How It Works
Fourier's Law of Heat Conduction
Heat conduction is the transfer of thermal energy through a material due to a temperature gradient. Fourier's Law states that heat flux is proportional to the negative temperature gradient:
Q = -k * A * (dT/dx)
Where:
- Q = Heat transfer rate (W or BTU/hr)
- k = Thermal conductivity (W/m-K)
- A = Cross-sectional area (m2)
- dT/dx = Temperature gradient (K/m)
The negative sign indicates heat flows from hot to cold (opposite to the temperature gradient direction).
Thermal Resistance Concept
Analogous to electrical resistance (Ohm's Law), thermal resistance relates temperature difference to heat flow:
Q = DeltaT / R_thermal
For different geometries:
- Flat Wall:
R = L / (k * A) - Cylinder:
R = ln(r2/r1) / (2*pi*k*L) - Sphere:
R = (1/r1 - 1/r2) / (4*pi*k)
Composite Walls (Multi-Layer)
For walls with multiple layers in series, thermal resistances add up:
R_total = R1 + R2 + R3 + ...
The heat flow through each layer is the same, but temperature drops are proportional to each layer's resistance. Interface temperatures can be calculated by applying Q * R for each layer.
Cylindrical Coordinates
For pipes and tubes, heat conducts radially. The area changes with radius, leading to a logarithmic temperature profile:
Q = 2*pi*k*L*(T1-T2) / ln(r2/r1)
Critical radius exists for insulated pipes: r_crit = k_insul / h_conv - below this radius, adding insulation can increase heat loss!
Spherical Coordinates
For spherical shells (tanks, vessels), heat conducts radially through an area that varies as r2:
Q = 4*pi*k*(T1-T2) / (1/r1 - 1/r2)
Key Factors Affecting Conduction
- Thermal Conductivity (k): Material property - metals have high k, insulators have low k
- Thickness (L): Thicker walls = more resistance = less heat transfer
- Area (A): Larger area = more heat transfer
- Temperature Difference: Driving force for heat transfer
Practical Applications
- Building Insulation: Minimize heat loss through walls, roofs
- Pipe Insulation: Prevent heat loss from steam/hot water lines
- Furnace Walls: Multi-layer refractory + insulation design
- Electronics Cooling: Heat sink and thermal interface design
Quick-Select Applications
Heat Conduction Calculator
Calculate heat transfer through flat walls, cylindrical pipes, spherical shells, and multi-layer composite walls using Fourier's Law.
Layers (from hot to cold side):
Thermal conductivity values for common materials. Click a row to use that value.
| Material | k (W/m-K) | k (BTU/hr-ft-F) | Category |
|---|
Results
Conduction Formulas
Flat Wall:
Q = k * A * (T1 - T2) / L
R = L / (k * A)
Cylindrical Wall:
Q = 2*pi*k*L*(T1-T2) / ln(r2/r1)
R = ln(r2/r1) / (2*pi*k*L)
Spherical Wall:
Q = 4*pi*k*(T1-T2) / (1/r1 - 1/r2)
R = (1/r1 - 1/r2) / (4*pi*k)
Series Resistance:
R_total = R1 + R2 + R3 + ...
Common Thermal Conductivities
| Material | k (W/m-K) | k (BTU/hr-ft-F) |
|---|---|---|
| Copper | 385 | 223 |
| Aluminum | 205 | 119 |
| Steel | 50 | 29 |
| Concrete | 1.7 | 0.98 |
| Brick | 0.72 | 0.42 |
| Fiberglass | 0.04 | 0.023 |
| Polyurethane | 0.025 | 0.014 |
| Aerogel | 0.015 | 0.009 |