Heat Transfer

Introduction

Heat transfer is the study of thermal energy transport caused by temperature differences and coupled transport processes. At university level, you use heat-transfer analysis to predict temperature fields, heat rates, thermal stresses, equipment performance, cooling requirements, insulation needs, and energy efficiency. The subject connects thermodynamics, fluid mechanics, materials science, and applied mathematics.

Heat is energy in transit across a system boundary because of a temperature difference. In engineering systems, the principal mechanisms are conduction, convection, and thermal radiation. Phase change, advection, and mass transfer can strongly modify the overall energy transport. Most real devices involve several mechanisms at the same time.

The kettle illustrates simultaneous modes: conduction through solids in contact, convection within the water and surrounding air, and radiation between surfaces and the environment.

Learning Goals

By the end of this aiMOOC, you should be able to formulate energy balances, apply Fourier's law, use the heat equation, build thermal-resistance networks, judge when lumped transient analysis is valid, select convection correlations using dimensionless groups, calculate radiative exchange, analyze heat exchangers with LMTD and effectiveness-NTU methods, and evaluate simplifying assumptions.

Fundamental Framework

Energy Balance and Heat Rate

A heat-transfer model begins with conservation of energy. For a control volume, a useful engineering statement is

$\frac{d E_{c v}}{d t} = \dot{Q} - \dot{W} + \sum_{i n} \dot{m} (h + \frac{V^{2}}{2} + g z) - \sum_{o u t} \dot{m} (h + \frac{V^{2}}{2} + g z) .$

For many heat-transfer problems, kinetic and potential energy terms are negligible, and the model reduces to a balance among heat transfer, work, enthalpy transport, and energy storage. The heat-transfer rate $\dot{Q}$ is measured in watts, while heat flux $q^{″} = \dot{Q} / A$ is measured in watts per square metre.

A strong solution states the system boundary, assumptions, sign convention, material properties, geometry, governing equations, and boundary or initial conditions before numbers are substituted.

Thermal Properties

Thermal conductivity $k$ measures a material's ability to conduct heat. Density $ρ$ and specific heat $c_{p}$ determine thermal energy storage. Their combination gives thermal diffusivity,

$α = \frac{k}{ρ c_{p}},$

which measures how quickly a temperature disturbance diffuses through a material.

In quantitative work, properties may depend on temperature, pressure, phase, composition, and direction. You should evaluate properties at a physically defensible reference temperature and state whether they are treated as constant.

Conduction

Fourier's Law

Conduction is thermal energy transport caused by microscopic interactions in matter. For an isotropic material, Fourier's law is

$𝐪^{″} = - k \nabla T .$

The minus sign means the heat-flux vector points toward decreasing temperature. For one-dimensional conduction through a plane wall,

$\dot{Q} = - k A \frac{d T}{d x} .$

Heat Diffusion Equation

Combining Fourier's law with an energy balance for a differential control volume gives the heat equation. For constant properties in a stationary solid with volumetric heat generation ${\dot{q}}^{‴}$ ,

$\frac{\partial T}{\partial t} = α \nabla^{2} T + \frac{{\dot{q}}^{‴}}{ρ c_{p}} .$

A complete problem also requires initial and boundary conditions. Common boundary conditions prescribe temperature, heat flux, convection, radiation, or an interface condition. The mathematical form changes with Cartesian, cylindrical, or spherical geometry.

Steady One-Dimensional Thermal Resistance

For a plane wall of thickness $L$ and constant $k$ ,

$R_{c o n d} = \frac{L}{k A}$

and

$\dot{Q} = \frac{T_{1} - T_{2}}{R_{c o n d}} .$

For a cylindrical wall between radii $r_{1}$ and $r_{2}$ ,

$R_{c o n d} = \frac{\ln (r_{2} / r_{1})}{2 π k L} .$

Thermal-resistance networks are analogous to electrical resistor networks. Series resistances add directly. Parallel paths divide the heat flow. Contact resistance can be important where nominally touching solids have microscopic gaps.

Extended Surfaces and Fins

Fins increase heat transfer by increasing exposed area. Their usefulness depends on conductivity, geometry, convection coefficient, and the temperature difference between the base and fluid. For a straight fin of uniform cross-section, the standard one-dimensional model leads to

$\frac{d^{2} θ}{d x^{2}} - m^{2} θ = 0, m^{2} = \frac{h P}{k A_{c}},$

where $θ = T - T_{\infty}$ . Fin efficiency compares the actual heat transfer with the ideal heat transfer that would occur if the whole fin were at the base temperature.

Transient Conduction

When temperatures vary with time, storage matters. The Biot number

$B i = \frac{h L_{c}}{k}$

compares internal conduction resistance with external convection resistance. If $B i$ is sufficiently small, commonly below about 0.1 for engineering use, the temperature inside a body can often be approximated as spatially uniform. The lumped-capacitance model then gives

$\frac{T - T_{\infty}}{T_{i} - T_{\infty}} = \exp (- \frac{h A_{s}}{ρ V c_{p}} t) .$

The Fourier number

$F o = \frac{α t}{L_{c}^{2}}$

is a dimensionless measure of diffusion time.

