Steady and Transient Heat Conduction

Introduction and Context

Heat conduction is the transfer of thermal energy through a stationary medium caused by a temperature gradient. Unlike convection, conduction does not require bulk motion of the material; unlike radiation, it does not require electromagnetic emission across a space. It is central to the design of heat exchangers, electronic packages, reactor components, building envelopes, cryogenic systems, thermal protection systems, and manufacturing processes.

A conduction problem is classified as steady when temperature at every location is independent of time, and transient when temperature changes with time. The distinction is not merely descriptive. Steady problems generally require a spatial solution of the heat equation, whereas transient problems require simultaneous consideration of spatial temperature variation and thermal storage.

At postgraduate level, conduction analysis also requires attention to variable properties, multidimensionality, internal heat generation, contact resistance, non-Fourier effects, numerical solution, uncertainty, and the validity of simplifying assumptions.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Formulate the heat-conduction equation from energy conservation.
2. Apply Fourier’s law and thermal-resistance concepts to steady systems.
3. Derive and use solutions for plane walls, cylinders, spheres, and fins.
4. Distinguish lumped-capacitance and distributed transient conduction.
5. Use Biot and Fourier numbers to select an appropriate transient model.
6. Interpret Heisler-chart and eigenfunction solutions.
7. Formulate conduction problems with internal heat generation and thermal contact resistance.
8. Select suitable analytical, numerical, and experimental methods for engineering design.

1. Physical Basis of Conduction

At the microscopic level, conduction results from molecular interaction, lattice vibration, and—in electrically conducting materials—electron transport. The macroscopic relation is Fourier’s law. In vector form,

where q is heat flux (W·m⁻²), k is thermal conductivity (W·m⁻¹·K⁻¹), and T is temperature (K). The negative sign indicates that heat flows toward decreasing temperature. For one-dimensional conduction in the x direction,

Fourier’s law is a constitutive law, not an energy balance. It is applicable when local thermal equilibrium exists and the characteristic length is much larger than the molecular mean free path. At microscale or ultrafast conditions, ballistic transport and non-Fourier models may be required.

Thermal conductivity depends on temperature, composition, porosity, moisture, pressure, and microstructure. Treating k as constant is appropriate only when its variation over the temperature range is acceptably small. For variable conductivity, integration is preferable to using an arithmetic average.

2. General Heat-Conduction Equation

Consider a stationary differential control volume with density ρ, specific heat c, thermal conductivity k, volumetric heat-generation rate q̇ (W·m⁻³), and temperature T. Conservation of energy gives

This is the general conduction equation for an isotropic material with no bulk advection. If k is constant,

where thermal diffusivity is

and has units m²·s⁻¹. Thermal diffusivity measures the rate at which a temperature disturbance spreads, while conductivity measures the ability to transport heat under a given gradient. A material can have high conductivity but moderate diffusivity if its volumetric heat capacity is also high.

In Cartesian coordinates,

For radial conduction in a cylinder and sphere, respectively,

A complete problem requires the governing equation, initial condition, boundary conditions, geometry, material properties, and any symmetry conditions.

3. Steady One-Dimensional Conduction

At steady state, ∂T/∂t = 0. For constant properties and no heat generation, the heat equation reduces to Laplace’s equation,

For a plane wall of thickness L and area A, with boundary temperatures T₁ and T₂,

The corresponding thermal resistance is

so that heat rate can be written as

This analogy is useful because resistances in series add directly. For a wall with convection at both surfaces,

and

Here h₁ and h₂ are convection coefficients (W·m⁻²·K⁻¹), and T∞ denotes bulk-fluid temperature. The overall conductance is often expressed as UA = 1/Rtotal.

Cylindrical and Spherical Walls

For radial conduction through a cylindrical wall of length L, inner radius r₁, outer radius r₂, and constant k,

For a spherical shell,

The logarithmic form for cylinders and reciprocal-radius form for spheres must not be replaced indiscriminately by the plane-wall expression. The plane-wall approximation is reasonable only when the wall thickness is small relative to the local radius.

