Heat Transfer Simulator

Beta Version 1.0 I 1/12/2025

What is this tool?

The Heat Transfer Simulator is a comprehensive web-based thermal modelling application that enables architects, engineers, and building scientists to create, simulate, and analyze 2D heat transfer models. It provides an integrated environment for geometry creation, material assignment, boundary condition specification, mesh generation, and steady-state heat transfer simulation directly in your browser.

Key Capabilities:

Visual geometry creation with precision drawing tools including structural steel sections
Thermal simulation using finite difference analysis
Material library with 49+ predefined building materials
Boundary condition management with 6 standard conditions
Interactive results visualization including temperature contours and heat flux vectors
U-value and PSI-value calculation for thermal performance analysis

This tool is provided as-is for thermal modeling purposes. Use at your own risk. It requires no installation and runs entirely in your web browser. For support or feature requests, please contact [email protected] for support or use the Support Ticket system.

Key Features

Geometry Creation

Precision Drawing Tools: Rectangle, polygon, and line drawing with snap-to-grid
Steel Section Library: 8 standard structural steel sections (H-Shape, I-Shape, Channel, Angles, T-Shape, SHS, RHS)
Precise Polygon Tool: Enter exact dimensions for each segment
Edge Management: Add, edit, and delete edges with visual feedback
Layer Organization: Multi-layer support with visibility control
DXF Import/Export: Import geometry from AutoCAD and other CAD software

Material & BC Management

49+ Predefined Materials: Complete thermal materials library organized by category
6 Built-in Boundary Conditions: Standard interior, exterior, and adiabatic conditions
Visual Feedback: Color-coded materials and BCs with interactive assignment
Custom Materials: Create and manage custom thermal materials with full property editing
Custom BCs: Define custom boundary conditions with specific parameters

Simulation & Analysis

Finite Difference Solver: Heat transfer simulation using Gauss-Seidel iteration
Temperature Visualization: Interactive contour plots with customizable ranges
Heat Flux Analysis: Vector field visualization of heat flow
U-Value Calculation: Automatic thermal transmittance calculation
PSI-Value Calculation: Linear thermal transmittance for thermal bridges (ISO 10211)
Convergence Monitoring: Real-time iteration tracking

System Requirements

Browser Compatibility

Chrome 90+ (Recommended)
Firefox 88+
Edge 90+
Safari 14+

Hardware

Minimum: 4GB RAM, modern dual-core processor
Recommended: 8GB+ RAM, quad-core processor for large models
Display: 1920x1080 or higher resolution recommended

Network

Internet connection required for initial load (CDN resources)
Offline use possible after initial load (application cached)

Access

The Heat Transfer Simulator is a standalone HTML file that runs entirely in your web browser:

Download Heat_Transfer_Simulator_V1_0.html
Double-click to open in your default browser
No installation or server required
All processing happens locally in your browser

Basic Workflow (7 Steps)

Step 1: Set Up Grid and Snapping

Top Control Bar:

Set grid spacing (default: 10mm)
Enable/disable grid visibility
Enable/disable snap-to-grid
Configure snap modes (Grid, Vertex, Edge, Ortho)

Recommended Settings:

Grid Spacing: 10mm (for precision)
Grid Visible: Yes
Snap to Grid: Yes
Snap Modes: Grid + Vertex enabled

Step 2: Draw Geometry

Drawing Tools (Toolbar):

Polygon Tool (default): Click points, right-click or click near first point to close
Precise Polygon Tool: Click points, enter dimensions for each segment
Rectangle Tool: Click two corners to create rectangle
Steel Section Tool: Select section type, set dimensions, click to place
Select Tool: Click to select layers for editing

Best Practices:

Work from outside to inside (exterior walls first)
Use consistent dimensions (multiples of grid spacing)
Close all polygons (no open shapes)
Avoid overlapping or gaps between materials

Example - Simple Wall:

1. Select Rectangle Tool
2. Click at (0, 0)
3. Click at (300, 2400) - creates 300mm × 2400mm wall
4. Layer automatically created

Step 3: Insert Steel Sections (Optional)

For structural steel elements:

