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This paper presents an optimization method to design a solar water heating (SWH) system based on life cycle cost (LCC). A genetic algorithm is employed to optimize its configuration and sizing as the optimization technique. To ensure that the optimal solution obtained from the proposed method is a practical design, three constraint conditions, including the energy balance, solar fraction, and available space to install solar collectors, have been set. In addition, the real devices available in the marketplace are considered in the optimization process that searches for optimal configuration and sizing, which is represented by the type and number of each component. By using the proposed method, a SWH system in an office building, South Korea has been designed and optimized. It is observed that a low solar fraction does not always present a decrease in the LCC. A trade-off between the equipment cost and the energy cost results in an optimal design of the SWH system that yields the minimum LCC.

Global energy consumption has increased steadily over the last few decades and has recently been marked by especially dramatic growth rates in many developing countries, such as China, India, and Brazil. Especially, heating accounts for 40% to 50% of the world’s energy demand and most of the energy supply for heating currently comes from fossil fuels [

The proper design of SWH systems is important to assure good performance and maximize the economic benefits of these systems. There are many studies in the literature that address the design method of these systems. These design methods can be broadly classified into two categories, namely, correlation-based methods and simulation-based methods [

To overcome these issues in the previously stated correlation and simulation based methods, linear and nonlinear optimization techniques and evolutionary search algorithms have also been applied for the design of SWH systems. Matrawy

As previously stated, studies on the optimal design for SWH systems have been increasing and are helpful in identifying the sizing of the SWH systems. However, the number of optimization methods is rather small compared to a wide spread range of correlation and simulation based design methods developed in the last few decades. Furthermore, most of the optimization methods have designed and determined a proper SWH system by examining the appropriate sizing of each component or value of the operation parameters through parametric studies based on objective functions, such as annual efficiency, solar fraction, life cycle savings, LCC, and payback period. In addition, the majority of the studies optimized SWH systems with a given configuration consisting of one type of device for each component. System performance and economic benefits, however, vary considerably depending on even one of these design variables and the relation among them. There are a number of system devices in the marketplace. Each device represents different technical characteristics, which may lead to variation in the energetic and economic performance of SWH systems. Thus, the optimal designs can be different compared to the original designs.

Therefore, this paper presents an optimization method to design a SWH system based on LCC by considering the real devices available in the marketplace. GA is employed to optimize its configuration and sizing as the optimization technique. In addition, this study has been set three constraint conditions, including energy balance, solar fraction, and available space to install solar collectors to ensure that the optimal solution obtained from the proposed method is a practical design. This paper is organized as follows:

A schematic diagram of an indirect forced SWH system with a flat plate solar collector array, a heat exchanger, a storage tank, and an auxiliary heater is shown in _{w}_{p,w}^{3}) and the specific heat of water (J/kg·°C); _{s}^{3}); _{TS}_{l}_{d}_{Ls}

Schematic diagram of the SWH system considered in this study.

The solar energy supplied to the tank (_{TS}_{u}_{c}^{2}); _{c,s}_{R}_{R}U_{L}_{T}^{2}); _{ho}_{a}

For identical collector modules in series, the intercept and slope of the efficiency curve can be estimated as [_{R}_{1}(τα)_{1} and _{R}_{1}_{L}_{1} are the intercept and the slope of the efficiency curve of a single collector; _{c}_{p,c}

To calculate _{Ts}_{r}_{hex} is product of the overall heat transfer coefficient and area of a heat exchanger (W/°C); _{hex,min}_{hex,max}_{hex,h}_{hex,c}

The capacity rates of the fluid on the hot and cold sides of the heat exchanger are given as follows:
_{c,p}_{hi}_{ho}_{ci}_{co}

To satisfy the desired hot water temperature and flow rate, the storage tank discharge flow rate is mixed with make-up water. By considering the mass and energy balance at the mixing junction the flow rate drawn from the tank is determined as:
_{s}_{l}_{l}_{m}

