# Analysis and Optimization Design of a Solar Water Heating System Based on Life Cycle Cost Using a Genetic Algorithm

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of SWH System

_{w}, C

_{p,w}are the density (kg/m

^{3}) and the specific heat of water (J/kg·°C); V

_{s}is the volume of a storage tank (m

^{3}); q

_{TS}, q

_{l}, q

_{d}, and q

_{Ls}are the solar energy supplied to a storage tank (W), the heat loss of a storage tank (W), the discharged heat to avoid overheating of a storage tank (W), and the solar energy extracted from the storage tank (W), respectively.

_{TS}) is the energy transferred from the useful heat gain of the collector array (q

_{u}) through a heat exchanger according to the differential temperature control. The solar useful gain of identical collector modules in series is [23]:

_{c}is the gross area of a single collector module (m

^{2}); N

_{c,s}is the number of identical collectors in series; F

_{R}(τα) and F

_{R}U

_{L}are the intercept and the slope of the efficiency curve of identical collector modules in series; I

_{T}is the solar irradiance on the tilted surface (W/m

^{2}); T

_{ho}is the hot stream outlet temperature of the heat exchanger (°C); T

_{a}is the outdoor dry-bulb temperature (°C); and the + sign indicates that the collector fluid circulates between the collector array and the hot side of an external heat exchanger only when solar useful heat gain becomes positive, respectively.

_{R}

_{1}(τα)

_{1}and F

_{R}

_{1}U

_{L}

_{1}are the intercept and the slope of the efficiency curve of a single collector; m

_{c}is the mass flow rate of the collector fluid (kg/s); and C

_{p,c}is the specific heat of the collector fluid (J/kg·°C).

_{Ts}, hot and cold stream outlet temperatures for the plate heat exchanger must be known. Both outlet temperatures can be determined by the effectiveness-number of heat transfer units (NTU-ε) analysis. The NTU-ε method uses three dimensionless parameters, such as the heat exchanger effectiveness (ε), number of exchanger heat transfer units (NTU), and capacity rate ratio (c

_{r}). For a given counter-flow heat exchanger, the three parameters can generally be expressed as [23]:

_{hex}is product of the overall heat transfer coefficient and area of a heat exchanger (W/°C); C

_{hex,min}and C

_{hex,max}are the smaller and larger values between the hot fluid capacity rate (C

_{hex,h}) and the cold fluid capacity rate (C

_{hex,c}).

_{c,p}is the number of parallel connections in the collector array. Therefore, the heat transfer rate (i.e., solar energy supplied to the tank), hot stream outlet temperature, and cold stream outlet temperature can be determined as:

_{hi}and T

_{ho}are the hot stream inlet and outlet temperatures of the heat exchanger (°C) and T

_{ci}and T

_{co}are the cold stream inlet and outlet temperatures of the heat exchanger (°C).

_{s}is the mass flow rate from the storage tank to the load (kg/s); m

_{l}is the mass flow rate of the desired hot water load (kg/s); T

_{l}is the desired hot water temperature (°C); and T

_{m}is the make-up water temperature (°C).

_{LS}) can be estimated as:

_{l}) to the ambient air can be expressed as:

_{s}and A

_{s}are the heat loss coefficient (W/m

^{2}·°C) and the surface area (m

^{2}) of a storage tank and T

_{amb}is the ambient temperature (°C).

_{s,max}), the surplus heat will be discharged to avoid overheating of the storage tank. The discharged flow rate and heat can be calculated as:

_{s,f}is the final storage tank temperature at the end of any time step.

_{hex}of 3000 W/°C is used. For the thermal performance of an auxiliary heater, a simple boiler is modeled with its overall efficiency and part load ratio from the device capacity and the energy required to meet the load.

## 3. Optimization Method of SWH System

#### 3.1. Decision Variable

^{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}is the type of solar collectors; N

_{coll}is the number of solar collectors; T

_{tank}is the type of storage tanks; T

_{aux}is the type of auxiliary heaters; and N

_{aux}is the number of auxiliary heaters.

