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Pitch value is an important consideration in residential gable roof design and construction. However, how roof pitch, coupled with air flows in attic space, affects the energy performance of building attics has been barely investigated. In this paper, a 2D unsteady computational fluid dynamics (CFD) model is employed to investigate the effects of roof pitch on air flow and heating load of both sealed and vented attics for gable-roof residential buildings. The simulation results show that air flow in the sealed attics is steady and asymmetric, while that in the vented attics is a combination of an essentially symmetric base flow and a periodically oscillating flow. For both the sealed and vented attic cases, the heating load is found to increase with the roof pitch, and the heat transfer of turbulent air flow in attic space can be satisfactorily correlated by a simple relationship between appropriately defined Nusselt number and Rayleigh number.

_{p}

specific heat, J kg^{−1} K^{−1}

elliptic relaxation function

gravitational acceleration, m s^{−2}

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

attic height, m

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

turbulence length scale, m

Nusselt number

pressure, Pa

_{atm}

atmospheric pressure, Pa

heat flux, Wm^{−2}

heat transfer rate, W/m

thermal resistance, K m^{2}/W

Rayleigh number

strain rate, s^{−1}

_{c}

ceiling thickness, m

temperature, K

_{0}

reference temperature, K

_{in}

inlet air temperature, K

velocity component, m/s

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

half width of attic, m

coordinates, m

thermal expansion coefficient, K^{−1}

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

thermal conductivity, W m^{−1} K^{−1}

_{c}

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

molecular viscosity, kg m^{−1} s^{−1}

_{t}

eddy viscosity, kg m^{−1} s^{−1}

density, kg m^{−3}

_{k}

effective turbulence Prandtl number for

_{ε}

effective turbulence Prandtl number for

_{T}

effective turbulence Prandtl number for

turbulence time scale, s

air

ceiling

ceiling-bottom

ceiling-top

roof

roof-bottom

roof-top

total

Pitched roofs are widely used in residential building construction. In general, houses in areas with more precipitations have steeper roofs. The steepness of a roof is quantitatively measured by roof pitch, which is usually expressed as a rational fraction between the vertical rise and the corresponding horizontal run distance.

For a given roof footprint, a steeper roof requires a larger area of roof material and forms a larger and higher attic space. Therefore, it is natural to expect that roof pitch, coupled with air flows in attic space, will affect the energy performance of a building. The primary objective of this study is to numerically investigate the effects of roof pitch on air flow and heating load of sealed and vented attics for gable-roof residential buildings, which can be reasonably represented by 2D numerical models.

Attic ventilation has been a common practice in the United States since 1940s [

Despite these long-established guidelines for attic ventilation in residential building construction and retrofit, the technical data used to derive these ventilation requirements are very limited [

The majority of the previous research on heat transfer in attic spaces is concerned with natural convection in triangular enclosures, as reviewed by Kamiyo

In the present study, the effects of roof pitch on the thermal performance of gable-roof residential buildings are investigated by simulating attics with roof pitches of 3/12, 5/12, 8/12, 12/12, and 18/12, respectively. For each roof pitch value, both a sealed attic case and a vented attic case are modeled. Similar to [

A schematic diagram of a cross-section plan of the physical model is shown in _{0}, that is specified to the outside ambient air temperature to correctly calculate the buoyancy effects. In all the cases reported in this study, _{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

where _{c} and _{c} are the thickness and thermal conductivity of the ceiling; _{ct} and _{cb} are the temperatures at the ceiling-top and ceiling-bottom, respectively; and the heat transfer coefficient _{c} is the reciprocal of the ceiling thermal resistance.

Schematic of the computational domain and boundary conditions for (

In this study, a ceiling-bottom temperature is specified to _{cb} = 293 K, while a heat transfer coefficient of _{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, ^{2}/W). Similarly, a roof-top temperature of _{rt} = 273 K and a heat transfer coefficient of _{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.

For all the cases investigated in this study, balanced vent areas are assumed. Both the left and right soffit vents are assumed to be 1 cm wide, while the ridge vent is assumed to 2 cm wide, resulting in a ventilation ratio of 1/200. For the sealed attic cases, as shown in _{atm}. Therefore, the obtained air flow is purely driven by the thermally induced buoyancy forces,

Following the URANS approach to turbulence, the distributions of time-averaged air velocity

where _{p}_{T}_{ij}

The turbulence model employed in this study is the v2f model [_{t}

where _{µ}

where

and the turbulence time scale and length scale are defined by

The model constants take the following default values [

The governing equations formulated above are solved by the commercial CFD software Ansys Fluent 13.0 [^{+} value for the first grid close to the walls is everywhere less than 1.

The modeling results presented in this paper are based on grids consisting of about 30,000–50,000 nodes and a time step size of 1 s. All the calculations start from initial conditions of zero velocity and uniform temperature. Within each time step, 20 iterations are executed. Numerical experiments show that decreasing the time step to 0.5 s or requiring 40 iterations in each time step generate negligible difference in solutions. In addition, a grid dependence test shows that refinement of the grids by doubling the node numbers results in less than 2% difference in the total mass flow rate and wall heat transfer rate results.

