Performance Analysis

Performance analysis of solar thermal systems requires rigorous evaluation of collector efficiency, radiation availability, and system-level thermal performance under varying meteorological conditions.

Solar Collector Efficiency

Instantaneous Efficiency Equation

The fundamental relationship governing flat-plate collector performance:

η = FR(τα) - FRUL[(Ti - Ta)/Gt]

Where:

• η = Instantaneous thermal efficiency (dimensionless)
• FR = Heat removal factor (dimensionless)
• (τα) = Transmittance-absorptance product (dimensionless)
• UL = Overall heat loss coefficient (W/m²·K or Btu/h·ft²·F)
• Ti = Inlet fluid temperature (°C or °F)
• Ta = Ambient temperature (°C or °F)
• Gt = Total incident solar radiation (W/m² or Btu/h·ft²)

This linear relationship plots as efficiency versus (Ti - Ta)/Gt, with y-intercept FR(τα) representing optical efficiency and slope -FRUL representing thermal losses.

Efficiency Curve Parameters

Performance testing per ASHRAE 93 determines:

Modified Efficiency for Operating Temperature

For elevated temperature applications:

η = η0 - a1(Tm - Ta)/Gt - a2(Tm - Ta)²/Gt

Where:

• Tm = Mean fluid temperature = (Ti + To)/2
• a1 = First-order heat loss coefficient (W/m²·K)
• a2 = Second-order (temperature-dependent) loss coefficient (W/m²·K²)

The quadratic term becomes significant for concentrating collectors or high-temperature flat-plate systems.

Solar Radiation Analysis

Total Incident Radiation Components

Gt = Gb·Rb + Gd·Rd + ρg·G·Rr

Where:

• Gb = Direct beam radiation on horizontal surface (W/m²)
• Rb = Ratio of beam radiation on tilted to horizontal surface
• Gd = Diffuse radiation on horizontal surface (W/m²)
• Rd = Ratio of diffuse radiation on tilted to horizontal surface
• ρg = Ground reflectance (typically 0.2, snow: 0.7)
• G = Total horizontal radiation
• Rr = Ratio of reflected radiation on tilted surface

Geometric Factor for Beam Radiation

Rb = cos(θ)/cos(θz)

Where:

• θ = Incidence angle on tilted surface
• θz = Solar zenith angle

cos(θ) = sin(δ)sin(φ)cos(β) - sin(δ)cos(φ)sin(β)cos(γ) + cos(δ)cos(φ)cos(β)cos(ω) + cos(δ)sin(φ)sin(β)cos(γ)cos(ω) + cos(δ)sin(β)sin(γ)sin(ω)

Where:

• δ = Solar declination angle
• φ = Latitude
• β = Collector tilt angle from horizontal
• γ = Surface azimuth angle (0° = south)
• ω = Hour angle

Typical Meteorological Year (TMY) Data

TMY3 datasets provide hourly values for performance simulation:

• Direct normal irradiance (DNI)
• Diffuse horizontal irradiance (DHI)
• Global horizontal irradiance (GHI)
• Ambient dry-bulb temperature
• Wind speed and direction
• Atmospheric pressure

Data sources include NREL National Solar Radiation Database (NSRDB) covering 1991-2005 representative meteorological conditions.

F-Chart Method Analysis

The f-chart correlation method estimates long-term solar thermal system performance without hour-by-hour simulation.

Dimensionless Parameters

X = FR’ULAc(Tref - Ta)Δt / L

X represents the ratio of collector heat losses to total heating load.

Y = FR’(τα)nAcH̄T / L

Y represents the ratio of absorbed solar energy to total heating load.

Where:

• FR’ = Collector heat exchanger efficiency factor × FR
• Ac = Collector area (m²)
• Tref = Reference temperature (typically 100°C for water heating, 11.6°C above Ta for space heating)
• Ta = Monthly average ambient temperature (°C)
• Δt = Total time in month (hours)
• L = Monthly total heating load (J or Btu)
• (τα)n = Transmittance-absorptance product at normal incidence
• H̄T = Monthly average daily radiation on collector surface per unit area (J/m² or Btu/ft²)

F-Chart Correlation for Liquid Systems

f = 1.029Y - 0.065X - 0.245Y² + 0.0018X² + 0.0215Y³

For standard liquid-based solar water heating and space heating systems with water thermal storage.

F-Chart Correlation for Air Systems

f = 1.040Y - 0.065X - 0.159Y² + 0.00187X² - 0.0095Y³

For air-based solar heating systems with pebble-bed or rock storage.

The monthly solar fraction f represents the portion of monthly load supplied by solar energy. Annual performance:

F = Σ(f·L) / Σ(L)

Correction Factors

Storage capacity correction: Csc = (X/XR)^(-0.25)

For storage mass different from reference (75 kg water per m² collector area).

Heat exchanger correction: FR’/FR

Accounts for temperature penalty in liquid-to-liquid heat exchangers between collector loop and storage.

Utilizability Method

The φ-f̄ (phi-f-bar) method provides improved accuracy by accounting for radiation distribution:

φ = exp[-a1(Tc,min - Ta)/(a0·It)]

Where:

• Tc,min = Minimum useful collector temperature
• It = Hourly total radiation on tilted surface
• a0, a1 = Collector efficiency parameters

Monthly utilizability f̄ correlates to critical radiation level and monthly radiation statistics.