#### 3.2. Objective Function

_{I}, C

_{M}, C

_{R}, C

_{E}, and C

_{S}represent the initial, maintenance, replacement, energy, and subsidy costs, respectively.

_{coll,j}, C

_{tank,j}, and C

_{aux,j}are the purchase price of the $j$th device of solar collectors, storage tanks, and auxiliary heaters; ${N}_{coll}$ and N

_{aux}are the installation number of the jth device of solar collectors and auxiliary heaters; and R

_{I}is a percentage of the supplementary cost against the direct purchase cost.

_{M}is a percentage of the annual maintenance cost against the initial cost; n

_{p}is the planning period; and i is the real discount rate.

_{R,c}and C

_{I,c}are the replacement costs and the initial costs of each component; n

_{l,c}is the lifetime of each component; and n

_{r,c}is the replacement times of each component.

_{ELE}and c

_{LNG}are the hourly electricity cost [KRW/kWh] and hourly LNG cost [KRW/m

^{3}] for a SWH system; F

_{ELE}and F

_{LNG}are the hourly electricity consumption [kWh] and hourly LNG consumption [m

^{3}]; and e

_{fuel}is the fuel price escalation rate.

_{coll,j}is the gross area of the jth device of solar collectors (m

^{2}); A

_{R,max}is the maximum capacity available to receive the subsidy cost (m

^{2}); and R

_{S}is a percentage of the subsidy cost against the initial cost (%).

#### 3.3. Constraint Conditions

_{aux}are determined by Equation (26) and the limits of N

_{coll}are restricted by Equations (27) and (28). Meanwhile, the limits of the decision variables (T

_{coll}, T

_{tank}, and T

_{aux}) that denote the types of components are set automatically at the number of types in the inputted data tables of each component.

- (a)
- Energy balance:$${Q}_{L,peak}\le {Q}_{aux,tot}$$
- (b)
- Solar fraction (penetration of the solar energy):$${F}_{S,min}\le {F}_{S}\le {F}_{S,max}$$
- (c)
- Available space to install the collector array:$${A}_{c,\text{}ins}\le {A}_{c,max}$$

_{L,peak}is the peak load (kW); Q

_{aux,j}is the heating capacity of the $j$th device of auxiliary heaters (kW); Q

_{L,year}is the annual hot water load (kWh); Q

_{aux,tot}is the total heating capacity of the auxiliary heaters (kW); F

_{S,min}and F

_{S,max}are the minimum and the maximum solar fractions (%); F

_{S}is the solar fraction of any SWH system; W

_{coll,j}and H

_{coll,j}are the width and the height of the j th module of solar collectors (m); β

_{coll}is the slope of the collector array (°); α

_{s,w}is the meridian altitude in winter (°); A

_{c,max}is the available space to install solar collectors (m

^{2}); and A

_{c,ins}is the installation area of the solar collectors (m

^{2}).

#### 3.4. Optimization Algorithm

## 4. Simulation Results and Discussion

#### 4.1. Simulation Parameters or Data

^{3}/day at 60 °C, respectively. The meteorological conditions during the year are illustrated in Figure 3. The average daily solar irradiance is 3.38 kW/m

^{2}, and the average hourly air temperature is 12.18 °C, respectively.

**Figure 3.**(

**a**) Hourly global horizontal solar irradiance, and (

**b**) outdoor air temperature and make-up water temperature over one year for Incheon, South Korea.

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 |

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 |

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 |

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 |

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 |

#### 4.2. Optimization Results of the Base Case

**Figure 4.**Evolution of (

**a**) the objective function and (

**b**) the solar fraction during the optimization process for the base case.

^{2}, a storage tank of 3.76 m

^{3}, and an auxiliary heater of 34.89 kW.

