o3o5-os4s/t5 s3.00 + .oa Pergamon Ress Ltd.
Comput. & Ops. Rrs. Vol. 12. No. 1, pp. 17-23. 1985 Printed in chc U.S.A.
OF SOLAR WATER HEATING SYSTEMS
LAYBK ABDBL-MALBKt of Industrial Engineering, College of Engineering, Rutgers University, PO Box 909, Piscataway, NJ 08854, U.S.A.
and JON T. CJBJ$ Polytcclmic of New York (Received February 1983) evaluation of solar water heating systems is very important in scope and pmposc-The determining their economic feasibility. The evaluation process of solar water heating systems is rather diicult. There are many random variables affecting the amount of energy deliverad from the sun to the users of the system. In this paper a model is developed to determine the average parcent of the energy occdcd by a howhold supplied by the stm. The system was mod&d as a queue with limited waiting room and state dependent arrival and departure rates. Ah&act-The paper addresses itself to stochastic modeling of solar water heating systans. War water hasting systems will be modellcd in order to estimate the pamontage of energy contributed by the sum to the total required load by system usars. The major tool in developing the model is Chapman-Kolmogorov equation. The model takes into considcratioo the affect of randomness of insolation, demand for hot water, and the saquanca in which they occur on the evaluatioo of the system pcrformancc.
this paper, a stochastic model is developed to evaluate the performance of solar water heating systems. By evaluation is meant the estimation of the percentage of energy needed by the users of a solar water heating system. The model is suitable for systems like the one shown in the schematic diagram. It consists mainly of a flat plate collector, a water tank, an auxiliary supply system, a pump, and control devices [7). The solar collector is used to transform the solar radiation into thermal energy. The collected energy is used to heat the water in the tank. It should be mentioned that not all the energy falling on the collector surface is absorbed. The amount absorbed by the collector depends on the temperature of the water entering the collector. When the demand for hot water occurs, the auxiliary supply may be automatically incorporated to provide extra energy if needed, The problems of evaluating such a system are: l The random nature of the insolation falling on the collector surface. l The randomness in demand for hot water. l The randomness in the state of the system (the state of the system is the number of thermal units stored in the water tank at a certain moment), which affects both the amount of energy absorbed by the collector and the amount of energy delivered to the system users at a certain moment. The Chapman-Kohnogorov equation is found to be a suitable model of the system taking into consideration the above sources of variability. In
tLayek Abdel-Malek is Assistant professor at Rutgers University, Ph.D. in Oporatioos Rcsaarch from Polytechnic Institute of N.Y. Before coming to Rutgars, he taught at Columbia University. His intarasts are in applications of operations ramarch in production and mechanical systcols.
$J. T. Chu is Professor of operations Rcscarch at Polytechnic Institute of New York. He has bean a rcprasantative of Operations Ruearch fiociety of America until 1981. Currently he is on sabbatical to anbancc understanding bctwcun the U.S. Coograss and academia. 17
LAYEK ABDEL-MALEK and JON T. CHU Collector Losses Load
The absorption of thermal units by the collector is assumed to occur in a discrete fashion. That is, the absorption of a thermal unit will be considered to take place in an incremental period of time. The ~nsumption of thermal units is assumed also to take place in a discrete fashion. The solar incidence on the collector surface and the demand rate for hot water are assumed to be independent of each other. The solar incident arrival process, and the demand for hot water process are assumed to be memoriless. EXISTING
The two widely used models for the evaluation of solar water heating systems are the TRNSYS (Transient Simulation System) and the f chart method. The TRNSYS [7j is a modular simulation of a solar heating device. It simulates the performance of the interconn~ted components indi~dually. The TRNSYS is very expensive to run on computer. Also, it simulates the performance of the collector and the water tank independently. However, the amount of energy absorbed by the collector at a certain moment depends on the temperature of the tank at that same moment. The f chart method  was developed to avoid lengthy calculation of the -TRNSY S. After 300 runs of the TRNSYS, it has been found that a correlation exists among the solar incidence, the solar collector losses, and f, which is the average annual fraction of energy provided by the sun to the solar water heating system user. Although the f chart method is widely used, its evaluation estimates are far from accurate . THE
The daytime performance of a solar water heating system is different from that of the nighttime, During daytime, the system experiences a birth-death process. Nevertheless, after sunset it experiences a pure death process. Thus, the evaluation of solar water heating system will be carried out by combining its daytime and nighttime performances. Before modeling the daytime and the nighttime performance, the following definitions are given: l The state of the system is the number of thermal units (integer) stored in the water tank at a certain moment. l The arrival rate is the average rate of energy absorbed by the collector per unit time. l The departure rate is the average energy dissipation rate from the water tank per unit time due to both demand for hot water and heat losses. Daytime performance Arrival raze. The arrival rate to the solar water heating system will be estimated using
Evaluation of solar water heating systems
a model developed by Hottel and Whiller  which is given by eqn (I).
