Thermal Bridge Calculation: Method, Formulas and a Worked Psi-Value
A thermal bridge calculation models heat flow through a junction, such as a slab edge, window frame or parapet. It returns the extra heat loss as a linear transmittance (ψ, in W/(m·K)) or a point transmittance (χ, in W/K), plus the coldest interior surface temperature for condensation checks. The steps are the same every time: define the geometry, assign conductivities and boundary conditions, solve, subtract the one-dimensional losses, and check the result against the rules that apply to your project.
This guide walks through that sequence, shows a worked psi-value with numbers, and explains where the calculated values go next.
TL;DR
A thermal bridge is a localized area where heat flow rises because of a change in material or geometry. The calculation quantifies that extra heat loss and the minimum surface temperature.
Linear bridges get ψ in W/(m·K). Isolated three-dimensional bridges get χ in W/K. Never convert one into the other by dividing by an arbitrary length.
ψ equals the modeled heat flow minus the heat loss already counted by U × area, divided by the junction length and temperature difference.
fRsi = (θsi − θe) / (θi − θe) uses the coldest surface point from the model.
Use numerical modeling for final values, catalogues for early design, and follow the method your local regulation names.
Table of Contents
What a thermal bridge calculation actually determines
A thermal bridge is a localized part of the building envelope where heat flow increases because the thermal resistance changes, either through a material penetration or a geometric transition. Passipedia’s definition of thermal bridges adds what the calculation is for: it determines the heat loss through linear or point transmittance, and the minimum internal surface temperature used to judge energy efficiency and condensation risk.
So you are really after two outputs from one model:
Heat loss. A ψ or χ value that feeds the whole-building energy balance.
Surface temperature. The coldest interior point, which becomes a temperature factor for mold and condensation checks.
The same Passipedia page states the whole-envelope transmission heat loss as the sum of three parts: planar elements (area × U), linear bridges (length × ψ) and point bridges (χ). That sum is why ψ must contain only the extra loss. If it included the planar loss too, the building model would count that heat twice.
What you need before you start
A clean model starts with a clean input list. Collect these before opening any solver.
Geometry. A section drawing of the junction at true dimensions, including every layer thickness, fixings, thermal breaks and frames. ISO 10211 covers 2D and 3D thermal bridge models, including their geometrical boundaries and subdivisions.
Material conductivities. Each material cell gets one homogeneous thermal conductivity. The 2007 edition of ISO 10211 describes dividing the construction into material cells, assigning those conductivities and applying boundary conditions. Use declared design values, not marketing figures.
Boundary conditions. Interior and exterior air temperatures and surface resistances. Two details matter here:
The standard assumes material properties do not change with temperature and that there are no heat sources inside the building element.
Surface resistances should follow the convention of the method you report under. Mixing conventions between the wall U-value and the junction model is a common source of wrong ψ values.
If you want a structured way to pick boundary conditions, our guide to selecting temperature boundary conditions starts from the dominant resistance in the path, which is also a sensible habit for a building junction: the surface films and the insulation layer usually dominate.
Cut-off planes. The model must stop at symmetry lines or at a sufficient distance from the central element. The same ISO text calls this distance dmin, and it must be large enough for the total heat flow to be accurate. Cutting the model too close to a slab edge or window frame is the easiest way to get a ψ that looks plausible and is wrong.
2D or 3D: choosing the model
Use a 2D model when the cross-section stays the same along the junction: a slab-to-wall edge, a parapet, a window head, jamb or sill running a long way. The result is a linear transmittance ψ.
Use a 3D model when the section changes in all three directions. In ISO terms, a three-dimensional geometrical model has a cross-section that changes perpendicular to each of the three orthogonal axes within the model boundary. Typical cases are a balcony connector, a post or anchor penetrating insulation, or a three-wall corner.
A point thermal bridge gets χ in W/K. Passipedia is explicit that linear bridges use ψ in W/(m·K) while isolated 3D bridges use χ in W/K, so do not divide a point-bridge result by an arbitrary length and call it ψ. If a repeating fixing is spread over a wall, the usual practice is to fold it into the wall’s U-value or report it as a count of χ values. Follow the method your regulation uses.
Calculating ψ from modeled heat flow
The principle: the model gives you the total heat flow through a piece of construction that contains the junction. Part of that flow would exist anyway through the plain wall or slab. Only the remainder belongs to the junction.