**Figure 5.**Variation of component size for the best and worst solutions in each generation for the base case.

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

${T}_{coll}$ | Type of the solar collector (–) | 4 |

${N}_{coll}$ | Number of the solar collectors (ea.) | 37 |

${T}_{tank}$ | Type of the storage tank (–) | 4 |

${T}_{aux}$ | Type of the auxiliary heater (–) | 4 |

${N}_{aux}$ | Number of the auxiliary heaters (ea.) | 1 |

${A}_{c,tot}$ | Total area of the solar collector (m^{2}) | 104.71 |

${V}_{s}$ | Volume of the storage tank (m^{3}) | 3.76 |

${Q}_{aux,tot}$ | Capacity of the auxiliary heaters (kW) | 34.89 |

${A}_{c,ins}$ | Installation area of the solar collectors (m^{2}) | 194.3 |

${Q}_{L,peak}$ | Peak hot water load (kW) | 27.35 |

${Q}_{L,year}$ | Annual hot water load (kWh/year) | 60,218 |

${I}_{T}$ | Annual solar irradiance on the collector array (kWh/year) | 137,495 |

${Q}_{u}$ | Annual useful heat gain of the collector array (kWh/year) | 39,986 |

${Q}_{Ts}$ | Annual solar energy supplied to the storage tank (kWh/year) | 37,332 |

${Q}_{l}$ | Annual heat loss of the storage tank (kWh/year) | 752 |

${Q}_{d}$ | Annual discharged heat from the storage tank (kWh/year) | 4 |

${Q}_{Ls}$ | Annual solar energy supplied by the storage tank (kWh/year) | 36,386 |

${Q}_{aux}$ | Annual auxiliary energy supplied by the heaters (kWh/year) | 23,832 |

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

${F}_{ELE}$ | Annual electricity consumption (kWh/year) | 1413 |

${F}_{S}$ | Annual solar fraction (%) | 60.42 |

${C}_{I}$ | Initial cost (1000 KRW) | 57,238 |

${C}_{M}$ | Maintenance cost (1000 KRW) | 20,129 |

${C}_{R}$ | Replacement cost (1000 KRW) | 41,302 |

${C}_{E}$ | Energy cost (1000 KRW) | 124,616 |

${C}_{S}$ | Subsidy cost (1000 KRW) | 28,619 |

${C}_{LCC}$ | Life cycle cost (1000 KRW) | 214,666 |

#### 4.3. Effect According to Variation of the Maximum Solar Fraction

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

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

${T}_{coll}$ (–) | 3 | 3 | 0 | 3 | 3 | 3 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |

${N}_{coll}$ (ea.) | 2 | 5 | 8 | 7 | 13 | 19 | 25 | 19 | 19 | 25 | 31 | 37 | 37 |

${T}_{tank}$ (–) | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 3 | 3 | 3 | 3 | 4 |

${T}_{aux}$ (–) | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

${N}_{aux}$ (ea.) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${A}_{c,tot}$ (m^{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 |

${V}_{s}$ (m^{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 |

${Q}_{aux,tot}$ (kW) | 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 |

${A}_{c,ins}$ (m^{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 |

${Q}_{L,peak}$ (kW) | 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 |

${Q}_{L,year}$ (kWh/year) | 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 |

${Q}_{Ts}$ (kWh/year) | 2462 | 5994 | 8740 | 11,159 | 14,698 | 17,754 | 21,185 | 23,695 | 26,625 | 30,374 | 33,440 | 35,901 | 37,332 |

${Q}_{l}$ (kWh/year) | −57 | −19 | 7 | 49 | 68 | 98 | 165 | 200 | 375 | 455 | 524 | 583 | 752 |

${Q}_{d}$ (kWh/year) | 0 | 0 | 0 | 0 | 0 | 3 | 1 | 7 | 0 | 0 | 5 | 21 | 4 |

${Q}_{Ls}$ (kWh/year) | 2523 | 6013 | 8731 | 11,103 | 14,624 | 17,622 | 20,986 | 23,385 | 26,195 | 29,848 | 32,744 | 34,966 | 36,386 |