Click Steel Section button (H-icon) in toolbar
Select section type from dropdown (H-Shape, I-Shape, Channel, etc.)
Adjust dimensions as needed (Height, Width, Thicknesses)
Choose material (Steel, Stainless Steel, Aluminum, Copper)
Click "Click to Place"
Click on canvas to position section
Section extends down and to the right from click point

Available Steel Sections:

Section

Default Size

Key Parameters

H-Shape

200×200mm

H, B, tw=8, tf=12

I-Shape

200×100mm

H, B, tw=5.5, tf=8.5

Channel

150×75mm

H, B, tw=6, tf=9

Unequal Angle

100×75mm

L1, L2, t=8

Equal Angle

75×75mm

L, t=6

T-Shape

100×100mm

H, B, tw=6, tf=10

SHS

100×100mm

H=B, t=5

RHS

150×100mm

H, B, t=5

Step 4: Assign Materials to Layers

Material Assignment:

Select a layer in the Layers panel (right sidebar)
Click the palette icon or choose material from Materials dropdown
Layer updates with material color
Repeat for all layers

Material Categories:

Metals (7 materials)
Insulation (8 materials)
Wood (6 materials)
Masonry (10 materials)
Glass (3 materials)
Plastics (4 materials)
Boards (3 materials)
Membranes (2 materials)
Air (2 materials)
Other (4 materials)

Visual Feedback:

Each material has a distinct color
Selected layer highlights in red
Material name and k-value display in Layers panel

Step 5: Assign Boundary Conditions to Edges

BC Assignment Process:

Select "Assign BC" mode from toolbar
Hover over edges to highlight them
Click on an edge to select it
Choose BC from dropdown in sidebar
Edge color updates to BC color
Repeat for all boundary edges

BC Types:

BC Type

h (W/m²·K)

Temp (°C)

Use

Interior, horizontal

7.692

Vertical walls

Interior, upwards

10.0

Ceiling surfaces

Interior, downwards

5.882

Floor surfaces

Exterior, air

25.0

Exposed exterior

Exterior, ventilated

7.692

Ventilated cavities

Adiabatic

0.0

N/A

Symmetry planes

Auto-Assignment:

Interior edges between materials automatically treated as continuous
Only assign BCs to exterior/boundary edges
Adiabatic for cut planes and symmetry boundaries

Step 6: Run Simulation

Simulation Panel (Settings):

Set grid resolution (default: 75×75)
Set convergence tolerance (default: 0.0001°C)
Set maximum iterations (default: 2000)
Click "Run Simulation"
Monitor convergence in real-time

Simulation Parameters:

Grid Resolution: 50 (fast) to 200 (accurate)
Convergence Tolerance: Maximum temperature change between iterations
Max Iterations: Safety limit to prevent infinite loops
Typical Time: 5-30 seconds for most models

Convergence Monitoring:

Iteration 100: Max ΔT = 0.543°C
Iteration 200: Max ΔT = 0.012°C
Iteration 287: Max ΔT = 0.00009°C ✓ CONVERGED

Step 7: Visualize Results

Visualization Options:

Toggle Contour: Show/hide temperature contours on canvas
View Results: Open detailed visualization modal
Results Report: Generate THERM-style report

View Results Modal:

Temperature Field: Color-coded temperature distribution with isotherms
Material Distribution: Thermal conductivity map (log scale)
Heat Flux Magnitude: Heat flow intensity visualization
Heat Flux Vectors: Arrows showing heat flow direction and magnitude

Analysis Tools:

U-Value Line: Draw line through assembly to calculate U-value
PSI-Value Line: 6-point measurement for linear thermal transmittance
Dimension Tool: Measure and annotate distances

Steel Section Tool

The Steel Section Tool allows rapid insertion of standard structural steel profiles commonly found in building construction.