When the Biot number is not small, spatial temperature gradients must be resolved using analytical series solutions, Heisler charts where appropriate, finite differences, finite elements, or other numerical methods.

Convection

Physical Picture and Newton's Law of Cooling

Convective heat transfer between a surface and a moving fluid combines molecular diffusion near the wall with energy transport by fluid motion. The engineering rate equation is

$\dot{Q} = h A_{s} (T_{s} - T_{\infty}),$

where $h$ is the convective heat-transfer coefficient. Unlike thermal conductivity, $h$ is not solely a material property; it depends on fluid properties, geometry, flow speed, surface condition, and the flow regime.

Forced, Natural, Internal, and External Convection

Forced convection uses a fan, pump, or imposed flow. Natural convection is driven by buoyancy caused by density differences in a gravitational field. Internal flow is bounded by walls, as in pipes and ducts. External flow develops over bodies such as plates, cylinders, turbine blades, and electronic components.

The thermal boundary layer is the region where temperature changes from the surface value toward the free-stream or bulk-fluid value. Its development is coupled to the velocity boundary layer through transport properties.

Dimensionless Groups and Correlations

Dimensionless analysis organizes convection data and guides correlation selection:

$R e = \frac{ρ V L}{μ}$ compares inertia with viscous effects.

$P r = \frac{ν}{α}$ compares momentum diffusivity with thermal diffusivity.

$N u = \frac{h L}{k_{f}}$ measures convection relative to pure conduction across a fluid layer.

$G r = \frac{g β (T_{s} - T_{\infty}) L^{3}}{ν^{2}}$ characterizes buoyancy relative to viscous effects.

$R a = G r P r$ is central to many natural-convection correlations.

A correlation such as $N u = f (R e, P r)$ is valid only for its stated geometry, boundary condition, property range, and flow regime. Extrapolating a correlation outside its validated range can create large errors.

Thermal Radiation

Blackbody Radiation and Real Surfaces

Thermal radiation is electromagnetic energy emitted by matter because of its temperature. It does not require a material medium. A blackbody is an ideal surface that absorbs all incident radiation and emits the maximum possible thermal radiation at a given temperature.

The Stefan-Boltzmann law for a blackbody is

$E_{b} = σ T^{4},$

where $σ$ is the Stefan-Boltzmann constant. A diffuse-gray real surface is often approximated by

$E = ε σ T^{4},$

where emissivity $ε$ lies between zero and one.

Exchange Between Surfaces

For a small gray surface exchanging radiation with large isothermal surroundings,

${\dot{Q}}_{r a d} = ε σ A (T_{s}^{4} - T_{s u r}^{4}) .$

For multiple surfaces, geometry matters through view factors. The view factor $F_{i j}$ is the fraction of radiation leaving surface $i$ that directly reaches surface $j$ . Reciprocity and summation rules reduce the number of independent factors in an enclosure.

For linearized combined convection and radiation, you can define an effective radiation coefficient $h_{r}$ and use

$\dot{Q} = A (h + h_{r}) (T_{s} - T_{s u r})$

when the reference temperatures are chosen consistently.

Heat Exchangers

Energy Balance and Overall Heat-Transfer Coefficient

A heat exchanger transfers energy between fluids, often without mixing them. For steady operation with negligible heat loss to the surroundings,

$\dot{Q} = {\dot{m}}_{h} c_{p, h} (T_{h, i n} - T_{h, o u t}) = {\dot{m}}_{c} c_{p, c} (T_{c, o u t} - T_{c, i n}) .$

The overall coefficient $U$ combines convection, wall conduction, fouling, and other resistances into a single relation,

$\dot{Q} = U A Δ T_{d r i v i n g} .$

LMTD Method

For a heat exchanger with known terminal temperatures,

$\dot{Q} = U A Δ T_{l m},$

where the log-mean temperature difference is

$Δ T_{l m} = \frac{Δ T_{1} - Δ T_{2}}{\ln (Δ T_{1} / Δ T_{2})} .$

Parallel-flow and counterflow arrangements have different temperature profiles. Counterflow often provides a larger mean temperature difference for the same inlet temperatures.

Effectiveness-NTU Method

When outlet temperatures are unknown, the effectiveness-NTU method is often more convenient. Define

$C_{h} = {\dot{m}}_{h} c_{p, h}, C_{c} = {\dot{m}}_{c} c_{p, c},$

$C_{m i n} = min (C_{h}, C_{c}), N T U = \frac{U A}{C_{m i n}},$

and

$ε = \frac{\dot{Q}}{{\dot{Q}}_{m a x}}, {\dot{Q}}_{m a x} = C_{m i n} (T_{h, i n} - T_{c, i n}) .$

The relation between effectiveness, NTU, and the heat-capacity-rate ratio depends on exchanger configuration.

Interactive Heat-Exchanger Simulation

Use the simulation to compare parallel and counterflow temperature profiles. Change flow rates and inlet conditions, then explain how the heat-capacity rates affect outlet temperatures and exchanger effectiveness.