${Q}_{aux}$ (kWh/year) | 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 |

${F}_{LNG}$ (m^{3}/year) | 6083 | 5715 | 5428 | 5186 | 4810 | 4497 | 4154 | 3904 | 3623 | 3244 | 2940 | 2704 | 2562 |

${F}_{ELE}$ (kWh/year) | 142 | 186 | 335 | 354 | 516 | 632 | 799 | 909 | 892 | 1076 | 1260 | 1425 | 1413 |

${F}_{S}$ (%) | 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 |

${C}_{I}$ (1000 KRW) | 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 |

${C}_{M}$ (1000 KRW) | 3986 | 4727 | 5394 | 5472 | 6702 | 8183 | 9973 | 10,867 | 12,501 | 14,750 | 17,000 | 19,249 | 20,129 |

${C}_{R}$ (1000 KRW) | 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 |

${C}_{E}$ (1000 KRW) | 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 |

${C}_{S}$ (1000 KRW) | 5668 | 6721 | 7670 | 7780 | 9529 | 11,635 | 14,179 | 15,450 | 17,774 | 20,972 | 24,170 | 27,368 | 28,619 |

${C}_{LCC}$ (1000 KRW) | 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 |

#### 4.4. Effect According to Variation of the Minimum Solar Fraction

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

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

${T}_{coll}$ (–) | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

${N}_{coll}$ (ea.) | 37 | 49 | 61 | 67 | 79 | 145 | 223 |

${T}_{tank}$ (–) | 4 | 4 | 4 | 7 | 9 | 8 | 9 |

${T}_{aux}$ (–) | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

${N}_{aux}$ (ea.) | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${A}_{c,tot}$ (m^{2}) | 104.71 | 138.67 | 172.63 | 189.61 | 223.57 | 410.35 | 631.09 |

${V}_{s}$ (m^{3}) | 3.76 | 3.76 | 3.76 | 6.21 | 9.58 | 6.92 | 9.58 |

${Q}_{aux,tot}$ (kW) | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 | 34.89 |

${A}_{c,ins}$ (m^{2}) | 194.3 | 257.3 | 320.3 | 351.8 | 414.8 | 761.3 | 1170.9 |

${Q}_{L,peak}$ (kW) | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 | 27.35 |

${Q}_{L,year}$ (kWh/year) | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 | 60,218 |

${Q}_{Ts}$ (kWh/year) | 37,332 | 41,410 | 44,318 | 47,618 | 51,206 | 58,867 | 65,290 |

${Q}_{l}$ (kWh/year) | 752 | 886 | 994 | 1138 | 1684 | 1922 | 2221 |

${Q}_{d}$ (kWh/year) | 4 | 36 | 78 | 49 | 43 | 402 | 607 |

${Q}_{Ls}$ (kWh/year) | 36,386 | 39,949 | 42,303 | 45,517 | 48,367 | 51,372 | 54,256 |

${Q}_{aux}$ (kWh/year) | 23,832 | 20,269 | 17,915 | 14,701 | 11,851 | 8846 | 5962 |

${F}_{LNG}$ (m^{3}/year) | 2562 | 2181 | 1928 | 1595 | 1291 | 966 | 660 |

${F}_{ELE}$ (kWh/year) | 1413 | 1712 | 1986 | 2064 | 2279 | 3630 | 5053 |

${F}_{S}$ (%) | 60.42 | 66.34 | 70.25 | 75.59 | 80.32 | 85.31 | 90.10 |

${C}_{I}$ (1000 KRW) | 57,238 | 70,030 | 82,822 | 96,197 | 117,025 | 180,589 | 270,529 |

${C}_{M}$ (1000 KRW) | 20,129 | 24,628 | 29,126 | 33,830 | 41,155 | 63,508 | 95,137 |

${C}_{R}$ (1000 KRW) | 41,302 | 48,506 | 55,710 | 66,797 | 82,622 | 114,957 | 169,068 |

${C}_{E}$ (1000 KRW) | 124,616 | 108,296 | 97,765 | 82,838 | 69,716 | 59,867 | 51,207 |

${C}_{S}$ (1000 KRW) | 28,619 | 35,015 | 41,411 | 48,098 | 58,513 | 90,294 | 110,214 |

${C}_{LCC}$ (1000 KRW) | 214,666 | 216,445 | 224,012 | 231,564 | 252,005 | 328,627 | 475,727 |

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${A}_{c}$ | gross area of a single collector module, m ^{2} |