Section Types

Solid Sections:

Type

Description

Parameters

H-Shape

Wide flange beam (equal flanges)

H, B, tw, tf

I-Shape

Standard I-beam (narrow flanges)

H, B, tw, tf

Channel

C-channel section

H, B, tw, tf

T-Shape

T-section beam

H, B, tw, tf

Unequal Angle

L-section with unequal legs

L1, L2, t

Equal Angle

L-section with equal legs

L, t

Hollow Sections:

Type

Description

Parameters

SHS

Square Hollow Section

H (=B), t

RHS

Rectangular Hollow Section

H, B, t

Default Dimensions

H-Shape Section (Wide Flange Beam)

Height (H): 200mm
Width (B): 200mm
Web thickness (tw): 8mm
Flange thickness (tf): 12mm

I-Shape Section (I-Beam)

Height (H): 200mm
Width (B): 100mm
Web thickness (tw): 5.5mm
Flange thickness (tf): 8.5mm

Channel Section (C-Channel)

Height (H): 150mm
Width (B): 75mm
Web thickness (tw): 6mm
Flange thickness (tf): 9mm

Unequal Angle Section

Leg 1 (longer): 100mm
Leg 2 (shorter): 75mm
Thickness (t): 8mm

Equal Angle Section

Leg width (both): 75mm
Thickness (t): 6mm

T-Shape Section

Height (H): 100mm
Width (B): 100mm
Web thickness (tw): 6mm
Flange thickness (tf): 10mm

Square Hollow Section (SHS)

Height & Width: 100mm × 100mm
Wall thickness (t): 5mm

Rectangular Hollow Section (RHS)

Height: 150mm
Width: 100mm
Wall thickness (t): 5mm

Material Options

Material

k (W/m·K)

Color

Steel

Dark Gray

Stainless Steel

Light Gray

Aluminum

237

Silver

Copper

380

Orange-Brown

Usage Instructions

Open Modal: Click the Steel Section button (H-icon) in the toolbar
Select Type: Choose section type from dropdown menu
Adjust Dimensions: Modify default values as needed
Select Material: Choose from Steel, Stainless Steel, Aluminum, or Copper
Initiate Placement: Click "Click to Place" button
Position Section: Click on canvas at desired location
Section Created: Geometry extends down and to the right from click point

Geometry Generation

Solid Sections (H, I, C, T, L shapes):

Created as single complex polygon
Traces the complete outline of the section
Heat flows through entire cross-section

Hollow Sections (SHS, RHS):

Created as 4 separate wall rectangles
Air cavity in center is not filled
Allows proper thermal simulation of hollow profiles

Thermal Bridge Analysis

Steel sections are significant thermal bridges due to their high conductivity. Use the PSI-Value Line tool to quantify the additional heat loss:

Place steel section in wall assembly
Run simulation
Use PSI-Value Line tool to measure linear thermal transmittance
Compare with 1D calculation to determine thermal bridge effect

Measurement Tools

U-Value Line

Measures thermal transmittance through a building assembly section.

Usage:

Select U-Value Line tool from toolbar
Click start point (typically on Interior BC)
Click end point (typically on Exterior BC)
U-value calculates and displays on the line

Calculation Method:

U = Q / (L × ΔT)

Where:

Q = Heat flow through the section (W/m)
L = Length of measurement line (m)
ΔT = Temperature difference (K)

PSI-Value Line

Measures linear thermal transmittance for thermal bridges per ISO 10211.

6-Point Measurement Process:

Point

Description

External start point (first construction)

External reference point (defines external offset)

External end point (second construction)

Internal start point (first construction)

Internal reference point (defines internal offset)

Internal end point (second construction)

Calculation:

ψ = L₂D - Σ(Uᵢ × lᵢ)

Where:

L₂D = 2D thermal conductance from simulation
Uᵢ = U-value of connecting element i
lᵢ = Length of connecting element i

Dimension Tool

Adds dimension annotations to the model.

Usage:

Select Dimension tool from toolbar
Click start point of measurement
Click end point of measurement
Click reference point to set dimension offset side
Dimension displays in meters

Material & Boundary Condition Library

Metals (7 materials)

Material

k (W/m·K)

Color

Aluminum

160.0

0.05

Silver

Steel

45.0

0.20

Dark Gray

Stainless Steel

17.0

0.15

Light Gray

Copper

380.0

0.05

Orange

Brass

120.0

0.05

Gold

Bronze

65.0

0.10

Brown

Zinc

112.0

0.05

Blue-Gray

Insulation (8 materials)

Material

k (W/m·K)