The Passive House New Zealand instructions for Therm put it this way: ψ is the heat loss left after deducting the one-dimensional U-value heat loss already accounted for by U multiplied by area. The result is an additional heat loss per unit junction length and per unit temperature difference.
In equation form, working per kelvin:
ψ = L₂D − Σ(Uᵢ × lᵢ)
L₂D is the thermal coupling coefficient from the model: total heat flow per metre of junction divided by the temperature difference, in W/(m·K).
Uᵢ is the one-dimensional U-value of each adjoining element, in W/(m²·K).
lᵢ is the length of that element’s surface within the model, measured with the same dimension convention you use in the building model, in m.
The dimension convention matters. ψ values calculated with internal dimensions are not interchangeable with those from external dimensions. Irish assessment guidance, for instance, says ψ depends on the junction geometry, the measurement conventions and the level of insulation at the junction, according to the DEAP guide. State which convention you used next to every value you report.
Worked example: an insulated slab edge
The numbers below are illustrative, chosen to show the arithmetic, not taken from a real project.
Setup. A 2D model of a concrete slab meeting an insulated wall, with a 20 K temperature difference between inside and outside. The model includes 3.0 m of wall surface (internal dimensions) and the slab section.
Item | Value |
Wall U-value (clear field) | 0.30 W/(m²·K) |
Wall length in model | 3.0 m |
Planar loss, U × l | 0.90 W/(m·K) |
Modeled total heat flow per metre | 21.2 W/m |
Temperature difference | 20 K |
Coupling coefficient L₂D | 1.06 W/(m·K) |
Step 1. Run the solver and read total heat flow: 21.2 W per metre of slab edge.
Step 2. Divide by the temperature difference: 21.2 / 20 = 1.06 W/(m·K).
Step 3. Subtract the planar loss already counted by the wall U-value: 1.06 − 0.90 = ψ = 0.16 W/(m·K).
Step 4. In the building model, multiply by the length of the junction. A 40 m slab edge adds 40 × 0.16 = 6.4 W/K to the transmission heat loss coefficient.
If the slab were also adjoined by a floor element with its own U-value, you would subtract that element’s U × l as well. Forgetting one adjoining element is the classic way to get a ψ that is too high.
Corners, parapets and windows
The workflow above is identical for every junction. What changes is where you cut the model and which elements you subtract.
Wall corners. Model a horizontal section through the corner. The geometry usually produces a negative ψ when using external dimensions, because the outside surface area is larger than the inside one and the dimension convention over-counts the loss. That is normal, not an error.
Parapets and roof edges. Model a vertical section, with the roof and wall both extended far enough that the cut-offs sit in undisturbed one-dimensional flow. Check that the insulation continuity line is drawn exactly as built: a gap of a few millimeters at the upstand can dominate the result.
Window installation. Model head, jamb and sill separately. Each has a different frame position and insulation overlap, so each gets its own ψ. The frame itself needs its own U-value, typically from a separate frame calculation or a manufacturer’s declared value, and the installation ψ covers only the interface between frame and wall.
In every case, vary one thing at a time. Move the window frame 30 mm toward the insulation and rerun. A parametric run tells you which detail decision matters, which is more useful than a single number.
Surface temperature and the fRsi check
The same model that gives you heat flow also gives a temperature field. Find the coldest point on the interior surface, then compute the temperature factor.
The Passive House Institute criteria give it as:
fRsi = (θsi − θe) / (θi − θe)
where θsi is the minimum internal surface temperature, θi the internal design temperature and θe the external design temperature.
Illustrative numbers: with θi = 20 °C, θe = −5 °C and a modeled θsi of 16.5 °C, fRsi = (16.5 + 5) / 25 = 0.86. A higher factor means a warmer surface relative to the temperature difference. Pass or fail limits come from your regulation, your climate and your humidity assumptions, so read the threshold from the document that governs your project rather than from a general article.
Numerical modeling, catalogues and validation
You have two routes to a ψ value:
Route | Where it fits | Trade-off |
Numerical modeling (2D/3D) | Final design, compliance documentation, novel details | Needs a validated solver and careful inputs |
Catalogue or tabulated values | Early design, quick comparisons | Less precise |
ISO 14683 gives guidelines for simplified methods such as catalogues. They are less precise than numerical modeling but useful for preliminary design.