${A}_{coll,j}$ | gross area of the jth device of solar collectors, m ^{2} |

${A}_{c,ins}$ | installation area of solar collectors, m ^{2} |

${A}_{c,tot}$ | total area of solar collectors, m ^{2} |

${A}_{c,max}$ | available space to install collector array, m ^{2} |

${A}_{R,max}$ | maximum capacity available to receive the subsidy cost, m ^{2} |

${A}_{s}$ | surface area of a storage tank, m ^{2} |

${C}_{hex,c}$ | capacity rate of fluid on cold side of a heat exchanger, W/°C |

${C}_{hex,h}$ | capacity rate of fluid on hot side of a heat exchanger, W/°C |

${C}_{hex,max}$ | maximum capacity rate, W/°C |

${C}_{hex,min}$ | minimum capacity rate, W/°C |

${C}_{p,w}$ | specific heat of water, J/kg·°C |

${C}_{p,c}$ | specific heat of collector fluid, J/kg·°C |

${C}_{aux,j}$ | purchase price of the jth auxiliary heater, KRW |

${C}_{coll,j}$ | purchase price of the jth solar collector, KRW |

${C}_{tank,j}$ | purchase price of the jth storage tank, KRW |

${C}_{E}$ | energy cost, KRW |

${C}_{I}$ | initial cost, KRW |

${C}_{I,c}$ | initial cost of each component, KRW |

${C}_{M}$ | maintenance cost, KRW |

${C}_{R}$ | replacement cost, KRW |

${C}_{R,c}$ | replacement cost of each component, KRW |

${C}_{S}$ | subsidy cost, KRW |

${C}_{LCC}$ | life cycle cost, KRW |

${c}_{ELE}$ | hourly electricity cost, KRW/kWh |

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

${c}_{r}$ | capacity rate ratio of a heat exchanger |

${e}_{fuel}$ | fuel price escalation rate, % |

${F}_{ELE}$ | hourly electricity consumption, kWh |

${F}_{LNG}$ | hourly LNG consumption, m ^{3} |

${F}_{R}$ | collector heat removal factor of identical collectors in series |

${F}_{R1}$ | collector heat removal factor of a collector |

${F}_{S}$ | solar fraction of any solar water heating system, % |

${F}_{S,min}$ | minimum solar fraction, % |

${F}_{S,max}$ | maximum solar fraction, % |

${H}_{coll,j}$ | height of the jth device of solar collectors, m |

${I}_{T}$ | hourly total solar radiation on the tilted collector array, W/m ^{2} |