Thermal Performance Modeling

Energy Balance on Collector

Q̇u = ṁcp(To - Ti) = AcFR[Gt(τα) - UL(Ti - Ta)]

Where:

• Q̇u = Useful energy gain rate (W or Btu/h)
• ṁ = Mass flow rate (kg/s or lb/h)
• cp = Specific heat of fluid (J/kg·K or Btu/lb·°F)
• Ac = Collector gross area (m²)

Storage Tank Energy Balance

McpTs,dTs/dt = Q̇u - Q̇load - Q̇loss

Where:

• M = Storage mass (kg)
• Ts = Storage temperature (°C)
• Q̇load = Load heat extraction rate (W)
• Q̇loss = Storage heat loss rate = UA(Ts - Tambient)

System Thermal Efficiency

ηsys = Q̇delivered / (Ac·Gt,avg)

Accounts for:

• Collector thermal efficiency
• Piping heat losses
• Heat exchanger penalties
• Storage losses
• Parasitic pumping energy

System Optimization

Collector Tilt Angle Optimization

For maximum annual energy collection:

βopt ≈ φ ± 10° to 15°

Seasonal optimization:

• Winter (heating dominant): β = φ + 15°
• Summer (domestic hot water): β = φ - 15°
• Year-round: β ≈ φ

Collector Area Sizing

Economic optimum typically occurs at solar fraction f = 0.4 to 0.7 for most climates.

Ac = (f·Lannual) / (FR(τα)·Hannual·ηsys)

Oversizing beyond f = 0.7 results in diminishing returns and summer stagnation concerns.

Storage Volume Optimization

ASHRAE recommendation:

Vstorage = 50 to 100 liters per m² collector area

Or 1.5 to 2.0 gallons per ft² collector area.

Larger storage (75-100 L/m²) suits:

• High daily load variation
• Multi-day storage capability
• Locations with intermittent solar availability

Smaller storage (50-75 L/m²) suits:

• Consistent daily loads
• High solar availability
• Cost-constrained applications

Flow Rate Optimization

ṁopt = 0.015 to 0.020 kg/s per m² collector area

Or approximately 0.02 gpm/ft² collector area.

Higher flow rates:

• Increase heat removal factor FR
• Reduce thermal stratification in storage
• Increase pumping energy

Lower flow rates:

• Reduce collector efficiency
• Enhance storage stratification
• Minimize pumping energy

Performance Testing Standards

ASHRAE 93: Methods of Testing to Determine the Thermal Performance of Solar Collectors

Key test procedures:

• Steady-state thermal efficiency testing
• Time constant determination
• Incident angle modifier testing
• Outdoor testing under natural sunlight

Test conditions:

• Minimum incident radiation: 790 W/m² (250 Btu/h·ft²)
• Maximum wind speed variation: ±1.5 m/s
• Ambient temperature stability: ±1.5°C
• Minimum four data points across inlet temperature range

SRCC OG-100: Solar Thermal Collector Certification

Provides certified ratings based on:

• Clear day outdoor tests
• Incident angle modifiers
• Thermal performance at standard conditions
• Durability testing (thermal shock, rain, stagnation)

System Performance Monitoring

Key Performance Indicators

Instrumentation Requirements

Minimum monitoring points:

• Collector inlet and outlet temperatures (±0.3°C accuracy)
• Storage tank temperature stratification (3-5 points vertically)
• Flow rate measurement (±2% accuracy)
• Incident radiation on collector plane (±5% accuracy)
• Ambient temperature
• Pump electrical consumption

Data logging interval: 5 to 15 minutes for detailed analysis; hourly for long-term monitoring.

Shading Analysis

Shading reduces collector output proportionally to shaded area. Critical assessment includes:

• Winter solstice sun path (lowest solar altitude)
• 9:00 AM to 3:00 PM solar window (critical collection period)
• Obstructions: trees, buildings, terrain features
• Self-shading in collector arrays (row spacing)

Minimum row spacing to avoid inter-row shading:

S = H / tan(α)

Where:

• S = Row spacing
• H = Collector height above ground
• α = Solar altitude angle at winter solstice noon

Simulation Tools

TRNSYS (Transient System Simulation)

Component-based simulation platform:

• Type 1b: Flat-plate collector with IAM
• Type 4: Storage tank with thermal stratification
• Type 56: Multi-zone building model
• Type 2b: Differential controller

Hourly timestep simulation using TMY3 data provides detailed annual performance prediction.

Simplified Calculation Methods

For preliminary sizing:

Ac = (Daily load × LCR) / (Daily insolation × ηsys)

Where LCR (Load-to-Collector Ratio) typically ranges from 40 to 80 MJ/m²·day for domestic hot water applications.

References:

• ASHRAE Handbook—HVAC Applications, Chapter 35: Solar Energy Equipment
• ASHRAE Standard 93: Methods of Testing to Determine the Thermal Performance of Solar Collectors
• Duffie, J.A., and Beckman, W.A., Solar Engineering of Thermal Processes, 4th Edition
• Klein, S.A., et al., “TRNSYS—A Transient System Simulation Program,” Solar Energy Laboratory, University of Wisconsin-Madison