Therefore, the solar energy supplied from the storage tank to the load (_{LS}_{l}_{s}_{s}^{2}·°C) and the surface area (m^{2}) of a storage tank and _{amb}

In this paper, an optimization method is developed to design a SWH system for low temperature applications (below 100 °C) such as a residential hot water system. So if the storage tank temperature is greater than the maximum allowable temperature (_{s,max}

The SWH system parameters from Equation (2) to Equation (16) are evaluated on the basis of the initial storage tank temperature at any time step. The final storage tank temperature at the end of any time step must be known because it will be the initial temperature for the next time step. The final storage tank temperature can be estimated as:
_{s,f}

In this optimization method, the SWH system is operated to meet the hot water demand using differential temperature control on an hourly basis. Hourly demands and weather conditions are required as input data. In the proposed method, the number of heat exchangers is fixed as one because this is the common configuration of forced circulation SWH systems in South Korea. Furthermore, a counter-flow type heat exchanger with _{hex}

The optimization method in this paper is developed to determine the optimal configuration and sizing for a SWH system composed of solar collectors, a storage tank, and auxiliary heaters. Here, the configuration means the combination of the selected types for each component and the sizing is computed using its unit capacity and quantity. The capacity units of a solar collector, a storage tank and an auxiliary heater are the area of a collector module (m^{2}), tank volume (m^{3}), and rated heating rate (kW), respectively. This study fixes the number of storage tanks at one because a single tank is generally used in SWH systems for low temperature applications. Therefore, a SWH system is expressed as a decision vector composed of five integer variables that represent the type and number of each component as shown below:
_{coll}_{coll}_{tank}_{aux}_{aux}

This design method obtains the optimal configuration and sizing of a SWH system by minimizing the LCC of the system, which includes all of the costs throughout the lifetime of the system. It can be formulated as follows [_{I}_{M}_{R}_{E}_{S}

The initial cost is related to the direct purchase cost of the main components and the supplementary cost, as follows:
_{coll,j}_{tank,j}_{aux,j}_{aux}_{I}

The maintenance cost is calculated as a percentage of the initial cost of a SWH system, described as follows:
_{M}_{p}

The replacement costs are incurred depending on each component’s lifetime during the planning period and are described as follows:
_{R,c}_{I,c}_{l,c}_{r,c}

The energy cost is computed by applying the electricity and liquid natural gas (LNG) escalation rate and is described as follows:
_{ELE}_{LNG}^{3}] for a SWH system; _{ELE}_{LNG}^{3}]; and _{fuel}

It is considered that part of the initial cost is backed by the government depending on the related regulations regarding the installation of renewable energy systems. According to the total gross area of the collector modules, the subsidy cost is calculated as follows:
_{coll,j}^{2}); _{R,max}^{2}); and _{S}

The constraints restrict each decision variable to take a value within the minimum and the maximum limits. In this paper, the decision variables present the type and number of main components for a SWH system. Most previous studies that optimized a particular SWH system consisting of only one model selected beforehand by researchers had to constraint the limits of each decision variable. However, when optimizing the types of component that has different capacities, it is difficult to set the maximum limits of decision variables that indicate the number of components to the specific values because the limits vary according to the device types. Therefore, this study has set the following inequality constraints, namely the energy balance, the solar fraction, and the available space to install the collectors that are used in the practical design problems of a SWH system. The limits of _{aux}_{coll}_{coll}_{tank}_{aux}

Energy balance:

Solar fraction (penetration of the solar energy):

Available space to install the collector array:

_{L,peak}

_{aux,j}

_{L,year}

_{aux,tot}

_{S,min}

_{S,max}

_{S}

_{coll,j}

_{coll,j}

_{coll}

_{s,w}

_{c,max}

^{2}); and

_{c,ins}

^{2}).

