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Optimal sizing of a solar water heating system based on a genetic algorithm for an aquaculture system
Mathematical and Computer Modelling 55 (2012) 1436–1449
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Optimal sizing of a solar water heating system based on a genetic algorithm for an aquaculture system Doaa M. Atia a,∗ , Faten H. Fahmy a , Ninet M. Ahmed a , Hassen T. Dorrah b a
Electronics Research Institute, Cairo, Egypt
b
Faculty of Engineering, Department of Electrical Power and Machines, Cairo University, Egypt
article
info
Article history: Received 24 August 2010 Received in revised form 21 September 2011 Accepted 20 October 2011 Keywords: Solar thermal energy Aquaculture system Optimization Genetic algorithm Economic Optimal sizing
abstract The most common use of thermal solar energy has been for water heating systems; this use has been commercialized in many countries in the world. This paper presents a model of a forced circulation solar water heating system for supplying a hot water at a certain temperature for an aquaculture system. The main components of the system are flat plate collector, storage tank, and a biogas auxiliary heater. The optimization problem is carried out using a genetic algorithm, which is one of the modern optimization techniques because of its evolutionary nature; it can handle any kind of objective function and constraint. Genetic algorithms don’t have mathematical requirements about the optimization problem; Also genetic algorithms are very effective in performing a global search (in probability), and provide a great flexibility. The optimal design of flat plate collector area using genetic algorithm is used to optimize the objective function considering the constraints required for the system. As the genetic algorithm is a discrete optimization tool, the number of variables in principle is unlimited. The economic analysis of such system is evaluated with the life cycle cost method. The collector area is equal to 63 m2 , at this value the solar fraction reached 98% which is a very high value. Also, sensitivity analysis to solar radiation variation, air temperature variation, and interest rate has been carried out. Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved.
1. Introduction Interest in using the energy of the Sun, together with other renewable energy sources (RES), is on the rise as fossil fuel prices become higher, supported by increasing public awareness on the need of preserving the environment. Due to an increase in conventional energy prices and environmental effects, such as air pollution, global heating, depletion of the ozone layer, and greenhouse effects, the use of solar energy has increased, following the energy crisis in 1973. On the other hands, solar energy is being seriously considered for satisfying a part of the energy demand in Egypt. Proper design of solar water heating system is important to ensure maximum benefit to the users, especially for a large system. Designing a solar hot water system involves appropriate sizing of different components based on predicted solar insolation and hot water demand [1]. It is increasingly becoming evident that the current pattern of rising conventional energy consumption cannot be sustained in the future due to the environmental consequences of the heavy dependence on fossil fuels and the depletion of fossil fuels. In recent years, global warming has emerged as the most serious environmental threat ever faced by mankind. Urban air pollution and acid rains are also major problems associated with large-scale use of fossil fuels [2]. Sizing of solar thermal water heating system components is very important issue from the viewpoint of economics [3].
∗
Corresponding author. Tel.: +20 2 33310512; fax: +20 2 33351631. E-mail addresses: [email protected], [email protected] (D.M. Atia).
0895-7177/$ – see front matter Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.10.022
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Nomenclature Ac Collector surface area (m2 ). Ap The area of the pond (m2 ). Ca The collector area dependent cost ($). CI Initial system cost ($). CF The collector area independent cost ($). CM & O Cost of maintenance and operation ($). CR Replacement cost ($). CS Salvage value ($). Cs The solar system cost ($). Cp Specific heat of water (4190 J/kg °C). crf Capital recovery factor. G Intensity of solar radiation (W/m2 ). FR The collector heat removal factor. F Fixed cost ($). hw Wind heat transfer coefficient (W/m2 °C). I Initial cost ($). kb The back insulation thermal conductivity (W/mK). Ke The edge insulation thermal conductivity (W/mK). Lb The back insulation thermal thickness (cm). Le The edge insulation thermal thickness (cm). Vs Storage tank volume (m3 ). Greek symbols
β εg ε εp σ α ατ ρ
m. N Pa qc ql qstl qr qe sdfd Ta Tdp Ti TP Tpm TS Ts Ul Ut Ub V
ωa ωP
Tilt angle degree. Glass emittance (%). The emissivity of the surface (0.97). Plate emittance (%). The Stefan–Boltzmann constant. Pond absorptance (90%) The transmittance absorptance product. Water density (kg/m3 ). Mass flow rate (kg/s). Number of glass cover. The ambient air pressure (101.3 kPa). Actual useful energy gain (W). Load energy. Storage tank losses. The radiation loss (W). The evaporation loss (W). Sinking fund deposit factor The ambient temperature (K). The dew point temperature (°C). The inlet temperature (K). The pond temperature (K). Mean plate temperature (K). The sky temperature (K). Average storage tank temperature (°C). Collector overall heat transfer coefficient (W/m2 °C). Top collector overall heat transfer coefficient (W/m2 °C). Back collector overall heat transfer coefficient (W/m2 °C). Variable cost ($). The humidity ratio of the ambient air above the pond. The saturation humidity ratio at the pond temperature.
A wide spread range of design methods for solar thermal systems has been developed in the last thirty years, ranging from the rather simple f -chart correlation method to sophisticated simulation packages like TRNSYS. However, the number
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Fig. 1. Schematic of a solar water heating system.
of optimization tools is rather small, and they are based on different objective functions like life cycle savings, pay back period, and internal rate of return. Most of these design methods and optimization tools analyze and compare systems with a given structure. Usually, the cost function is expressed in terms of the solar collector area, the solar fraction f , the capital cost and the auxiliary energy cost. As an example, the relative areas method is based on correlation to results obtained by an f -chart and it offers a quick and reasonably accurate calculation of the optimum collector area based on life cycle cost analysis [4,5] In recent years, some optimization methods that are conceptually different from traditional mathematical programming techniques have been developed. These methods are labeled as modern or nontraditional methods of optimization. Most of these methods are based on certain characteristics and behavior of biological and molecular systems, swarms of insects, and neurobiological systems. Many practical optimum design problems are characterized by mixed continuous–discrete variables, and discontinuous and no convex design spaces. Genetic algorithms (GA) are well suited for solving such problems, and in most cases they can find the global optimum solution with a high probability. Although the GA was first presented systematically by Holland [6], the basic ideas of analysis and design are based on the concepts of biological evolution. Genetic algorithms are based on the principles of natural genetics and natural selection. The basic elements of GA are reproduction, crossover and mutation, which are used in the genetic search procedure [7]. This paper presents the optimal sizing of solar thermal hot water system for an aquaculture system using genetic algorithm. Also, sensitivity analysis is carried out. 2. Aquaculture system Aquaculture is defined by the US National Marine Fisheries Service as the cultivation of freshwater or marine species under controlled conditions for human consumption. According to the Food and Agriculture Organization of the United Nations (FAO), it is an important industry in many countries because it is a vital part of the economy and a source of healthy food. The steady growth of the aquaculture industry suggests that it is a major source of income for many developed and developing countries. The growth of any industry also opens up many new employment positions, varying from minimum wage earning factory workers to people working in executive and management positions, earning far greater salaries [8]. The aquaculture system refers to the water container and everything that goes with it to make it a useful place to keep plants or animals for whatever purpose the culturist has in mind. Most systems include some means of water supply and drainage. In addition, aeration, shade, and a number of other factors may add parts to a system [9]. Fig. 1 shows an aquaculture farm consists of several rectangular ponds connected to a water source, which is sea or river, according to the farm type. 3. Mathematical model of solar thermal system A schematic diagram of a direct circulation solar thermal water heating system is shown in Fig. 2. In this system, a pump is used to circulate potable water from a storage tank to the collectors when there is enough available solar irradiance to increase its temperature and then return the heated water to the storage tank until it is needed. In cases of cloudy days and night hours a biogas heater is used as an auxiliary heater, is placed in series with the storage tank in the load supply line to meet the temperature requirement of the load. Storage tank temperature is an important parameter because it affects the system performance. The energy balance is used to describe well mixed storage tank, expressed as follows [1].
