Heat Transfer: Conduction

Conduction is the mode of heat transfer that occurs through a solid (or between solids in contact) because of microscopic energy exchange. In metals it is dominated by mobile electrons; in nonmetals it is mainly lattice vibration. From an engineering standpoint, conduction is often the quiet workhorse behind insulation design, heat sinks, furnace walls, and temperature control of parts during manufacturing.

This article focuses on practical conduction analysis: Fourier’s law, steady-state thermal resistance networks, fins, and transient conduction using lumped capacitance and Heisler charts.

Fourier’s Law: The Starting Point

Conduction is quantified by Fourier’s law. In one dimension it states that heat flows from high temperature to low temperature in proportion to the temperature gradient:

$q_x=-k\frac{dT}{dx}$

where:

• $q_x$ is the heat flux (W/m²) in the $x$ direction,
• $k$ is the thermal conductivity (W/m·K),
• $T$ is temperature (K or °C),
• the negative sign enforces flow down the gradient.

For a plane wall of thickness $L$ with constant $k$ and steady temperatures $T_1$ and $T_2$ on its faces, Fourier’s law integrates to:

$q=kA\frac{T_1-T_2}{L}$

Here, $q$ is the heat rate (W) and $A$ is the cross-sectional area normal to heat flow.

Two immediate implications matter in real design:

1. Material matters: high $k$ (copper, aluminum) spreads heat well; low $k$ (foams, ceramics, air gaps) insulates.
2. Geometry matters: doubling thickness halves conduction heat rate for the same temperature difference.

Steady-State Conduction and Thermal Resistance

Most day-to-day engineering calculations reduce conduction to a thermal resistance form, analogous to electrical circuits:

$q=\frac{\Delta T}{R_{th}}$

For a plane wall:

$R_{cond}=\frac{L}{kA}$

This representation becomes powerful when multiple layers or paths are involved.

Composite Walls (Series Resistance)

When heat flows through layers stacked in the direction of heat flow (for example, brick plus insulation), resistances add:

$R_{total}=R_1+R_2+\cdots$

Then:

$q=\frac{T_{hot}-T_{cold}}{R_{total}}$

This makes it straightforward to see where temperature drops occur. The layer with the largest $R$ takes the largest portion of the temperature difference, which is why thin layers of low-$k$ insulation can dominate overall performance.

Parallel Conduction Paths

If heat can flow through multiple paths side-by-side (for example, insulation interrupted by metal studs), resistances combine like parallel electrical resistors:

$\frac{1}{R_{total}}={\sum }_i\frac{1}{R_i}$

Parallel paths are a common source of “thermal bridging,” where a small fraction of highly conductive material carries disproportionate heat, reducing overall insulating effectiveness.

Cylinders and Spheres (Radial Conduction)

Many practical systems involve pipes, tanks, and wires. For radial conduction through a hollow cylinder (inner radius $r_1$, outer radius $r_2$, length $L$):

$R_{cond,cyl}=\frac{\ln (r_2/r_1)}{2\pi kL}$

The logarithmic dependence means insulation effectiveness on pipes depends on radius. This leads to a notable concept in insulation design: for a cylinder with external convection, there can be a critical radius where adding a small amount of insulation initially increases heat loss by increasing surface area faster than it adds resistance. In practice, thick enough insulation almost always reduces losses, but the geometry warns against assuming “more insulation always immediately helps” for small wires and tubes.

Extended Surfaces (Fins): Boosting Heat Transfer

Fins are added to increase heat transfer by increasing surface area. While fins are often discussed in convection terms, their performance is fundamentally governed by conduction within the fin. Heat must conduct from the base into the fin before it can be convected away.

A fin is effective when:

• the fin material has high thermal conductivity (to reduce temperature drop along the fin),
• convection at the surface is not so weak that added area is wasted,
• the fin geometry provides meaningful extra area without excessive thickness or mass.

