Training Solar Thermal Systems — Collectors, Geometry & Storage Flat-Plate Collector Theory — HWB Equation & Efficiency
1 / 2

Flat-Plate Collector Theory — HWB Equation & Efficiency

60 min Solar Thermal Systems — Collectors, Geometry & Storage

Flat-Plate Collector Theory — HWB Equation & Efficiency

The Hottel–Whillier–Bliss (HWB) equation is the fundamental steady-state energy model for flat-plate solar collectors. It relates the useful heat output $\dot{Q}_u$ to the incident irradiance $G_T$, the optical efficiency $(F_R \tau\alpha)$, and the thermal loss coefficient $F_R U_L$. The HWB equation enables collector selection, system sizing, and performance prediction from standard test parameters.

1. Flat-Plate Collector Energy Balance

On the absorber plate, the instantaneous energy balance per unit area:

$$\dot{q}_u = G_T (\tau\alpha) - U_L (T_m - T_a)$$

where $G_T$ is the total irradiance on the tilted collector surface (W/m²), $\tau$ is the glazing transmittance, $\alpha$ is the absorber absorptance, $U_L$ is the overall loss coefficient (W/m²·K), and $(T_m - T_a)$ is the mean plate-to-ambient temperature difference. Introducing the collector heat-removal factor $F_R$ (accounting for fluid-to-plate temperature distribution), the HWB equation for the whole collector:

$$\dot{Q}_u = A_c F_R [G_T (\tau\alpha) - U_L (T_{in} - T_a)]$$

$F_R$ is the ratio of actual useful heat gain to the gain if the whole absorber were at inlet temperature $T_{in}$. Typical $F_R = 0.85{-}0.98$ for well-designed collectors.

2. Collector Efficiency and the HWB Plot

Instantaneous efficiency $\eta = \dot{Q}_u / (A_c G_T)$:

$$\eta = F_R (\tau\alpha) - F_R U_L \frac{T_{in} - T_a}{G_T} = \eta_0 - U_{L,eff} X$$

where $X = (T_{in} - T_a)/G_T$ is the reduced temperature parameter (m²·K/W). The efficiency curve is a straight line on the $\eta$-vs-$X$ plot: $y$-intercept $= F_R(\tau\alpha)$ (optical efficiency), slope $= -F_R U_L$ (thermal loss coefficient). Stagnation temperature (no flow, $\dot{Q}_u = 0$):

$$T_{stag} = T_a + \frac{G_T \cdot F_R(\tau\alpha)}{F_R U_L}$$

Typical stagnation temperatures: flat-plate 150–200°C; evacuated tube 200–280°C; concentrating trough 300–400°C.

3. Evacuated Tube Collectors

Evacuated tubes eliminate convective losses ($U_{conv} \approx 0$), reducing $U_L$ from ~4–6 W/m²·K (flat-plate) to ~1–2 W/m²·K. The HWB equation still applies with the same form; the dramatic reduction in slope gives much higher efficiency at elevated inlet temperatures ($X > 0.05$ m²·K/W), making evacuated tubes preferable for industrial process heat and solar cooling (absorption chiller driving temperatures of 70–90°C).

4. Fluid Flow Rate Effect

Increasing mass flow rate $\dot{m}$ lowers $T_{out} - T_{in}$ and $T_m$, thereby reducing thermal losses and raising $F_R$ toward its maximum value of 1. The heat-removal factor:

$$F_R = \frac{\dot{m} c_p}{A_c U_L} \left[1 - \exp\left(-\frac{A_c U_L}{\dot{m} c_p}\right)\right]$$

The dimensionless group $A_c U_L / (\dot{m} c_p)$ is the number of heat-transfer units (NTU); optimal NTU $\approx 1.5{-}3$ for solar collectors.

Worked Example — Flat-Plate Collector Performance

$A_c = 2$ m², $F_R(\tau\alpha) = 0.76$, $F_R U_L = 5.2$ W/m²·K, $G_T = 800$ W/m², $T_{in} = 50°$C, $T_a = 20°$C. $\dot{Q}_u = 2[800(0.76) - 5.2(50-20)] = 2[608 - 156] = 904$ W. $\eta = 904/(2\times800) = 56.5\%$. $T_{stag} = 20 + 0.76\times800/5.2 = 20 + 116.9 = 137°$C. ✓

Solar Collector Performance — HWB Equation
Useful gain Q_u =452W/m²
Efficiency η =56.5%
Reduced temp X =0.0375m²K/W
Stagnation T =137°C

Practice Problems

1. An evacuated tube collector has $F_R(\tau\alpha) = 0.72$ and $F_R U_L = 1.4$ W/m²·K. Compare its efficiency to a flat-plate collector ($F_R(\tau\alpha) = 0.76$, $F_R U_L = 5.0$ W/m²·K) at $X = 0.08$ m²·K/W. Which collector is preferable at this operating point?
2. Derive the expression for $F_R$ from first principles using the fin efficiency model for a tube-and-sheet absorber. Show how $F_R \to 1$ as $\dot{m} \to \infty$ and $F_R \to 0$ as $U_L \to \infty$.
3. A solar water heater requires 300 L/day at 60°C (from 20°C mains). Average daily solar irradiation $H = 18$ MJ/m²·day, $F_R(\tau\alpha) = 0.74$, $F_R U_L = 4.8$ W/m²·K, average $T_a = 15°$C. Estimate the required collector area using the daily energy balance.
Solar Geometry & Irradiation Modelling