ρ Vs cp
dTs dt
= qc − ql − qstl ,
(1)
where ρ is the water density (kg/m3 ), Vs is the storage tank volume (m3 ), Ts is the average storage tank temperature (°C), Cp is the specific heat of water (4190 J/kg °C), qc is the actual useful energy gain, ql is the load energy, and qstl is the storage tank losses.
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Fig. 2. An aquaculture farm description. Table 1 MSC-32 flat plate solar collector. 80 °C
Average plate temperature Cover
Plate
Tubes
Back and edge insulation
Material Number of cover plates Glass emittance Material Conductivity Absorptance Emittance Thickness Number of tubes Space between liquid tube Tube outer diameter Tube inner diameter Flow rate Back Thickness Insulation Conductivity Edge Thickness Insulation Conductivity
Dimensions of collector
Glass 2 0.88 Copper 400 W/mK 0.95 0.1 0.02 cm 10 0.15 m 0.01 m 0.007 m 0.06 kg/s 0.03 0.03 0.03 m 0.03 W/mK 2.491 × 1.221 m2
3.1. Flat plate collector modeling The solar useful heat gain rate (qc ) from the collector array is calculated by [1] qc = FR Ac [ατ G − Ul (Ti − Ta )]+ ,
(2)
where qc represents the actual useful energy gain (W), FR is the collector heat removal factor, G is the intensity of solar radiation (W/m2 ), Ac is the collector surface area (m2 ), ατ is the transmittance absorptance product, Ul is the collector overall heat transfer coefficient (W/m2 °C), Ta is the ambient temperature (°C), and Ti is the inlet temperature. Where the + sign indicates that only positive values of qc are considered in the analysis. This implies that hot water from the collector enters the tank only when the solar useful heat gain becomes positive [1]. The flat plate collector specification used in the calculation is shown in Table 1. The collector heat removal factor, FR , is the ratio of the actual useful energy gain of a collector to the maximum possible useful gain if the whole collector surface were at the fluid inlet temperature. It is defined as [10] FR =
m. cp Ac Ul
[
1 − exp −
AUl F ′ m. cp
]
,
(3)
where m. is the mass flow rate (kg/s) and Cp is the specific heat of water (4190 J/kg °C). The overall heat loss from a solar collector consists of top heat loss through the cover systems, and back heat loss, edge heat loss which are heat loss through the back and edge insulation of the collector. With the assumption that all
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the losses are based on a common mean plate temperature Tpm , the overall heat loss from the collector can be represented as [11,12] Ul = Ut + Ub + Ue
(4)
−1
N
Ut =
Tpm
Ub = Ue =
Kb Lb Ke Le
T C −T e pm a
+
1 hw
+
2 σ (Tpm + Ta )(Tpm + Ta ) 2
(εp + 0.00591Nhw)−1 +
N +f
2N +f −1+0.133εp
εg
(5)
−N
(Tpm − Ta )
(6)
(Tpm − Ta ),
(7)
where Ul is the collector overall loss coefficient and the subscripts t, b, and e represent for the top, back, and edge contribution, respectively. The value of FR and Ul used in the calculation are calculated in a detailed program using MATLAB software; their values are 0.6646 and 10.76 W/m2 °C respectively. The values for radiation and air temperature used in this calculation are average values so the final results are one value for FR and Ul and these are used in the sizing calculation. 4. Load profile In solar heating system design, it is necessary to estimate the long-term (annual and/or monthly) average heating loads. The water heating load or the amount of energy required to warm water from the inlet cold water to a desired temperature is dependent on several factors such as hot water consumption rate, cold water inlet and desired hot water set temperatures, location and orientation of the system. 4.1. Modeling of an aquaculture pond Non-covered body of water exposed to heat exchange with the atmosphere by way of four mechanisms which are
• • • •
Evaporation losses. Convection losses. Radiation losses. Conduction losses.
A numerical model based on energy balance was developed to simulate the thermal behavior of the open-pond system. Calculation of these losses can be based on simplified equations that are generally applicable to water bodies [13]. Heat exchange by conduction with the ground is normally so small as to be negligible. We assumed uniform temperature for the entire pond, although we know that there are some temperature differences through the depth, and thus applied a wellmixed model; using of the aeration system in the pond will make this assumption acceptable because the system used to circulate water in addition to adding dissolved oxygen [10]. 4.2. Evaporation losses The evaporation heat loss is the largest loss component and it is given by qe = Ap Pa [35V + 43 (Tp − Ta )1/3 ] (ωp − ωa ),
(8)
where qe is the evaporation loss (W), V is the wind speed in (m/s) in the vicinity of the pond, Pa is the ambient air pressure (101.3 kPa), TP is the pond temperature, Ta is the ambient temperature, ωP is the saturation humidity ratio at the pond temperature, ωa is the humidity ratio of the ambient air above the pond, and Ap is the area of the pond [10,11]. 4.3. Convection losses Heat losses due to convection to the ambient air can be expressed as [13,14] qc = qe × 0.0006
Tp − Ta . ωp − ωa
(9)
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Fig. 3. The solar gain and losses of the pond during a typical day in the summer season.
4.4. The net radiation losses The loss results from the surface of the pond to the sky can be expressed as [13,14] qr = εσ Ap [(Tp + 273)4 − Ts4 ]
(10)
where qr is the radiation loss, ε is the emissivity of the surface, σ is the Stefan–Boltzmann constant, TS is the sky temperature K, and Tp is the pond temperature. 4.5. Solar radiation heat gain Heat gain due to the absorption of solar radiation by the pond is given by [13,14] qs = α Ac G,
(11)
where α is the pond absorptance (0.9). 4.6. Calculation of pond heating load The pond heating load is the total heat losses by the three mechanisms described above less any heat gain from incident solar radiation as below: ql = qe + qc + qr − qs .
(12)
Figs. 3 and 4 represent the three losses components and the solar gain in summer and winter, respectively. The evaporation loss represents the largest component of pond losses followed by the convection losses and finally the radiation losses. As the air temperature decrease, the losses of the pond increase. The solar gain follows the solar radiation variation in the desired location. The energy required is inversely proportional to air temperature. 5. Meteorological data To get the optimum design of the solar thermal system, it is important to collect meteorological data (solar irradiance and air temperature) for the site under consideration. Fig. 5 shows the monthly solar irradiance variation. These data which used are a practical data obtained for Mersa Matruh in Egypt [15]. The latitude and longitude of Mersa Matruh are 31° 33′ and 27° 22′ respectively. The values are an average values for one day over the year. The wind speed, humidity ratio, solar irradiance, and solar irradiance variation over the year have been cleared in Table 2. 6. Conventional sizing using a heuristic approach The mass flow rate of hot water circulating in the collector heating system can be calculated from the heat load equation. The heat load equation is represented by ql = m. cp 1T .
