# A9.4.6.2 Double-Layer Roof

The *U-factor* of *metal building roofs* that are insulated with double layers of fiberglass insulation (see Figure A9.4.6.2-1) shall be calculated using the procedure outlined in this section. The procedure assumes the insulation is compressed over the purlin and there may be a thermal spacer block present.

There are six steps in the calculation process:

Step 1—Characterize the thermal conductivity of the fiberglass.

Step 2—Determine the *U-factor* for the insulation in the cavity.

Step 3—Determine the *U-factor* over the structural framing member.

Step 4—Area weight the *U-factors* calculated in Steps 2 and 3.

Step 5—Determine the *U-factor* from the finite element analysis results.

Step 6—Determine the *U-factor* for any *continuous insulation* if present.

**Step 1:** The thermal conductivity of the fiberglass batt insulation is represented by a thermal curve of the form in Equation A9.4-12:

(A9.4-12) |

where

k | = | thermal conductivity, Btu/h•ft^{2}•°F |

ρ | = | density, lb/ft^{3} |

A | = | 0.014917 |

B | = | 0.0004377 |

C | = | 0.0056897 |

**Step 2:** Assume the double-layer fiberglass batt forms a parabolic profile defined by Equation A9.4-13:

(A9.4-13) |

The presence of two layers of fiberglass adds complexity because each layer has distinct reference properties (see Table A9.4.6.1). As the double layers are compressed, the thickness of each layer needs to be determined by considering that each layer achieves the same compressive force. Instead of having a closed-form analytical solution that predicts the *U-factor* for the cavity, the double-layer *system* requires that the parabolic profile be numerically integrated. The compression of the double-layer *system* is presented in Figure A9.4.6.2-2.

The thickness of the second layer (*Y*_{2}) is described by Equation A9.4-14:

(A9.4-14) |

where

Y_{c} | = | compressed thickness of the double layers, ft |

ρ_{o}_{1} | = | reference density of first layer, lb/ft^{3} |

ρ_{o}_{2} | = | reference density of second layer, lb/ft^{3} |

W_{1} | = | reference weight of first layer, lb/ft^{2} |

W_{2} | = | reference weight of second layer, lb/ft^{2} |

The solutions to Equation A9.4-14 are Equation A9.4-15a and A9.4-15b:

(A9.4-15a) |

(A9.4-15b) |

where

a | = | 1 |

b | = | |

C | = |

Select the smaller value of *Y*_{2,a} and *Y*_{2,b} as *Y*_{2}, *Y*_{1} shall be calculated as the difference between *Y _{c}* and

*Y*

_{2}

*.*Next, the

*R-values*for the two compressed layers of insulation shall be calculated and converted to a

*U-factor.*This process shall be repeated along the entire profile and the results numerically integrated using maximum 0.04167 ft increments.

It is important to note that Equation A9.4-14 does not apply when the two layers of insulation are the same material. In this case, each compressed layer has the same thickness, which simplifies the *U-factor* calculations. The numerical integration still needs to be completed to determine the *U _{co}.*

**Step 3:** Determine the *U-factor* over the structural framing member. The variable *Y _{o}* represents the thickness of the thermal spacer block and the thickness of the compressed insulation. The density of the compressed insulation is determined by Equation A9.4-16:

(A9.4-16) |

where

ρ_{c} | = | density of the compressed insulation over the framing member, lb/ft^{3} |

t_{c} | = | thickness of the compressed insulation over the framing member, ft |

The *thermal resistance* of the compressed insulation is determined by Equation A9.4-17:

(A9.4-17) |

Determine the overall framing *U-factor* (*U _{fo}*) at the structural framing member, including the air film resistances, using Equation A9.4-18:

(A9.4-18) |

where

U_{fo} | = | U-factor over the structural framing member, Btu/h•ft^{2}•°F |

R_{TB} | = | R-value of the thermal spacer block, h•ft^{2}•°F/Btu |

R_{c} | = | R-value of the compressed insulation, h•ft^{2}•°F/Btu |

**Step 4:** Determine the overall area-weighted *U-factor* for the entire *system* using Equation A9.4-19:

(A9.4-19) |

where

U_{es} | = | area weighted U-factor for the entire system, Btu/h•ft^{2}•°F |

**Step 5:** Calculate the adjusted overall *U-factor* (*U _{adj}*) using Equation A9.4-20:

(A9.4-20) |

where

U_{adj} | = | adjusted overall U-factor represented by correlation with the finite element modeling, Btu/h•ft^{2}•°F |

**Step 6:** If there is any *continuous insulation* present, calculate the overall *U-factor* using Equation A9.4-21:

(A9.4-21) |

**Figure A9.4.6.2-1 Geometry of double layers of fiberglass batts.**

where

X | = | distance from edge of purlin or girt, ft |

Y | = | distance from edge of roof panel or wall panel, ft |

L | = | length from edge of purlin or girt to centerline of cavity, ft |

w_{f} | = | width of purlin or girt flange, ft |

Y_{o} | = | distance between purlin or girt and the roof panel or wall panel, ft |

Y_{m} | = | distance from edge of roof panel or wall panel at the cavity centerline, ft |

**Figure A9.4.6.2-2 Compression of double layers of fiberglass insulation.**

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