 0.604 of the 24 hr/day load. (10) Calculate the life cycle savings L from eqn (4.2), L=$5.14x104 which represents a cost effective investment (initial capital cost = $2.04 x 105). To illustrate the degree to which this open-loop singlepass design is thermally superior to conventional designs, we perform tile calculation for the same industrial hot water application designed as a multi-pass, closed-loop

system, as exemplified by the ~-f chart[5] or f-chart[5] methods. In this instance we have used the f-chart rather than the 4~-f chart method to calculate the annual solar fraction. The reasons for this are: (1) The solar fraction is relatively insensitive to the load distribution pattern if M/A >-75kg/m2[9]. (2) If the load characteristics overlap those permitted for the f-chart method, then use of the f-chart method is recommended since it produces results of comparable accuracy with significantly less computation than required by the @-f chart method[9]. The conventional system design has storage and collectors thermally coupled, operating in closed loop, with a flow rate per collector area of 0.015 kgl(sec-m2). We take the same total collector area of 970 m 2 and follow the conventional recommendation of 75 kg of storage per m2 of collector, so that M =7.275 × 104kg=72.75 m 3 water (note that this is markedly larger than the 49.8 m3 required for the optimal single-pass open-loop design). Were the same storage volume of 49.8 m3 assumed in the f-chart calculation, the annual solar fraction of the conventional design would be even lower than the value cited below. The monthly insolation and ambient temperature data for Sede Boqer, Israel are taken from[10] and are presented in Table 2. The result of this calculation is an annual solar fraction of 0.,i72, In this specific case, then, the single-pass openloop design not only delivers 28 per cent more useful energy than the conventional design, but employs 32 per cent less storage (that storage not having to be pressurized to boot).

et al.

M. COLLARES-PEREIRA

128

Table 2. Monthly average insolation and ambient temperature data for Sede Boqer, Israeli10] Month

Daily horizontal insolation

Ambient temperature (°C)

(Md/m2)

JAN

11.9

9.6

FEB

14.4

11.6

MAR

17.5

13.6

APR

22.0

17.0

MAY

25.3

20.9

JUN

28.8

24.2

JUL

28.0

24.8

AUG

25.9

25.3

SEP

22.1

23.6

OCT

17.8

20.6

NOV

13,9

16.0

DEC

10.3

11,4

One key reason for the thermal superiority of the one-pass open-loop design is the fact that all energy stored on a given day is depleted by the nighttime load by collector turn-on time the following day. The more conventional system design, with well mixed storage assumed, can never fully deplete the energy stored on a given day by collector turn-on tim~ the following day. The reader can repeat calculations of the type presented immediately above for a variety of applications, climates and collector types. He will find that the same basic point emerges: namely, the thermal and therefore economic superiority of the single-pass open-loop design for solar industrial hot water systems with storage. One additional point to be noted is the sensitivity of the life cycle savings L to the exact optimum values of A and M. In fact, L typically exhibits a rather broad maximum, so that in the vicinity of the optimum system size, there are small changes in L for substantial changes in A. As an illustration of this point, we plot in Fig. 2 the life cycle savings as a function of collector area (at fixed M = Mmax) for the solved example above. In this case, variations in A up to 10 per cent about Aop, give rise to changes in L of the order of a per cent.

6. HEAT EXCHANGERS, CONTROLS AND PARASITICS

When a heat exchanger is to be used, the crucial formulae above include an additional factor to account for the heat exchanger penalty, and an additional variable, the heat exchanger size, must be taken into account. For systems with a heat exchanger, the equation for instantaneous collector efficiency is obtained from eqn (2.2) by the heat exchanger penalty factor F~

~=F, FR[rlo U(T~"IT"'"h)].

(6.1l

The heat exchanger penalty factor[1 l] F, is given by

F, = +(FR(UA),,) [1 m,C~

1]

(6.2)

where the subscript c denotes collector; the subscript p denotes process; (mCp)m~,=the minimum ((rhCp),., (rhCp)~); (mCp)max=the maximum ((mCp)c, (n~Cn)p; e is the heat exchanger effectiveness, which for single pass counterflow is given by 1 - exp(B) e - 1 - Cexp(B)

(6.3)