(13)
The mass flow rate of hot water circulating in the collector heating system can be calculated from the heat load equation and from the technical data of the solar collector tables, the collector area is 59 m2 [16].
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Fig. 4. The solar gain and losses of the pond during a typical day in the winter season.
Fig. 5. Monthly solar radiation variation.
7. Genetic algorithm Genetic algorithms offer a new and powerful approach to the optimization problems, and their use is made possible by the increasing availability of high performance computers at relatively low costs. These algorithms have recently found extensive applications in solving global optimization searching problems when the closed-form optimization technique cannot be applied. Genetic algorithms (GA) are parallel and global search techniques that emulate natural genetic operators. GA is more likely to converge toward the global solution because it, simultaneously, evaluates many points in the parameter
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Table 2 Meteorological data in Mersa Matruh. Month
Irradiation (kW/m2 )
Air temperature (°C)
Relative humidity (%)
Wind speed (m/s)
January February March April May June July August September October November December
4.31 5.23 5.65 6.33 6.30 6.35 6.42 6.43 6.23 5.28 4.47 3.96
13.2 13.7 15.3 17.4 20.05 23.25 24.75 25.4 24.15 21.9 18.3 14.8
66.9 76.3 76.8 78.2 74.3 72.3 71.7 74.8 76.4 73.0 68.6 64.8
6.06 6.06 6.29 5.7 4.88 5.38 5.19 4.72 4.41 4.26 4.77 5.85
Annual mean
6.62
19.3
72.8
5.29
space. It does not need to assume that the search space is differentiable or continuous. The advantages of using GA are that they require no knowledge or gradient information about the response surface; they are resistant to becoming trapped in local optima, and they can be employed for a wide variety of optimization problems. On other hand GA could have trouble in finding the exact global optimum and they require a large number of fitness function evaluations [17]. 7.1. Advantages of genetic algorithms
• • • • • •
Concept is easy to understand. Modular and separate from the application. Support multi-objective optimization. Good for ‘‘noisy’’ environments. Always provide an answer; the answer gets better with time. Inherently parallel, and easily distributed.
7.2. Components of a GA
• • • • • • • •
A problem to solve, and . . . . Encoding technique (gene, chromosome). Initialization procedure (creation). Evaluation function (environment). Selection of parents (reproduction). Genetic operators (mutation, recombination, crossover). Parameter settings (practice and art). Simple genetic algorithm.
The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over successive generations, the population ‘‘evolves’’ toward an optimal solution. 7.3. Simple genetic algorithm steps 1. 2. 3. 4. 5. 6.
Initialize population; Evaluate population; While termination criteria are not satisfied: Select parents for reproduction; Perform recombination and mutation; Evaluate population.
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Fig. 6. The flow chart of the genetic algorithm.
8. Optimal sizing using a genetic algorithm The optimization technique with genetic algorithm is used to find mainly the optimal size of flat plate collector and auxiliary heater based on minimum cost of the system. A life-cycle analysis is performed in order to obtain the total cost (or life-cycle cost). The period of economic analysis is taken as 20 years (i.e., life of the system), whereas the auxiliary heater has been sized to satisfy the whole load without solar radiation. GA differs from the traditional methods of optimization in the following respects: 1. A population of points (trial design vectors) is used for starting the procedure instead of a single design point. 2. GA uses only the values of the objective function. The derivatives are not used in the search procedure. 3. In GA, the design variables are represented as strings of binary variables that correspond to the chromosomes in natural genetics. Thus the search method is naturally applicable for solving discrete and integer programming problems. For continuous design variables, the string length can be varied to achieve any desired resolution. 4. The objective function value corresponding to a design vector plays the role of fitness in natural genetics. 5. In every new generation, a new set of strings is produced by using randomized parents selection and crossover from the old generation (old set of strings) [7]. 8.1. Economic analysis of a heating system The solar thermal system cost represents by the annualized life cycle cost method is as below Cs = CI + CM & O + CR − CS
(14)
CI = CF + Ca A,
(15)
where Cs is the solar system thermal cost, CI is the initial system cost, CM & O represents the cost of maintenance and operation, equal to 2% of initial value, CR represents the replacement cost, CS is salvage value, equal to 20% of initial cost [18, 19], where CF is the collector area independent cost (12 000$), and Ca is the collector area dependent cost (250 $/m2 ). The auxiliary system cost is given by [20] Caux = I + F + V ,
(16)
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Table 3 Genetic algorithm parameters. Parameter
Value
Population size Selection function Crossover function Crossover Elites Population size
20 Uniform Uniform 80% 2 20
Fig. 7. Output energy generated from collector and auxiliary heater with collector area variation.
where I is initial cost, equal to 45 $/kW, F is fixed cost, which has the value of 2.7% of initial cost, and V is variable cost, equal to 1.2 $/MWh . The total system cost is TC = Cs + Caux .
(17)
To convert to an annualized value of money there are two factors have been used which are capital recovery factor (crf) and sinking fund deposit factor (sdfd) respectively, using the interest rate (i) [20] crf =
i
(18)
1 + (1 + i)−n
sdfd =
i
(1 + i)n − 1
.
(19)
8.2. Problem description and sizing procedures The flowchart of genetic algorithm which is used in sizing calculation is depicted in Fig. 6. The economic minimization model is as below: Minimize LCC =
n −
CI i + CRi + CM & Oi − CS i .
(20)
j =1
The system constraint is the energy balance, which must be satisfied during the year. The energy balance ensures continuity for feeding the load demands over the year. The constraint model is as follows: Subject to PL = Pth + Paux .
(21)
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Fig. 8. Total system cost variation with collector area variation. 100 area=10 m2 90 area=20m2 80
area=30m2
Solar fraction (%)
70
area=40m2
60
area=50m2
50
area=60m2
40
area=63m2
30 20 10 0 April
May
June
July
Aug
Sept
Months Fig. 9. Solar fraction variation with collector areas during the summer season. (Constant air temperature and variable values of solar irradiance.)
9. Results and discussion Genetic algorithm toolbox in MATLAB is used to calculate the optimal design. The genetic algorithm uses three main types of rules at each step to create the next generation from the current population:
• Selection rules; select the individuals, called parents, that contribute to the population at the next generation. • Crossover rules; combine two parents to form children for the next generation. • Mutation rules; apply random changes to individual parents to form children. A population is an array of individuals. The same individual can appear more than once in the population. At each iteration, the genetic algorithm performs a series of computations on the current population to produce a new population. Each successive population is called a new generation. Parents and children To create the next generation, the genetic algorithm selects certain individuals in the current population, called parents, and uses them to create individuals in the next generation, called children. Typically, the algorithm is more likely to select parents that have better fitness values. The following outline summarizes how the genetic algorithm works: 1. The algorithm begins by creating a random initial population. 2. The algorithm then creates a sequence of new populations. At each step, the algorithm uses the individuals in the current generation to create the next population. The program input are solar irradiance, air temperature, wind speed load demand, solar thermal energy, and auxiliary energy. Two m-file programs are written and entered to the GA toolbox. Genetic algorithm toolbox in MATLAB is used
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100
area=10m2 90
Solar fraction (%)
area=20m2 80
area=30m2
70
area=40m2
60
area=50m2 area=60m2
50
area=63m2 40 30 20 10 0
Oct
Nov
Dec Jan Months
Feb
Mar
Fig. 10. Solar fraction variation with collector areas during winter season. (Constant air temperature and variable values of solar irradiance.)
