# Effects of Roof Pitch on Air Flow and Heating Load of Sealed and Vented Attics for Gable-Roof Residential Buildings

^{*}

## Abstract

**:**

## Nomenclature

c_{p} | specific heat, J kg ^{−1} K^{−1} |

f | elliptic relaxation function |

g | gravitational acceleration, m s ^{−2} |

h | heat transfer coefficient, W m ^{−2} K^{−1} |

H | attic height, m |

k | turbulence kinetic energy, m ^{2} s^{−2} |

L | turbulence length scale, m |

Nu | Nusselt number |

p | pressure, Pa |

p_{atm} | atmospheric pressure, Pa |

q | heat flux, Wm ^{−2} |

Q | heat transfer rate, W/m |

R | thermal resistance, K m ^{2}/W |

Ra | Rayleigh number |

S | strain rate, s ^{−1} |

t_{c} | ceiling thickness, m |

T | temperature, K |

T_{0} | reference temperature, K |

T_{in} | inlet air temperature, K |

u | velocity component, m/s |

turbulence velocity variance scale, m ^{2} s^{−2} | |

W | half width of attic, m |

x,y | coordinates, m |

#### Greek symbols

β | thermal expansion coefficient, K ^{−1} |

ε | turbulence dissipation rate, m ^{2} s^{−3} |

λ | thermal conductivity, W m ^{−1} K^{−1} |

λ_{c} | ceiling thermal conductivity, W m ^{−1} K^{−1} |

µ | molecular viscosity, kg m ^{−1} s^{−1} |

µ_{t} | eddy viscosity, kg m ^{−1} s^{−1} |

ρ | density, kg m ^{−3} |

σ_{k} | effective turbulence Prandtl number for k |

σ_{ε} | effective turbulence Prandtl number for ε |

σ_{T} | effective turbulence Prandtl number for T |

τ | turbulence time scale, s |

#### Subscripts

a | air |

c | ceiling |

cb | ceiling-bottom |

ct | ceiling-top |

r | roof |

rb | roof-bottom |

rt | roof-top |

t | total |

## 1. Introduction

## 2. Numerical Model

_{0}, that is specified to the outside ambient air temperature to correctly calculate the buoyancy effects. In all the cases reported in this study, T

_{0}= 267 K is assumed. In order to correctly account for the thermal resistances of the ceiling and roofs, which are excluded from the computational domain, convection-type boundary conditions are applied to both the ceiling and roof boundaries. For example, energy balance across the ceiling thickness gives

_{c}and λ

_{c}are the thickness and thermal conductivity of the ceiling; T

_{ct}and T

_{cb}are the temperatures at the ceiling-top and ceiling-bottom, respectively; and the heat transfer coefficient h

_{c}is the reciprocal of the ceiling thermal resistance.

**Figure 1.**Schematic of the computational domain and boundary conditions for (

**a**) sealed attics and (

**b**) vented attics.

_{cb}= 293 K, while a heat transfer coefficient of h

_{c}= 0.284 W/m

^{2}K is adopted to approximate a ceiling insulation level of R-20 (simulating the thermal resistance of a layer of 15 cm thick glass fiber; in this paper, the R-value is used according to the U.S. convention, i.e., R-1 = 1 h·ft²·°F/Btu = 0.176110 Km

^{2}/W). Similarly, a roof-top temperature of T

_{rt}= 273 K and a heat transfer coefficient of h

_{r}= 4.73 W/m

^{2}K (equivalent to an insulation level of R-1.2) are specified to the roof boundaries to simulate a typical condition of a 2 cm plywood roof covered by snow. It is obvious that the assumption of uniform ceiling-bottom and roof-top temperatures is an ideal simplification to the real thermal conditions of the modeled attics. However, for typical conditions when the indoor temperature only slightly varies along the ceiling-bottom and the roof-top is covered by snow, the simplified situation considered here still provides good approximation to the reality.

_{atm}. Therefore, the obtained air flow is purely driven by the thermally induced buoyancy forces, i.e., the stack effect. At the soffit vents, the inlet air is assumed to enter at the ambient temperature of 267 K with a turbulent intensity of 1%.

_{p}, λ, β, σ

_{T}, S

_{ij}are air density, viscosity, specific heat, thermal conductivity, thermal expansion coefficient, Prandtl number, and strain rate, respectively. The unsteady terms in the above equations account for all unsteadiness that does not belong to the turbulence, i.e., the unsteadiness that is not represented by the turbulence model [30].

_{t}, is defined by

_{µ}is a constant. The eddy viscosity is determined from the following unsteady transport equations for the turbulence kinetic energy k, dissipation rate ε, velocity variance scale , and elliptic relaxation function f:

^{+}value for the first grid close to the walls is everywhere less than 1.