Variable Thermal Conductivity

For a plane wall with conductivity k(T) and no heat generation, the steady heat rate is constant and

If k(T) = k₀(1 + βT), where β has units K⁻¹, then

Temperatures in this expression must be used consistently with the definition of β; absolute temperature is required when the property correlation is defined on an absolute scale.

4. Internal Heat Generation

Internal generation occurs in nuclear fuel, electrical conductors, chemical reactors, batteries, and curing materials. For a plane wall with uniform generation q̇, half-thickness L, symmetry at x = 0, and prescribed surface temperature Ts, the steady equation is

Using dT/dx = 0 at x = 0 and T = Ts at x = L gives

The maximum temperature occurs at the symmetry plane:

If the surface loses heat by convection to a fluid at T∞, the surface temperature satisfies

and the maximum temperature becomes

The first term after T∞ represents the convective temperature rise; the second represents the internal conduction rise. This decomposition is valuable in design because it identifies whether increasing h or k will be more effective.

5. Fins and Extended Surfaces

Fins increase heat-transfer area when the convection coefficient is relatively low. For a straight fin of uniform cross-section, constant k, negligible radiation, and convection coefficient h, define excess temperature θ = T − T∞. The one-dimensional fin equation is

where

Here P is the perimeter exposed to convection and Ac is the cross-sectional area normal to conduction. The equation assumes negligible temperature variation across each cross-section, which requires a sufficiently small cross-sectional Biot number.

For an infinitely long fin, θ/θb = exp(−mx), and the heat rate is

For an adiabatic-tip fin of length L,

Fin efficiency is

where Af is fin surface area. A fin is not automatically beneficial: its added convection must exceed the heat that would have left from the base area it covers. The practical measure is fin effectiveness, the ratio of heat transfer with the fin to heat transfer from the unfinned base area.

6. Transient Conduction

Transient conduction involves thermal energy storage. A general initial-boundary-value problem consists of the heat equation, an initial temperature field, and boundary conditions that may be prescribed temperature, prescribed heat flux, convection, radiation, or contact conditions.

Two dimensionless groups are particularly important. The Fourier number is

where t is time and Lc is a characteristic length. It measures the extent of thermal diffusion. The Biot number is

It compares internal conduction resistance with external convection resistance. For a body of characteristic volume V and convecting area As, the standard characteristic length is Lc = V/As. In slab, cylinder, or sphere solutions, a half-thickness or radius is often used instead; the definition must be stated because the numerical Biot value depends on the chosen length.

Lumped-Capacitance Method

If internal temperature gradients are negligible, the body has a spatially uniform temperature T(t). The energy balance is

For constant properties and constant T∞,

or equivalently,

when Lc = V/As. The lumped approximation is generally reliable when Bi ≤ 0.1. This criterion is a guideline, not an absolute law; accuracy also depends on geometry, property variation, and the desired tolerance.

Example: Lumped Cooling of a Metal Sphere

A steel sphere of radius 0.015 m is initially at 200 °C and is cooled in air at 25 °C. Let h = 80 W·m⁻²·K⁻¹, k = 45 W·m⁻¹·K⁻¹, ρ = 7800 kg·m⁻³, and c = 460 J·kg⁻¹·K⁻¹. Determine the approximate time to reach 80 °C.

For a sphere, Lc = V/As = r/3 = 0.005 m. Thus

Bi = hLc/k = (80 × 0.005)/45 = 0.0089,

so lumped analysis is justified. The temperature ratio is

θ/θi = (80 − 25)/(200 − 25) = 0.3143.

The time constant is

Therefore τ = (7800 × 460 × 0.005)/80 = 224.25 s. Since θ/θi = exp(−t/τ),

t = −τ ln(0.3143) = 259 s, approximately 4.3 min.

The result should not be reported without the Biot check; the check establishes why a single body temperature is acceptable.