100 area=10m2
90
area=20m2 80
area=30m2 area=40m2
Solar fraction (%)
70
area=50m2 60
area=60m2 area=63m2
50 40 30 20 10 0 April
May
Jun
Jul
Aug
Sept
Month Fig. 11. Solar fraction variation with collector areas during summer. (Constant solar irradiance and variable air temperature.)
to calculate the optimal area. After many trials the optimal area is equal to be 63 m2 . The output program parameters are presented in Table 3. Crossover rate determines the probability that the crossover operator will be applied to a particular chromosome during a generation. The program gives the area of the flat plate collector is equal to 63 m2 , at this area the solar fraction is equal to 98% which satisfies the load. Fig. 7 shows the output energy generated from the collector and auxiliary heater with collector area variation; as the collector area increases, the amount of energy required from the biogas auxiliary heater decreases, which affects the total system cost as depicted in Fig. 8. The aquaculture system has two operating seasons during the year (summer and winter). The solar fraction, f , is defined as the ratio of the useful solar energy supplied to the system to the energy needed to heat the water if no solar energy is used. In other words, f is a measure of the fractional energy savings relative to that used for an auxiliary system. Figs. 9 and 10 show the variation of solar fraction with different collector areas and solar radiation in summer and winter respectively. As the collector area increases, the solar fraction increases, but it depends on the system cost. The average value of the solar
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Solar fraction (%)
70
area=40m2
60
area=50m2
50
area=60m2
40
area=63m2
30 20 10 0
Oct
Nov
Dec
Jan
Feb
Mar
Month Fig. 12. Solar fraction variation with collector areas during winter. (Constant solar irradiance and variable air temperature.) 6 area=10m2 area=20m2 area=30m2
5.5
area=40m2
System cost (*10^3) ($)
area=50m2 5
area=60m2 area=63m2
4.5
4
3.5
3
April
May
June
July
Aug
Sept
Months Fig. 13. System cost variation with different collector areas in summer.
fraction in summer is higher than that in winter due to the high value of solar radiation in summer for a 63 m2 collector is able to supply 90–99% of the hot water demand from May to September. Figs. 11 and 12 present the variation of air temperature effect on the solar fraction in summer and winter, respectively. January has the minimum value of solar fraction due to the low values of air temperature and solar radiation over the winter season. The value of solar fraction in winter is in range of 44.9–92.8%, and in summer it is 68.1–99%. The sensitivity analysis are carried out by varying the input parameters (solar radiation, air temperature, and interest rate). Figs. 13 and 14 present the effect of solar radiation variation over the year on the cost function. In winter season, the auxiliary heater is used to provide more energy to heat the water to the desired temperature, so the variation of cost is increased depending on the value of the auxiliary heater energy. Table 4 shows the effect of interest rate variation on the system cost: as the interest rate increases, the system cost are increased. In Egypt, the value of interest rate is equal to (=11.6%.) the system cost variation at different collector areas in summer and winter respectively. 10. Conclusion Optimum sizing of system components is important in design of a large solar thermal water heating system. In this paper a design of a solar thermal water heating system for supplying an aquaculture system with the required hot water demand was presented. A methodology of sizing the solar thermal water heating system using genetic algorithm was proposed. Genetic
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6.5
area=10m2 6
area=20m2 area=30m2
5.5
area=40m2
System cost*10^3 ($)
area=50m2 5
area=60m2 area=63m2
4.5
4
3.5
3
Oct
Nov
Dec
Jan
Feb
Mar
Months Fig. 14. System cost variation with different collector areas in winter. Table 4 Cost variation with interest rate. Interest rate (%)
Cost ($)
5 8 10 12 15 20
2601.2 3382.1 3933.6 4352.2 5328.4 6686.8
algorithm has been suggested in order to determine the optimal sizing of the solar thermal system according to minimize the objective function considering the different constraints and to give the optimal area of the flat plate collector. The collector area is equal to 63 m2 , at this value the solar fraction reached to 98%, which is very high value. The solar radiation has an obvious effect on the solar fraction and system cost, specially when the collector area increases. The sensitivity analysis was carried out by varying the input parameters (solar radiation, air temperature, and interest rate). References [1] Govind N. Kulkarni, Shireesh B. Kedare, Santanu Bandyopadhyay, Determination of design space and optimization of solar water heating systems, Solar Energy 81 (2007) 958–968. [2] Soteris Kalogirou, Thermal performance, economic and environmental life cycle analysis of thermosiphon solar water heaters, Solar Energy 83 (2009) 39–48. [3] Govind N. Kulkarni, Shireesh B. Kedare, Santanu Bandyopadhyay, Design of solar thermal systems utilizing pressurized hot water storage for industrial applications, Solar Energy 82 (2008) 686–699. [4] V. Badescu, Optimum size and structure for solar energy collection systems, Energy 31 (2006) 1819–1835. [5] V. Badescu, Optimum fin geometry in flat plate solar collector systems, Energy Conversion and Management 47 (2006) 2397–2413. [6] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975. [7] Singiresu S. Rao, Engineering Optimization, Theory and Practice, fourth ed., John Wiley & Sons, Inc., Hoboken, New Jersey, Canada, 2009. [8] Molly McShea, Jo-Ellen Sullivan, Design of an aquaculture database for INCOPESCA July 5, 2007, Report, July 5, 2007. [9] Jim Szyper, Backyard Aquaculture in Hawaii a Practical Manual, University of Hawaii Windward Community College and Hawaii Institute of Marine Biology, 1980. [10] Joshua Reed Plaisted, Life cycle performance evaluation of an air-based solar thermal system, M.Sc. Wisconsin Madison University, 2000. [11] M. Bojic, S. Kalogirou, K. Petronijevic, Simulation of a solar domestic water heating system using a time marching model, Renewable Energy 27 (2002) 441–452. [12] Jae-Mo Koo, Development of a flat-plate solar collector design program, M.Sc. Wisconsin Madison University, 1999. [13] John Gelegenis, Paschalis Dalabakis, Andreas Ilias, Heating of a fish wintering pond using low-temperature geothermal fluids, Porto Lagos, Greece, Geothermics 35 (2006) 87–103. [14] J. Duffie, W. Beckman, Solar Engineering of Thermal Processes, second ed., John Wiley & Sons Interscience, New York, 1991. [15] Egyptian solar radiation atlas, Cairo, Egypt, 1998. [16] Arab Solar Energy and Technology Company. Available at: www.asetegypt.com. [17] D.E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Addison Wesley, 1990. [18] M. Loomans, H. Visser, Application of the genetic algorithm for the optimization of large solar hot water systems, Solar Energy 72 (2002) 427–439. [19] Jasbir Arora, Introduction to Optimum Design, second ed., Academic Press, 2004. [20] J. Parrilla, W. Fichtner, The impact of transportation costs on the profitability of heat and power generation with wood products, in: Proc. of International Conference on Renewable Energies and Power Quality, ICREPQ’09, Valencia, Spain, April 2009.
