Multidimensional and Transient Conduction

Introduction and Context

Heat conduction is often introduced through one-dimensional, steady models such as a plane wall, cylinder, or sphere. Those models are powerful, but real components frequently have heat flow in two or three directions and may experience temperatures that change with time. Examples include electronic packages, turbine blades, building corners, weldments, heat-treatment loads, cryogenic vessels, and castings.

This chapter develops methods for such situations. Two broad questions are addressed:

1. How can heat transfer through multidimensional geometries be estimated without solving the complete temperature field analytically?
2. How does the temperature of a body change when its thermal state is initially nonuniform or when external conditions vary with time?

Shape factors provide a convenient engineering approximation for many steady multidimensional problems. Numerical methods—including finite-difference and finite-volume approaches—provide more general solutions. For transient conduction, the Biot number determines whether internal temperature gradients are important, while the Fourier number measures the progress of thermal diffusion.

Learning Objectives

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

• distinguish one-, two-, and three-dimensional conduction;
• use conduction shape factors to estimate heat transfer in selected geometries;
• formulate and interpret the multidimensional heat equation;
• construct finite-difference equations for steady and transient conduction;
• explain the physical meaning of the Biot and Fourier numbers;
• apply the lumped-capacitance method and assess its validity;
• use analytical transient solutions and charts for spatially varying systems;
• select suitable numerical methods and evaluate their limitations;
• interpret conduction results in practical engineering applications.

1. Governing Principles of Multidimensional Conduction

Fourier’s law in vector form is

where q⃗ is heat flux, W/m²; k is thermal conductivity, W/(m·K); T is temperature, K or °C for temperature differences; and ∇T is the temperature gradient, K/m. The negative sign indicates that heat flows toward decreasing temperature.

For a stationary control volume with constant properties, volumetric heat generation q̇ and no fluid motion, energy conservation gives

where ∇² is the Laplacian operator, q̇ is heat generation, W/m³, and α is thermal diffusivity, m²/s:

Here ρ is density, kg/m³, and c_p is specific heat, J/(kg·K). The equation assumes isotropic material behavior and constant k, ρ, and c_p. In Cartesian coordinates,

For steady conduction without generation, the equation reduces to Laplace’s equation:

If heat generation is present but the process is steady, the equation is Poisson’s equation:

The dimensionality of a problem depends on the number of spatial coordinates required to describe the temperature field. A wall may be one-dimensional if its thickness is small compared with its other dimensions and edge effects are negligible. A wall corner, however, generally requires two-dimensional analysis. A component with variation in all three coordinates requires a three-dimensional model.

Boundary and Initial Conditions

A conduction equation is not complete until boundary and, for transient problems, initial conditions are specified. Common boundary conditions are:

• Specified temperature: T = T_s on a boundary.
• Specified heat flux: −k ∂T/∂n = q″_s, where n is the outward normal direction.
• Convection: −k ∂T/∂n = h(T_s − T_∞).
• Radiation: −k ∂T/∂n = εσ(T_s⁴ − T_sur⁴).
• Insulation or symmetry: ∂T/∂n = 0.

The convection condition uses h, the convection coefficient, W/(m²·K), and T_∞, the surrounding-fluid temperature. Radiation is nonlinear because of the fourth-power temperature terms; absolute temperature must be used in kelvin.

2. Shape Factors for Multidimensional Steady Conduction

2.1 Concept and Definition

A shape factor, S, summarizes the effect of geometry on steady conduction. For a homogeneous body with constant thermal conductivity and boundaries maintained at temperatures T₁ and T₂, the heat-transfer rate is written as

where Q̇ is heat-transfer rate, W; k is thermal conductivity, W/(m·K); and S is the conduction shape factor, m. The units of S must be metres so that kS∆T has units of watts.

The corresponding thermal resistance is

with units K/W. Shape factors are not universal constants. They depend on geometry, boundary locations, and the definition of the isothermal surfaces. They are most useful when the principal boundaries are approximately isothermal and the material is homogeneous.

For a one-dimensional plane wall of area A and thickness L,

For radial conduction through a cylinder of length L between radii r₁ and r₂,

For radial conduction through a sphere,

These expressions reproduce the familiar one-dimensional resistances. More complicated two-dimensional shapes require tabulated shape factors, analytical approximations, or numerical calculation.