## 3. Results and Discussion

#### 3.1. Sealed Attics

**Figure 2.**Predicted (

**left**) steady streamlines (in kg/m s) and (

**right**) isotherms (in K) in sealed attics with roof pitches of 3/12, 5/12, 8/12, 12/12, and 18/12, respectively.

**Figure 3.**Predicted steady profiles of (

**a**) horizontal velocity and (

**b**) temperature along the central vertical line x = 0 for sealed attics with different roof pitches.

**Figure 4.**Predicted steady temperatures at the ceiling (

**a**) and roof (

**b**) boundaries for sealed attics with different roof pitches.

**Table 1.**Predicted heat transfer through the ceiling and roof boundaries for sealed attics with different roof pitches.

Pitch | Heat Gain from Left Ceiling (W/m) | Heat Gain from Right Ceiling (W/m) | Heat Loss from Left Roof (W/m) | Heat Loss from Right Roof (W/m) | Q_{t} (W/m) |
---|---|---|---|---|---|

3/12 | 17.21 | 16.79 | 15.78 | 18.21 | 34.00 |

5/12 | 17.33 | 16.87 | 15.98 | 18.22 | 34.20 |

8/12 | 17.57 | 17.15 | 17.31 | 17.41 | 34.72 |

12/12 | 17.98 | 17.60 | 17.71 | 17.86 | 35.58 |

18/12 | 18.30 | 17.93 | 17.99 | 18.24 | 36.23 |

_{ct}− T

_{rb}. Although it is possible to obtain this temperature difference by subtracting a curve in Figure 4a by the one corresponding to the same roof pitch in Figure 4b and then taking average over the ceiling range, we prefer an alternative approach, i.e., evaluating T

_{ct}− T

_{rb}by the classical method of thermal resistance analysis. An advantage of the thermal resistance approach is that the thermal resistance of attic air flow can be obtained as well, which itself is also of engineering significance.

_{t}, equals the sum of the thermal resistances of the sub-systems

_{t}is the total heat transfer rate, as listed in the last column of Table 1. Since the ceiling and roof insulations are specified to be R-20 and R-1.2, respectively, Equations (15,16) can be used to obtain all the thermal resistances, and the obtained results are listed in Table 2. It should be noted that the R

_{r}column in Table 2 is obtained by scaling R-1.2 by the roof-ceiling area ratio, (W

^{2}+ H

^{2})

^{1/2}/W, so that all the resistances are defined based on unit ceiling area. Table 2 shows that the thermal resistance of the attic air is around R-5 level and decreases about 31% from the 3/12 pitched attic to the 18/12 pitched attic.

**Table 2.**Thermal resistances, in R-value (R-1 = 0.17611 Km

^{2}/W), for sealed attics with different roof pitches.

Pitch | R_{t} (Total) | R_{c} (Ceiling) | R_{r} (Roof) | R_{a} (Air) |
---|---|---|---|---|

3/12 | 26.721 | 20 | 1.164 | 5.556 |

5/12 | 26.565 | 20 | 1.108 | 5.457 |

8/12 | 26.167 | 20 | 0.998 | 5.179 |

12/12 | 25.535 | 20 | 0.849 | 4.686 |

18/12 | 25.077 | 20 | 0.666 | 4.411 |

**Table 3.**Representative ceiling-top and roof-bottom temperatures together with derived Nusselt and Rayleigh numbers for sealed attics with different roof pitches.

Pitch | T_{ct} (K) | T_{rb} (K) | Nu =
HQ_{t}/2W(T_{ct} − T_{rb})λ | Ra =
ρ^{2}c_{p}gβH^{3}(T_{ct} − T_{rb})/λµ |
---|---|---|---|---|

3/12 | 278.03 | 273.87 | 42.57 | 6.14 × 10^{8} |

5/12 | 277.94 | 273.83 | 72.23 | 2.81 × 10^{9} |

8/12 | 277.71 | 273.76 | 122.08 | 1.11 × 10^{10} |

12/12 | 277.34 | 273.66 | 201.43 | 3.47 × 10^{10} |

18/12 | 277.05 | 273.53 | 321.64 | 1.12 × 10^{11} |

**Figure 6.**Correlation for heat transfer in sealed attics developed based on the numerical predictions.

#### 3.2. Vented Attics

**Figure 7.**Temporal evolution of (

**a**) heat gain from the ceiling boundary; (

**b**) heat gain from the roof boundaries; and (

**c**) mass flow rate across the attic space for vented attics with different roof pitches.

**Figure 8.**Periodical oscillation in vertical velocity at the location of x = 0 and y = H/2 for vented attics with different roof pitches.

**Figure 9.**Snapshots of predicted streamlines (in kg/m s) at various times in the vented attic with a 12/12 pitch.

**Figure 10.**Snapshots of predicted isotherms (in K) at various times in the vented attic with a 12/12 pitch.

**Figure 11.**Predicted (

**left**) streamlines (in kg/m s) and (

**right**) isotherms (in K) at t = 10,000 s in vented attics with roof pitches of 3/12, 5/12, 8/12, 12/12, and 18/12, respectively.