This study uses the real coded GA [

Meanwhile, as described in

The proposed optimization method was applied for the design and optimal configuration sizing of a SWH system for an office building in Incheon at Latitude 36° N and Longitude 125° E, South Korea.

In the present study, three different hot water consumptions of weekday, Saturday and Sunday are distributed during a day according to the hot water load profile of the typical office building [^{3}/day at 60 °C, respectively. The meteorological conditions during the year are illustrated in ^{2}, and the average hourly air temperature is 12.18 °C, respectively.

Hourly hot water consumptions over one day in a case study building.

(

The SWH system of the case study is comprised of three main components, namely, five types of solar collectors, ten types of storage tanks, and eight types of auxiliary heaters. The technical and economical characteristics of the solar collectors, storage tanks, and auxiliary heaters used in the optimization design are shown in

All devices shown in

Technical and economic parameters of the solar collectors for the case study.

Parameters | Types | ||||
---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | |

Useful gain (kWh/m^{2} day) |
2.228 | 2.361 | 2.417 | 2.444 | 2.556 |

Intercept of the collector efficiency (–) | 0.7200 | 0.7208 | 0.7445 | 0.7043 | 0.7203 |

Negative of the slope of the collector efficiency (W/m^{2}·°C) |
4.09 | 4.7999 | 4.8483 | 4.5368 | 3.9488 |

Flow rate of the fluid at standard condition (kg/s) | 0.0400 | 0.0373 | 0.0381 | 0.0368 | 0.0533 |

Overall height (m) | 2.00 | 2.00 | 2.00 | 2.00 | 2.40 |

Overall width (m) | 1.00 | 1.00 | 1.00 | 0.99 | 1.18 |

Lifetime (years) | 20 | 20 | 20 | 20 | 20 |

Purchase cost (1,000 KRW/ea.) | 520 | 530 | 545 | 540 | 820 |

Technical and economic parameters of the storage tanks for the case study.

Parameters | Types | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

Tank volume (m^{3}) |
0.44 | 0.96 | 1.72 | 2.65 | 3.76 | 4.91 | 5.54 | 6.21 | 6.92 | 9.58 |

Heat loss coefficient (W/°C) | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |

Overall height (m) | 1.22 | 1.22 | 1.52 | 2.00 | 2.44 | 2.44 | 2.44 | 2.44 | 3.05 | 3.05 |

Overall diameter (m) | 0.68 | 1.00 | 1.20 | 1.30 | 1.40 | 1.60 | 1.70 | 1.80 | 1.70 | 2.00 |

Lifetime (years) | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 |

Purchase cost (1,000,000 KRW/ea.) | 6.60 | 7.15 | 9.49 | 10.73 | 12.65 | 15.88 | 17.33 | 18.02 | 18.98 | 24.20 |

Technical and economic parameters of the auxiliary heaters for the case study.

Parameters | Types | |||||||
---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |

Rated heating capacity (kW) | 15.12 | 18.61 | 23.26 | 29.08 | 34.89 | 58.15 | 81.41 | 116.30 |

Rated efficiency (%) | 83 | 84 | 85 | 86 | 86 | 82 | 83 | 83 |

Lifetime (years) | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 |

Purchase cost (1000 KRW/ea.) | 807 | 844 | 909 | 964 | 1,039 | 2,291 | 2,565 | 3,207 |

Design parameters and assumptions required for the optimization process are summarized in

Optimization design parameters considered in the case study.