Fin Efficiency and Practical Meaning

Because a fin is cooler at the tip than at the base, not all fin area operates at the base temperature. Fin efficiency ${\eta }_f$ captures this:

${\eta }_f=\frac{\text{actual heat transfer from fin}}{\text{heat transfer if entire fin were at base temperature}}$

Engineers use fin efficiency to decide whether adding more fin length helps. Long, thin fins on low-$k$ materials often have low efficiency, meaning extra length adds little heat dissipation. By contrast, aluminum heat sinks work well because aluminum conducts heat effectively along the fin.

Fin Applications

• Electronics cooling: heat sinks conduct heat away from components and spread it to fin surfaces.
• Heat exchangers: finned tubes increase area on the air side where convection is weaker.
• Engines and compressors: fins remove heat from cylinders when forced liquid cooling is absent.

Transient Conduction: When Temperatures Change with Time

Steady-state assumptions fail whenever a part is heating up, cooling down, or cycling. Transient conduction describes time-dependent temperature fields.

The central idea is energy storage: solids have thermal mass. The characteristic property is thermal diffusivity:

$\alpha =\frac{k}{\rho c_p}$

where $\rho$ is density and $c_p$ is specific heat. A high $\alpha$ material spreads temperature changes quickly (many metals); a low $\alpha$ material responds slowly (many polymers and insulations).

Lumped Capacitance Method (Uniform Temperature Approximation)

The lumped capacitance model treats the solid as having a uniform temperature at any instant. It is appropriate when internal conduction is much faster than surface heat transfer, so temperature gradients inside the body are negligible.

The criterion is the Biot number:

$Bi=\frac{h{L}_c}{k}$

where $h$ is the convection coefficient and $L_c$ is a characteristic length (often volume divided by surface area). A common rule is that lumped capacitance is acceptable when $Bi\lesssim 0.1$.

When valid, the temperature follows an exponential approach to the surrounding fluid temperature $T_{\infty }$:

$\frac{T(t)-T_{\infty }}{T_i-T_{\infty }}=\exp \left(-\frac{hA}{\rho c_{p}V}t\right)$

This is widely used for quenching small parts, estimating cool-down time of thin components, or predicting sensor response when the sensor body is small and conductive.

Heisler Charts: Transient Conduction with Internal Gradients

When $Bi$ is not small, internal gradients matter and the lumped model fails. Heisler charts provide practical solutions for transient conduction in standard shapes (plane walls, cylinders, spheres) with convection boundary conditions, without requiring full analytic series calculations.

Heisler charts relate:

• dimensionless temperature ratio $(T-T_{\infty })/(T_i-T_{\infty })$,
• dimensionless time (Fourier number) $Fo=\alpha t/L_c^2$,
• Biot number $Bi$.

With these charts, engineers can estimate:

• centerline temperature as a function of time (critical for thermal stress and material limits),
• temperature distribution within the solid at a given time,
• total heat transferred up to time $t$.

They remain valuable for quick, defensible estimates in thermal processing, food heating and cooling, and cooldown of thicker mechanical parts where gradients are unavoidable.

Putting It Together: How Conduction Analysis Is Used

In real projects, conduction problems typically combine several of these elements:

• A multilayer wall analysis uses Fourier’s law and thermal resistance to estimate heat loss, then checks temperatures at interfaces for condensation risk or material limits.
• A heat sink design relies on conduction within the base and fins, then balances fin geometry with convection performance.
• A transient heating step might start with lumped capacitance for small parts; if gradients are expected, Heisler charts provide center temperature history and heating time.

A useful workflow is:

1. Start with a steady-state resistance model to understand heat flow magnitude.
2. Evaluate whether time dependence matters; if so, check $Bi$.
3. Use lumped capacitance for small $Bi$, or Heisler charts for standard shapes when gradients matter.
4. Revisit geometry and materials: increasing $k$, increasing thickness for insulation, or adding fins often changes the design more effectively than tweaking operating conditions.

Why Conduction Still Deserves Careful Attention

Conduction is often treated as “simple,” but small modeling choices can cause large errors: ignoring thermal bridges, assuming constant $k$ when it varies strongly with temperature, or using lumped capacitance outside its range. A solid grasp of Fourier’s law, thermal resistance, fins, and transient tools like Heisler charts lets you make fast estimates that are not just convenient, but technically sound.