**Figure 12.**Predicted temperatures at the ceiling (

**a**) and roof (

**b**) boundaries at t = 10,000 s in vented attics with different roof pitches.

**Table 4.**Thermal resistances, in R-value (R-1 = 0.17611 Km

^{2}/W), for vented attics with different roof pitches.

Pitch | R_{t,c} (Ceiling-Side Total) | R_{a,c} (Ceiling-Side Air) | R_{t,r} (Roof-Side Total) | R_{a,r} (Roof-Side Air) |
---|---|---|---|---|

3/12 | 28.480 | 8.480 | 19.127 | 17.963 |

5/12 | 27.176 | 7.176 | 13.148 | 12.040 |

8/12 | 26.541 | 6.541 | 9.070 | 8.072 |

12/12 | 26.229 | 6.229 | 6.693 | 5.844 |

18/12 | 25.964 | 5.964 | 4.784 | 4.118 |

**Table 5.**Representative ceiling-top and roof-bottom temperatures together with derived Nusselt and Rayleigh numbers for vented attics with different roof pitches.

Pitch | T_{ct} (K) | Nu = HQ_{c}/2W(T_{ct}− T_{in})λ | Ra = ρ^{2}c_{p}gβH^{3}(T_{ct}− T_{in})/λµ | T_{rb} (K) | Nu = HQ_{r}/2W(T_{rb}− T_{in})λ | Ra =
ρ^{2}c_{p}gβH^{3}(T_{rb} − T_{in})/λµ |
---|---|---|---|---|---|---|

3/12 | 274.74 | 27.91 | 1.14 × 10^{9} | 272.63 | 13.18 | 8.29 × 10^{8} |

5/12 | 273.87 | 54.91 | 4.69 × 10^{9} | 272.49 | 32.78 | 3.75 × 10^{9} |

8/12 | 273.41 | 96.42 | 1.79 × 10^{10} | 272.34 | 78.16 | 1.49 × 10^{10} |

12/12 | 273.17 | 152.05 | 5.83 × 10^{10} | 272.24 | 162.05 | 4.95 × 10^{10} |

18/12 | 272.97 | 238.12 | 1.90 × 10^{11} | 272.16 | 345.02 | 1.64 × 10^{11} |

**Figure 14.**Correlations for heat transfer on (

**a**) the ceiling side and (

**b**) the roof side within vented attics developed based on numerical predictions.

## 4. Conclusions

- (1) all the investigated roof pitches, air flow in the sealed attics is steady and asymmetric, while that in the vented attics tend to be an essentially symmetric base flow superimposed by a periodical oscillation.
- (2) both the sealed and vented attics, the heating load monotonically increases with the roof pitch. As the roof pitch increases from 3/12 to 5/12, 8/12, 12/12, and 18/12, the attic heating load is predicted to increase by 0.6%, 2.1%, 4.6%, 6.6%, respectively, for the sealed attics and to increase by 4.8%, 7.3%, 8.6%, and 10%, respectively, for the vented attics.
- (3) mass flow rate of the ventilating air through the vented attics increases monotonically with the roof pitch.As the roof pitch increases from 3/12 to 5/12, 8/12, 12/12, and 18/12, the mass flow rate of the ventilating air is predicted to increase by 23%, 53%, 83%, and 124%, respectively.
- (4) heat transfer of turbulent air flow in both the sealed and vented attics can be satisfactorily correlated by a simple relationship between appropriately defined Nusselt number and Rayleigh number.

- (1) Investigating wind effects. The air flows presented in this study are purely driven by buoyancy. Such buoyancy-driven cases are corresponding to a worst-case scenario, because real attic ventilation is generally enhanced by winds.
- (2) Including moisture transfer. The contribution of the latent heat associated with moisture transfer may be significant for the energy performance of attics in humid climates, especially in summer times.

## Acknowledgments

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## Share and Cite

**MDPI and ACS Style**

Wang, S.; Shen, Z.
Effects of Roof Pitch on Air Flow and Heating Load of Sealed and Vented Attics for Gable-Roof Residential Buildings. *Sustainability* **2012**, *4*, 1999-2021.
https://doi.org/10.3390/su4091999

**AMA Style**

Wang S, Shen Z.
Effects of Roof Pitch on Air Flow and Heating Load of Sealed and Vented Attics for Gable-Roof Residential Buildings. *Sustainability*. 2012; 4(9):1999-2021.
https://doi.org/10.3390/su4091999

**Chicago/Turabian Style**

Wang, Shimin, and Zhigang Shen.
2012. "Effects of Roof Pitch on Air Flow and Heating Load of Sealed and Vented Attics for Gable-Roof Residential Buildings" *Sustainability* 4, no. 9: 1999-2021.
https://doi.org/10.3390/su4091999