6.3 Solar Heat Gain (SHGC), Fenestration & Cooling Load Temperature Difference / Radiant Time Series (CLTD/RTS)

6.3 Solar Heat Gain (SHGC), Fenestration & Cooling Load Temperature Difference / Radiant Time Series (CLTD/RTS)

Solar radiation passing through fenestration and striking opaque exterior envelope surfaces represents one of the largest and most dynamic components of building cooling loads. Unlike steady-state winter heating calculations, cooling load calculations must account for the dual nature of solar radiation: direct optical transmission through glazing, and transient thermal storage within opaque building materials. Engineering methods developed by ASHRAE—including the traditional Cooling Load Temperature Difference (CLTD / SCL / CLF) method and the modern Radiant Time Series (RTS) method—translate instantaneous thermal heat gains into actual cooling loads experienced by the air conditioning system.

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1. Fenestration Heat Transfer & Solar Optics

Heat gain through glazed fenestration assemblies (windows, curtain walls, skylights) consists of two concurrent, independent heat transfer mechanisms:

1. Conductive/Convective Heat Transmission: Driven by the dry-bulb temperature difference across the glass.
2. Solar Radiation Heat Gain: Driven by direct beam, diffuse sky, and ground-reflected solar irradiance striking the glass surface.

+-----------------------------------------------------------------------------------------+
| FENESTRATION TOTAL HEAT GAIN MECHANISMS                                                 |
+-----------------------------------------------------------------------------------------+
| Incident Solar Irradiance (E_t)                                                         |
|     |                                                                                   |
|     +---> [Reflected Outward] (rho * E_t)                                               |
|     |                                                                                   |
|     +---> [Directly Transmitted] (tau * E_t) -----------------------------> Room Space  |
|     |                                                                            |      |
|     +---> [Absorbed in Glass] (alpha * E_t) --+-> [Inward Re-radiation/Conv] --->+      |
|                                               +-> [Outward Re-radiation/Conv]           |
|                                                                                         |
| Conduction Across Glass:  q_cond = U * A * (T_outdoor - T_indoor)                       |
+-----------------------------------------------------------------------------------------+

Total Fenestration Heat Gain Equation

$$q_{\text{fen}} = q_{\text{conduction}} + q_{\text{solar}}$$ $$q_{\text{fen}} = U \cdot A \cdot (T_{\text{outdoor}} - T_{\text{indoor}}) + A \cdot \text{SHGC} \cdot \text{IAC} \cdot E_t$$

Where:

• $q_{\text{fen}} = \text{Total instantaneous fenestration heat gain } (\text{Btu/hr})$
• $U = \text{Overall fenestration assembly } U\text{-factor, including frame } (\text{Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)})$
• $A = \text{Total fenestration gross area } (\text{ft}^2)$
• $\text{SHGC} = \text{Solar Heat Gain Coefficient of the glazing assembly (dimensionless, } 0.0\text{ to }1.0)$
• $\text{IAC} = \text{Interior Attenuation Coefficient (accounting for interior blinds, shades, or drapes)}$
• $E_t = \text{Total incident solar irradiance on the surface } (\text{Btu/(hr}\cdot\text{ft}^2))$

Solar Heat Gain Coefficient (SHGC) vs. Legacy Shading Coefficient (SC)

• SHGC: Defined as the fraction of incident solar radiation admitted through a window, both via direct transmission and inward release of absorbed heat: $\text{SHGC} = \tau + N_i \alpha$.
• Shading Coefficient (SC): Legacy metric defined as the ratio of solar heat gain through a given glazing to that through standard 1/8-inch clear double-strength sheet glass (DSA, $\text{SHGC}_{\text{ref}} = 0.86$ to $0.87$).

$$\text{SHGC} = \text{SC} \times 0.86 \quad \iff \quad \text{SC} = \frac{\text{SHGC}}{0.86} \approx 1.163 \times \text{SHGC}$$

Fenestration Shading & Overhang Geometry

External architectural shading devices (horizontal overhangs, vertical fins) block direct solar beam radiation during high sun angles. The Projection Factor (PF) characterizes horizontal overhang depth:

$$\text{PF} = \frac{B}{H}$$

Where $B$ is the horizontal overhang projection width (ft) and $H$ is the vertical distance from the window sill to the bottom of the overhang (ft).