2.2 Approximate Additivity and Conduction Paths

In some geometries heat spreads through several distinct regions. If the regions can be treated as parallel conduction paths, their heat rates may be added:

This requires care. The same temperature difference must apply to each path, and the paths should not strongly interact. Shape-factor addition is an engineering approximation, not a general law for arbitrary geometries.

Corners and edges often increase heat flow relative to a one-dimensional estimate because heat spreads laterally. In building envelopes, these effects are called thermal bridges. A one-dimensional wall calculation can therefore underestimate local heat loss and surface temperature depression.

2.3 Worked Example: Shape-Factor Estimate

A thick refractory wall contains a two-dimensional thermal bridge. Experimental or tabulated geometric analysis gives a shape factor of S = 0.85 m. The bridge material has k = 0.40 W/(m·K), and the bounding isothermal surfaces are at 500°C and 80°C. Estimate the steady heat-transfer rate.

Using

with ∆T = 500 − 80 = 420 K,

Thus, Q̇ ≈ 143 W. The unit check is W/(m·K) × m × K = W. This result applies only if the stated shape factor correctly represents the actual boundary geometry and the material properties are adequately approximated as constant.

3. Numerical Methods for Multidimensional Conduction

When geometry, material properties, internal generation, or boundary conditions are too complex for shape factors, the governing equation is solved numerically. The most common approaches are finite difference, finite volume, and finite element methods.

3.1 Finite-Difference Formulation

Consider a uniform two-dimensional grid with spacing ∆x and ∆y. For steady, constant-k conduction without generation, the central-difference equation at an interior node is

For a square grid, ∆x = ∆y = ∆, this becomes

Thus, at steady state, an interior temperature equals the average of its four neighboring temperatures. This is a discrete statement of energy conservation.

For a three-dimensional cubic grid,

These equations assume uniform conductivity, equal grid spacing, and no heat generation. For nonuniform k or nonuniform spacing, conductance-weighted equations must be used.

3.2 Heat Generation

For a square grid with uniform k and uniform generation q̇, the steady finite-difference equation is

where T_E, T_W, T_N, and T_S are neighboring nodal temperatures. The generation term has units K because q̇∆²/k gives [W/m³][m²]/[W/(m·K)] = K.

3.3 Transient Explicit Scheme

For a two-dimensional square grid with no generation, an explicit time-marching equation is

where

Here ∆t is the time step, s, and the superscript n denotes the time level. For the simple two-dimensional explicit scheme, stability requires approximately

For a three-dimensional cubic grid, the corresponding restriction is approximately Fo∆ ≤ 1/6. These limits apply to the stated uniform-grid, constant-property explicit schemes. Implicit schemes are more stable and permit larger time steps, but require solution of algebraic equations at every time step.

3.4 Boundary Nodes with Convection

At a convective boundary, conduction from the solid to the surface must equal convection to the fluid:

A finite-volume formulation is especially convenient because it applies an energy balance to each control volume. Conductances to neighboring nodes and the boundary convection conductance are assembled into a linear algebraic system. This approach conserves energy locally and handles irregular material interfaces more reliably than a casually applied finite-difference equation.

3.5 Solution Procedures and Verification

The algebraic equations may be solved by Gauss–Seidel iteration, successive over-relaxation, or a direct matrix solver. A calculation should be checked using:

• residual reduction, ensuring that each node satisfies the discretized energy balance;
• grid refinement, comparing results on progressively finer meshes;
• global energy balance, comparing total input and output heat rates;
• comparison with limiting one-dimensional or analytical solutions;
• sensitivity to uncertain values of k, h, contact resistance, and generation.

A visually smooth temperature contour is not evidence of correctness. Numerical errors can remain hidden unless conservation and grid independence are tested.

4. Dimensionless Groups for Transient Conduction

4.1 Biot Number

The Biot number compares internal conduction resistance with external convection resistance:

where h is the convection coefficient, L_c is a characteristic length, and k is the solid conductivity. For lumped-capacitance analysis, the standard characteristic length is

where V is body volume, m³, and A_s is the convecting surface area, m².

A small Biot number means that internal conduction is fast relative to surface convection, so the body remains nearly uniform in temperature. A commonly accepted criterion is

for lumped analysis. This criterion concerns internal gradients; it does not mean that the body temperature changes slowly.

4.2 Fourier Number

The Fourier number measures the ratio of elapsed time to the characteristic diffusion time:

where t is time. Large Fo indicates that thermal diffusion has progressed substantially. Unlike Bi, Fo is a time-dependent quantity.