Color

Fiberglass

0.040

0.90

Light Pink

Mineral Wool

0.038

0.90

Gray-Pink

XPS

0.029

0.90

Light Blue

EPS

0.033

0.90

White

Polyurethane

0.023

0.90

Yellow

Polyisocyanurate (PIR)

0.022

0.90

Orange

Cellulose

0.039

0.90

Beige

Spray Foam

0.026

0.90

Light Yellow

Wood (6 materials)

Material

k (W/m·K)

Color

Wood-Softwood

0.12

0.90

Light Brown

Wood-Hardwood

0.16

0.90

Dark Brown

Wood-Plywood

0.13

0.90

Tan

Wood-OSB

0.13

0.90

Orange-Brown

Wood-Particleboard

0.17

0.90

Medium Brown

Wood-MDF

0.14

0.90

Gray-Brown

Masonry (10 materials)

Material

k (W/m·K)

Color

Concrete-Normal

1.40

0.90

Gray

Concrete-Lightweight

0.80

0.90

Light Gray

Concrete Block-Hollow

0.60

0.90

Medium Gray

Concrete Block-Solid

1.00

0.90

Dark Gray

Brick-Common

0.70

0.90

Red

Brick-Face

1.00

0.90

Dark Red

Stone-Granite

2.80

0.90

Dark Gray

Stone-Limestone

1.50

0.90

Beige

Stone-Sandstone

1.70

0.90

Tan

Mortar

0.80

0.90

Light Gray

Glass (3 materials)

Material

k (W/m·K)

Color

Glass-Clear

1.00

0.84

Light Blue

Glass-Low-E

1.00

0.10/0.84*

Blue

Glass-Tinted

1.00

0.84

Dark Blue

*Low-E glass: εfront=0.10, εback=0.84

Plastics (4 materials)

Material

k (W/m·K)

Color

PVC

0.17

0.90

White

EPDM Rubber

0.25

0.90

Black

Polyethylene

0.42

0.90

Translucent

Acrylic

0.20

0.90

Clear

Boards (3 materials)

Material

k (W/m·K)

Color

Gypsum Board

0.16

0.90

Off-White

Plaster

0.50

0.90

White

Cement Board

0.25

0.90

Gray

Membranes (2 materials)

Material

k (W/m·K)

Color

Air Barrier Membrane

0.20

0.90

Light Green

Vapor Barrier

0.16

0.90

Light Blue

Air (2 materials)

Material

k (W/m·K)

Color

Air-Cavity

0.026

1.00

Very Light Blue

Air-Still

0.024

1.00

Pale Blue

Built-in Boundary Conditions

BC Name

h (W/m²·K)

Temp (°C)

Use

Interior, upwards

10.0

Ceiling surfaces

Interior, downwards

5.882

Floor surfaces

Interior, horizontal

7.692

Vertical walls

Exterior, air

25.0

Exposed exterior

Exterior, ventilated

7.692

Ventilated cavity

Adiabatic

0.0

N/A

Symmetry planes

Keyboard Shortcuts

General Shortcuts

Key

Action

Esc

Cancel current operation / Return to select tool

Delete

Delete selected layer(s)

View Shortcuts

Key

Action

Scroll

Zoom in/out

Shift+Drag

Pan view

Drawing Shortcuts

Key

Action

Right-click

Finish polygon

Click near first point

Close polygon

Common Issues & Troubleshooting

Issue: "Simulation already running" Error

Symptoms:

Cannot start new simulation
Button appears disabled or shows warning

Solutions:

Wait for current simulation to complete
Refresh the page to reset simulation state
Clear simulation results using "Remove Results" button

Issue: Steel Section Not Appearing

Symptoms:

Click on canvas but no geometry appears
Console shows errors

Solutions:

Ensure you clicked "Click to Place" button in modal first
Check that modal closed after clicking the button
Click within the canvas boundaries
Verify material exists in database (Steel, Stainless Steel, Aluminum, Copper)

Issue: Boundary Conditions Not Assigning

Symptoms:

Edge doesn't change color when clicked
BC doesn't appear on edge

Solutions:

Ensure "Assign BC" mode is active (button highlighted)
Click directly on edge line, not nearby
Hover over edge first to see highlight
Check that both Interior and Exterior BCs are assigned before simulation

Issue: No Contours Displayed After Simulation

Symptoms:

