# A9.4.6.3 Single-Layer in Cavity and Double-Layer Walls

The *U-factor* of *metal building walls* that are insulated with a single-layer in cavity or multiple layers of mineral fiber insulation (see Figure A9.4.6.3) shall be calculated using the procedure outlined in this section. For double-layer *walls*, the procedure assumes that the outer layer of insulation is compressed between the *wall* panel and girt. There may also be a thermal spacer block or *continuous insulation* present. Air *spaces* may also exist depending on the specific drape profiles.

There are nine steps in the calculation process:

Step 1—Characterize the thermal conductivity of the mineral fiber insulation.

Step 2—Define the parabolic profiles for each insulation layer.

Step 3—Calculate the *R-values* for insulation and air *spaces* in cavity both outside and inside insulation layers, including air films.

Step 4—Calculate the *R-value* inside the girt and adjacent to the web.

Step 5—Calculate the *R-value* outside the girt.

Step 6—Add the *R-values* inside and outside the girt, including air films.

Step 7—Calculate the overall insulation assembly using the *R-values* in Steps 3 and 6.

Step 8—Calculate the *U-factor* from the finite element analysis results.

Step 9—Calculate the *U-factor* for any *continuous insulation* if present.

**Step 1:** The thermal conductivity of the mineral fiber insulation is represented by a thermal curve of the form in Equation A9.4-22:

(A9.4-22) |

where

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

ρ_{o} |
= | nominal density, lb/ft^{3} |

δ_{o} |
= | nominal thickness, ft |

y |
= | thickness of insulation, ft |

A |
= | 0.014917 |

B |
= | 0.0004377 |

C |
= | 0.00056897 |

**Step 2:** Assume that each layer of mineral fiber has a parabolic profile defined by Equation A9.4-23:

(A9.4-23) |

where

x |
= | distance from edge of girt, ft |

y |
= | distance from edge of wall panel, ft |

Y_{o} |
= | insulation thickness at x = 0, ft |

Y_{m} |
= | insulation thickness at x = X, ft_{m} |

**Step 3:** Calculate *R-values* for the insulation and air *spaces* in the cavity both inside and outside insulation layers, including air films.

Because the configuration can possibly consist of both mineral fiber insulation and an air *space*, the composite is given by Equation A9.4-24:

(A9.4-24) |

where *k _{a}* is the thermal conductivity of air in Btu•ft/h•ft

^{2}•°F.

The trapezoidal integration method is used to evaluate the integral and calculate *R* and is given by Equation A9.4-25:

(A9.4-25) |

where

x_{k} |
= | point to analyze along the x-axis, ft |

x_{k+}_{1} |
= | point ahead of the point being analyzed, ft |

y_{k} |
= | thickness at point being analyzed, ft |

y_{k+}_{1} |
= | thickness at point ahead of the point being analyzed, ft |

The integral represents the combined *R-value* of the air *space* and insulation over the region 0 < *x < X _{a}*. Because the thermal conductivity of air is independent of the thickness, Equation 9.4.25 can be simplified using the air

*space*mean thickness (

*Y*) to produce Equation A9.4-26:

_{a}(A9.4-26) |

However, if the air *space* is characterized by convection instead of conduction then the term *Y _{a}/k_{a}* can be replaced by the

*R-value*for convection (R-0.92 h•ft

^{2}•°F/Btu for

*walls*). Adding the inside and outside layers is expressed in Equation A9.4-27:

(A9.4-27) |

Add the air film resistances at the exterior (*R _{AT}*) and interior (

*R*), which are defined as Equation A9.4-28:

_{AB}(A9.4-28) |

where *h _{AB}* is the air film heat transfer coefficient at the exterior in Btu/h•ft

^{2}•°F.

(A9.4-29) |

where *h _{AT}* is the air film heat transfer coefficient at the interior in Btu/h•ft

^{2}•°F.

The sum of the *R-values* for the insulation and air films beyond the girt are expressed in Equation A9.4-30.