For analytical solutions involving a plane wall, infinite cylinder, or sphere, the characteristic length in Fo is often the half-thickness, radius, or sphere radius, respectively. The definition must therefore be stated with the geometry.

5. Lumped-Capacitance Analysis

If Bi ≤ 0.1, the body can be approximated as having a spatially uniform temperature T(t). An energy balance gives

Assuming constant properties, constant h, and constant ambient temperature, integration from initial temperature T_i gives

Using L_c = V/A_s and α = k/(ρc_p), this may be written as

where θ = T − T_∞ and θ_i = T_i − T_∞. The product BiFo is equal to hA_st/(ρVc_p) when the same characteristic length is used.

The heat transferred from the body over time t is obtained from the decrease in internal energy:

This expression assumes no internal heat generation and negligible radiation or other heat-transfer modes.

Worked Example: Cooling of a Small Metal Part

A steel sphere of radius 0.015 m is initially at 200°C and is suddenly exposed to air at 25°C. Take h = 60 W/(m²·K), k = 45 W/(m·K), ρ = 7800 kg/m³, and c_p = 470 J/(kg·K). Find its temperature after 300 s and determine whether lumped analysis is appropriate.

For a sphere, V/A_s = r/3, so

The Biot number is

Since Bi < 0.1, lumped analysis is appropriate. The exponential coefficient is

Therefore,

The dimensionless temperature is

Hence,

The approximate temperature after 300 s is 91°C. The result is physically reasonable: the part has moved substantially toward the air temperature but has not reached it.

6. Spatially Varying Transient Conduction

When Bi > 0.1, internal gradients cannot be neglected. The temperature depends on position and time, and solutions are commonly expressed using dimensionless temperature:

For simple geometries with uniform initial temperature, constant properties, and convection at the surface, separation-of-variables solutions are available. A plane wall, infinite cylinder, or sphere solution has the general form

The coefficients C_n and eigenvalues λ_n depend on geometry, boundary condition, and Biot number. The eigenvalues are found from transcendental equations. For a plane wall of half-thickness L with convection at both surfaces, one common eigenvalue relation is

where Bi = hL/k with L equal to the half-thickness. The exact coefficient expressions depend on whether the local or average temperature is required. Consequently, Heisler charts or validated tabulated solutions are often used in undergraduate engineering practice.

For sufficiently large Fourier number, the first term dominates:

This approximation is useful when higher modes have decayed significantly, but the coefficient and eigenvalue must correspond to the selected geometry and temperature definition. A centerline temperature cannot be substituted for an average temperature.

Heisler Charts and Their Use

Heisler charts provide dimensionless centerline temperature, average temperature, and heat-transfer information for standard geometries. A sound procedure is:

1. Identify the geometry and characteristic length.
2. Calculate Bi and Fo using the chart’s definitions.
3. Select the chart for centerline or average temperature.
4. Read θ* and convert back to dimensional temperature.
5. Check whether the chart assumptions apply.

The charts assume constant thermal properties, initially uniform temperature, a constant ambient temperature, and a specified convection boundary. They do not automatically account for radiation, contact resistance, phase change, temperature-dependent properties, or internal generation.

7. Practical Case: Thermal Treatment of an Aluminum Slab

Illustrative engineering case. An aluminum slab enters a cooling line after solution heat treatment. The slab must cool rapidly enough to meet a production cycle, but excessive surface-to-center temperature differences can cause distortion.

Background and Problem

The slab is 0.10 m thick, initially at 500°C, and exposed on both broad faces to air at 30°C. Its approximate properties are k = 180 W/(m·K), ρ = 2700 kg/m³, c_p = 900 J/(kg·K), and h = 80 W/(m²·K). Edge losses are initially neglected. Determine whether the slab can be treated as lumped during early cooling.

Analysis

For a plane wall cooled on both faces, the relevant characteristic length for the lumped Biot test is V/A_s = half-thickness = 0.05 m. Thus,

Since Bi < 0.1, the slab temperature is nearly uniform under the stated assumptions. Its thermal diffusivity is

At t = 600 s,

The product BiFo is approximately 0.395. Therefore,

The estimated mean and nearly uniform temperature are

Approach and Engineering Interpretation

The lumped result predicts a temperature of about 347°C after 10 minutes. Although this is useful for production-cycle estimation, a complete design should also examine edge effects, variation in h over the slab, radiation, temperature-dependent properties, and mechanical distortion. If the slab is supported by fixtures, contact conductance may create additional local gradients.

