Insulation R-value upgrade: the UA reduction

Why the first layer of insulation matters most

The relation is a before-and-after pair. Divide the assembly’s area by its existing resistance and the leak as built appears; divide the same area by the upgraded resistance and the leak as improved appears; the reduction is their difference. Both intermediate figures are returned alongside it, because the pair tells a retrofit story the difference alone hides — a modest reduction can mean either a good assembly barely improved or a terrible one transformed.

The reciprocal is where intuition fails and this page earns its keep. Doubling an assembly’s resistance halves its leak; doubling it again removes only half of what remains. Each added increment of R buys less than the one before — diminishing returns are not a rule of thumb here but the shape of division itself. That is why an uninsulated cavity is the bargain of the housing stock and why over-stuffing an already-thick loft mostly buys bragging rights.

The units are a genuine trap, and the pack’s field help defuses it: these are SI R-values, in square-metre-kelvins per watt, while the ratings printed on batts and boards in North American merchants are imperial and about five and a half times larger — divide an imperial rating by 5.678 before entry, exactly as the input help instructs. An undivided imperial figure describes an assembly resistant beyond any real wall, and the input band is sized to refuse it.

Both resistances describe the WHOLE assembly, not a product label. An existing wall is already a stack — films of still air, sheathing, cladding, plaster, whatever sits in the cavity — and its R is the stack’s total; the upgraded figure is the same stack with the new layer counted in. The sibling payback page takes the added layer’s own rating instead and does this addition internally; this page expects the totals, which is the difference between the two near-twins’ inputs.

Area is the lever multiplying everything. The same resistance jump on a sprawling roof plane moves the ledger by far more than on a sliver of wall, which is why triage reads area-times-improvement rather than improvement alone. The reduction this page returns feeds the annual-saving sibling, where the climate multiplies it into kilowatt-hours and the tariff into money — no economics happens here, by design.

When this calculation is used

• Turning a proposed batt, board or blown-fill upgrade on one assembly into the conductance reduction the saving and payback arithmetic consume.
• Comparing candidate assemblies for the same money — the roof’s reciprocal gap against the wall’s, area included, before quotes are gathered.
• Rebuilding a whole-house ledger after a measure: the after-figure replaces the assembly’s line in the roll-up page.
• Demonstrating diminishing returns to a client or a household — the same increment applied to a poor and a good assembly, side by side.

Worked example

Run the pack’s declared anchor upgrade: a loft-sized area whose modest as-built SI resistance is raised to well over double by the new layer — the reference retrofit the payback sibling prices from its own inputs.

The engine returns the conductance before, the conductance after, and the reduction between them. Read the pair first: the before-figure is the assembly’s standing leak, and the fraction of it the reduction removes is the honest headline of the retrofit.

Every figure in this example is produced by the certified engine when the calculator loads, checked against the signed pack’s declared test vectors; nothing on this page stores an answer. Re-run with the same increment added to the already-upgraded assembly and watch the reduction shrink — the reciprocal’s diminishing return, live.

What each input represents

Assumptions and limits

• One resistance describes the whole assembly uniformly: thermal bridging through studs, rafters and fixings must already be averaged into the R entered, or the reduction will flatter the measure.
• Heat flow is steady-state and one-dimensional through the plane; corners, junctions and air movement within the cavity are outside the relation.
• The reduction is pure conductance, deliberately unpriced — climate, system efficiency and tariff belong to the saving and payback siblings.
• Moisture, compression and installation quality can leave an installed layer below its rated resistance; the engine computes from the figures entered, not from the workmanship.

What the guards protect against

• An upgraded resistance that fails to exceed the existing one is refused — the pack ships the refusal as a declared test vector — because the relation describes an improvement, and a negative or zero reduction dressed as a result would let a data-entry slip masquerade as physics.
• Both resistances are held to the band real SI assemblies occupy, which is what catches an imperial rating entered undivided — a figure several times too large that would understate every leak it touched.
• The area must be positive: an assembly with no surface has no conductance to reduce, so a zeroed field is refused rather than returning a vacuous nothing-changed.

Provenance

Envelope conductance from thermal resistance: component UA as area over R-value, reductions as the difference of reciprocals

The assembly’s area is divided by the existing SI R-value for the conductance before, by the upgraded R-value for the conductance after, and the reduction is returned as their difference alongside both intermediates.

Screening and reference material, to be checked against the governing standard and a qualified engineer; not a design determination. The signed pack carries its own citation, which displays from the verified leaf when the calculator loads.