where: Q,(r) is the average amount of heat collected per unit time; F, is a constant known as efficiency factor;+4 is the collector area; H, is the rate at which solar radiation is falling on the collector surface in thermal units per unit area per unit time; t is the solar t~nsrnit~n~ of the t~nspa~nt covers of the collector; a is the solar absorptance of the collector plate; V, is heat loss coefficient; Ti is the temperature of the fluid entering the collector at a certain moment; T, is the time dependent ambient temperature. If we assume that T, is approximately equal to the temperature of the water entering the tank from the main, the state of the system at time z (i(r)) may be expressed as: i(t) = iw,C,,(Ti - TJ
where M,,,is the mass of water in the tank; C, is the specific heat of the water at constant pressure. ?JIUS, from equations (I) and (2) one can write Q&l) = l(t)fl - Ci(t)] = &(t) where: A(t) = Fg4H,7a
The departure rare. The dissipation of the thermal units in the water tank at a certain moment (t) takes place due to demand for hot water and heat losses through the walls of the water tank. Assume that the users average demand for hot water is k(t). Assume further that the favorite temperature at which the user seeks hot water is constant at Tk. Tk is also the maximum temperature to which the water in the tank is allowed to rise for technical reasons. Thus the average consumption of thermal units from the water tank at state i and time t per unit time is MXt) = th(r)C,(T, - Td + C,A,i(t) = i(r)(M(t)
where: ti(f) is the mass of hot water needed/unit time at time (I) (demand rate); M(f) is ~(r)~~~; C, is the average heat transfer coefficient per unit time per unit area of the water tank walls; A, is the water tank surface area. Water sank thermal capuciry k. The temperature of the water in the tank is not aliowed to exceed a certain limit T, because of technical reasons. These technical reasons relate to the change in the state of the water in the tank due to the change of its temperature, and to protect the water tank from explosion [s]. Thus the rna~rn~ number of thermal units which can be stored in a certain tank may be expressed as k = M,C,(T, - TJ.
The Chapman-Kolmogorov equation is used to determine the system state probabilities. That is the percentage of time that the water tank spends in each state. More clearly, P&) is the proportion of time during the daylight hours that the system has i thermal units stored in the water tank (i = 0, 1,2,, ‘. , k). The equations for the system
LAYEK ABDEL-MALEK and JON ‘I’. CHU
(6) Pkt + At) = P&)[l - nj@I(r)][l + Pi-,(Mo)~(~)[l
- M~l)Ll(f)] - M-,(W(r)]
+ P~+,(f)~j+,(f)d(r)~l - ~,+,(O~(O] + Pi(t)r&(t)A(t)M,(t)A(r) 15 i s k - 1 Pk(t + At) - P#)[l
Before solving the above equations, let us assume the following: (i) A*(t)= Li. That is, the A,(?) is independent of time. The actual and the approximate average insolation on a flat plate facing south is shown in the graph below .
(ii) M,(t) = Me The consumption of thermal units is independent of time. Consequently, using equations (6x8) one can derive the following equations:
Now, we consider a certain period, let us say one month, during which we assume that the average insolation is constant. The expected amount of energy during the daytime supplied for the users from the sun per unit time is Qs = Md - i ti~Cp(Tk - TJPi i-l
where: M,, = thC,( Tk - To)< Equation (1 l), can be also written as Q,=M
where E(i) is the expected amount of thermal units in the water tank (i).