Contents lists available at SciVerse ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Optimal sizing of a solar water heating system based on a genetic algorithm for an aquaculture system Doaa M. Atia a,∗ , Faten H. Fahmy a , Ninet M. Ahmed a , Hassen T. Dorrah b a
Electronics Research Institute, Cairo, Egypt
b
Faculty of Engineering, Department of Electrical Power and Machines, Cairo University, Egypt
article
info
Article history: Received 24 August 2010 Received in revised form 21 September 2011 Accepted 20 October 2011 Keywords: Solar thermal energy Aquaculture system Optimization Genetic algorithm Economic Optimal sizing
abstract The most common use of thermal solar energy has been for water heating systems; this use has been commercialized in many countries in the world. This paper presents a model of a forced circulation solar water heating system for supplying a hot water at a certain temperature for an aquaculture system. The main components of the system are flat plate collector, storage tank, and a biogas auxiliary heater. The optimization problem is carried out using a genetic algorithm, which is one of the modern optimization techniques because of its evolutionary nature; it can handle any kind of objective function and constraint. Genetic algorithms don’t have mathematical requirements about the optimization problem; Also genetic algorithms are very effective in performing a global search (in probability), and provide a great flexibility. The optimal design of flat plate collector area using genetic algorithm is used to optimize the objective function considering the constraints required for the system. As the genetic algorithm is a discrete optimization tool, the number of variables in principle is unlimited. The economic analysis of such system is evaluated with the life cycle cost method. The collector area is equal to 63 m2 , at this value the solar fraction reached 98% which is a very high value. Also, sensitivity analysis to solar radiation variation, air temperature variation, and interest rate has been carried out. Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved.
1. Introduction Interest in using the energy of the Sun, together with other renewable energy sources (RES), is on the rise as fossil fuel prices become higher, supported by increasing public awareness on the need of preserving the environment. Due to an increase in conventional energy prices and environmental effects, such as air pollution, global heating, depletion of the ozone layer, and greenhouse effects, the use of solar energy has increased, following the energy crisis in 1973. On the other hands, solar energy is being seriously considered for satisfying a part of the energy demand in Egypt. Proper design of solar water heating system is important to ensure maximum benefit to the users, especially for a large system. Designing a solar hot water system involves appropriate sizing of different components based on predicted solar insolation and hot water demand [1]. It is increasingly becoming evident that the current pattern of rising conventional energy consumption cannot be sustained in the future due to the environmental consequences of the heavy dependence on fossil fuels and the depletion of fossil fuels. In recent years, global warming has emerged as the most serious environmental threat ever faced by mankind. Urban air pollution and acid rains are also major problems associated with large-scale use of fossil fuels [2]. Sizing of solar thermal water heating system components is very important issue from the viewpoint of economics [3].
∗
Corresponding author. Tel.: +20 2 33310512; fax: +20 2 33351631. E-mail addresses: [email protected], [email protected] (D.M. Atia).
0895-7177/$ – see front matter Crown Copyright © 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.10.022
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Nomenclature Ac Collector surface area (m2 ). Ap The area of the pond (m2 ). Ca The collector area dependent cost ($). CI Initial system cost ($). CF The collector area independent cost ($). CM & O Cost of maintenance and operation ($). CR Replacement cost ($). CS Salvage value ($). Cs The solar system cost ($). Cp Specific heat of water (4190 J/kg °C). crf Capital recovery factor. G Intensity of solar radiation (W/m2 ). FR The collector heat removal factor. F Fixed cost ($). hw Wind heat transfer coefficient (W/m2 °C). I Initial cost ($). kb The back insulation thermal conductivity (W/mK). Ke The edge insulation thermal conductivity (W/mK). Lb The back insulation thermal thickness (cm). Le The edge insulation thermal thickness (cm). Vs Storage tank volume (m3 ). Greek symbols
β εg ε εp σ α ατ ρ
m. N Pa qc ql qstl qr qe sdfd Ta Tdp Ti TP Tpm TS Ts Ul Ut Ub V
ωa ωP
Tilt angle degree. Glass emittance (%). The emissivity of the surface (0.97). Plate emittance (%). The Stefan–Boltzmann constant. Pond absorptance (90%) The transmittance absorptance product. Water density (kg/m3 ). Mass flow rate (kg/s). Number of glass cover. The ambient air pressure (101.3 kPa). Actual useful energy gain (W). Load energy. Storage tank losses. The radiation loss (W). The evaporation loss (W). Sinking fund deposit factor The ambient temperature (K). The dew point temperature (°C). The inlet temperature (K). The pond temperature (K). Mean plate temperature (K). The sky temperature (K). Average storage tank temperature (°C). Collector overall heat transfer coefficient (W/m2 °C). Top collector overall heat transfer coefficient (W/m2 °C). Back collector overall heat transfer coefficient (W/m2 °C). Variable cost ($). The humidity ratio of the ambient air above the pond. The saturation humidity ratio at the pond temperature.
A wide spread range of design methods for solar thermal systems has been developed in the last thirty years, ranging from the rather simple f -chart correlation method to sophisticated simulation packages like TRNSYS. However, the number
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Fig. 1. Schematic of a solar water heating system.
of optimization tools is rather small, and they are based on different objective functions like life cycle savings, pay back period, and internal rate of return. Most of these design methods and optimization tools analyze and compare systems with a given structure. Usually, the cost function is expressed in terms of the solar collector area, the solar fraction f , the capital cost and the auxiliary energy cost. As an example, the relative areas method is based on correlation to results obtained by an f -chart and it offers a quick and reasonably accurate calculation of the optimum collector area based on life cycle cost analysis [4,5] In recent years, some optimization methods that are conceptually different from traditional mathematical programming techniques have been developed. These methods are labeled as modern or nontraditional methods of optimization. Most of these methods are based on certain characteristics and behavior of biological and molecular systems, swarms of insects, and neurobiological systems. Many practical optimum design problems are characterized by mixed continuous–discrete variables, and discontinuous and no convex design spaces. Genetic algorithms (GA) are well suited for solving such problems, and in most cases they can find the global optimum solution with a high probability. Although the GA was first presented systematically by Holland [6], the basic ideas of analysis and design are based on the concepts of biological evolution. Genetic algorithms are based on the principles of natural genetics and natural selection. The basic elements of GA are reproduction, crossover and mutation, which are used in the genetic search procedure [7]. This paper presents the optimal sizing of solar thermal hot water system for an aquaculture system using genetic algorithm. Also, sensitivity analysis is carried out. 2. Aquaculture system Aquaculture is defined by the US National Marine Fisheries Service as the cultivation of freshwater or marine species under controlled conditions for human consumption. According to the Food and Agriculture Organization of the United Nations (FAO), it is an important industry in many countries because it is a vital part of the economy and a source of healthy food. The steady growth of the aquaculture industry suggests that it is a major source of income for many developed and developing countries. The growth of any industry also opens up many new employment positions, varying from minimum wage earning factory workers to people working in executive and management positions, earning far greater salaries [8]. The aquaculture system refers to the water container and everything that goes with it to make it a useful place to keep plants or animals for whatever purpose the culturist has in mind. Most systems include some means of water supply and drainage. In addition, aeration, shade, and a number of other factors may add parts to a system [9]. Fig. 1 shows an aquaculture farm consists of several rectangular ponds connected to a water source, which is sea or river, according to the farm type. 3. Mathematical model of solar thermal system A schematic diagram of a direct circulation solar thermal water heating system is shown in Fig. 2. In this system, a pump is used to circulate potable water from a storage tank to the collectors when there is enough available solar irradiance to increase its temperature and then return the heated water to the storage tank until it is needed. In cases of cloudy days and night hours a biogas heater is used as an auxiliary heater, is placed in series with the storage tank in the load supply line to meet the temperature requirement of the load. Storage tank temperature is an important parameter because it affects the system performance. The energy balance is used to describe well mixed storage tank, expressed as follows [1].