Whichever solver you use, validate it. A European review of thermal bridge handling recommends that software be checked against the EN ISO 10211 validation cases. Run those cases once, keep the output, and you have evidence for any reviewer who asks.
Where the values go: regulations by region
The calculation method is international. The way results are applied is local.
International. ISO 10211:2017 specifies 2D and 3D numerical models for total heat flows and internal surface temperatures, and can be used to derive linear and point transmittances. The ISO page lists it as confirmed in 2022.
United Kingdom. In SAP, thermal bridge heat loss is computed from junction lengths and ψ values. Conventions follow BRE Report BR 497 and Information Paper IP 1/06, and under those routes calculated values are increased by 0.02 or 25%, whichever is larger, except in Scotland, according to SAP 2009 Appendix K. That document is dated, so check the current SAP edition for present rules.
Ireland. Psi-values enter DEAP according to its own conventions, described in the DEAP guide linked above.
United States and Canada. No single national ψ method applies, and building envelope guides generally treat the clear-field U-value and the junction effects together, so confirm which values your energy code or program accepts.
Also keep in mind that researchers have found existing approaches for implementing thermal bridge effects have not kept pace with rising insulation requirements. As walls get better insulated, junctions make up a larger share of total loss, which is a reason to model them rather than default to a flat percentage.
Common mistakes
Double counting. Not subtracting U × l for every adjoining element.
Mixed conventions. Internal-dimension ψ used in an external-dimension model, or the reverse.
Short models. Cut-off planes inside the disturbed zone.
Wrong surface resistances. Using generic values where the standard or program prescribes different ones, especially in corners behind furniture.
Treating χ as ψ. Spreading a point-bridge result over an arbitrary length.
Skipping a mesh check. Halve the cell size near the junction and confirm ψ moves by a negligible amount.
For the physics underneath, our heat transfer model guide covers conduction, convection and radiation methods and the maths behind them. If you want to cross-check the radiative part of a cavity or surface film, see thermal radiation calculation.
Our perspective
Most thermal bridge errors we see are not solver errors. They are bookkeeping errors: a missing subtraction, a mismatched convention, a cut-off plane drawn too close. A tool that keeps conduction, convection and radiation in one place, with the material properties stored next to the model, removes a lot of those slips. That is how we built the Jewlz Engineering Toolkit. It combines conduction, convection and radiation analysis with a built-in property database, so you are not retyping conductivities from three different datasheets. It handles variable material properties across a wide temperature range, which helps when you want to see how far a constant-conductivity assumption sits from reality in a cold-climate detail or an industrial envelope. Our aim is to let small teams and students run professional-grade thermal analysis without a licensing barrier. For a junction, we suggest building the model, checking it against a hand calculation like the one above, then varying the detail to see which decision moves ψ most. Joel
Run your own junction analysis
If you want to test a detail before committing to it, the Jewlz Engineering Toolkit gives you thermal analysis for conduction, convection and radiation in a downloadable Excel-based tool, with a built-in material property database. You can start from the Jewlz Engineering Toolkit page and see how our thermal analysis software handles heat flow in your geometry.
Before you rely on any result for a compliance submission, confirm which software and method your authority accepts, and keep your validation run on file.
FAQ
What is the difference between psi and chi values?
Psi (ψ) describes a linear thermal bridge, a junction that runs along a length, and is given in W/(m·K). Chi (χ) describes a point thermal bridge, an isolated three-dimensional feature, and is given in W/K. Do not divide a χ result by an arbitrary length to create a ψ.
Do I need a 3D model for every thermal bridge?
No. A junction whose cross-section stays constant along its length, such as a slab edge or a window head, can be modeled in 2D. You need 3D when the cross-section changes in all three directions, as with a balcony connector or an anchor passing through the insulation.
How do I check a thermal bridge model is right?
Run the solver on the EN ISO 10211 validation cases, then compare your junction result with a hand calculation of the planar losses. Refine the mesh near the junction and confirm ψ barely changes. Also check that your cut-off planes sit at symmetry lines or far enough from the junction.
Can I use catalogue values instead of calculating?
Often yes for preliminary design. ISO 14683 describes simplified methods such as catalogues, which are less precise than numerical modeling. For final documentation, or any detail not found in a catalogue, use a numerical model and the method named by your regulation.
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