Parameters | Value |
---|---|

Slope of collector array (°) | 35 |

Azimuth of collector array (°) | 0 |

Meridian altitude in winter season (°) | 29 |

Desired hot water temperature (°C) | 60 |

Maximum allowable storage tank temperature (°C) | 100 |

Temperature of the environment surrounding the storage tank (°C) | 20 |

Specific heat of collector fluid (J/kg·°C) | 3560 |

Specific heat of water (J/kg·°C) | 4180 |

Density of collector fluid (kg/m^{3}) |
1043 |

Density of water (kg/m^{3}) |
1000 |

Product of the overall heat transfer coefficient and area of a heat exchanger (W/°C) | 3000 |

Maximum number of collectors in series (ea.) | 6 |

Project lifetime (years) | 40 |

Real discount rate (%) | 2.91 |

Nominal interest rate (%) | 6.00 |

Inflation rate (%) | 3.00 |

Electricity cost escalation rate (%) | 4.00 |

Gas cost escalation rate (%) | 4.00 |

Maximum capacity available to receive the subsidy cost (m^{2}) |
500 |

Area available to install solar collectors (m^{2}) |
600 |

Supplementary cost ratio against the purchase cost (%) | 30 |

Maintenance cost ratio against the initial cost (%) | 1.5 |

Subsidy cost ratio against the initial cost (%) | 50 |

Electricity and liquid natural gas tariffs.

Classification | Value | ||
---|---|---|---|

Electricity | Basic charge | 6160 | |

Energy charge (KRW/kWh) | Summer (June, July and August) | 105.7 | |

Spring/Fall (March, April, May, September, and October) | 65.2 | ||

Winter (November, December, January and February) | 92.3 | ||

Natural gas | Energy charge (KRW/MJ) | Summer (May, June, July, August and September) | 19.26 |

Spring/Fall (April, October and November) | 19.28 | ||

Winter (December, January, February and March) | 19.46 |

A case study was conducted to find the optimal SWH system that represents the minimum LCC without restriction for the solar fraction using the proposed optimization method. The minimum and maximum solar fractions of the base case were set as 0% and 100%, respectively.

Evolution of (

Meanwhile, the optimization algorithm converges toward a solar fraction of 60.42% in the base case. As indicated in

The variation in component size for the best and worst solutions during 60 generations can be observed in ^{2}, a storage tank of 3.76 m^{3}, and an auxiliary heater of 34.89 kW.

Meanwhile, using an Intel(R) Core(TM) i5–2310 @2.90GHz_CPU and 4 GB memory computer, the developed method requires approximately three minutes to optimize the SWH system of the case study. This result indicates that the proposed design method can obtain an optimal SWH system within a short computation time.

Variation of component size for the best and worst solutions in each generation for the base case.

Characteristics of the optimal SWH system for the base case.

Variable | Description | Value |
---|---|---|

Type of the solar collector (–) | 4 | |

Number of the solar collectors (ea.) | 37 | |

Type of the storage tank (–) | 4 | |

Type of the auxiliary heater (–) | 4 | |

Number of the auxiliary heaters (ea.) | 1 | |

Total area of the solar collector (m^{2}) |
104.71 | |

Volume of the storage tank (m^{3}) |
3.76 | |

Capacity of the auxiliary heaters (kW) | 34.89 | |

Installation area of the solar collectors (m^{2}) |
194.3 | |

Peak hot water load (kW) | 27.35 | |

Annual hot water load (kWh/year) | 60,218 | |

Annual solar irradiance on the collector array (kWh/year) | 137,495 | |

Annual useful heat gain of the collector array (kWh/year) | 39,986 | |

Annual solar energy supplied to the storage tank (kWh/year) | 37,332 | |

Annual heat loss of the storage tank (kWh/year) | 752 | |

Annual discharged heat from the storage tank (kWh/year) | 4 | |

Annual solar energy supplied by the storage tank (kWh/year) | 36,386 | |

Annual auxiliary energy supplied by the heaters (kWh/year) | 23,832 | |

Annual LNG consumption (m^{3}/year) |
2562 | |

Annual electricity consumption (kWh/year) | 1413 | |

Annual solar fraction (%) | 60.42 | |

Initial cost (1000 KRW) | 57,238 | |

Maintenance cost (1000 KRW) | 20,129 | |

Replacement cost (1000 KRW) | 41,302 | |

Energy cost (1000 KRW) | 124,616 | |

Subsidy cost (1000 KRW) | 28,619 | |

Life cycle cost (1000 KRW) | 214,666 |

Given that the minimum solar fraction is set 0% and the maximum solar fraction is increased to 100% in 5% increments, variation in the LCC and solar fraction of the optimal SWH systems at each maximum solar fraction is shown in

From

Variation of costs and solar fractions at the different maximum solar fractions.