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2. Sol-Air Temperature ($T_{sol\text{-}air}$)

Solar radiation striking opaque exterior walls and roofs increases surface temperature significantly above ambient air dry-bulb temperature. The Sol-Air Temperature ($T_{sol\text{-}air}$) is a fictitious outdoor air temperature that, in the absence of all radiation exchanges, would produce the same total rate of heat transmission into the surface as the combined actual outdoor air temperature, incident solar radiation, and longwave radiant exchange with the sky and surroundings.

Sol-Air Temperature Derivation

Balancing heat flux at the exterior surface:

$$q/A = h_o (T_o - T_s) + \alpha I_t - \varepsilon \Delta R = h_o (T_{sol\text{-}air} - T_s)$$ $$T_{sol\text{-}air} = T_o + \frac{\alpha \cdot I_t}{h_o} - \frac{\varepsilon \cdot \Delta R}{h_o}$$

Where:

• $T_o = \text{Outdoor ambient dry-bulb temperature } ({}^\circ\text{F})$
• $\alpha = \text{Surface solar absorptance (dimensionless, } 0.2\text{ for bright white to }0.95\text{ for black asphalt/tar)}$
• $I_t = \text{Total incident solar radiation (direct + diffuse) } (\text{Btu/(hr}\cdot\text{ft}^2))$
• $h_o = \text{Exterior surface combined convective and radiative heat transfer coefficient } (\approx 3.0\text{ Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)})$
• $\varepsilon = \text{Surface hemispherical thermal emittance } (\approx 0.90\text{ for most building materials)}$
• $\Delta R = \text{Longwave radiation correction factor between surface and sky } (\text{Btu/(hr}\cdot\text{ft}^2))$
  • Horizontal Roofs: $\Delta R \approx 20\text{ Btu/(hr}\cdot\text{ft}^2)$ (giving $\frac{\varepsilon \Delta R}{h_o} \approx 7^\circ\text{F}$ for horizontal surfaces facing open sky)
  • Vertical Walls: $\Delta R = 0\text{ Btu/(hr}\cdot\text{ft}^2)$ (assuming ground and surroundings are at air temperature, giving $\frac{\varepsilon \Delta R}{h_o} = 0^\circ\text{F}$)

+-----------------------------------------------------------------------------------------+
| SOL-AIR TEMPERATURE PRACTICAL SHORTCUT FORMULAS (h_o = 3.0 Btu/hr-ft²-°F)              |
+-----------------------------------------------------------------------------------------+
| Horizontal Flat Roofs:     T_sol-air = T_o + (alpha * I_t / 3.0) - 7°F                  |
| Vertical Exterior Walls:   T_sol-air = T_o + (alpha * I_t / 3.0)                        |
+-----------------------------------------------------------------------------------------+

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3. Cooling Load Temperature Difference (CLTD) Method

The CLTD / SCL / CLF method is a simplified hand-calculation procedure that accounts for the transient thermal storage and time-delay effects of building envelope assemblies without requiring complex differential equations.

1. Opaque Roofs and Walls Cooling Load

$$q_{\text{envelope}} = U \cdot A \cdot \text{CLTD}_{\text{corrected}}$$

Where $\text{CLTD}_{\text{corrected}}$ modifies tabular baseline CLTD values (which are tabulated for standard conditions: $T_{\text{indoor}} = 78^\circ\text{F}$, maximum outdoor temperature $T_{o,\text{max}} = 95^\circ\text{F}$, daily range $DR = 21^\circ\text{F}$):

$$\text{CLTD}_{\text{corrected}} = \text{CLTD}_{\text{table}} + (78 - T_{\text{indoor}}) + (T_{o,m} - 85)$$

Where the mean outdoor daily temperature $T_{o,m}$ is:

$$T_{o,m} = T_{o,\text{max}} - \frac{DR}{2}$$

2. Glazing Conduction & Solar Load via CLTD and SCL

$$q_{\text{glazing, conduction}} = U \cdot A \cdot \text{CLTD}_{\text{corr, glass}}$$ $$q_{\text{glazing, solar}} = A \cdot \text{SC} \cdot \text{SCL}$$

Where $\text{SCL}$ is the Solar Cooling Load factor (in $\text{Btu/(hr}\cdot\text{ft}^2)$), accounting for solar intensity, latitude, orientation, and room thermal storage mass.

