Thermal Conductivity Equations

Why This Matters

Thermal conductivity equations form the mathematical backbone of heat transfer analysis—and you'll see them everywhere in Heat Mass Transport. Whether you're designing insulation systems, analyzing heat exchangers, or predicting temperature distributions in composite materials, these equations connect the fundamental physics of energy transport to real engineering applications. You're being tested on your ability to apply Fourier's Law, understand material-dependent behavior, and work with resistance networks in both simple and complex geometries.

Don't just memorize the equations—know why thermal conductivity varies between materials, how different phases (solids, liquids, gases) conduct heat through distinct mechanisms, and when to apply effective property models versus simple bulk values. The concepts here link directly to conduction analysis, boundary conditions, and system design problems you'll encounter on exams and in practice.

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The Foundational Law: Fourier's Equation

All thermal conductivity analysis builds from one governing relationship: heat flows from hot to cold, and the rate depends on both material properties and the temperature gradient. This is the starting point for every conduction problem you'll solve.

Fourier's Law of Heat Conduction

• Defines the heat flux as proportional to the negative temperature gradient—the negative sign ensures heat flows from high to low temperature, consistent with the second law of thermodynamics
• Mathematical form: $q = -k \frac{dT}{dx}$, where $q$ is heat flux (W/m²), $k$ is thermal conductivity (W/m·K), and $\frac{dT}{dx}$ is the temperature gradient
• Foundation for all steady-state conduction problems—this equation gets integrated with appropriate boundary conditions to solve for temperature profiles and heat transfer rates

Thermal Conductivity Definition

• Material property quantifying heat transfer ability—measured in $\text{W/m·K}$, representing the heat flow rate through a unit thickness per unit temperature difference
• Ranges across orders of magnitude: from ~0.01 W/m·K for insulating gases to ~400 W/m·K for copper, making material selection critical in thermal design
• Appears in the denominator of thermal resistance—higher $k$ means lower resistance to heat flow and faster thermal equilibration

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Resistance Network Analysis

When solving practical problems, converting conductivity into resistance allows you to treat thermal systems like electrical circuits. Series and parallel combinations become straightforward once you master this framework.

Thermal Resistance and Conductance

• Thermal resistance: $R = \frac{L}{kA}$, where $L$ is thickness, $k$ is conductivity, and $A$ is cross-sectional area—units are K/W
• Conductance is the reciprocal: $C = \frac{kA}{L}$, representing ease of heat flow rather than opposition to it
• Essential for composite walls and insulation design—resistances add in series, conductances add in parallel, enabling quick analysis of layered systems

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Phase-Dependent Mechanisms

Thermal conductivity varies dramatically between solids, liquids, and gases because the physical mechanism of energy transport differs in each phase. Understanding these mechanisms helps you predict behavior and select appropriate models.

Thermal Conductivity of Solids (Debye Model)

• Heat conduction occurs via phonons—quantized lattice vibrations that carry thermal energy through the crystalline structure
• Temperature dependence follows characteristic curves: conductivity typically increases at low temperatures (fewer phonon-phonon collisions), peaks, then decreases at high temperatures
• Debye model predicts $k \propto T^3$ at very low temperatures—critical for cryogenic applications and understanding fundamental solid-state physics

Thermal Conductivity of Liquids

• Intermediate values between gases and solids—typically 0.1–0.7 W/m·K, governed by intermolecular forces and molecular spacing
• Generally decreases with temperature (except water below 130°C)—as thermal motion disrupts the quasi-ordered structure that facilitates energy transfer
• Viscosity correlation exists—more viscous liquids often have lower thermal conductivity due to restricted molecular motion, important for heat exchanger fluid selection

Thermal Conductivity of Gases (Kinetic Theory)

• Lowest conductivity of all phases—typically 0.01–0.05 W/m·K because molecules are widely spaced and energy transfer requires collisions
• Kinetic theory predicts: $k \propto \sqrt{T}$ and $k \propto \frac{1}{\sqrt{M}}$, where $M$ is molecular weight—lighter gases conduct better
• Nearly independent of pressure at moderate conditions—because mean free path and density effects cancel out, simplifying many gas-phase calculations

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Temperature and Composition Effects

Real materials don't have constant thermal conductivity—it varies with temperature and, for mixtures, with composition. Accounting for these variations is essential for accurate heat transfer predictions.

Temperature Dependence of Thermal Conductivity

• Metals generally increase in conductivity with temperature at low $T$, then plateau or decrease—due to competing phonon and electron scattering mechanisms
• Non-metals (insulators) typically decrease with temperature—phonon-phonon scattering (Umklapp processes) becomes dominant at higher temperatures
• Empirical correlations or property tables required—linear approximations like $k = k_0(1 + \beta T)$ work for limited ranges, but check validity bounds

Wiedemann-Franz Law for Metals

• Links thermal and electrical conductivity: $\frac{k}{\sigma} = LT$, where $L$ is the Lorenz number ($\approx 2.44 \times 10^{-8} \text{ W·Ω/K}^2$) and $T$ is absolute temperature
• Explains why good electrical conductors are good thermal conductors—free electrons carry both charge and thermal energy in metals
• Useful for estimating $k$ when electrical conductivity data is available—particularly valuable for alloys where thermal data may be scarce

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Heterogeneous and Complex Media

Engineering materials are rarely homogeneous—composites, porous media, and multi-phase systems require effective property models. These situations appear frequently in design problems and exams.

Thermal Conductivity of Composite Materials

• Effective conductivity depends on component arrangement—series (layers perpendicular to heat flow) vs. parallel (layers aligned with heat flow) give bounding estimates
• Volume fraction weighting: parallel bound is $k_{eff} = \sum \phi_i k_i$; series bound is $\frac{1}{k_{eff}} = \sum \frac{\phi_i}{k_i}$—real composites fall between these limits
• Anisotropic behavior common—fiber-reinforced composites conduct differently along vs. across fiber directions, requiring directional conductivity values

Effective Thermal Conductivity in Porous Media

• Combines solid matrix and fluid contributions—neither phase alone determines overall behavior
• Porosity ($\varepsilon$) is the key parameter—higher porosity means more fluid phase, shifting effective conductivity toward fluid properties
• Maxwell and other mixing models provide estimates: $k_{eff} = k_s \frac{2k_s + k_f - 2\varepsilon(k_s - k_f)}{2k_s + k_f + \varepsilon(k_s - k_f)}$ for dilute spherical inclusions

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Quick Reference Table

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Self-Check Questions

1. Mechanism comparison: Both the Debye model and kinetic theory explain thermal conductivity—what is the fundamental energy carrier in each, and which applies to which phase?

2. Equation application: Given a composite wall with three layers of different materials, how would you set up the thermal resistance network to find total heat transfer rate?

3. Property prediction: Using the Wiedemann-Franz Law, if a metal's electrical conductivity doubles, what happens to its thermal conductivity (assuming constant temperature)?

4. Compare and contrast: Why does thermal conductivity generally increase with temperature for gases but decrease for crystalline insulators? Explain the underlying mechanisms.

5. FRQ-style: A porous insulation material has porosity $\varepsilon = 0.4$, solid conductivity $k_s = 1.0$ W/m·K, and air conductivity $k_f = 0.025$ W/m·K. Explain why the effective conductivity will be closer to the series bound than the parallel bound, and estimate which phase dominates heat transfer.