Distributed-Parameter Solutions

When Bi > 0.1, internal gradients matter. Exact solutions for canonical geometries are commonly expressed as eigenfunction series. For a plane wall of half-thickness L, initially uniform temperature, and symmetric convection at both surfaces, the dimensionless centerline temperature has the form

where the eigenvalues ζn satisfy

for the convention Bi = hL/k. The coefficients An depend on the geometry and boundary condition. Similar series exist for cylinders and spheres, with eigenvalue equations involving Bessel functions for cylinders and spherical functions for spheres. Heisler charts provide graphical approximations for center temperature, local temperature, and total heat transfer for standard geometries.

A common engineering procedure is to use the first-term approximation after sufficient time has elapsed:

The approximation is valid only when the omitted terms are small. It is unsafe to apply it at very small Fourier numbers without checking convergence.

7. Boundary and Interface Conditions

Common boundary conditions include:

• First kind: prescribed temperature, T = Ts.
• Second kind: prescribed heat flux, −k∂T/∂n = q″.
• Third kind: convection, −k∂T/∂n = h(Ts − T∞).
• Radiative condition: −k∂T/∂n = εσ(Ts⁴ − Tsur⁴), often nonlinear.
• Symmetry: ∂T/∂n = 0 at a geometric or thermal symmetry plane.

At a perfect interface, temperature and heat flux are continuous. With contact resistance Rc,

For an interface area Ac, Rc is often represented as Rc = 1/(hcAc), where hc is the contact conductance. Contact conductance depends on surface roughness, pressure, hardness, oxidation, interstitial material, and temperature. It should be measured or obtained from a validated correlation rather than assumed negligible.

8. Numerical Methods and Advanced Considerations

Analytical solutions are limited to simple geometry and boundary conditions. Finite-difference, finite-volume, and finite-element methods are used for irregular geometries, temperature-dependent properties, phase change, moving boundaries, and coupled conduction–convection–radiation.

For an explicit one-dimensional finite-difference scheme with constant α and grid spacing Δx,

where FoΔx = αΔt/(Δx)². Stability requires, for the simple one-dimensional interior scheme, FoΔx ≤ 1/2. This is a numerical-stability condition, not a physical law. Implicit and Crank–Nicolson schemes permit larger time steps, but they still require accuracy, convergence, and verification checks.

Best practice includes grid refinement, time-step refinement, residual monitoring, energy-balance checking, comparison with limiting analytical solutions, and sensitivity analysis for uncertain k, h, contact resistance, and boundary temperature. Verification asks whether the equations were solved correctly; validation asks whether the equations and properties represent the real system.

At microscale, the classical Fourier model can overpredict the speed of thermal propagation. Cattaneo–Vernotte models introduce a finite relaxation time, while phonon-based and Boltzmann-transport approaches may be necessary for nanostructures. These models should be adopted only when characteristic length and time scales justify them.

9. Applied Case: Thermal Management of an Electronic Module

Background

Consider an illustrative power-electronic module mounted on an aluminium heat spreader. The semiconductor generates 60 W during operation. The allowable junction temperature is 125 °C, while ambient air is 30 °C. The thermal path includes semiconductor-to-case contact, a thermal interface material, aluminium spreading, and convection from an attached finned heat sink.

Problem

The design must determine whether the heat sink can maintain the junction below its limit during continuous operation. The principal uncertainties are interface contact resistance, convection coefficient, and nonuniform heat spreading.

Analysis

A first model uses a steady resistance network:

where Rjc is junction-to-case resistance, Rcs is case-to-sink resistance, and Rsa is sink-to-air resistance. If the total resistance is limited to

then the sum of all resistances must not exceed 1.583 K·W⁻¹. Suppose measured or manufacturer data give Rjc = 0.40 K·W⁻¹ and Rcs = 0.20 K·W⁻¹. The heat sink must therefore provide no more than 0.983 K·W⁻¹.

A transient start-up calculation is also needed. If the module has a significant thermal mass, its junction temperature may remain below the steady value for several minutes but later exceed it. A compact RC network can approximate this behavior, but a finite-element model is preferable when the heat source is localized and spreading resistance is important.