During the nighttime the system experiences a pure death process with a random initial state. That is, after sunset the water in the tank will be losing thermal units only due to
Evaluation of solar water heating systems
the demand for hot water and the heat losses through the tank walls. The amount of heat stored in the tank at the moment of sunset is random. It varies from 0, I, 2, . . . , k thermal units. As we assumed before, the consumption of the stored thermal units occurs in a discrete fashion. Also, we are going to assume that the energy stored in the water tank after sunset will be consumed before the successive sunshine. Thus, it is plausible to assume that the user depletes the energy in the water tank first, then the auxiliary supply compensates for the extra need for energy. In the following we will introduce a sub-mode1 to estimate the expected time duration during which the energy stored in the tank from the sun will last. Define a pure death process with an initial state Vz’which is a random variable. Letting Adf(r) = 0 in eqns (6)-(8) one arrives at:
Taking the Laplace transform of eqn (13) we obtain sP(s) - P(0) = [Alp(s)
where P(0) is the initial probability vector at t = 0 (time of sunset) which is
n0 1 _f_ ; ; . 1
Now, let us assume that the p.d.f. (probability density function) of the consumption of a thermal unit is J(t), which has a mean Mi, then P,(t) = F,(t) = 1 - c ‘f(r) dz.
Thus, the expected time that the death process spends in state i, q, is given by: f,-- I0
Thus from eqns (14H16) the expected time for the process to start from state n to reach its absorbing-state (the tank becomes empty of thermal units), is
r, = $i&.
But, as we said before, n is a random variable which varies from 0 to k. Consequently, the expected time for the process to reach its absorbing state, 5, is given by:
Pa E P, during the daytime.
LAYEK ABDEL-MALEKand JON T. CHU
The evaluation formula The percentage of energy during a month period for example, supplied from the sun to the household for hot water is given by
where L is the amount of energy sought by the household during one month (estimates are available for L); E,, is the amount of energy provided by the auxiliary supply during one month. Which is E,=E,+E,
Ed, E,,,, are the amounts of energy provided from the auxiliary supply during one month in the daytime and the nighttime respectively. Ed = 3OM,(l - E(i)/k)h
where h is the average length of daytime during a given month Enn= 30[(24 - h) - ?]Mw.
Using eqns (20)-(22) in eqn (19) one can evaluate the system for the considered month. Example The evaluation of a standard water heating system in Madison, Wisconsin is considered here. This system has been evaluated on 8 January 1976 by The TRNSYS [I. The TRNSYS result was 66% of the energy needed for the household is supplied from the sun. Below are the system information and the evaluation results using our model. System information (9 Location and weather data Latitude 40” place Madison, Wisconsin, U.S.A. Insolation average 5.05 Megajoule/meteti/day. (ii) Collector data Area = 6.5 mete? Average collector loss coefficient ( Vc)= 14.4 Kilojoule/hr/meteti collector cover transmittance (7) = 0.82 collector plate absorption (a) = 0.94 collector efficiency factor (F,) = 0.92. (iii) Tank data Volume = .39 (meter)3 Height = 1.65 meter overall loss coefficient = 15 kilojoule/(metr)2/hr. Load data (iv) 300 kilogram of water per day Delivery temperature 60°C Inlet temperature 15°C.
Using eqns (12), (18)-(20), (22) the fraction of energy delivered by the sun during the 8th of January 1977 is found to be 63.2%. The result differs from that obtained by TRNSYS model (66%). This may be attributed to the fact that the TRNSYS simulates systems component individually.
Evaluation of solar water heating systems
The evaluation of solar water heating system using the development model does not require tedious numerical calculation as required by the TRNSYS. It uses as input information available data about the climatic conditions. Also, it takes into consideration the effect of different factors such as heat losses through the reservoir walls, and the change in the absorption rate of the collector with change of the state of the system. Using the developed model, the optimum size of water tank and the area of the collector in a certain location can be determined. The effect of both the water tank volume and the areas of the collector on the lifetime of the system collector can also be estimated. REFERENCES
1. L. Abdel-Malek, Optimum design of solar water heating systems. Ph.D. Thesis, Polytechnic institutes of New York (1980). 2. H. Bae et ol., Optimization models for the economic design of wind power systems. Solar Energy 20, Oxford (1978). 3. W. Beckman ef al., Solar Heating Design Using 1 Chrt. Wiley, New York (1977). 4. A. M&et, Applied Sottii Energy. Addison Wesley, Reading, Mass. (1977). 5. National Bureau of Standards, InfemtediateStan&& for S&r Domestic Hot Water System. Distributed by Polytechnic Institute of New York (1977). 6. A. Safeir, A stochastic model for pmdicting solar system performance. Solar Energy 25, (1980). 7. Solar Energy Laboratory, A transie?tt sin&x&n progrm, Report 38, University of Wisconsin (1976). 8. H. Wagner, Prittctpks of Uperutions Research. Prentice-Halt, En&w& Cliffs, New Jersey.