Outcome and Learning Points

The initial screening calculation shows that internal conduction is much less resistant than external convection. A lumped model is therefore justified for the broad faces. However, the model does not guarantee uniform temperature near corners or contact points. A two-dimensional numerical model is appropriate if distortion, residual stress, or local metallurgical history is important.

8. Selecting a Solution Method

A practical decision sequence is:

Finite-element software can represent complex geometry and coupled thermal–mechanical behavior, while finite-volume software is particularly strong for local conservation and conjugate heat transfer. The software does not replace physical modeling: incorrect boundary conditions or material data can produce precise but misleading answers.

9. Limitations, Safety, and Professional Practice

Property Variation

The constant-property equations are approximations. Metals may experience substantial changes in thermal conductivity and specific heat over broad temperature ranges. If properties vary strongly, the numerical model should update them with temperature, or an effective-property method should be justified.

Radiation and Nonlinear Boundaries

At elevated temperatures, radiation may be comparable to or greater than convection. The radiative heat flux is

where ε is emissivity, dimensionless, and σ = 5.670 × 10^(−8) W/(m²·K⁴). Surface and surrounding temperatures must be in kelvin. Linearizing radiation using an effective coefficient can simplify a calculation, but the chosen reference temperature should be reported.

Phase Change and Internal Generation

Melting, solidification, evaporation, and chemical reaction require latent heat or source terms. A simple conduction model is not adequate when phase boundaries move significantly. Similarly, batteries, nuclear fuel, electrical conductors, and biological tissues may have spatially varying heat generation.

Numerical Reliability and Governance

Thermal predictions can affect pressure-vessel integrity, fire safety, battery safety, medical devices, and aircraft structures. Engineers should document assumptions, mesh quality, convergence criteria, property sources, uncertainty ranges, and validation evidence. A simulation should not be used to justify operation outside certified temperature or material limits without appropriate review.

10. Contemporary Developments

Modern thermal analysis increasingly combines numerical conduction models with sensor data, reduced-order models, and optimization algorithms. Digital twins may update a model using measured temperatures, while machine-learning surrogates can accelerate repeated design calculations. These tools are valuable only when trained and validated against physically consistent data. Conservation laws, dimensional checks, uncertainty quantification, and independent validation remain essential.

For design practice, the most useful trend is not replacing heat-transfer fundamentals but integrating them with automated meshing, adaptive time stepping, parameter estimation, and multiphysics simulation. A compact lumped model may provide rapid control decisions, while a detailed numerical model is reserved for design verification and failure analysis.

Chapter Summary

Multidimensional conduction is governed by the heat equation, with the Laplacian representing spatial temperature curvature and thermal diffusivity controlling the rate of thermal response. Shape factors express steady heat transfer through selected multidimensional geometries as Q̇ = kS∆T and are useful when boundaries and material properties satisfy their assumptions.

Numerical methods convert the governing differential equation into algebraic energy balances. Finite-difference methods are simple for regular grids; finite-volume methods emphasize local conservation; and finite-element methods handle complex geometry effectively. Mesh refinement, residual checks, and global energy balances are essential verification practices.

The Biot number compares internal conduction and external convection resistance. When Bi ≤ 0.1, lumped capacitance usually provides an accurate temperature history. The Fourier number measures the progress of thermal diffusion. When internal gradients matter, analytical series, Heisler charts, or numerical transient models are required. Engineering judgment must account for variable properties, radiation, generation, phase change, contact resistance, and uncertainty.

Key Terms

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

Conduction shape factor: Geometry-dependent parameter S used in Q̇ = kS∆T.

Fourier number: Dimensionless time αt/L_c² measuring diffusion progress.

Lumped capacitance: Approximation in which a body has a spatially uniform temperature.

Thermal diffusivity: α = k/(ρc_p), measuring the rate of thermal diffusion.

Laplacian: Spatial operator ∇²T describing temperature curvature.

Poisson equation: Steady conduction equation with internal heat generation.

Laplace equation: Steady, generation-free conduction equation.

Thermal bridge: Region that provides enhanced heat flow through or around an insulation layer.

Finite difference: Numerical method based on approximating derivatives at discrete nodes.

Finite volume: Numerical method based on energy balances over control volumes.

Finite element: Numerical method using weighted residuals and element interpolation.

Explicit scheme: Time-marching method using known values from the previous time level.