Approach and Solution

The engineering workflow is: measure or estimate interface resistance; calculate a conservative steady resistance budget; select fin area and airflow; solve the two- or three-dimensional temperature field; perform mesh and time-step studies; and test the prototype under worst-case ambient and power conditions. If the steady result is marginal, reducing contact resistance may be more effective than increasing fin area.

Outcome and Learning Points

The resistance budget identifies the allowable sink performance before detailed design begins. It also reveals that thermal interfaces can dominate the result even when the heat sink material has high conductivity. The case demonstrates the importance of separating steady-state adequacy from transient start-up behavior and of validating assumptions experimentally.

10. Limitations, Safety, and Professional Practice

Conduction models can fail when material properties vary strongly with temperature, radiation is significant, phase change occurs, heat generation is nonuniform, or fluid flow changes the boundary coefficient. “Adiabatic” and “isothermal” surfaces are idealizations that require physical justification.

Thermal design has safety implications. Overlooking hot spots can cause insulation degradation, fire, battery failure, biological injury, or structural weakening. Designers should document property sources, uncertainty margins, operating envelopes, sensor calibration, and failure modes. Computational results should never be presented as measurements, and proprietary or safety-critical data should be governed by appropriate quality-assurance procedures.

Contemporary Developments

Current research includes topology-optimized heat sinks, additively manufactured lattice structures, nanostructured thermal interfaces, thermal metamaterials, phase-change heat spreaders, microchannel cooling, and digital twins that combine sensor data with reduced-order models. Machine-learning surrogates can accelerate design exploration, but they do not remove the need for conservation laws, physically meaningful training data, uncertainty quantification, and out-of-distribution checks.

Chapter Summary

Steady conduction is governed by spatial energy balance with no thermal storage, while transient conduction includes the accumulation of internal energy. Fourier’s law relates heat flux to temperature gradient, and the heat equation follows from conservation of energy. Plane, cylindrical, and spherical geometries possess different resistance expressions. Internal heat generation produces parabolic temperature fields in simple geometries, and fins enhance heat transfer when their effectiveness justifies the added material and surface.

The Biot number determines whether internal temperature gradients are important, while the Fourier number measures the progress of thermal diffusion. Lumped-capacitance analysis is appropriate when Bi ≤ 0.1; otherwise distributed solutions, charts, or numerical methods are required. Contact resistance, variable properties, nonlinear radiation, and multidimensional spreading often control practical accuracy. Reliable design combines analytical estimates, numerical verification, experimental validation, uncertainty assessment, and safe operating margins.

Key Terms

Fourier’s law: Constitutive relation linking heat flux to the temperature gradient.

Thermal conductivity: Material property measuring conductive heat-transfer capability.

Thermal diffusivity: α = k/(ρc), a measure of the rate of thermal response.

Steady conduction: Conduction in which temperature is independent of time.

Transient conduction: Conduction involving temperature variation with time.

Heat flux: Heat-transfer rate per unit area, measured in W·m⁻².

Volumetric heat generation: Thermal energy generated per unit volume and time.

Thermal resistance: Temperature difference divided by heat rate.

Contact resistance: Resistance caused by imperfect thermal contact between surfaces.

Biot number: Ratio of internal conduction resistance to external convection resistance.

Fourier number: Dimensionless measure of thermal diffusion time.

Lumped-capacitance method: Transient model assuming a uniform body temperature.

Eigenvalue: Parameter determined by a boundary-condition equation in series solutions.

Fin: Extended surface used to increase convective heat transfer.

Fin efficiency: Actual fin heat transfer divided by ideal isothermal-fin transfer.

Internal heat generation: Energy production within a material volume.

Thermal spreading resistance: Resistance associated with nonuniform, multidimensional heat flow.

Verification: Demonstration that a computational solution correctly solves the formulated equations.

Validation: Demonstration that the model represents the physical system adequately.