Simulation completes but no temperature visualization
Toggle Contour button doesn't show results

Solutions:

Click "Toggle Contour" button to enable display
Check that simulation actually converged (no error messages)
Verify geometry has valid boundary conditions
Try "View Results" button for detailed visualizations

Issue: Simulation Won't Converge

Symptoms:

Maximum iterations reached
Final residual above tolerance

Solutions:

Increase max iterations in Settings (try 5000)
Increase grid resolution for complex geometry
Check boundary conditions are correctly assigned
Verify material properties are reasonable (no zero conductivity)
Simplify model geometry if very complex

Issue: Layers Panel Shows Errors

Symptoms:

Layer cards don't display properly
Material colors missing

Solutions:

Refresh the page
Check browser console for JavaScript errors
Ensure materials are properly assigned to layers
Delete and recreate problematic layers

Best Practices

Geometry Creation

Start Simple: Create basic geometry first, add complexity gradually
Use Actual Scale: Model at 1:1 scale in millimeters
Clean Geometry: No overlapping faces, no gaps between materials
Close Polygons: Ensure all shapes are properly closed

Steel Section Placement

Consider Thermal Bridging: Steel has high conductivity (k=45 W/m·K)
Use Thermal Breaks: Add insulation around steel sections where possible
Check Dimensions: Verify section sizes match actual specifications
Multiple Sections: Place each section separately for complex frames

Material Assignment

Systematic Approach: Work from outside to inside
Verify Properties: Check k-values match manufacturer data
Use Validation: Run model validation before simulation

Boundary Conditions

Minimum Requirements: At least one Interior and one Exterior BC
Adiabatic Planes: Use for symmetry and cut boundaries
Correct Direction: Interior upwards for ceilings, downwards for floors

Simulation Settings

Use Case

Grid Resolution

Tolerance

Max Iterations

Quick Check

50×50

0.001

1000

Standard Analysis

100×100

0.0001

2000

Detailed Study

150×150

0.00001

5000

Frequently Asked Questions

Can I use this tool offline?

Partially. After the first load (which requires internet for CDN resources), the application works offline. However, you cannot reload the page without internet.

What units does the tool use?

Millimeters (mm) for dimensions, °C for temperature, W/m·K for conductivity, and W/m²·K for heat transfer coefficients.

How do I add structural steel sections?

Click the Steel Section button (H-icon) in the toolbar, select the section type, adjust dimensions, choose material, click "Click to Place", then click on the canvas to position it.

Can hollow sections be modeled correctly?

Yes. SHS and RHS sections are created as 4 separate wall elements, leaving an air cavity in the center for accurate thermal simulation.

How accurate are the simulation results?

Results are suitable for preliminary analysis and comparative studies. Accuracy depends on mesh resolution, material properties, and boundary conditions. For critical applications, verify with dedicated FEA software.

How do I calculate PSI-values for thermal bridges?

Use the PSI-Value Line tool which requires 6 points to define external and internal measurement planes per ISO 10211. The tool calculates the linear thermal transmittance automatically.

What if my model won't converge?

Common causes: (1) missing boundary conditions, (2) unrealistic material properties, (3) grid too coarse. Solutions: check BCs, verify materials, increase resolution and iterations.

Can I export my model?

Yes. Use the DXF Export button for geometry, JSON Save for complete project including materials and BCs, or THERM XML export for compatibility with LBNL THERM software.

Underlying Mathematical Equations

The Heat Transfer Simulator solves the steady-state heat conduction equation using the finite difference method. This section describes the governing equations, boundary conditions, and numerical formulation.

Governing Equation: Steady-State Heat Conduction

The fundamental equation governing heat transfer by conduction in a solid is Fourier's law combined with energy conservation. For a two-dimensional domain D in steady state (no time dependence), the governing partial differential equation is:

∇·(k∇T) = 0  in D

Expanded in Cartesian coordinates (x, y):

∂/∂x(k·∂T/∂x) + ∂/∂y(k·∂T/∂y) = 0

Where:

T(x,y) = Temperature field [°C or K]
k(x,y) = Thermal conductivity [W/(m·K)]
∇ = Gradient operator (del operator)
∇· = Divergence operator

Physical Meaning: This equation states that in steady state, the net heat flux into any infinitesimal volume must be zero (conservation of energy with no storage term).