ρ Vs cp
dTs dt
= qc − ql − qstl ,
(1)
where ρ is the water density (kg/m3 ), Vs is the storage tank volume (m3 ), Ts is the average storage tank temperature (°C), Cp is the specific heat of water (4190 J/kg °C), qc is the actual useful energy gain, ql is the load energy, and qstl is the storage tank losses.
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Fig. 2. An aquaculture farm description. Table 1 MSC-32 flat plate solar collector. 80 °C
Average plate temperature Cover
Plate
Tubes
Back and edge insulation
Material Number of cover plates Glass emittance Material Conductivity Absorptance Emittance Thickness Number of tubes Space between liquid tube Tube outer diameter Tube inner diameter Flow rate Back Thickness Insulation Conductivity Edge Thickness Insulation Conductivity
Dimensions of collector
Glass 2 0.88 Copper 400 W/mK 0.95 0.1 0.02 cm 10 0.15 m 0.01 m 0.007 m 0.06 kg/s 0.03 0.03 0.03 m 0.03 W/mK 2.491 × 1.221 m2
3.1. Flat plate collector modeling The solar useful heat gain rate (qc ) from the collector array is calculated by [1] qc = FR Ac [ατ G − Ul (Ti − Ta )]+ ,
(2)
where qc represents the actual useful energy gain (W), FR is the collector heat removal factor, G is the intensity of solar radiation (W/m2 ), Ac is the collector surface area (m2 ), ατ is the transmittance absorptance product, Ul is the collector overall heat transfer coefficient (W/m2 °C), Ta is the ambient temperature (°C), and Ti is the inlet temperature. Where the + sign indicates that only positive values of qc are considered in the analysis. This implies that hot water from the collector enters the tank only when the solar useful heat gain becomes positive [1]. The flat plate collector specification used in the calculation is shown in Table 1. The collector heat removal factor, FR , is the ratio of the actual useful energy gain of a collector to the maximum possible useful gain if the whole collector surface were at the fluid inlet temperature. It is defined as [10] FR =
m. cp Ac Ul
[
1 − exp −
AUl F ′ m. cp
]
,
(3)
where m. is the mass flow rate (kg/s) and Cp is the specific heat of water (4190 J/kg °C). The overall heat loss from a solar collector consists of top heat loss through the cover systems, and back heat loss, edge heat loss which are heat loss through the back and edge insulation of the collector. With the assumption that all
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the losses are based on a common mean plate temperature Tpm , the overall heat loss from the collector can be represented as [11,12] Ul = Ut + Ub + Ue
(4)
−1
N
Ut =
Tpm
Ub = Ue =
Kb Lb Ke Le
T C −T e pm a
+
1 hw
+
2 σ (Tpm + Ta )(Tpm + Ta ) 2
(εp + 0.00591Nhw)−1 +
N +f
2N +f −1+0.133εp
εg
(5)
−N
(Tpm − Ta )
(6)
(Tpm − Ta ),
(7)
where Ul is the collector overall loss coefficient and the subscripts t, b, and e represent for the top, back, and edge contribution, respectively. The value of FR and Ul used in the calculation are calculated in a detailed program using MATLAB software; their values are 0.6646 and 10.76 W/m2 °C respectively. The values for radiation and air temperature used in this calculation are average values so the final results are one value for FR and Ul and these are used in the sizing calculation. 4. Load profile In solar heating system design, it is necessary to estimate the long-term (annual and/or monthly) average heating loads. The water heating load or the amount of energy required to warm water from the inlet cold water to a desired temperature is dependent on several factors such as hot water consumption rate, cold water inlet and desired hot water set temperatures, location and orientation of the system. 4.1. Modeling of an aquaculture pond Non-covered body of water exposed to heat exchange with the atmosphere by way of four mechanisms which are
• • • •
Evaporation losses. Convection losses. Radiation losses. Conduction losses.
A numerical model based on energy balance was developed to simulate the thermal behavior of the open-pond system. Calculation of these losses can be based on simplified equations that are generally applicable to water bodies [13]. Heat exchange by conduction with the ground is normally so small as to be negligible. We assumed uniform temperature for the entire pond, although we know that there are some temperature differences through the depth, and thus applied a wellmixed model; using of the aeration system in the pond will make this assumption acceptable because the system used to circulate water in addition to adding dissolved oxygen [10]. 4.2. Evaporation losses The evaporation heat loss is the largest loss component and it is given by qe = Ap Pa [35V + 43 (Tp − Ta )1/3 ] (ωp − ωa ),
(8)
where qe is the evaporation loss (W), V is the wind speed in (m/s) in the vicinity of the pond, Pa is the ambient air pressure (101.3 kPa), TP is the pond temperature, Ta is the ambient temperature, ωP is the saturation humidity ratio at the pond temperature, ωa is the humidity ratio of the ambient air above the pond, and Ap is the area of the pond [10,11]. 4.3. Convection losses Heat losses due to convection to the ambient air can be expressed as [13,14] qc = qe × 0.0006
Tp − Ta . ωp − ωa
(9)
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Fig. 3. The solar gain and losses of the pond during a typical day in the summer season.
4.4. The net radiation losses The loss results from the surface of the pond to the sky can be expressed as [13,14] qr = εσ Ap [(Tp + 273)4 − Ts4 ]
(10)
where qr is the radiation loss, ε is the emissivity of the surface, σ is the Stefan–Boltzmann constant, TS is the sky temperature K, and Tp is the pond temperature. 4.5. Solar radiation heat gain Heat gain due to the absorption of solar radiation by the pond is given by [13,14] qs = α Ac G,
(11)
where α is the pond absorptance (0.9). 4.6. Calculation of pond heating load The pond heating load is the total heat losses by the three mechanisms described above less any heat gain from incident solar radiation as below: ql = qe + qc + qr − qs .
(12)
Figs. 3 and 4 represent the three losses components and the solar gain in summer and winter, respectively. The evaporation loss represents the largest component of pond losses followed by the convection losses and finally the radiation losses. As the air temperature decrease, the losses of the pond increase. The solar gain follows the solar radiation variation in the desired location. The energy required is inversely proportional to air temperature. 5. Meteorological data To get the optimum design of the solar thermal system, it is important to collect meteorological data (solar irradiance and air temperature) for the site under consideration. Fig. 5 shows the monthly solar irradiance variation. These data which used are a practical data obtained for Mersa Matruh in Egypt [15]. The latitude and longitude of Mersa Matruh are 31° 33′ and 27° 22′ respectively. The values are an average values for one day over the year. The wind speed, humidity ratio, solar irradiance, and solar irradiance variation over the year have been cleared in Table 2. 6. Conventional sizing using a heuristic approach The mass flow rate of hot water circulating in the collector heating system can be calculated from the heat load equation. The heat load equation is represented by ql = m. cp 1T .
