# The Impact of Roof Pitch and Ceiling Insulation on Cooling Load of Naturally-Ventilated Attics

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

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

_{0}= 305.15 K is assumed.

_{cb}and T

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

_{c}is determined by the thermal conductivity of ceiling material ${\lambda}_{\text{c}}$ divided by ceiling thickness ${t}_{\text{c}}$. It should be note that h

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

_{c}, which may be expressed in R-value, i.e., R-1 = 1 h·ft

^{2}·°F/Btu = 0.176110 Km

^{2}/W.

_{cb}= 295.15 K, while a heat transfer coefficient of h

_{c}varying between 4.733 W/m

^{2}K to 0.142 W/m

^{2}K is adopted to approximate a ceiling insulation level between R-1.2 (R

_{c}= 1.2 × 0.176110 Km

^{2}/W = 0.211 Km

^{2}/W) and R-40 (R

_{c}= 7.044 Km

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

_{rt}= 345.15 K and a heat transfer coefficient of h

_{r}= 4.733 W/m

^{2}K (equivalent to an insulation level of R-1.2) are specified to the roof boundary to simulate a condition of a 3 cm plywood roof.

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

_{c}) and roof heat gain (Q

_{r}) results are less than 2% (Table 1).

**Table 1.**Summary of grid and time-step dependence tests for a 5/12 attic with R-20 ceiling insulation.

Case | A | B | C |
---|---|---|---|

Total nodes | 26,508 | 26,508 | 53,430 |

Total elements | 46,835 | 46,835 | 88,715 |

Time step (s) | 1 | 0.5 | 1 |

Iterations per time step | 20 | 40 | 20 |

Q_{c} (W/m) | 13.85555 | 13.85551 | 13.58413 |

Q_{r} (W/m) | 216.5511 | 216.5506 | 215.6445 |

$\dot{m}$ (kg/s m) | 0.00923552 | 0.00923555 | 0.00908235 |

**Figure 2.**Predicted profiles of (

**a**) horizontal velocity along the vertical line x = 2 m and (

**b**) temperature along the symmetric line x = 0 for gird and time-step dependence test cases defined in Table 1.

**Figure 3.**Comparison of velocity distribution between (

**a**) experimental measurement [19]; (

**b**) large-eddy prediction [20]; and (

**c**) present study (streamlines in kg/m·s), together with (

**d**) the isotherms (in K) predicted by the present study, for the validation case of turbulent mixed convection in a square cavity.

**Figure 4.**Comparison of modeling predictions with experimental data for the validation case: (

**a**) vertical velocity along the section of y = 0.502 m; (

**b**) horizontal velocity along the section of x = 0.502 m; (

**c**) temperature along the section of y = 0.502 m; (

**d**) temperature along the section of x = 0.502 m.

## 3. Results and Discussion

**Figure 5.**Predicted (

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

**right**) isotherms (in K) for attics with R-20 ceiling insulation.

**Figure 6.**Predicted profiles of (

**a**) horizontal velocity along the vertical line x = 2 m and (

**b**) temperature along the symmetric line x = 0 for attics with R-20 ceiling insulation.

**Figure 7.**Predicted (

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

**right**) isotherms (in K) for attics 3/12 with R-1.2 ceiling insulation.

**Figure 8.**Predicted profiles of (

**a**) horizontal velocity along the vertical line x = 2 m and (

**b**) temperature along the symmetric line x = 0 for attics with R-1.2 ceiling insulation.

_{c}), the heat gain from roof (Q

_{r}), and the ventilating mass flow rate ($\dot{m}$) are direct modeling outputs.