Characteristics of the optimal SWH systems for the different maximum solar fractions.

Parameter | Maximum Solar Fraction | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

5% | 10% | 15% | 20% | 25% | 30% | 35% | 40% | 45% | 50% | 55% | 60% | 65% | |

3 | 3 | 0 | 3 | 3 | 3 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |

2 | 5 | 8 | 7 | 13 | 19 | 25 | 19 | 19 | 25 | 31 | 37 | 37 | |

0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | |

4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

^{2}) |
3.96 | 9.90 | 16.00 | 13.86 | 25.74 | 37.62 | 50 | 53.77 | 53.77 | 70.75 | 87.73 | 104.71 | 104.71 |

^{3}) |
0.44 | 0.44 | 0.44 | 0.96 | 0.44 | 0.44 | 0.96 | 0.96 | 2.65 | 2.65 | 2.65 | 2.65 | 3.76 |

34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | |

^{2}) |
7.4 | 18.4 | 29.7 | 25.7 | 47.8 | 69.8 | 92.7 | 99.8 | 99.8 | 131.3 | 162.8 | 194.3 | 194.3 |

27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | |

60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | |

2462 | 5994 | 8740 | 11,159 | 14,698 | 17,754 | 21,185 | 23,695 | 26,625 | 30,374 | 33,440 | 35,901 | 37,332 | |

−57 | −19 | 7 | 49 | 68 | 98 | 165 | 200 | 375 | 455 | 524 | 583 | 752 | |

0 | 0 | 0 | 0 | 0 | 3 | 1 | 7 | 0 | 0 | 5 | 21 | 4 | |

2523 | 6013 | 8731 | 11,103 | 14,624 | 17,622 | 20,986 | 23,385 | 26,195 | 29,848 | 32,744 | 34,966 | 36,386 | |

57,695 | 54,205 | 51,487 | 49,115 | 45,594 | 42,596 | 39,232 | 36,833 | 34,023 | 30,370 | 27,474 | 25,252 | 23,832 | |

^{3}/year) |
6083 | 5715 | 5428 | 5186 | 4810 | 4497 | 4154 | 3904 | 3623 | 3244 | 2940 | 2704 | 2562 |

142 | 186 | 335 | 354 | 516 | 632 | 799 | 909 | 892 | 1076 | 1260 | 1425 | 1413 | |

4.19 | 9.99 | 14.50 | 18.44 | 24.28 | 29.26 | 34.85 | 38.83 | 43.50 | 49.56 | 54.37 | 58.06 | 60.42 | |

11,335 | 13,441 | 15,339 | 15,560 | 19,057 | 23,269 | 28,359 | 30,900 | 35,548 | 41,944 | 48,340 | 54,736 | 57,238 | |

3986 | 4727 | 5394 | 5472 | 6702 | 8183 | 9973 | 10,867 | 12,501 | 14,750 | 17,000 | 19,249 | 20,129 | |

11,444 | 12,630 | 13,699 | 14,188 | 15,793 | 18,165 | 21,395 | 22,826 | 27,812 | 31,414 | 35,016 | 38,618 | 41,302 | |

280,705 | 264,059 | 251,502 | 240,506 | 223,934 | 210,091 | 195,048 | 184,011 | 171,145 | 154,502 | 141,301 | 131,163 | 124,616 | |

5668 | 6721 | 7670 | 7780 | 9529 | 11,635 | 14,179 | 15,450 | 17,774 | 20,972 | 24,170 | 27,368 | 28,619 | |

301,802 | 288,136 | 278,264 | 267,946 | 255,957 | 248,073 | 240,596 | 233,154 | 229,232 | 221,638 | 217,487 | 216,398 | 214,666 |

Variation of component sizes at the different maximum solar fractions.