(13)
The mass flow rate of hot water circulating in the collector heating system can be calculated from the heat load equation and from the technical data of the solar collector tables, the collector area is 59 m2 [16].
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Fig. 4. The solar gain and losses of the pond during a typical day in the winter season.
Fig. 5. Monthly solar radiation variation.
7. Genetic algorithm Genetic algorithms offer a new and powerful approach to the optimization problems, and their use is made possible by the increasing availability of high performance computers at relatively low costs. These algorithms have recently found extensive applications in solving global optimization searching problems when the closed-form optimization technique cannot be applied. Genetic algorithms (GA) are parallel and global search techniques that emulate natural genetic operators. GA is more likely to converge toward the global solution because it, simultaneously, evaluates many points in the parameter
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Table 2 Meteorological data in Mersa Matruh. Month
Irradiation (kW/m2 )
Air temperature (°C)
Relative humidity (%)
Wind speed (m/s)
January February March April May June July August September October November December
4.31 5.23 5.65 6.33 6.30 6.35 6.42 6.43 6.23 5.28 4.47 3.96
13.2 13.7 15.3 17.4 20.05 23.25 24.75 25.4 24.15 21.9 18.3 14.8
66.9 76.3 76.8 78.2 74.3 72.3 71.7 74.8 76.4 73.0 68.6 64.8
6.06 6.06 6.29 5.7 4.88 5.38 5.19 4.72 4.41 4.26 4.77 5.85
Annual mean
6.62
19.3
72.8
5.29
space. It does not need to assume that the search space is differentiable or continuous. The advantages of using GA are that they require no knowledge or gradient information about the response surface; they are resistant to becoming trapped in local optima, and they can be employed for a wide variety of optimization problems. On other hand GA could have trouble in finding the exact global optimum and they require a large number of fitness function evaluations [17]. 7.1. Advantages of genetic algorithms
• • • • • •
Concept is easy to understand. Modular and separate from the application. Support multi-objective optimization. Good for ‘‘noisy’’ environments. Always provide an answer; the answer gets better with time. Inherently parallel, and easily distributed.
7.2. Components of a GA
• • • • • • • •
A problem to solve, and . . . . Encoding technique (gene, chromosome). Initialization procedure (creation). Evaluation function (environment). Selection of parents (reproduction). Genetic operators (mutation, recombination, crossover). Parameter settings (practice and art). Simple genetic algorithm.
The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over successive generations, the population ‘‘evolves’’ toward an optimal solution. 7.3. Simple genetic algorithm steps 1. 2. 3. 4. 5. 6.
Initialize population; Evaluate population; While termination criteria are not satisfied: Select parents for reproduction; Perform recombination and mutation; Evaluate population.
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Fig. 6. The flow chart of the genetic algorithm.
8. Optimal sizing using a genetic algorithm The optimization technique with genetic algorithm is used to find mainly the optimal size of flat plate collector and auxiliary heater based on minimum cost of the system. A life-cycle analysis is performed in order to obtain the total cost (or life-cycle cost). The period of economic analysis is taken as 20 years (i.e., life of the system), whereas the auxiliary heater has been sized to satisfy the whole load without solar radiation. GA differs from the traditional methods of optimization in the following respects: 1. A population of points (trial design vectors) is used for starting the procedure instead of a single design point. 2. GA uses only the values of the objective function. The derivatives are not used in the search procedure. 3. In GA, the design variables are represented as strings of binary variables that correspond to the chromosomes in natural genetics. Thus the search method is naturally applicable for solving discrete and integer programming problems. For continuous design variables, the string length can be varied to achieve any desired resolution. 4. The objective function value corresponding to a design vector plays the role of fitness in natural genetics. 5. In every new generation, a new set of strings is produced by using randomized parents selection and crossover from the old generation (old set of strings) [7]. 8.1. Economic analysis of a heating system The solar thermal system cost represents by the annualized life cycle cost method is as below Cs = CI + CM & O + CR − CS
(14)
CI = CF + Ca A,
(15)
where Cs is the solar system thermal cost, CI is the initial system cost, CM & O represents the cost of maintenance and operation, equal to 2% of initial value, CR represents the replacement cost, CS is salvage value, equal to 20% of initial cost [18, 19], where CF is the collector area independent cost (12 000$), and Ca is the collector area dependent cost (250 $/m2 ). The auxiliary system cost is given by [20] Caux = I + F + V ,
(16)
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Table 3 Genetic algorithm parameters. Parameter
Value
Population size Selection function Crossover function Crossover Elites Population size
20 Uniform Uniform 80% 2 20
Fig. 7. Output energy generated from collector and auxiliary heater with collector area variation.
where I is initial cost, equal to 45 $/kW, F is fixed cost, which has the value of 2.7% of initial cost, and V is variable cost, equal to 1.2 $/MWh . The total system cost is TC = Cs + Caux .
(17)
To convert to an annualized value of money there are two factors have been used which are capital recovery factor (crf) and sinking fund deposit factor (sdfd) respectively, using the interest rate (i) [20] crf =
i
(18)
1 + (1 + i)−n
sdfd =
i
(1 + i)n − 1
.
(19)
8.2. Problem description and sizing procedures The flowchart of genetic algorithm which is used in sizing calculation is depicted in Fig. 6. The economic minimization model is as below: Minimize LCC =
n −
CI i + CRi + CM & Oi − CS i .
(20)
j =1
The system constraint is the energy balance, which must be satisfied during the year. The energy balance ensures continuity for feeding the load demands over the year. The constraint model is as follows: Subject to PL = Pth + Paux .
(21)
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Fig. 8. Total system cost variation with collector area variation. 100 area=10 m2 90 area=20m2 80
area=30m2
Solar fraction (%)
70
area=40m2
60
area=50m2
50
area=60m2
40
area=63m2
30 20 10 0 April
May
June
July
Aug
Sept
Months Fig. 9. Solar fraction variation with collector areas during the summer season. (Constant air temperature and variable values of solar irradiance.)
9. Results and discussion Genetic algorithm toolbox in MATLAB is used to calculate the optimal design. The genetic algorithm uses three main types of rules at each step to create the next generation from the current population:
• Selection rules; select the individuals, called parents, that contribute to the population at the next generation. • Crossover rules; combine two parents to form children for the next generation. • Mutation rules; apply random changes to individual parents to form children. A population is an array of individuals. The same individual can appear more than once in the population. At each iteration, the genetic algorithm performs a series of computations on the current population to produce a new population. Each successive population is called a new generation. Parents and children To create the next generation, the genetic algorithm selects certain individuals in the current population, called parents, and uses them to create individuals in the next generation, called children. Typically, the algorithm is more likely to select parents that have better fitness values. The following outline summarizes how the genetic algorithm works: 1. The algorithm begins by creating a random initial population. 2. The algorithm then creates a sequence of new populations. At each step, the algorithm uses the individuals in the current generation to create the next population. The program input are solar irradiance, air temperature, wind speed load demand, solar thermal energy, and auxiliary energy. Two m-file programs are written and entered to the GA toolbox. Genetic algorithm toolbox in MATLAB is used
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100
area=10m2 90
Solar fraction (%)
area=20m2 80
area=30m2
70
area=40m2
60
area=50m2 area=60m2
50
area=63m2 40 30 20 10 0
Oct
Nov
Dec Jan Months
Feb
Mar
Fig. 10. Solar fraction variation with collector areas during winter season. (Constant air temperature and variable values of solar irradiance.)