Fourier's Law of Heat Conduction

The heat flux vector q (heat flow per unit area) is related to the temperature gradient by Fourier's law:

q = -k∇T

In component form:

qₓ = -k·∂T/∂x
qᵧ = -k·∂T/∂y

Where:

q = Heat flux vector [W/m²]
qₓ, qᵧ = Heat flux components in x and y directions [W/m²]
The negative sign indicates heat flows from hot to cold (down the temperature gradient)

Magnitude of Heat Flux:

|q| = √(qₓ² + qᵧ²) = k·|∇T|

Material Properties

Thermal Conductivity k:

Measures a material's ability to conduct heat
Isotropic materials: k is scalar (same in all directions)
In Heat Transfer Simulator: k varies spatially but is constant within each material layer
Higher k → better heat conductor (metals)
Lower k → better insulator (fiberglass, foam)

Typical Values:

Metals:      k = 15 - 400 W/(m·K)
Masonry:     k = 0.5 - 3.0 W/(m·K)
Wood:        k = 0.1 - 0.2 W/(m·K)
Insulation:  k = 0.02 - 0.05 W/(m·K)

Boundary Conditions

The solution requires boundary conditions on the boundary ∂D of the domain. Three types are used:

1. Dirichlet Boundary Condition (Temperature Specified)

T = T₀  on ∂Dᴅ

Temperature is prescribed directly
Example: T = 20°C on interior surface
Strong enforcement (replaces equation at boundary nodes)

2. Neumann Boundary Condition (Heat Flux Specified)

q·n = -k·∂T/∂n = q₀  on ∂Dₙ

Where:

n = Outward unit normal vector to boundary
∂T/∂n = Temperature gradient normal to boundary
q₀ = Prescribed heat flux [W/m²]

Special Case - Adiabatic Boundary:

∂T/∂n = 0  on ∂Dₐ

No heat flux crosses boundary
Used for symmetry planes and insulated boundaries

3. Robin (Convective) Boundary Condition

q·n = -k·∂T/∂n = h(T - T∞)  on ∂Dᵣ

Where:

h = Heat transfer coefficient [W/(m²·K)]
T∞ = Ambient (fluid) temperature [°C]
T = Surface temperature [°C]

Physical Meaning: Heat flux at surface is proportional to temperature difference between surface and ambient fluid (Newton's law of cooling).

Typical h Values:

Still air (interior):        h = 5 - 10 W/(m²·K)
Moving air (interior):       h = 10 - 20 W/(m²·K)
Exterior (sheltered):        h = 10 - 15 W/(m²·K)
Exterior (exposed/windy):    h = 20 - 40 W/(m²·K)

Finite Difference Discretization

Temperature Approximation: The continuous temperature field T(x,y) is approximated on a regular grid:

T(x,y) ≈ Tᵢ,ⱼ  at grid point (i,j)

Where:

i, j = Grid indices in x and y directions
Δx, Δy = Grid spacing in x and y directions
Tᵢ,ⱼ = Temperature at grid node (i,j) (unknown)

Central Difference Approximation:

For interior nodes, the Laplacian is approximated using central differences:

∂²T/∂x² ≈ (Tᵢ₊₁,ⱼ - 2Tᵢ,ⱼ + Tᵢ₋₁,ⱼ) / Δx²
∂²T/∂y² ≈ (Tᵢ,ⱼ₊₁ - 2Tᵢ,ⱼ + Tᵢ,ⱼ₋₁) / Δy²

For uniform grid (Δx = Δy = h):

Tᵢ,ⱼ = (Tᵢ₊₁,ⱼ + Tᵢ₋₁,ⱼ + Tᵢ,ⱼ₊₁ + Tᵢ,ⱼ₋₁) / 4

This is the 5-point stencil for Laplace's equation.

Material Interface Handling:

At interfaces between materials with different conductivities:

keff = 2·k₁·k₂ / (k₁ + k₂)

This harmonic mean ensures proper heat flux continuity across material boundaries.

Solution Method: Gauss-Seidel Iteration

Heat Transfer Simulator uses the Gauss-Seidel iterative method to solve the discretized system.