+-----------------------------------------------------------------------------------------+
| THERMAL MASS TIME LAG & DAMPING EFFECT                                                  |
+-----------------------------------------------------------------------------------------+
| Peak Solar Irradiance on Roof occurs at Solar Noon (12:00 PM)                           |
|   - Light Metal Roof (Low Mass):    Peak Cooling Load occurs at 1:00 PM  (1-hr lag)     |
|   - Heavy Concrete Roof (High Mass): Peak Cooling Load occurs at 6:00 PM (6-hr lag)     |
|   *Thermal mass shifts the peak hour and flattens (damps) peak cooling demand!         |
+-----------------------------------------------------------------------------------------+

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4. Radiant Time Series (RTS) Method & Periodic Response Factors

The Radiant Time Series (RTS) method is the modern ASHRAE standard for non-residential cooling load calculations. It replaces older transfer function techniques by separating heat gains into convective and radiant components.

1. Convective vs. Radiant Heat Gain Split

All heat gains entering a space are partitioned into two distinct physical fractions:

• Convective Heat Gain ($f_c$): Heat transferred directly to room air by convection. Becomes instantaneous cooling load immediately.
• Radiant Heat Gain ($f_r = 1 - f_c$): Radiant energy (longwave and shortwave) that passes through air without heating it, striking interior surfaces (floor, partitions, furniture) where it is absorbed and stored.

2. Conduction Time Series Factors (CTSF)

For opaque walls and roofs, the 24-hour conduction heat gain ($q_{e,\theta}$) at hour $\theta$ is calculated from historical sol-air temperatures using Conduction Time Series Factors ($c_j$):

$$q_{e,\theta} = A \sum_{j=0}^{23} c_j \cdot U \cdot \left(T_{sol\text{-}air,\theta - j} - T_{\text{indoor}}\right)$$

Where $\sum_{j=0}^{23} c_j = 1.00$. The series factors $c_j$ define the precise thermal damping and time delay of the assembly mass.

3. Radiant Time Factors (RTF)

Radiant heat gains ($q_{r,\theta}$) generated across the preceding 24 hours are converted into hourly cooling loads ($Q_{r,\theta}$) using zone Radiant Time Factors ($r_j$):

$$Q_{r,\theta} = \sum_{j=0}^{23} r_j \cdot q_{r,\theta - j}$$

Where $\sum_{j=0}^{23} r_j = 1.00$. Total zone sensible cooling load ($Q_{\text{sensible},\theta}$) at hour $\theta$ is the sum of the instantaneous convective gains and the radiant cooling load:

$$Q_{\text{sensible},\theta} = q_{\text{convective},\theta} + Q_{r,\theta}$$

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5. NCEES Reference Handbook Navigation Tactics

• Fenestration Formulas: Search "Fenestration" or "SHGC" in Section 7 (HVAC & Refrigeration Applications) to find the overall window heat gain equation and typical $U$-factors and $\text{SHGC}$ values for double-pane low-e glass.
• Sol-Air Temperature: Search "Sol-Air" to access the exact equation $T_{sol\text{-}air} = T_o + (\alpha I_t / h_o) - (\varepsilon \Delta R / h_o)$.
• CLTD Table Adjustments: Search "CLTD" to find the temperature correction formula $\text{CLTD}_{\text{corr}} = \text{CLTD} + (78 - T_i) + (T_{o,m} - 85)$.