100 area=10m2
90
area=20m2 80
area=30m2 area=40m2
Solar fraction (%)
70
area=50m2 60
area=60m2 area=63m2
50 40 30 20 10 0 April
May
Jun
Jul
Aug
Sept
Month Fig. 11. Solar fraction variation with collector areas during summer. (Constant solar irradiance and variable air temperature.)
to calculate the optimal area. After many trials the optimal area is equal to be 63 m2 . The output program parameters are presented in Table 3. Crossover rate determines the probability that the crossover operator will be applied to a particular chromosome during a generation. The program gives the area of the flat plate collector is equal to 63 m2 , at this area the solar fraction is equal to 98% which satisfies the load. Fig. 7 shows the output energy generated from the collector and auxiliary heater with collector area variation; as the collector area increases, the amount of energy required from the biogas auxiliary heater decreases, which affects the total system cost as depicted in Fig. 8. The aquaculture system has two operating seasons during the year (summer and winter). The solar fraction, f , is defined as the ratio of the useful solar energy supplied to the system to the energy needed to heat the water if no solar energy is used. In other words, f is a measure of the fractional energy savings relative to that used for an auxiliary system. Figs. 9 and 10 show the variation of solar fraction with different collector areas and solar radiation in summer and winter respectively. As the collector area increases, the solar fraction increases, but it depends on the system cost. The average value of the solar
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D.M. Atia et al. / Mathematical and Computer Modelling 55 (2012) 1436–1449 100 area=10m2 90 area=20m2 80 area=30m2
Solar fraction (%)
70
area=40m2
60
area=50m2
50
area=60m2
40
area=63m2
30 20 10 0
Oct
Nov
Dec
Jan
Feb
Mar
Month Fig. 12. Solar fraction variation with collector areas during winter. (Constant solar irradiance and variable air temperature.) 6 area=10m2 area=20m2 area=30m2
5.5
area=40m2
System cost (*10^3) ($)
area=50m2 5
area=60m2 area=63m2
4.5
4
3.5
3
April
May
June
July
Aug
Sept
Months Fig. 13. System cost variation with different collector areas in summer.
fraction in summer is higher than that in winter due to the high value of solar radiation in summer for a 63 m2 collector is able to supply 90–99% of the hot water demand from May to September. Figs. 11 and 12 present the variation of air temperature effect on the solar fraction in summer and winter, respectively. January has the minimum value of solar fraction due to the low values of air temperature and solar radiation over the winter season. The value of solar fraction in winter is in range of 44.9–92.8%, and in summer it is 68.1–99%. The sensitivity analysis are carried out by varying the input parameters (solar radiation, air temperature, and interest rate). Figs. 13 and 14 present the effect of solar radiation variation over the year on the cost function. In winter season, the auxiliary heater is used to provide more energy to heat the water to the desired temperature, so the variation of cost is increased depending on the value of the auxiliary heater energy. Table 4 shows the effect of interest rate variation on the system cost: as the interest rate increases, the system cost are increased. In Egypt, the value of interest rate is equal to (=11.6%.) the system cost variation at different collector areas in summer and winter respectively. 10. Conclusion Optimum sizing of system components is important in design of a large solar thermal water heating system. In this paper a design of a solar thermal water heating system for supplying an aquaculture system with the required hot water demand was presented. A methodology of sizing the solar thermal water heating system using genetic algorithm was proposed. Genetic
D.M. Atia et al. / Mathematical and Computer Modelling 55 (2012) 1436–1449
1449
6.5
area=10m2 6
area=20m2 area=30m2
5.5
area=40m2
System cost*10^3 ($)
area=50m2 5
area=60m2 area=63m2
4.5
4
3.5
3
Oct
Nov
Dec
Jan
Feb
Mar
Months Fig. 14. System cost variation with different collector areas in winter. Table 4 Cost variation with interest rate. Interest rate (%)
Cost ($)
5 8 10 12 15 20
2601.2 3382.1 3933.6 4352.2 5328.4 6686.8
algorithm has been suggested in order to determine the optimal sizing of the solar thermal system according to minimize the objective function considering the different constraints and to give the optimal area of the flat plate collector. The collector area is equal to 63 m2 , at this value the solar fraction reached to 98%, which is very high value. The solar radiation has an obvious effect on the solar fraction and system cost, specially when the collector area increases. The sensitivity analysis was carried out by varying the input parameters (solar radiation, air temperature, and interest rate). References [1] Govind N. Kulkarni, Shireesh B. Kedare, Santanu Bandyopadhyay, Determination of design space and optimization of solar water heating systems, Solar Energy 81 (2007) 958–968. [2] Soteris Kalogirou, Thermal performance, economic and environmental life cycle analysis of thermosiphon solar water heaters, Solar Energy 83 (2009) 39–48. [3] Govind N. Kulkarni, Shireesh B. Kedare, Santanu Bandyopadhyay, Design of solar thermal systems utilizing pressurized hot water storage for industrial applications, Solar Energy 82 (2008) 686–699. [4] V. Badescu, Optimum size and structure for solar energy collection systems, Energy 31 (2006) 1819–1835. [5] V. Badescu, Optimum fin geometry in flat plate solar collector systems, Energy Conversion and Management 47 (2006) 2397–2413. [6] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975. [7] Singiresu S. Rao, Engineering Optimization, Theory and Practice, fourth ed., John Wiley & Sons, Inc., Hoboken, New Jersey, Canada, 2009. [8] Molly McShea, Jo-Ellen Sullivan, Design of an aquaculture database for INCOPESCA July 5, 2007, Report, July 5, 2007. [9] Jim Szyper, Backyard Aquaculture in Hawaii a Practical Manual, University of Hawaii Windward Community College and Hawaii Institute of Marine Biology, 1980. [10] Joshua Reed Plaisted, Life cycle performance evaluation of an air-based solar thermal system, M.Sc. Wisconsin Madison University, 2000. [11] M. Bojic, S. Kalogirou, K. Petronijevic, Simulation of a solar domestic water heating system using a time marching model, Renewable Energy 27 (2002) 441–452. [12] Jae-Mo Koo, Development of a flat-plate solar collector design program, M.Sc. Wisconsin Madison University, 1999. [13] John Gelegenis, Paschalis Dalabakis, Andreas Ilias, Heating of a fish wintering pond using low-temperature geothermal fluids, Porto Lagos, Greece, Geothermics 35 (2006) 87–103. [14] J. Duffie, W. Beckman, Solar Engineering of Thermal Processes, second ed., John Wiley & Sons Interscience, New York, 1991. [15] Egyptian solar radiation atlas, Cairo, Egypt, 1998. [16] Arab Solar Energy and Technology Company. Available at: www.asetegypt.com. [17] D.E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Addison Wesley, 1990. [18] M. Loomans, H. Visser, Application of the genetic algorithm for the optimization of large solar hot water systems, Solar Energy 72 (2002) 427–439. [19] Jasbir Arora, Introduction to Optimum Design, second ed., Academic Press, 2004. [20] J. Parrilla, W. Fichtner, The impact of transportation costs on the profitability of heat and power generation with wood products, in: Proc. of International Conference on Renewable Energies and Power Quality, ICREPQ’09, Valencia, Spain, April 2009.