This chapter develops conductive heat transfer from first principles, explaining the microscopic mechanisms of conduction in gases, liquids, and solids, the meaning of thermal properties, and the derivation and interpretation of the energy conservation equation in solids. Scale analysis is introduced to identify conduction regimes, followed by steady, quasi-steady, and unsteady conduction behavior. Emphasis is placed on physical interpretation and modeling implications in CFD and thermal simulations.

Context and Role of Conduction

Conduction is the most fundamental heat transfer mechanism:

It exists in all materials
It is always present, even when convection and radiation dominate
It governs heat transfer inside solids and stagnant fluids

In CFD and thermal modeling:

Conduction controls wall temperatures
It couples fluid and solid domains (conjugate heat transfer)
It determines thermal inertia and response times

Physical Mechanisms of Conduction

3.1 Conduction in Gases

In gases:

Molecules move randomly in all directions
Collisions exchange kinetic energy
A temperature gradient means one region has, on average, more energetic molecules

Even when the gas is macroscopically at rest:

Molecules crossing an imaginary surface carry energy
More energy flows from the hot side than the cold side
This net energy transfer is conduction

Key insight:
Conduction in gases is a diffusive process, driven by molecular motion, not bulk flow.

3.2 Conduction in Liquids

Liquids behave similarly to gases but:

Molecules are closer together
Collisions are more frequent

As a result:

Energy spreads faster
Thermal conductivity is higher than in gases

The mechanism remains molecular diffusion of energy.

3.3 Conduction in Solids

Solids differ fundamentally because atoms:

Cannot translate freely
Are bound in a lattice

Two main mechanisms exist:

Lattice vibrations (phonons)

Atoms vibrate around equilibrium positions
Vibrations propagate through bonds like coupled springs
Dominant in non-metals

Free electrons

Present in metals
Electrons transport kinetic energy rapidly
Explains high thermal conductivity of metals

Specific Internal Energy and Heat Capacity

4.1 Specific Internal Energy

Specific internal energy is:

Energy stored per unit mass
Associated with microscopic motion and interactions

Under local thermodynamic equilibrium:

It depends primarily on temperature
Pressure dependence is negligible for solids and liquids

4.2 Heat Capacity (Physical Meaning)

Heat capacity measures:

How much energy is required to change temperature

Physically:

Materials with high heat capacity absorb more energy with smaller temperature rise
They have greater thermal inertia

For solids:

Heat capacity is often treated as constant over moderate temperature ranges
This simplifies modeling without significant loss of accuracy

Conductive Heat Flux and Thermal Conductivity

5.1 Fourier’s Law (Physical Interpretation)

The conductive heat flux:

Is proportional to the temperature gradient
Flows from hot to cold regions

Thermal conductivity represents:

The material’s ability to spread energy spatially
High conductivity → small temperature gradients for given heat flow
Low conductivity → strong thermal resistance

5.2 Material Dependence

Typical trends:

Gases → very low conductivity
Liquids → moderate conductivity
Solids → wide range
- Metals: very high
- Insulators: very low

Thermal conductivity often depends on temperature:

Important for large temperature differences
Often neglected for small variations

5.3 Anisotropic Conduction

In some materials:

Heat spreads differently along different directions

Examples:

Crystalline solids
Composite materials
Fiber-reinforced structures

In modeling:

Conductivity becomes a tensor
Directional properties must be defined carefully
Tensor must remain symmetric and positive definite

Energy Conservation in a Solid

6.1 Integral Perspective

Energy conservation states:

Change of stored energy equals
Heat entering through boundaries
Plus heat generated internally

This form is useful for:

Global energy balances
Lumped thermal models

6.2 Differential (Local) Perspective

At each point in the solid:

Temperature changes due to
- Divergence of conductive heat flux
- Internal heat sources

This local view is essential for:

Temperature field prediction
CFD and FEA solvers

6.3 Thermal Diffusivity

Thermal diffusivity combines:

Thermal conductivity (energy transport ability)
Volumetric heat capacity (energy storage ability)

Physical meaning:

Measures how fast temperature disturbances propagate
High diffusivity → fast thermal response
Low diffusivity → slow response

Metals: high diffusivity
Plastics and insulators: low diffusivity

Scale Analysis and Conduction Regimes

Scale analysis helps determine:

Whether conduction is steady or unsteady
Whether spatial gradients are important

Key concepts:

Characteristic length
Characteristic time
Relative importance of storage vs diffusion

This leads to identification of regimes:

Quasi-steady conduction
Fully unsteady conduction
Lumped behavior (uniform temperature)

Quasi-Steady and Thermal Resistance Concept

When:

Temperature adjusts quickly compared to boundary changes

Then:

Time derivatives can be neglected
Conduction behaves as steady

This leads to the thermal resistance analogy:

Temperature difference ↔ voltage
Heat rate ↔ current
Resistance ↔ material and geometry effect

Very useful for:

Multilayer walls
Contact resistance
Simplified engineering estimates

Unsteady Conduction Behavior

Two limiting behaviors exist:

Fast unsteady regime

Temperature varies rapidly
Strong gradients near boundaries
Thermal penetration depth concept applies

Highly conducting solids

Temperature remains nearly uniform
Lumped capacity models valid

Understanding which regime applies is more important than solving the exact equations.

Boundary Conditions and Practical Modeling

From the application chapter perspective:

Common thermal boundary conditions:

Prescribed temperature
Prescribed heat flux
Convection (external environment)
Radiation
Mixed convection–radiation

For thin walls:

Full meshing may be unnecessary
Thermal resistance or shell conduction can capture physics efficiently

Study Priorities

If time is limited, the most important concepts to look into:

Physical meaning of conduction in solids
Thermal conductivity vs heat capacity
Thermal diffusivity interpretation
Energy conservation at a point
Steady vs unsteady conduction regimes
Thermal resistance concept

Key Takeaways

Conduction is energy diffusion due to microscopic motion.
Different materials conduct heat via different mechanisms.
Thermal conductivity controls heat flow; heat capacity controls temperature response.
Thermal diffusivity governs how fast temperature changes propagate.
Energy conservation provides both global and local views.
Correct regime identification simplifies modeling dramatically.

Heat transfer Chapter 2: Conductive heat transfer