C =Oi~C,,)m~,/(titC,,)..... and B =(UA)Hx(C- 1) /(mC~),,m, where the subscript HX denotes heat

with

exchanger. Apart from the additional multiplicative factor F, all formulae remain the same as for direct systems. With regard to controls, the collector field is turned on if and only if the absorbed insolation exceeds the heat loss. More precisely, the thermostat should be set to activate the pump whenever the collector temperature exceeds the value T~o (plus a small increment for stability of operation and to insure that the collector output exceeds the expenditure for pumping power). Since the storage tank is to be unpressurized, system safety considerations for the open-loop singlepass design would call for a sensor at the top of the storage tank that would turn off the collector pump whenever storage is full. Similarly, a sensor at the bottom of the tank should insure that when storage is depleted before collector turn-on, fluid is drawn from a secondary reserve (e.g. mains). One also wants to achieve good mixing in storage so as

Design and optimization of solar industrial hot water systems with storage 60--

,

I

I

I

I

I

5O

//// :.e.

40

//

//

// 30

0 o~

:,o

.3 io

I 200

i

I 400

,

I 600

~

I 800

~

1 I000

J

1200

COLLECTOR AREA (m2)

Fig. 2. Economics as function of collector area (at fixed storage mass) for the example in Section 5 (day-night storage).

129

storage facility (i.e. starting from zero storage), the economics of weekend storage may be different in the two cases. The results derived below will in any event afford the user the calculational tools with which to assess the cost effectiveness of weekend storage in both no-storage loops and systems with day-night storage. Consider a solar retrofit for a 7 day/week process load, designed in accordance with the ideas developed above and in/l]. Let us call the net yearly useful energy delivered by that system Q. Now consider the same application, but for a plant that is shut down every weekend for N days. Were no alteration made in the system design, and assuming that weekepd days are as likely to have given insolation and ambient temperature values as all other days, then (N/7)Q is dumped energy since the collectors stagnate for N days a week. Hence the actual yearly useful energy delivered by the system is

(7-N)Q/7. to insure that energy delivered above the process inlet temperature To is rendered useful. This can be achieved, amongst other ways, by delivering heate~i water to the bottom of the tank. With regard to parasitics, the annual parasitic energy requirement Qpump for the above design can be estimated from the yearly average collector operating time ~'op as

By adding storage that is" sufficient to hold all the process fluid that passes through the collectors during the weekend, the initially lost energy, (N/7)Q, becomes useful energy (except for storage losses). Going one step further, we now note that weekend storage enables us to reduce collector area by a factor N/7, i.e. Aw,h weeke.a..... ge = ((7 -- N)/7)A~.,,,L w,h . . . . . . ke,~ . . . . .

Qpump = ('TopPI "j"(8.64 × 104)P2)(365/109)

ge

(6.4)

where zoo is in s/day; Qpump is in GJ/yr; P~ (in W) is the power rating of the collector pump; and P2 (in W) is the power rating of the pump drawing fluid from storage to the process. Since the flow ratos for the single-pass designs are considerably lower than the flow rates in more conventional designs (e.g. a factor of 6 in the solved example of Section 5), parasitic losses will also be noticeably lower. 7. WEEKEND STORAGE

Since many industrial process heat plants do close on weekends, the design and optimization of systems with weekend storage is also an important problem. The basic design ideas for a weekend storage facility will be the same as those for day-night storage as developed above. Namely, a one-pass op6n-loop design, where collectors act as preheaters and always heat the coldest fluid possible. Storage size is to be increased to accomodate the fluid mass that passes through the collectors during their weekend operating time, with the collector flow rate and collector area to be modified in accordance with the principles developed below. The basic system design diagram of Fig. 1 applies here too. Similarly, the presentation above that deals with storage losses, heat exchangers, etc. applies equally well to the ideas developed below. As such, they will not explicitly be included in the analysis that follows. In no-storage systems, the introduction of ,weekend storage is a qualitatively "new" step, whereas in systems with day-night storage, one is considering enlarging an already existing or planned storage facility. Since the incremental cdst of storage (i.e. starting with a planned storage facility) may be different than the price of a new

