Thermal Heat Conduction Fourier's Law Slab Heat Flux Calculator
Calculate 1D steady-state conductive heat transfer rate (Q), heat flux (q''), thermal resistance, R-value, and U-factor across planar slabs using Fourier's Law.
Slab Geometry & Thermal Properties
Residential cavity wall and attic insulation
Conductive Heat Transfer Outputs
104.0 W354.86 BTU/hr
89.42 kcal/hr
16.48 BTU/(hr·ft²)
Gradient: 1300.0 K/m
High-efficiency thermal barrier / insulator. Effectively curtails conductive heat flux, critical for HVAC thermal envelope performance and cryogenic containment.
Fourier's Law of Heat Conduction: Theoretical Foundations
Formulated in 1822 by the French mathematician and physicist Jean-Baptiste Joseph Fourier in his seminal treatise Théorie analytique de la chaleur, Fourier's Law serves as the governing rate equation for conductive heat transfer. Heat conduction represents the microscopic transfer of kinetic energy between adjacent atoms, molecules, or free electrons within a stationary medium driven by a temperature difference.
In its most general three-dimensional differential vector notation, Fourier's Law is expressed as:
Where q" is the local heat flux vector in Watts per square meter (W/m²), k is the thermal conductivity of the material in Watts per meter-Kelvin (W/(m·K)), and ∇T is the spatial temperature gradient vector. The negative sign is a mathematical consequence of the Second Law of Thermodynamics: heat must spontaneously flow from high-temperature regions toward lower-temperature regions (down the temperature gradient).
| Variable Symbol | Description | SI Standard Unit | Imperial Unit | Physical Significance |
|---|---|---|---|---|
| Q | Total Heat Transfer Rate | Watts (W = J/s) | BTU/hr | Total thermal energy flowing through the boundary per unit time |
| q" | Conductive Heat Flux | W/m² | BTU/(hr·ft²) | Areal thermal power density normal to the direction of heat flow |
| k | Thermal Conductivity | W/(m·K) | BTU/(hr·ft·°F) | Intrinsic transport coefficient governing thermal energy diffusion |
| L | Slab Thickness | Meters (m) | Inches / Feet | Conductive diffusion path length across the planar boundary |
| A | Cross-Sectional Area | Square Meters (m²) | Square Feet (ft²) | Total surface area perpendicular to the heat transfer vector |
| ΔT | Temperature Difference | Kelvins (K) or °C | Rankine (°R) or °F | Thermodynamic driving potential between the two boundary faces |
One-Dimensional Planar Slab Derivation & Assumptions
For heat conduction across a large flat wall or slab of uniform thickness L and surface area A, where the face at x = 0 is maintained at constant temperature T₁ and the face at x = L is maintained at constant temperature T₂, the three-dimensional heat diffusion equation reduces to a straightforward boundary value problem:
Integrating twice with respect to x yields a strictly linear temperature distribution within the solid:
Differentiating T(x) gives a constant temperature gradient dT/dx = -(T₁ - T₂)/L = -ΔT / L. Substituting this into Fourier's 1D rate equation Q = -k · A · (dT/dx) yields the final algebraic solution:
- • Steady-State Flow: ∂T/∂t = 0. Temperatures at any location do not vary over time.
- • 1D Planar Geometry: Edge effects are negligible because slab thickness is small relative to width and height (L « √A).
- • Zero Internal Heat Generation: No chemical, nuclear, or electrical resistance volumetric heating (q_gen = 0).
- • Homogeneous & Isotropic Medium: Thermal conductivity k is identical in all directions and uniform throughout.
While thermal conductivity k varies slightly with temperature in real substances, engineering practice computes k at the arithmetic mean film temperature:
T_avg = (T₁ + T₂) / 2
This mean-value evaluation produces less than 1% divergence from exact non-linear integral solutions for moderate temperature differentials.