Algorithm:

Initialize: T⁰ᵢ,ⱼ = initial guess (e.g., average of boundary temperatures)

For iteration n = 1, 2, 3, ...
  For each interior node (i,j):
    Tⁿ⁺¹ᵢ,ⱼ = (Tⁿ⁺¹ᵢ₋₁,ⱼ + Tⁿᵢ₊₁,ⱼ + Tⁿ⁺¹ᵢ,ⱼ₋₁ + Tⁿᵢ,ⱼ₊₁) / 4
  
  Compute residual: rₘₐₓ = maxᵢ,ⱼ|Tⁿ⁺¹ᵢ,ⱼ - Tⁿᵢ,ⱼ|
  
  If rₘₐₓ < tolerance:
    CONVERGED
    Exit

Key Features:

Uses most recent values immediately (Tⁿ⁺¹ for already-computed nodes)
Generally faster convergence than Jacobi method
No matrix factorization required
Low memory requirements

Convergence:

Guaranteed for well-posed heat conduction problems
Typical convergence: 100-2,000 iterations
Default tolerance: 0.0001°C

U-Value Calculation

Overall U-Value Definition

The U-value (thermal transmittance) quantifies the overall rate of heat transfer through a building assembly under steady-state conditions.

Basic Definition:

U = Q/(A·ΔT)  [W/(m²·K)]

Where:

U = U-value (thermal transmittance) [W/(m²·K)]
Q = Total heat transfer rate [W]
A = Area of assembly [m²]
ΔT = Temperature difference between interior and exterior [K]

Physical Meaning: The U-value represents the heat flux (W/m²) per unit temperature difference. Lower U-values indicate better insulation.

Heat Transfer Rate Calculation

For finite difference implementation, sum over boundary nodes:

Q = Σᵢ hᵢ·Aᵢ·(Tᵢ - T∞,ᵢ)

Where:

hᵢ = Heat transfer coefficient at node i [W/(m²·K)]
Aᵢ = Area associated with node i [m²]
Tᵢ = Surface temperature at node i [°C or K]
T∞,ᵢ = Ambient temperature at node i [°C or K]

R-Value (Thermal Resistance)

R = 1/U  [m²·K/W]

For Multi-Layer Assembly:

R_total = Rₛᵢ + R₁ + R₂ + ... + Rₙ + Rₛₑ

Where:

Rᵢ = Thermal resistance of layer i = dᵢ/kᵢ
dᵢ = Thickness of layer i [m]
kᵢ = Thermal conductivity of layer i [W/(m·K)]

PSI-Value (Linear Thermal Transmittance)

The PSI value (ψ) quantifies additional heat loss due to thermal bridging at junctions, edges, and geometric discontinuities.

Definition of PSI (ψ):

ψ = L₂ᴅ - Σᵢ (Uᵢ·lᵢ)  [W/(m·K)]

Where:

ψ = Linear thermal transmittance (PSI value) [W/(m·K)]
L₂ᴅ = 2D thermal conductance of complete junction [W/(m·K)]
Uᵢ = U-value of connecting element i [W/(m²·K)]
lᵢ = Length of connecting element i in 2D model [m]

Physical Meaning: PSI represents the additional heat flow due to multidimensional effects not captured in 1D U-value calculations. A positive ψ indicates a thermal bridge (extra heat loss), while negative ψ indicates thermal improvement.

PSI Value Calculation Procedure

Step 1: Calculate 2D Heat Flow Run 2D simulation of complete junction:

L₂ᴅ = Q₂ᴅ/ΔT

Step 2: Calculate 1D Component Heat Flows For each connecting element:

Qᵢ,₁ᴅ = Uᵢ·lᵢ·ΔT

Step 3: Calculate PSI Value

ψ = (Q₂ᴅ - Σᵢ Qᵢ,₁ᴅ)/ΔT = L₂ᴅ - Σᵢ (Uᵢ·lᵢ)

Typical PSI Values:

Junction Type

Good

Moderate

Poor

Wall-Floor

< 0.15

0.15-0.40

> 0.40

Wall-Roof

< 0.10

0.10-0.30

> 0.30

Window Frame

< 0.03

0.03-0.10

> 0.10

Corner (External)