and still obtain a yearly useful energy (7-N)Q/7. That is, the introdtiction of weekend storage enables us to reduce collector area and still obtain the same yearly useful energy as for the system without weekend storage (except for storage losses). This is the first key observation of this section and will be quantified below. If the price of storage (per unit energy delivered) is then less than the price of collectors (per unit energy delivered), it will be more cost effective to introduce weekend storage and reduce collector area. We shall demonstrate in the following section's solved example that at current (Israeli) prices for collectors and hot water storage, weekend storage is indeed a cost effective measure. Once weekend storage is introduced, the system design must account for the fact that the process fluid during plant operation comes from the sum of weekend storage and weekday collector operation. The flow rate through the collectors is accordingly reduced so as to prevent storage exceeding its full level (mass dumping). As in the preceding sections, one will be able to introduce arbitrarily sized weekend storage and still perform the associated calculations for flow rate, area and yearly collectible energy. The appropriate equations are now presented. We consider a constant load that operates 7-N days/week. The initial system design can either be of the no-storage variety of[l] or of the type with day-night storage, as developed above. In both instances, weekend storage refers to the storage added to an already existing or planned storage facility. Several conditions will determine our initial design for the system with weekend storage. First, the mass of weekend storage, Mw,, is to be equal to the mass of process

M. COLLARES-PEREIRAet al.

130

fluid that passes through the collectors during the weekend, Mwe = Nrhc~'op.

(7.1)

Second, on each of the- 7-N weekdays, process fluid is drawn from weekend storage and the already existing system design (no-storage or day-night storage designs) simultaneously. The sum of these two contributions should equal the mass of fluid drawn by the process, thpzo = rh,.rop+M~e/(7-N)

(7.2)

where m, is the process flow rate (kg/s), and % is the daily duration of load demand (in see/day). Equations (7.1) and (7.2) can be solved for Mwo: M ~ = ( N[7)(7-N)rhprp

sary (eqn 7.3), Mwe = 3.09 x lO4 kg = 30.9 m 3 water and a collector flow rate (eqn 7.4) the = 0.843 kg/s. (The fact that top is about 2 per cent larger than r;, leads to negligible energy dumping in this case.) Eqs (3.2) and (3.5) are used to calculate Amax = 478 m 2 and Adumo= 378 m 2.

The optimal collector area is again determined from eqn (4.7) as

(7.3) Aopt = 346 m 2

and the collector flow rate is then and corresponds to (eqn (2.11)) rh,, : (7-N)~hprp/(7rop).

(7.4) Qco, = 1484 G J/yr.

The yearly collectible energy is calculated with eqn (2.11) and the conditions for checking on upper tolerance temperature and temperature dumping (eqns 3.2 and 3.5) still apply. The equations and procedure of Section 4 for determining optimal collector area and storage size still apply as well.

Again, Aop, is noted to be less than Amax, and since Aaump is greater than Aopt temperature dumping will not be a consideration. Evaluation of eqn (4.6) shows (OL/OM)a > 0, so that Mwe = 30.9 m3 and Aop, = 346 m 2 are indeed optimal. Storage losses are now evaluated with eqs (2.13) and (2.14):

8. SOLVED EXAMPLE: WEEKEND STORAGE

We consider the industrial plant described in the solved example of Section 5, but now change the problem in two ways: (a) the plant does not operate 24 hr/day but rather from 07:00 to 17:00 only (so that in this case ~-~= 10hr/day); and (b) the plant does not operate 7 days/week, but rather shuts down every Saturday (one day a week). All collector and economic parameters remain the same as in Section 5 (see Table 1). For the sake of simplicity and of more clearly isolating the contribution of weekend storage, this solved example illustrates the optimization procedure for adding weekend storage to an initially no-storage system design. Our design method is, h3wever, general and equally well applies to systems with day-night storage. Let us first design the system without storage, so that a given system design delivers 6/7 of the energy previously calculated (i.e. as calculated for a load of 7 days/week). Using eqn (4.7) for a no-storage system with rh~ = thp = 1 kg/s, we obtain Aopt = 282 m 2

Q = 1114 GJ/yr

L = $1.23 × 104.

We also check to ensure that this collector area corresponds to no energy dumping by also determining the maximum value of A with no dumping (eqn 2.'12 of[l]), Adump=337m 2 (no storage). (Were Aopt>Ad . . . . [1] describes the correct procedure for calculating Aopt, but only in this special case of no storage.) As our second design, with weekend storage, we consider a system with the largest weekend storage neces-

At,°k = 56.31 m2 Ut,,k = 0.42 W/mZK

"/~, = 49.7°C

Tony= Tin Qio~s= 23 GJ/yr. The net yearly useful energy delivered by the system is then Q = Qco,,- Q,o~s- Qdump = 1848-23 - 0 = 1461 GJ/yr with life cycle savings L = $1.45 × 104. The optimal system design with weekend storage turns out to be a superior investment relative to the system with no weekend storage. To again illustrate the broad economic optimum that occurs here, we present in Fig. 3 a plot of L vs A at fixed Mwo=30.9m 3 water. For purposes of comparison we also plot in Fig. 3 the corresponding L vs A curve for the system with no weekend storage (calculations performed as described in[l]). The above solved example considered the case of one day of weekend shutdown. For the case of two or more days of weekend storage, where the availability of space for very large storage tanks and/or economic considerations may render more than one day of storage as unfeasible, the user should consider the following alternative: Size storage for one day of weekend, and have the collectors switch into closed-loop operation (reheat-

Design and optimization of solar industrial hot water systems with storage 15

Q/.~.