(A9.4-30) |

**Step 4:** Calculate the *R-values* inside the girt and adjacent to the web.

The *R-values* inside the girt are the air *space* (RAUP) added in series with the insulation (R2UP); their combined value is then added in parallel to RPUP. Depending on the thickness of the air *space*, it can be modeled as conduction as shown in Equation A9.4-31:

(A9.4-31) |

where

H3 |
= | thickness of the air space, ft |

k_{a} |
= | thermal conductivity of air, Btu•ft/h•ft^{2}•°F |

When appropriate, the air *space* can be modeled as convection, which is a constant R-0.92 h•ft^{2}•°F/Btu for *walls.*

The *R-value* for R2UP is expressed in Equation A9.4-32. The insulation thickness is also not limited by the girt height and can extend beyond it.

(A9.4-32) |

where *H4* is the thickness of the mineral fiber at *x* = 0 in feet.

The *R-value* of the web (RPUP) is calculated using 26.2 h•ft^{2}•°F/Btu as the thermal conductivity of the girt in Equation A9.4-33:

(A9.4-33) |

where

k_{p} |
= | thermal conductivity of the girt, Btu•ft/h-ft^{2}•°F |

Web Height | = | height of the girt, ft |

The addition of the air *space* and insulation in series are combined to be in parallel with the girt, which is expressed as Equation A9.4-34:

(A9.4-34) |

Equation A9.4-34 can be rearranged and solved for RUP as presented in Equation A9.4-35:

(A9.4-35) |

Because the thickness of the girt is significantly less than the flange width (*L _{j}*), Equation A9.4-35 can be simplified as Equation A9.4-36. However it is important to note that RUP will be close to 2 or lower (depending on how the air is modeled) because of the significant effects of the steel girt:

(A9.4-36) |

**Step 5:** Calculate the *R-value* outside the girt.

Typical *construction* above the girt consists of a thermal spacer block and compressed mineral fiber insulation. These two insulations are in series, and the total *R-value* (*R _{OPI}*) is expressed as Equation A9.4-37. If there is thermal break tape present it is included as the third insulation in this series.

(A9.4-37) |

where

H1 |
= | thickness of thermal spacer block, ft |

H2 |
= | thickness of compressed mineral fiber insulation, ft |

k_{f} |
= | thermal conductivity of the thermal spacer block, Btu•ft/h•ft^{2}•°F |

k_{I} |
= | thermal conductivity of the compressed mineral fiber insulation, Btu•ft/h•ft^{2}•°F |

The impact of the thermal bridging associated with the outside of the girt and the insulation is to reduce the *thermal resistance* of the insulation. The reduction is calculated using Equation A9.4-38:

(A9.4-38) |

**Step 6:** Add the *R-values* inside and outside the girt including air films.

The total *thermal resistance* associated with the girt is the sum of the *R-values* inside and outside the girt as shown in Equation A9.4-39:

(A9.4-39) |

The next calculation is to add the inside and outdoor air film coefficients using Equation A9.4-40:

(A9.4-40) |

**Step 7:** Calculate the overall insulation assembly using the *R-values* in Steps 3 and 6.

The overall insulation *system R-value* is determined using Equation A9.4-41:

(A9.4-41) |

**Step 8:** Calculate the *U-factor* from the finite element analysis results.

The overall *U-factor* for the insulation assembly is determined using Equation A9.4-42:

(A9.4-42) |

where

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

**Step 9:** Calculate the overall *U-factor* for any *continuous insulation* if present.

If there is any *continuous insulation* present, first calculate the *R-value* adjacent to the flange using Equation A9.4-43:

(A9.4-43) |

where

R_{BFci} |
= | thermal resistance of continuous insulation adjacent to the flange, h•ft^{2}•°F/Btu |

R_{ci} |
= | thermal resistance of the continuous insulation, h•ft^{2}•°F/Btu |

h_{ci} |
= | thickness of the continuous insulation, ft |

Next, calculate the area-weighted *R-value* for the *continuous insulation* using Equation A9.4-44:

(A9.4-44) |

where

R_{oci} |
= | overall thermal resistance of continuous insulation in h•ft^{2}•°F/Btu |

Finally, calculate the overall *U-factor* using Equation A9.4-45:

(A9.4-45) |

**Figure A9.4.6.3 Geometry of single cavity layer or double-layer wall.**

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