Review Questions

Short Answer Questions

1. State Fourier’s law and explain the significance of its negative sign.
2. Define thermal conductivity and thermal diffusivity, including their units.
3. What physical comparison does the Biot number represent?
4. State the criterion commonly used to justify lumped-capacitance analysis.
5. Why are cylindrical and spherical conduction resistances different from plane-wall resistance?

Long Answer Questions

1. Derive the one-dimensional steady temperature distribution in a plane wall with uniform volumetric heat generation and a prescribed surface temperature.
2. Explain the construction and use of a thermal-resistance network containing conduction, convection, and contact resistances.
3. Derive the lumped-capacitance equation for a body cooled by convection and discuss its assumptions.
4. Explain fin efficiency, fin effectiveness, and the physical conditions under which fins are advantageous.
5. Compare analytical, finite-difference, finite-volume, and finite-element approaches to transient conduction.

Analytical / Application Questions

1. A plane wall has thickness 0.08 m, area 12 m², conductivity 1.6 W·m⁻¹·K⁻¹, and surface temperatures 180 °C and 40 °C. Calculate the steady heat-transfer rate and thermal resistance, neglecting edge effects.
2. A solid sphere is cooled in a fluid. Explain how you would decide between lumped and distributed analysis, and identify the measurements needed to validate the prediction.
3. An electronic device has a localized heat source and a temperature-dependent interface resistance. Propose a model hierarchy from preliminary calculation to validated three-dimensional simulation, explaining the assumptions at each stage.

Multiple Choice Questions with Answers

Question 1

Which equation represents Fourier’s law in one-dimensional conduction?

A. q_x = k dT/dx

B. q_x = -k dT/dx

C. q_x = h(T_s-T_∞)

D. q_x = ρc dT/dt

Correct Answer: B. q_x = -k dT/dx

Question 2

For a plane wall with constant k, thickness L, and area A, the conduction resistance is:

A. L/(kA)

B. kA/L

C. 1/(hA)

D. ln(L)/(2πkA)

Correct Answer: A. L/(kA)

Question 3

A body has Biot number 0.04. Which transient method is generally appropriate?

A. Radiation-only analysis

B. Lumped capacitance

C. Steady spherical analysis

D. Incompressible-flow analysis

Correct Answer: B. Lumped capacitance

Question 4

The Fourier number primarily measures:

A. The ratio of convection to conduction resistance

B. The progress of thermal diffusion with time

C. The ratio of radiation to convection

D. The ratio of heat generation to heat flux

Correct Answer: B. The progress of thermal diffusion with time

Question 5

For a uniformly generating plane wall with half-thickness L and prescribed surface temperature, where is the maximum temperature?

A. At either surface

B. At one-quarter thickness

C. At the symmetry plane

D. Outside the wall

Correct Answer: C. At the symmetry plane

Question 6

The thermal diffusivity of a material is defined as:

A. kρc

B. k/(ρc)

C. ρc/k

D. h/k

Correct Answer: B. k/(ρc)

Question 7

For radial conduction through a cylindrical wall, the resistance contains which geometric term?

A. L/(kA) only

B. ln(r₂/r₁)/(2πkL)

C. (r₂-r₁)/(4πk)

D. 1/(hA) only

Correct Answer: B. ln(r₂/r₁)/(2πkL)

Question 8

In a simple explicit one-dimensional finite-difference conduction scheme, instability may occur when αΔt/(Δx)² is:

A. Less than zero

B. Exactly zero

C. Greater than one-half

D. Equal to the Biot number

Correct Answer: C. Greater than one-half

Question 9

The purpose of a fin is primarily to:

A. Reduce the available convective area

B. Increase heat-transfer area

C. Eliminate thermal conductivity

D. Force the body to reach steady state instantly

Correct Answer: B. Increase heat-transfer area

Question 10

A thermal-resistance budget for an electronic device is useful because it:

A. Replaces all experimental validation

B. Identifies the allowable resistance of each thermal path

C. Applies only when there is no convection

D. Makes contact resistance irrelevant

Correct Answer: B. Identifies the allowable resistance of each thermal path