%

/

.~_ I0

\

/"

x\\,

/" t / / / .// / / / // / // / /I ./

z

/

u

5

// //

/

/

/

./

I00

200

~

131

collectors and energy, weekend storage is shown to be one of the more cost effective measures in system design. The present paper does not address the practical problems of building and operating a solar industrial hot water plant. The best design is worthless if one fails to pay sufficient attention to such problems as freeze protection (when necessary), pressure relief valves, expansion loops, collector degradation, nighttime cool down losses, etc. These practical problems are discussed in[12], and the reader is encouraged to consult it before designing a system.

400

COLLECTOR AREA (m 2)

Fig. 3. Economics as function of collector area (at fixed storage mass) for the example in Section 8 (weekend storage). (a) With weekend storage, (b) without weekend storage. ing the storage mass) from the second day of the weekend until the end of the weekend. The calculation of the useful energy delivered after the first day of the weekend could be performed as follows. Determine the average storage temperature at the end of the first day of weekend from the formalism described above, and then use that as input to a calculational tool such as ~-f chart[5,9] to perform the calculation for the remainder of the weekend. 9. SUMMARY

We have detailed a step-by-step procedure for designing and optimizing solar industrial process hot water systems with day-night and/or weekend storage (i.e. constant load applications). The single-pass open-loop design calls for a thermal decoupling of collectors from storage, so that collectors always heat the coldest fluid possible and stored energy is fully depleted by the nighttime load. The results are in the form of closed-form equations with which one can calculate the net yearly useful energy delivered by a given system design and can optimize for collector area and storage size. We have also included a presentation for including the effects of collector operating time, heat exchangers and secondary losses. One solved example illustrates the thermal superiority of the single-pass open-loop design to that of the more conventional type. Although only one solved example is presented toward this end, the reader can use the method described above to verify that, for a variety of applications, climates and collector types, the single-pass open-loop design indeed delivers more useful energy for the same collector area and storage size. For systems that require weekend storage, the key observation is that weekend storage effectively "replaces" a certain fraction of the total collector area (of the system without weekend storage), at the same time not decreasing the yearly useful energy delivered by the system (except for the usually small effect of storage losses). The method developed for day-night storage is then extended to designing systems with weekend storage, again' for arbitrary collector area and storage size. At the current (Israeli) costs of hot water storage,

Acknowledgement--Research funded by the Belfer Foundationfor Energy Research, Ramat Aviv, Israel. REFERENCES

1. J. M. Gordon and A. Rabl, Design, analysis and optimization of solar industrial process heat plants without storage. Solar Energy 28, 519 (1982). 2. M. Collares-Pereira and A. Rabl, Derivation of method for predicting long term average energy delivery of solar collectors. Solar Energy 23, 223 (1979). 3. J. M. Gordon and Y. Zarmi, Thermosyphon systems: single vs multi-pass. Solar Energy 27, 441 (1981). 4. A. Mertol, W. Place, T. Webster and R. Greif, Detailed lo3p model (DLM) analysis of liquid solar thermosiphons with heat exchangers. Solar Energy 27, 367 (1981). 5. J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal Processes. Wiley, New York (1980). 6. C. F. Kutscher and R. Davenport, Preliminary operational results of the low-temperature solar industrial process heat field tests. SERI Rep. SERI/TR-632-385 (1980). 7. Solar Heating and Cooling Systems: Operational Results, Conf. Proc., 27-30 Nov. 1979, SERI Rep. SERI/TP-245-430 (1980). 8. A. Rabl, Yearly average performance of the principal solar collector types. Solar Energy 27, 215 (1981). 9. K. A. Pearson, S. A. Klein and J. A. Duffle, Generalized design method for solar hot water heating systems. Am. Section ISES Meeting, Philadelphia, Penn., May 1981. 10. D. Faiman, J. M. Gordon and D. Govaer, Optimization of a solar water heating system for a Negev Kibbutz. Israel L Techn. 17, 19 (1979). 11. F. de Winter, Heat exchanger penalties in double loop solar water heating systems. Solar Energy 17, 335 (1975). 12. C. F. Kutscher, Design considerations for solar industrial process heat systems. SERI Rep. SERI/TR-632-783 (1981). 13. 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 (1963). 14. J. M. Gordon and M. Hochman, On the random nature of solar radiation. ASCU Rep. 82/10, Sede Boqer, Israel Solar Energy, in press (1984). t5. J. M. Gordon, M. Hochman and Y. Zarmi, Appropriate functions for utilizability. ASCU Rep. 82/6, Sede Boqer, Israel. Solar Energy. To be published. APPENDIX