Roof Pitch | 3/12 | 5/12 | 8/12 | 12/12 | 18/12 |
---|---|---|---|---|---|

Q_{c} (W/m) | 8.14 | 7.92 | 7.61 | 7.57 | 7.39 |

Q_{r} (W/m) | 182.4 | 214.3 | 260.4 | 307.1 | 391.4 |

$\dot{m}$ (kg/s·m) | 0.0074 | 0.0094 | 0.0127 | 0.0154 | 0.0193 |

T_{ct} (K) | 309.5 | 309.1 | 308.6 | 308.5 | 308.2 |

T_{av} (K) | 316.8 | 316.1 | 315.1 | 314.8 | 315.0 |

T_{rb} (K) | 335.5 | 333.8 | 331.4 | 328.9 | 324.5 |

Roof Pitch | 3/12 | 5/12 | 8/12 | 12/12 | 18/12 |
---|---|---|---|---|---|

Q_{c} (W/m) | 14.22 | 13.86 | 13.18 | 13.11 | 12.75 |

Q_{r} (W/m) | 184.6 | 216.6 | 262.2 | 308.9 | 393.2 |

$\dot{m}$ (kg/s·m) | 0.0073 | 0.0092 | 0.0125 | 0.0153 | 0.0192 |

T_{ct} (K) | 307.7 | 307.4 | 306.8 | 306.7 | 306.4 |

T_{av} (K) | 316.8 | 316.1 | 315.0 | 314.8 | 315.0 |

T_{rb} (K) | 335.4 | 333.7 | 331.3 | 328.8 | 324.4 |

Roof Pitch | 3/12 | 5/12 | 8/12 | 12/12 | 18/12 |
---|---|---|---|---|---|

Q_{c} (W/m) | 22.46 | 21.69 | 20.49 | 20.30 | 19.62 |

Q_{r} (W/m) | 188.14 | 219.5 | 264.6 | 311.2 | 395.5 |

$\dot{m}$ (kg/s·m) | 0.0071 | 0.0091 | 0.0124 | 0.0151 | 0.0190 |

T_{ct} (K) | 305.0 | 304.7 | 304.2 | 304.1 | 303.8 |

T_{av} (K) | 316.7 | 316.0 | 315.0 | 314.7 | 315.0 |

T_{rb} (K) | 335.2 | 333.6 | 331.2 | 328.7 | 324.3 |

Roof Pitch | 3/12 | 5/12 | 8/12 | 12/12 | 18/12 |
---|---|---|---|---|---|

Q_{c} (W/m) | 45.11 | 43.04 | 40.85 | 40.77 | 40.21 |

Q_{r} (W/m) | 199.0 | 227.6 | 271.4 | 318.2 | 402.6 |

$\dot{m}$ (kg/s·m) | 0.0069 | 0.0086 | 0.0119 | 0.0145 | 0.0185 |

T_{ct} (K) | 297.5 | 297.4 | 297.3 | 297.3 | 297.3 |

T_{av} (K) | 316.3 | 315.8 | 314.8 | 314.6 | 314.9 |

T_{rb} (K) | 334.6 | 333.1 | 330.8 | 328.3 | 323.9 |

_{ct}) and the average roof-bottom temperature (T

_{rb}) listed in Table 2, Table 3, Table 4 and Table 5 are determined in terms of the ceiling thermal resistance (R

_{c}) and the roof thermal resistance (R

_{r}), respectively, i.e.:

_{av}) is determined based on the overall energy balance of the attic air, i.e.:

## 4. Conclusions

- (1)
- Air flows in the attics are steady and exhibit a general streamline pattern that is qualitatively insensitive to the investigated variations of roof pitch and ceiling insulation.
- (2)
- For all the ceiling insulation levels investigated, the attic cooling load decreases by around 9% when the roof pitch increases from 3/12 to 8/12, and keeps essentially unchanged for roof pitches between 8/12 and 18/12. At the same time, along with the pitch increase, both the heat gain from the roof and the mass flow rate of the ventilating air increase by over 100%.
- (3)
- The attic cooling load increases remarkably with the decrease of ceiling insulation. For the 5/12 pitch, for example, the cooling load increases by 75%, 170%, and 440%, respectively, as the ceiling insulation drops from R-40 to R20, R10, and R-1.2. In the meantime, the heat gain from the roof increases by less than 10%, while the mass flow rate of the ventilating air decreases by less than 10%, regardless of the roof pitch. Therefore it is clear that ceiling insulation plays a dominant role in controlling cooling load of attic spaces in summer time, compared to ventilating factors.
- (4)
- Both the mass flow rate of the ventilating air and the cooling load of the attic can be satisfactorily correlated by simple relationships in terms of appropriately defined Rayleigh and Nusselt numbers.

## Acknowledgments

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**MDPI and ACS Style**

Wang, S.; Shen, Z.; Gu, L. The Impact of Roof Pitch and Ceiling Insulation on Cooling Load of Naturally-Ventilated Attics. *Energies* **2012**, *5*, 2178-2196.
https://doi.org/10.3390/en5072178

**AMA Style**

Wang S, Shen Z, Gu L. The Impact of Roof Pitch and Ceiling Insulation on Cooling Load of Naturally-Ventilated Attics. *Energies*. 2012; 5(7):2178-2196.
https://doi.org/10.3390/en5072178

**Chicago/Turabian Style**

Wang, Shimin, Zhigang Shen, and Linxia Gu. 2012. "The Impact of Roof Pitch and Ceiling Insulation on Cooling Load of Naturally-Ventilated Attics" *Energies* 5, no. 7: 2178-2196.
https://doi.org/10.3390/en5072178