Implicit scheme: Time-marching method requiring solution of equations at the new time level.

Characteristic length: Geometry-dependent length used in dimensionless analysis.

Shape factor resistance: Thermal resistance R_cond = 1/(kS).

Heisler chart: Graphical solution for transient conduction in standard geometries.

Thermal diffusivity: Property governing how rapidly temperature disturbances penetrate a material.

Convection boundary condition: −k∂T/∂n = h(T_s − T_∞).

Grid independence: Condition in which further mesh refinement produces negligible result changes.

Review Questions

5 Short Answer Questions

1. What physical comparison is represented by the Biot number?
2. State the assumptions required for the lumped-capacitance method.
3. Define a conduction shape factor and give its SI unit.
4. Why is grid refinement necessary in a numerical conduction calculation?
5. Distinguish between the Fourier number and the Biot number.

5 Long Answer Questions

1. Derive the lumped-capacitance temperature relation from an energy balance on a solid body.
2. Explain how the two-dimensional finite-difference equation follows from the steady heat equation.
3. Discuss the meaning, use, and limitations of shape factors in multidimensional conduction.
4. Compare finite-difference, finite-volume, and finite-element methods for thermal analysis.
5. Explain how analytical transient solutions and Heisler charts are applied to a plane wall.

3 Analytical / Application Questions

1. A solid has k = 12 W/(m·K), h = 90 W/(m²·K), volume 0.002 m³, and exposed area 0.08 m². Determine its characteristic length and Biot number. Should lumped analysis be considered?
2. A two-dimensional square grid has four neighboring temperatures of 100°C, 80°C, 60°C, and 40°C. With no generation and constant conductivity, determine the interior-node temperature using the steady finite-difference equation.
3. A cooling process includes strong radiation, temperature-dependent conductivity, and a corner region with significant lateral heat flow. Recommend a modeling approach and justify which simplifying assumptions should be avoided.

Multiple Choice Questions with Answers

Question 1

Which equation defines the conduction shape-factor method for two isothermal boundaries?

A. Q̇ = hA_s(T_s − T_∞)

B. Q̇ = kS(T₁ − T₂)

C. Q̇ = ρVc_p∆T

D. Q̇ = εσAT⁴

Correct Answer: B. Q̇ = kS(T₁ − T₂)

Question 2

For a body with volume 0.003 m³ and convecting area 0.06 m², the characteristic length V/A_s is:

A. 0.005 m

B. 0.02 m

C. 0.05 m

D. 20 m

Correct Answer: C. 0.05 m

Question 3

A Biot number of 0.02 generally indicates that:

A. Internal temperature gradients are likely small

B. Radiation is necessarily dominant

C. The body is at steady state

D. Thermal conductivity is negligible

Correct Answer: A. Internal temperature gradients are likely small

Question 4

The Fourier number primarily measures:

A. The ratio of radiation to convection

B. The progress of thermal diffusion with time

C. The ratio of density to conductivity

D. The amount of heat generation

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

Question 5

For a square two-dimensional grid with no generation, an interior finite-difference temperature equals:

A. The sum of neighboring temperatures

B. The product of neighboring temperatures

C. The average of the four neighboring temperatures

D. The highest neighboring temperature

Correct Answer: C. The average of the four neighboring temperatures

Question 6

The stability limit for the basic explicit two-dimensional square-grid scheme is approximately:

A. Fo∆ ≤ 1/4

B. Fo∆ ≤ 1/2

C. Fo∆ ≤ 1

D. Fo∆ ≤ 4

Correct Answer: A. Fo∆ ≤ 1/4

Question 7

In lumped-capacitance analysis, the body temperature is assumed to be:

A. Zero everywhere

B. Uniform throughout the body at each instant

C. Equal to the surface temperature only

D. Sinusoidal in time

Correct Answer: B. Uniform throughout the body at each instant

Question 8

For a sphere of radius r, the characteristic length V/A_s is:

A. r

B. 2r

C. r/2

D. r/3

Correct Answer: D. r/3

Question 9

In the relation α = k/(ρc_p), α is the:

A. Convection coefficient

B. Thermal diffusivity

C. Surface emissivity

D. Shape factor

Correct Answer: B. Thermal diffusivity

Question 10

At high temperatures, radiative heat transfer calculations require surface temperatures expressed in:

A. Degrees Celsius only

B. Degrees Fahrenheit only

C. Kelvin

D. Any scale without conversion

Correct Answer: C. Kelvin