Four issues are addressed here: (1) mass dumping; (2) temperature dumping; (3) the validity of performing calculations based on yearly average values for collector operating time; and (4) the importance of accuracy in predictions of collector operating time. (a) Mass dumping Mass dumping will occur whenever the net fluid mass delivered to storage exceeds the storage capacity, i.e. (rh,- mp)~-or> M.

M. COLLARES-PEREIRAet al.

132

The first important question to be answered is then: if mass dumping is a possibility (due to undersized storage relative to rh,,), might it not be better to lower &, so that mass dumping does not occur, at the same time decreasing collector efficiency (since A is fixed here and efficiency decreases when ih,,/A is lowered). In the case where mass dumping is a possibility (and in reality the collector pump will be turned off while useful solar energy is still available, so storage will not overflow), calculation of the energy so discarded can be performed from considerations of the hour-by-hour distribution of solar radiation on the collector aperture[2, 13, 14]. The sacrifice in collector efficiency for the alternative case of lowering rh, so that mass dumping does not occur but a lower value of th,/A is used can be calculated from eqn (2.11). When these comparative calculations are performed for a variety of collector types and climates, it turns out that the thermally superior strategy is the one in which rh, is reduced to avoid mass dumping. Hence in any attempt to determine the optimal combination of collector area, flow rate and storage size (for constant load applications), one should not be considering the option of mass dumping, but rather simply reducing collector flow rate accordingly.

where we have defined

f3 = rhc/(m~ - thp) FR~oA ( , h , - rh~,)C,,'

(1) Solve the heat balance equation governing the well-mixed time-dependent storage mass for those days on which temperature dumping is most likely to occur. (2) Determine the maximum temperature of the storage mass and set it equal to Tp (which determines the onset of temperature dumping). (3) Solve the final resulting equation for Adump.

(A4)

The solution to eqn (A2) for T,(t) is t

T,(t)- Tm=(t-to) ~e f (t'-to) ~3 '(I(t')- Xi.)dt'.

(A5)

Now T,(t) achieves its maximum value T'j'~'~(at t =- t,,~,~) when

d (T,(t) - Ti~) O, which with eqn (A2) yields T max- Tin = (d¢)(l(tm,,d - Xio).

(b) Temperature dumping Temperature dumping refers to the case in which the entire storage mass is heated above the process inlet temperature Tp (assuming that all energy delivered above Tp is not useful). For reasons stated in Section 4, we are not, at this stage, concerned with accurate predictions of the amount of energy dumped since for most practical system designs Aopt< Adomp. Rather, we simply want to determine the value of A at which the onset of temperature dumping occurs. To determine the collector area Aj.m~ at which the onset of temperature dumping occurs, we proceed as follows:

(A3)

(A6)

tm,~ can be determined by differentiating eqn (A5) with respect to t and equating to zero, yielding the implicit solution tmax

l(tm~,~)- Xin = /3(tin.×-- to) e f

(t -- to) e' ~(I(t)-- Xi.) dt.

(AT)

t.