Variation of costs and solar fraction at the different minimum solar fractions.

Variation of component sizes at the different minimum solar fractions.

It may be observed from

Characteristics of the optimal SWH systems for the different minimum solar fractions.

Parameter | Minimum Soar Fraction | ||||||
---|---|---|---|---|---|---|---|

60 | 65 | 70 | 75 | 80 | 85 | 90 | |

4 | 4 | 4 | 4 | 4 | 4 | 4 | |

37 | 49 | 61 | 67 | 79 | 145 | 223 | |

4 | 4 | 4 | 7 | 9 | 8 | 9 | |

4 | 4 | 4 | 4 | 4 | 4 | 4 | |

1 | 1 | 1 | 1 | 1 | 1 | 1 | |

^{2}) |
104.71 | 138.67 | 172.63 | 189.61 | 223.57 | 410.35 | 631.09 |

^{3}) |
3.76 | 3.76 | 3.76 | 6.21 | 9.58 | 6.92 | 9.58 |

34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | |

^{2}) |
194.3 | 257.3 | 320.3 | 351.8 | 414.8 | 761.3 | 1170.9 |

27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | |

60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | |

37,332 | 41,410 | 44,318 | 47,618 | 51,206 | 58,867 | 65,290 | |

752 | 886 | 994 | 1138 | 1684 | 1922 | 2221 | |

4 | 36 | 78 | 49 | 43 | 402 | 607 | |

36,386 | 39,949 | 42,303 | 45,517 | 48,367 | 51,372 | 54,256 | |

23,832 | 20,269 | 17,915 | 14,701 | 11,851 | 8846 | 5962 | |

^{3}/year) |
2562 | 2181 | 1928 | 1595 | 1291 | 966 | 660 |

1413 | 1712 | 1986 | 2064 | 2279 | 3630 | 5053 | |

60.42 | 66.34 | 70.25 | 75.59 | 80.32 | 85.31 | 90.10 | |

57,238 | 70,030 | 82,822 | 96,197 | 117,025 | 180,589 | 270,529 | |

20,129 | 24,628 | 29,126 | 33,830 | 41,155 | 63,508 | 95,137 | |

41,302 | 48,506 | 55,710 | 66,797 | 82,622 | 114,957 | 169,068 | |

124,616 | 108,296 | 97,765 | 82,838 | 69,716 | 59,867 | 51,207 | |

28,619 | 35,015 | 41,411 | 48,098 | 58,513 | 90,294 | 110,214 | |

214,666 | 216,445 | 224,012 | 231,564 | 252,005 | 328,627 | 475,727 |

Meanwhile, it may be noted from

The optimization design for a SWH system is a complicated process that uses mathematical models with many meteorological, technical, and economic variables. Thus, it has been difficult for traditional design techniques in the past to obtain satisfactory results within a reasonable computation time. In this paper, a GA has been employed to optimize the configuration and sizing of the SWH system on the basis of LCC. Through a numerical example of the SWH system for an office building in Incheon, South Korea, the effectiveness of the proposed method has been demonstrated. It was found that the LCC of the SWH system decreases first. Then, its decreasing speed becomes slow gradually and reaches the minimum cost. Finally, it increases sharply with the increase in capacity and solar fraction. This indicates that the global optimum SWH system was derived from the optimum solar fraction to maximize the economic benefits under given design conditions. Therefore, it could be helpful to determine the optimal configuration and sizing of the SWH system by comparing the feasible designs obtained by using the proposed method instead of simply adjusting the solar fraction depending only on the designer’s experience and intuition. Future work includes a further improvement of the proposed method to reflect the parameters, such as the slope and azimuth of collectors, flow rates on the hot and cold side of the heat exchanger, and operation conditions that affect the energetic and economic performance of the SWH system.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2014R1A6A3A01059739).