For sealed attics with different roof pitches and subject to the boundary conditions depicted in

The predicted steady streamlines and isotherms are shown in

However, as the Grashof number was increased above a critical value, the flow was observed to undergo a supercritical pitchfork bifurcation, in which case one of two possible mirror-image asymmetric flows was obtained, since the symmetric flow was no longer physically stable. The existence of multiple solutions above the pitchfork bifurcation point (including both stable and unstable solutions) necessitates that particular care be exercised during numerical simulation, e.g., setting sufficient iterations to allow finite numerical perturbations to grow, so that the solution can deviate from an unstable symmetric result and eventually converge to one of the two stable asymmetric results.

For all the roof pitches investigated, both the flow and temperature fields shown in

Predicted (

Predicted steady profiles of (

Predicted steady temperatures at the ceiling (

For the given ceiling-bottom temperature of 293 K, a decrease in the ceiling-top temperature indicates a greater temperature difference across the ceiling thickness and thus an increase in the attic heat loss, reflecting the enhanced heat transfer effect associated with a stronger natural convection. The decrease of the average roof-bottom temperature with the increasing roof pitch shown in

The results shown in

The integrated heat transfer rates across the ceiling and roof boundaries are compared in

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) | _{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 |

It is also clear from

A correlation for heat transfer dominated by natural convection is typically expressed as a relationship between Nusselt number (Nu) and Rayleigh number (Ra). In case of heat transfer in attic spaces, both the dimensionless numbers should be defined based on a representative constant temperature difference across the attic space, _{ct} − _{rb}. Although it is possible to obtain this temperature difference by subtracting a curve in _{ct} − _{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.

The thermal resistance diagram for the sealed attics, as shown in _{t}, equals the sum of the thermal resistances of the sub-systems

and can be evaluated by

where _{t} is the total heat transfer rate, as listed in the last column of _{r} column in ^{2} + ^{2})^{1/2}/

Diagram of thermal resistance for sealed attics.

Thermal resistances, in R-value (R-1 = 0.17611 Km^{2}/W), for sealed attics with different roof pitches.

Pitch | _{t} (Total) |
_{c} (Ceiling) |
_{r} (Roof) |
_{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 |

In terms of the resistance diagram (

More experimental or numerical data are needed to fully test the applicable domain of this correlation, including if it is approximately valid for situations beyond uniform thermal conditions, e.g., those with slight variations in roof-top and ceiling-bottom temperatures.

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

Pitch | _{ct} (K) |
_{rb} (K) |
Nu =
_{t}/2_{ct} _{rb} |
Ra =
^{2}_{p}gβH^{3}(_{ct} _{rb} |
---|---|---|---|---|

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} |

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

For vented attics with different roof pitches and subject to the boundary conditions depicted in

Different from the sealed attic cases in which the steady solutions are obtained, the converged solutions for the vented attics actually include periodical oscillation, which is reflected by the weak waves on some of the curves in

Temporal evolution of (

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

In order to clearly show a detailed picture of this periodical oscillation, snapshots of the velocity and temperature fields at selected instances within one oscillation cycle for the 12/12 pitch case are given in

Although more theoretical and experimental studies are needed to determine the fundamental mechanism that generates this periodical oscillation, we consider such oscillation as physically meaningful phenomenon, instead of a reflection of numerical singularity, based on the following observations. Firstly, the periodical oscillation is independent of the selection of turbulence model. It is found that essentially the same periodical oscillation also appears in solutions based on the SST

The predicted temperature distributions at the ceiling and roof boundaries at

The overall thermal performance of the vented attics can be quantified based on a thermal resistance analysis, similar to the analysis for the sealed attics. The thermal resistance diagram for the vented attics is shown in

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

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

Predicted (

Predicted temperatures at the ceiling (

Diagram of thermal resistance for vented attics.

Thermal resistances, in R-value (R-1 = 0.17611 Km^{2}/W), for vented attics with different roof pitches.

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

In terms of the resistance diagram (

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

Pitch | _{ct} (K) |
Nu = _{c}_{ct}_{in}) |
Ra = ^{2}_{p}gβH^{3}(_{ct}_{in})/ |
_{rb} (K) |
Nu = _{r}/2_{rb}_{in}) |
Ra =
^{2}_{p}gβH^{3}(_{rb} _{in})/ |
---|---|---|---|---|---|---|

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} |

Accordingly defined Nusselt numbers and Rayleigh numbers for both the ceiling and roof sides are listed in

for the ceiling side air flow, and

for the roof side air flow. Both the correlations are expected to be valid for vented attics under conditions similar to those specified in this study. More experimental or numerical data are needed to test if these correlations can be generalized to cover a wider parameter range.

Correlations for heat transfer on (

In this study, the buoyancy-driven turbulent air flows of the sealed and vented attics under winter conditions are simulated in terms of the CFD model. In particular, the impacts of roof pitch on air flow and heating load for sealed and vented attics are investigated. The findings from the numerical results are summarized as follows:

(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.

It should be noted that the above-mentioned conclusions are subject to the limitations of the modeling assumptions adopted in this study. This research can be furthered in the following aspects:

(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.

This study was partially supported by the Faculty Seed Grants from the Research Council of the University of Nebraska-Lincoln (2009–2010).