$i$ | real discount rate, % |

${m}_{c}$ | mass flow rate of the collector fluid, kg/s |

${m}_{d}$ | mass flow rate of the discharged water from a storage tank, kg/s |

${m}_{s}$ | mass flow rate from the storage tank to the load, kg/s |

${m}_{l}$ | mass flow rate of the desired hot water load, kg/s |

${N}_{c,s}$ | number of identical collectors in series |

${N}_{coll}$ | number of the jth device of solar collectors |

${N}_{aux}$ | number of the jth device of auxiliary heaters |

$NTU$ | number of exchanger heat transfer units |

${n}_{p}$ | planning period, year |

${n}_{l,c}$ | lifetime of each component, year |

${n}_{r,c}$ | replacement times of each component |

${Q}_{aux,j}$ | heating capacity of the jth device of auxiliary heaters, kW |

${Q}_{aux,tot}$ | total heating capacity of the auxiliary heaters, kW |

${Q}_{aux,year}$ | annual auxiliary heating energy, kWh |

${Q}_{L,peak}$ | peak hot water load, kW |

${Q}_{L,year}$ | annual hot water load, kWh |

${q}_{aux}$ | auxiliary heating energy, W |

${q}_{d}$ | discharged heat to avoid overheating of a storage tank, W |

${q}_{l}$ | heat loss of a storage tank, W |

${q}_{Ls}$ | solar energy extracted from the storage tank to the load, W |

${q}_{Ts}$ | solar energy supplied to a storage tank, W |

${q}_{u}$ | solar useful heat gain of identical collectors in series, W |

${R}_{I}$ | a percentage of the supplementary cost against the direct purchase cost, % |

${R}_{M}$ | a percentage of the annual maintenance cost against the initial cost, % |

${R}_{S}$ | a percentage of subsidy cost against the initial cost, % |

${T}_{a}$ | outdoor dry-bulb temperature, °C |

${T}_{amb}$ | ambient temperature, °C |

${T}_{ci}$ | cold stream outlet temperature of a heat exchanger, °C |

${T}_{co}$ | cold stream inlet temperature of a heat exchanger, °C |

${T}_{hi}$ | hot stream inlet temperature of a heat exchanger, °C |

${T}_{ho}$ | hot stream outlet temperature of a heat exchanger, °C |

${T}_{l}$ | desired hot water temperature, °C |

${T}_{m}$ | make-up water temperature, °C |

${T}_{s}$ | storage tank temperature at the beginning of the time step, °C |

${T}_{s,f}$ | storage tank temperature at the end of the time step, °C |

${T}_{s,max}$ | maximum allowable storage tank temperature, °C |

${T}_{aux}$ | type of auxiliary heater |

${T}_{coll}$ | type of solar collector |

${T}_{tank}$ | type of storage tank |

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

${U}_{L1}$ | collector overall heat loss coefficient of a collector, W/m ^{2}·°C |

${U}_{s}$ | heat loss coefficient of a storage tank, W/m ^{2}·°C |

$U{A}_{hex}$ | product of the overall heat transfer coefficient and area of a heat exchanger, W/°C |

$UP{A}_{ELE}^{\text{*}}$ | uniform present value factor adjusted to reflect the electricity price escalation rate |

$UP{A}_{LNG}^{*}$ | uniform present value factor adjusted to reflect the LNG price escalation rate |

$UP{A}_{fuel}^{*}$ | uniform present value factor adjusted to reflect the fuel price escalation rate |

${V}_{s}$ | storage tank volume, m ^{3} |

${W}_{coll,j}$ | width of the jth device of solar collectors, m |

${\text{\alpha}}_{s,w}$ | meridian altitude in winter, ° |

${\text{\beta}}_{coll}$ | slope of the collector array, ° |

$\text{\epsilon}$ | effectiveness of a heat exchanger |

${\text{\rho}}_{w}$ | density of water, kg/m ^{3} |

$\left(\text{\tau \alpha}\right)$ | product of the transmittance and the absorptance of identical collectors in series |

${\left(\text{\tau \alpha}\right)}_{1}$ | product of the transmittance and the absorptance of a collector |

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Ko, M.J. Analysis and Optimization Design of a Solar Water Heating System Based on Life Cycle Cost Using a Genetic Algorithm. *Energies* **2015**, *8*, 11380-11403.
https://doi.org/10.3390/en81011380

**AMA Style**

Ko MJ. Analysis and Optimization Design of a Solar Water Heating System Based on Life Cycle Cost Using a Genetic Algorithm. *Energies*. 2015; 8(10):11380-11403.
https://doi.org/10.3390/en81011380

**Chicago/Turabian Style**

Ko, Myeong Jin. 2015. "Analysis and Optimization Design of a Solar Water Heating System Based on Life Cycle Cost Using a Genetic Algorithm" *Energies* 8, no. 10: 11380-11403.
https://doi.org/10.3390/en81011380