The Thermal Circuit Analogy: R-Values, R_th, and U-Factors
Thermal transport is mathematically isomorphic to electrical current flow under Ohm's Law (I = ΔV / R_e). By defining the temperature difference ΔT as the thermal driving potential and the heat flow rate Q as the thermal current, the conduction equation can be cast into electrical network notation:
1. Absolute Thermal Resistance (R_th)
Measures total opposition to heat transfer across a specific physical component of cross-sectional area A. In series multi-layer composite walls, total resistance is purely additive: R_total = Σ R_th,i.
2. Area-Specific R-Value (R)
Standard architectural insulation index independent of wall area. Imperial R-value (US) relates via R_US = R_SI × 5.678263, expressed in (hr·ft²·°F)/BTU.
Upgrading residential walls or attic cavities to meet building energy codes? Calculate prescriptive climate zone thicknesses and heating cost reductions with our Attic & Wall Insulation R-Value Calculator.
3. Thermal Transmittance (U-Factor)
Represents the rate of heat transfer per unit area per degree of temperature difference. Lower U-factors signify superior thermal insulation performance in architectural windows and facades.
Thermal Conductivity (k) Spectrum Across Materials
Thermal conductivity spans over four orders of magnitude between high-conductivity structural metals and high-performance nanoporous aerogels. The underlying microscopic physical mechanisms differ fundamentally across states of matter:
| Material Class | Typical k Range [W/(m·K)] | Dominant Heat Transport Carrier | Representative Examples |
|---|---|---|---|
| Pure Metals | 100 – 430 | Free conduction electrons (Wiedemann-Franz law) | Silver (429), Copper (385), Gold (315), Aluminum (205) |
| Metal Alloys | 12 – 60 | Electrons hindered by impurity scattering & lattice disorder | Carbon Steel (50), Stainless Steel 304 (15), Inconel (11) |
| Non-Metallic Solids & Ceramics | 0.5 – 3.0 | Lattice vibrational waves (phonons) | Concrete (1.4), Glass (0.96), Red Brick (0.72), Soil (0.5) |
| Liquids (Non-Metallic) | 0.1 – 0.7 | Short-range molecular vibrational collision coupling | Water (0.606), Engine Oil (0.145), Glycol (0.256) |
| Gases (At Atmospheric Pressure) | 0.015 – 0.18 | Random molecular translations and kinetic collisions | Air (0.026), Argon (0.018), Carbon Dioxide (0.016), He (0.15) |
| Thermal Insulators & Foams | 0.012 – 0.045 | Suppression of gas convection + low solid volume fraction | Aerogel (0.015), Polyurethane (0.024), EPS (0.036), Batt (0.04) |
Step-by-Step Engineering Case Studies
Examine these two practical thermal engineering calculations contrasting an industrial steam pipe boiler wall with architectural residential wall insulation:
For high-temperature industrial combustion chambers where outer casing losses are shared between solid conduction and exterior thermal radiation, calculate fourth-power ambient heat losses with our Stefan-Boltzmann Radiant Blackbody Energy Solver.
- Given Parameters:
- Thickness L = 20 mm = 0.020 m
- Area A = 4.5 m², Material: Carbon Steel (k = 50.0 W/(m·K))
- Hot Face T₁ = 300°C, Outer Face T₂ = 80°C
- 1. Calculate Temperature Difference:
- ΔT = 300 - 80 = 220 K
- 2. Compute Heat Transfer Rate (Q):
- Q = (50.0 × 4.5 × 220) / 0.020 = 49,500 / 0.020 = 2,475,000 W = 2,475 kW
- 3. Compute Conductive Heat Flux (q"):
- q" = Q / A = 2,475,000 / 4.5 = 550,000 W/m²
- 4. Absolute Thermal Resistance (R_th):
- R_th = 0.020 / (50.0 × 4.5) = 0.0000889 K/W
- Given Parameters:
- Thickness L = 100 mm = 0.100 m
- Area A = 25.0 m², Material: PUR Foam (k = 0.024 W/(m·K))
- Inside T₁ = 22°C, Winter Outside T₂ = -8°C
- 1. Calculate Temperature Difference:
- ΔT = 22 - (-8) = 30 K
- 2. Compute Heat Transfer Rate (Q):
- Q = (0.024 × 25.0 × 30) / 0.100 = 18.0 / 0.100 = 180.0 W
- 3. Compute R-Value and U-Factor:
- R_SI = 0.100 / 0.024 = 4.17 m²·K/W
- R_US = 4.167 × 5.678 = R-23.7 (hr·ft²·°F/BTU)
- U = 1 / 4.167 = 0.240 W/(m²·K)
Frequently Asked Questions (FAQ)
What is Fourier's Law of Thermal Conduction?