One then determines Adunmas follows. Calculate t .... and hence l(tm,O with eqn (A.7). Take T~ "~ = Tp as the point for the onset of any energy dumping. Use the explicit A dependence of e (eqs A4 and 2.3) to solve for Ad,,mv, yielding

,,,c,,

Adump = -F'U

r

[ v(r,-r,.)ll

InllL - trlo~.x)-~i.)JJ

(A8)

This can be alternatively expressed as

The governing heat balance equation is

.,,¢, [,_[

ITo-T I ]l

d(M(t)C,(T,(t) - ri,)) = FenoA(l(t) - Xi,) - thi,Cp(T,(t) - Tin) (A1) where M(t)=(m,-rhp)(t-to) is the time-dependent storage mass, with to denoting collector turn-on time. The l.h.s, of eqn (AI) is the rate of change of the heat content of storage. The r.h.s, is the difference between solar input and load draw. l(t) is the time-dependent insolatidn on the collector aperture, and T,(t) is the time-dependent temperature of the storage mass. On what days is temperature dumping most likely to occur'? There are extraordinary conditions under which one could imagine the onset of temperature dumping at collector areas that are just large enough to produce collector outlet temperatures above T, under peak insolation. For example, consider days which are so totally overcast in the morning that the collectors are not turned on at all, followed by sudden total clearing around solar noon with peak insolation lm~,~ incident on the collector aperture. In this case, an empty tank is suddenly filled with hot water above T,, part of which is immediately drawn to the load, thus leading to temperature dumping. The actual frequency of occurrence of such extreme cases is sufficiently small that they can be neglected for our purposes. Rather, the calculation of interest will be one for the clearest days of the year, on which lit) (of eqn AI) is a symmetric function peaking at I ..... at solar noon. Using then the specific linear form for M(t), we recast eqn (A1) as (t - to)d(T,(t) - Ti.)+/3(T,(t)-

Tin) = E(l(t) - X,,)

(A2)

where

l(tm,O -- Xio Ira,x-- Xi, -~ I. (A10) The solution for t .... and l(tm,~) requires iterative (numerical) solution of eqn (A7) and is in general not a simple calculation. One furthermore has to know the l(t) appropriate to the particular collector type. To help simplify matters to a form amenable to easy use, we have solved (numerically) eqn (A7) for functional forms of l(t) that are appropriate to clear day insolation on the principal solar collector types and for system design parameters of practical interest (e.g. r~,p= 8-12 hr/day). We furthermore impose the (arbitrary but practical) constraint that one is not concerned about temperature dumping until it constitutes on the order of I per cent of the yearly collectible energy Q~,,,. Applying the criteria stated above, we then find that a ~ 0.8 yields a conservative expression for Aj,,mp, i.e. using a ~ 0.8 in eqn (A9). the onset of temperature dumping (for practical system design parameters) is predicted accurately for tracking collectors in the clearest climates (the "worst" case). Hence in most design problems cqn (A9) with a = 0.8 will afford quite a conservative or safe estimate (lower limit) for A,~,,mp: as such we recommend it as a simple approximate criterion for calculating Ad,,mr. Fortunately, this point for the onset of temperature dumping often corresponds to the case of either: (a) collector areas greater than the limit imposed by an upper tolerance temperature; or (b) collector areas larger than typical economically optimum areas.

Design and optimization of solar industrial hot water systems with storage (c) Yearly average top How valid is it to use one yearly average value for collector operating time 7°0 in the calculations prescribed above? There will clearly be days that are longer and sunnier than the yearly average, for which storage sized according to eqn (2.9) will be undersized. On such days the collectors would be turned off while useful insolation were still available, and collectible energy would hence be discarded (mass dumping). It turns out, however, that the energy so discarded constitutes a negligible fraction of the yearly collectible energy. To substantiate this claim we used both actual hourly solar radiation data for Israeli14] and the month-by-month correlations of[2] (for a variety of climates in the latter case) to calculate the yearly collectible energy so discarded when m, is determined from eqn (2.10) based on the yearly average ~oo. For a variety of collector types, the energy discarded is of the order of 1-2 per cent of the yearly collectible energy (this energy dumping occurring primarily during the summer months).

133

(d) Accuracy of r,,f, How critical is the accuracy of eqn (2.8) in predicting roo in terms of system design? Since eqn (2.6) was derived as accurate fits to yearly collectible energy, its derivative (~'op)may be a less accurate prediction of the real r,,r. In fact comparisons of eqn (2.8) against Israeli radiation data[15] and the correlation for r,,r of[2] indicate that eqn (2.8) tends to consistently underestimate zop, typically by about 10 per cent. In our design method, underestimating top leads to overestimates of the required storage capacity M. Hence the error introduced by use of eqn (2.8) will not lead to an error in the calculation of Q (since the negligible amount of mass dumping on days with r,,p larger than the yearly average will then be even more negligible) and at worst leads to small oversizing of storage. Of course, if and when more accurate and easily usable correlations for yearly r,,p values become available, they should replace eqn (2.8)'in our formalism.