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6. Worked Computational Examples

Example 1: Sol-Air Temperature Calculation for a Flat Commercial Roof

A dark-colored, flat asphalt membrane roof (solar absorptance $\alpha = 0.90$, emittance $\varepsilon = 0.90$) is exposed to peak solar radiation at solar noon in Phoenix, AZ. The following conditions apply:

• Outdoor dry-bulb temperature: $T_o = 104^\circ\text{F}$
• Total incident solar radiation: $I_t = 310\text{ Btu/(hr}\cdot\text{ft}^2)$
• Outside convective heat transfer coefficient: $h_o = 3.0\text{ Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)}$
• Longwave roof radiation correction factor: $\Delta R = 20\text{ Btu/(hr}\cdot\text{ft}^2)$

Calculate: (a) The sol-air temperature of the dark roof, and (b) The sol-air temperature if the roof is retrofitted with a bright white "cool roof" elastomeric coating having $\alpha = 0.25$.

Solution:

1. Sol-air temperature for the dark asphalt roof: $$T_{sol\text{-}air} = T_o + \frac{\alpha \cdot I_t}{h_o} - \frac{\varepsilon \cdot \Delta R}{h_o}$$ $$T_{sol\text{-}air} = 104 + \frac{0.90 \times 310}{3.0} - \frac{0.90 \times 20}{3.0}$$ $$T_{sol\text{-}air} = 104 + 93.0 - 6.0 = 191.0^\circ\text{F}$$
2. Sol-air temperature for the white "cool roof": $$T_{sol\text{-}air,\text{cool}} = 104 + \frac{0.25 \times 310}{3.0} - \frac{0.90 \times 20}{3.0}$$ $$T_{sol\text{-}air,\text{cool}} = 104 + 25.83 - 6.0 = 123.83^\circ\text{F}$$

Conclusion: Retrofitting with a high-albedo cool roof reduces the effective driving temperature difference across the roof by $67.2^\circ\text{F}$, drastically reducing roof conductive heat gain.

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Example 2: Peak Fenestration Cooling Load Calculation

A west-facing office window assembly has a total area of $250\text{ ft}^2$. The fenestration consists of double-pane low-e tinted glazing with the following properties:

• Overall assembly $U$-factor: $U = 0.35\text{ Btu/(hr}\cdot\text{ft}^2\cdot{}^\circ\text{F)}$
• Solar Heat Gain Coefficient: $\text{SHGC} = 0.38$
• Interior mini-blinds with an Interior Attenuation Coefficient: $\text{IAC} = 0.70$
• Outdoor design dry-bulb: $T_o = 96^\circ\text{F}$
• Indoor design setpoint: $T_i = 74^\circ\text{F}$
• Peak 4:00 PM incident solar irradiance on west facade: $E_t = 215\text{ Btu/(hr}\cdot\text{ft}^2)$

Calculate: (a) Fenestration conductive heat gain, (b) Fenestration solar radiation heat gain, and (c) Total instantaneous peak fenestration heat gain.

Solution:

1. Calculate conductive heat gain: $$q_{\text{cond}} = U \cdot A \cdot (T_o - T_i) = 0.35 \times 250\text{ ft}^2 \times (96 - 74)^\circ\text{F} = 0.35 \times 250 \times 22 = 1,925\text{ Btu/hr}$$
2. Calculate solar radiation heat gain: $$q_{\text{solar}} = A \cdot \text{SHGC} \cdot \text{IAC} \cdot E_t = 250\text{ ft}^2 \times 0.38 \times 0.70 \times 215\text{ Btu/(hr}\cdot\text{ft}^2)$$ $$q_{\text{solar}} = 250 \times 0.266 \times 215 = 14,297.5\text{ Btu/hr}$$
3. Calculate total instantaneous fenestration heat gain: $$q_{\text{fen}} = q_{\text{cond}} + q_{\text{solar}} = 1,925 + 14,297.5 = 16,222.5\text{ Btu/hr} = 1.352\text{ Tons}$$

(Notice that solar radiation accounts for 88.1% of the total window cooling load!)