The authors declare no conflict of interest.

gross area of a single collector module, m^{2}

gross area of the ^{2}

installation area of solar collectors, m^{2}

total area of solar collectors, m^{2}

available space to install collector array, m^{2}

maximum capacity available to receive the subsidy cost, m^{2}

surface area of a storage tank, m^{2}

capacity rate of fluid on cold side of a heat exchanger, W/°C

capacity rate of fluid on hot side of a heat exchanger, W/°C

maximum capacity rate, W/°C

minimum capacity rate, W/°C

specific heat of water, J/kg·°C

specific heat of collector fluid, J/kg·°C

purchase price of the

purchase price of the

purchase price of the

energy cost, KRW

initial cost, KRW

initial cost of each component, KRW

maintenance cost, KRW

replacement cost, KRW

replacement cost of each component, KRW

subsidy cost, KRW

life cycle cost, KRW

hourly electricity cost, KRW/kWh

hourly liquid natural gas (LNG) cost, KRW/m^{3}

capacity rate ratio of a heat exchanger

fuel price escalation rate, %

hourly electricity consumption, kWh

hourly LNG consumption, m^{3}

collector heat removal factor of identical collectors in series

collector heat removal factor of a collector

solar fraction of any solar water heating system, %

minimum solar fraction, %

maximum solar fraction, %

height of the

hourly total solar radiation on the tilted collector array, W/m^{2}

real discount rate, %

mass flow rate of the collector fluid, kg/s

mass flow rate of the discharged water from a storage tank, kg/s

mass flow rate from the storage tank to the load, kg/s

mass flow rate of the desired hot water load, kg/s

number of identical collectors in series

number of the

number of the

number of exchanger heat transfer units

planning period, year

lifetime of each component, year

replacement times of each component

heating capacity of the

total heating capacity of the auxiliary heaters, kW

annual auxiliary heating energy, kWh

peak hot water load, kW

annual hot water load, kWh

auxiliary heating energy, W

discharged heat to avoid overheating of a storage tank, W

heat loss of a storage tank, W

solar energy extracted from the storage tank to the load, W

solar energy supplied to a storage tank, W

solar useful heat gain of identical collectors in series, W

a percentage of the supplementary cost against the direct purchase cost, %

a percentage of the annual maintenance cost against the initial cost, %

a percentage of subsidy cost against the initial cost, %

outdoor dry-bulb temperature, °C

ambient temperature, °C

cold stream outlet temperature of a heat exchanger, °C

cold stream inlet temperature of a heat exchanger, °C

hot stream inlet temperature of a heat exchanger, °C

hot stream outlet temperature of a heat exchanger, °C

desired hot water temperature, °C

make-up water temperature, °C

storage tank temperature at the beginning of the time step, °C

storage tank temperature at the end of the time step, °C

maximum allowable storage tank temperature, °C

type of auxiliary heater

type of solar collector

type of storage tank

collector overall heat loss coefficient of identical collectors in series, W/m^{2}·°C

collector overall heat loss coefficient of a collector, W/m^{2}·°C

heat loss coefficient of a storage tank, W/m^{2}·°C

product of the overall heat transfer coefficient and area of a heat exchanger, W/°C

uniform present value factor adjusted to reflect the electricity price escalation rate

uniform present value factor adjusted to reflect the LNG price escalation rate

uniform present value factor adjusted to reflect the fuel price escalation rate

storage tank volume, m^{3}

width of the

meridian altitude in winter, °

slope of the collector array, °

effectiveness of a heat exchanger

density of water, kg/m^{3}

product of the transmittance and the absorptance of identical collectors in series

product of the transmittance and the absorptance of a collector