Fourier's Law is the fundamental constitutive equation of heat conduction. It states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area through which the heat flows: Q = -k · A · (dT/dx). For a 1D uniform planar slab with steady boundary temperatures, it integrates to Q = (k · A · ΔT) / L.
What is the difference between heat transfer rate (Q) and heat flux (q")?
Heat transfer rate (Q) is the total thermal power transferred across an entire surface, expressed in Watts (Joules per second) or BTU/hr. Heat flux (q") is the heat transfer rate per unit area perpendicular to the heat flow (q" = Q / A), measured in W/m² or BTU/(hr·ft²).
What is the relationship between thermal resistance, R-value, and U-factor?
Absolute thermal resistance R_th = L / (k · A) in K/W represents total resistance to heat flow. Area-specific thermal resistance (R-value) is R = L / k in m²·K/W (or hr·ft²·°F/BTU in imperial). The overall heat transfer coefficient (U-factor or thermal transmittance) is the reciprocal of the total area resistance: U = 1 / R = k / L in W/(m²·K).
Why is there a negative sign in the differential form of Fourier's Law?
The negative sign in q" = -k · (dT/dx) satisfies the Second Law of Thermodynamics, dictating that heat spontaneously flows down the temperature gradient from higher temperature to lower temperature regions. When temperature decreases along the positive x-direction, dT/dx is negative, rendering the heat flux q" positive.
How does thermal conductivity (k) vary across different materials?
Thermal conductivity varies across four orders of magnitude: pure metals like copper (~385 W/m·K) and aluminum (~205 W/m·K) have high conductivity due to free electron transport; structural materials like concrete (~1.4 W/m·K) and glass (~0.96 W/m·K) rely on lattice vibrations (phonons); and insulative foams (~0.024-0.040 W/m·K) and aerogels (~0.015 W/m·K) suppress gas molecular collisions and radiation.
What assumptions govern 1D steady-state conduction in a slab?
The mathematical model assumes: (1) Steady-state conditions (temperatures do not change with time, ∂T/∂t = 0), (2) One-dimensional planar heat flow (edge effects and lateral leakage are negligible), (3) Homogeneous and isotropic material properties, (4) Constant thermal conductivity k independent of temperature, and (5) No internal thermal energy generation (q_gen = 0).
How do you convert between SI R-values and US Imperial R-values?
1 RSI (m²·K/W) equals approximately 5.678263 US R-value (hr·ft²·°F/BTU). Conversely, 1 US R-value equals approximately 0.17611 RSI. For example, an imperial building insulation rating of R-13 corresponds to an RSI of 13 / 5.678263 ≈ 2.29 m²·K/W.
What is the thermal circuit analogy for conduction?
The thermal circuit analogy equates heat transfer to electrical conduction via Ohm's Law: heat rate Q corresponds to electric current I, temperature difference ΔT corresponds to electric potential (voltage) ΔV, and thermal resistance R_th = L / (k · A) corresponds to electrical resistance R = ρ · L / A. Thus, Q = ΔT / R_th, identical to I = ΔV / R.
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