< 0.05

0.05-0.15

> 0.15

Units: W/(m·K)

Heat Flux Post-Processing

Flux Calculation:

At each grid node, temperature gradients are computed using central differences:

∂T/∂x ≈ (Tᵢ₊₁,ⱼ - Tᵢ₋₁,ⱼ) / (2·Δx)
∂T/∂y ≈ (Tᵢ,ⱼ₊₁ - Tᵢ,ⱼ₋₁) / (2·Δy)

Heat Flux Components:

qₓ = -k·∂T/∂x
qᵧ = -k·∂T/∂y

Flux Magnitude and Direction:

|q| = √(qₓ² + qᵧ²)
θ = atan2(qᵧ, qₓ)  (angle from x-axis)

Summary of Numerical Formulation

Grid: Divide domain into regular rectangular grid
Material Assignment: Assign thermal conductivity k to each cell
Boundary Conditions: Apply temperature or convection BCs at boundaries
Discretization: Replace derivatives with finite differences
Solve: Gauss-Seidel iterations until convergence
Post-Process: Compute heat flux and U-value

Advantages of Finite Difference Method:

Simple implementation
Fast for rectangular domains
Low memory requirements
Easy parallelization
Direct handling of material interfaces

FEAScript Integration

The Heat Transfer Simulator can export models for advanced analysis using FEAScript, an open-source JavaScript finite element library.

Export Workflow to FEAScript

For complex geometries requiring unstructured meshing or multi-physics analysis, export your model and continue in FEAScript:

Step 1: Export Geometry

Complete your geometry in Heat Transfer Simulator
Use DXF Export to save geometry file
Import DXF into Gmsh to generate .msh mesh file

Step 2: Set Up FEAScript Model

import { FEAScriptModel, importGmshQuadTri, plotSolution } 
  from "https://core.feascript.com/dist/feascript.esm.js";

// Load mesh exported from Heat Transfer Simulator
const meshContent = await (await fetch("exported_model.msh")).text();
const meshFile = new File([meshContent], "exported_model.msh");
const parsedMesh = await importGmshQuadTri(meshFile);

// Configure heat conduction solver
const model = new FEAScriptModel();
model.setSolverConfig("heatConductionScript");
model.setMeshConfig({
  parsedMesh, 
  meshDimension: "2D", 
  elementOrder: "linear"
});

// Apply boundary conditions (use Gmsh physical group tags)
model.addBoundaryCondition("1", ["thermal", "convection", h, T_ambient]);
model.addBoundaryCondition("2", ["thermal", "convection", h, T_interior]);

// Solve and visualize
const { solutionVector, nodesCoordinates } = model.solve();
plotSolution(solutionVector, nodesCoordinates, model.solverConfig, 
  model.meshConfig.meshDimension, "contour", "resultsCanvas");

Step 3: Advanced Analysis

FEAScript enables capabilities beyond Heat Transfer Simulator:

Unstructured triangular/quadrilateral meshes for complex shapes
Higher-order elements for improved accuracy
Custom solver configurations
Integration with Node.js for batch processing

When to Use FEAScript

Use Case

Recommended Tool

Quick thermal bridge analysis

Heat Transfer Simulator

Standard U-value/PSI-value

Heat Transfer Simulator

Complex curved geometries

Export to FEAScript

Batch processing multiple models

FEAScript API

Custom web application

FEAScript API

Transient heat transfer

FEAScript

FEAScript Resources

Website: https://feascript.com
Heat Conduction Tutorial: https://feascript.com/tutorials/heat-conduction-2d-fin.html
GitHub: https://github.com/FEAScript/FEAScript-core
License: MIT (free for all uses)

Standards and References

Referenced Standards

ISO 10211: Thermal bridges in building construction - Heat flows and surface temperatures - Detailed calculations

ISO 14683: Thermal bridges in building construction - Linear thermal transmittance - Simplified methods and default values

ISO 6946: Building components and building elements - Thermal resistance and thermal transmittance - Calculation methods

EN 673: Glass in building - Determination of thermal transmittance (U value) - Calculation method

PreviousASHRAE 169 Climate Zone Tool

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