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Phase-Change Heat Transfer

Boiling and condensation can transfer large heat rates because latent heat is involved. In pool boiling, increasing surface superheat can move the system through natural convection, nucleate boiling, transition boiling, and film boiling. The critical heat flux marks a dangerous change in surface behavior for many applications. Condensation can occur as a film or as droplets; surface condition and wettability affect the heat-transfer coefficient.

Phase-change models require care because heat transfer may couple strongly to interfacial motion, nucleation, pressure, surface roughness, and two-phase flow.

Combined Modes, Modeling, and Design

Real systems are usually conjugate: conduction in solids interacts with convection in fluids, while surfaces exchange radiation. A complete model may therefore solve multiple fields and interface conditions simultaneously.

Useful engineering strategies include dimensional analysis, scale analysis, thermal circuits, analytical solutions for canonical geometries, numerical discretization, uncertainty analysis, and experiments. Finite-difference and finite-element methods solve conduction problems with complex geometry, while computational fluid dynamics can couple fluid flow and energy transport.

A model is only as reliable as its assumptions and inputs. You should perform limiting checks, unit checks, mesh or time-step studies for numerical work, sensitivity analysis, and comparison with benchmark solutions or measurements.

Engineering Applications

Heat transfer is central to electronic cooling, battery thermal management, heat pumps, refrigeration, power plants, chemical processing, building envelopes, aerospace thermal protection, solar energy systems, engines, biomedical devices, cryogenics, additive manufacturing, data centres, and industrial furnaces.

A recurring design trade-off is that increasing heat transfer may require larger area, higher flow rate, additional pumping power, more expensive materials, or tighter tolerances. Engineering design therefore considers thermal performance together with cost, safety, reliability, manufacturability, maintenance, and environmental impact.

Interactive Tasks

Quiz: Test Your Knowledge

Which law gives the conductive heat flux in terms of a temperature gradient? (Fourier law) (!Stefan Boltzmann law) (!Newton second law) (!Ideal gas law)

What does a small Biot number indicate for a transient solid? (Internal temperature gradients are small) (!Radiation is the only heat transfer mode) (!Fluid inertia dominates viscosity) (!The material has zero heat capacity)

Which quantity is a material property for an isotropic solid? (Thermal conductivity) (!Convective heat transfer coefficient) (!Nusselt number for a flow) (!Overall heat transfer coefficient)

Which dimensionless group compares convection with conduction across a fluid layer? (Nusselt number) (!Reynolds number) (!Mach number) (!Froude number)

What drives natural convection in a gravitational field? (Buoyancy caused by density differences) (!A mechanical fan) (!A vacuum pump) (!Only molecular vibration in a solid)

Which ideal surface emits the maximum thermal radiation at a given temperature? (Blackbody) (!Adiabatic wall) (!Perfect insulator) (!Transparent membrane)

What does a radiation view factor describe? (The geometric fraction of radiation reaching another surface) (!The thermal conductivity of a wall) (!The viscosity of a fluid) (!The heat capacity of a fin)

When is the LMTD method especially convenient for a heat exchanger? (When terminal temperatures are known) (!When no temperature data are available) (!When radiation is the only transfer mode) (!When both fluids have zero flow rate)

What is the purpose of a fin on a heat transfer surface? (To increase effective heat transfer area) (!To eliminate temperature differences) (!To make thermal conductivity zero) (!To prevent every form of radiation)

Which check is essential when applying an empirical convection correlation? (Verify its geometry and validity range) (!Assume it is universal) (!Ignore the fluid properties) (!Set every dimensionless group to one)

Memory Game

Fourier law	Relates conductive heat flux to the temperature gradient
Biot number	Compares internal conduction resistance with external convection resistance
Nusselt number	Measures convective enhancement relative to conduction in a fluid
Emissivity	Measures how strongly a real surface emits compared with a blackbody
LMTD	Represents the logarithmic mean temperature driving force in a heat exchanger
Thermal diffusivity	Measures the rate at which temperature disturbances diffuse through a material

Drag and Drop

Match the correct terms.	Topic
Temperature gradient	Fourier conduction
Boundary layer	Convective transport
View factor	Radiative geometry
Heat capacity rate	Heat exchanger analysis
Lumped temperature	Small Biot transient model

...

Crossword Puzzle

Conduction	Which heat-transfer mechanism is described by Fourier law?
Convection	Which mechanism couples surface diffusion with fluid motion?
Radiation	Which mechanism can transfer thermal energy through a vacuum?
Fourier	Which scientist is associated with the classical conduction law?
Nusselt	Which dimensionless number compares convection with conduction?
Emissivity	Which property compares real-surface emission with blackbody emission?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Thermal audit: Identify at least five heat-transfer paths in a room or laboratory and classify the dominant mechanism for each one.
Material comparison: Compare thermal conductivities of three engineering materials and explain which would be suitable for a heat sink, insulation, and a structural thermal barrier.
Cooling curve: Record the cooling of a warm liquid over time, plot temperature against time, and discuss whether a lumped model appears reasonable.
Heat transfer diagram: Create an annotated image of a household or laboratory device showing where conduction, convection, and radiation occur.

Standard

Composite wall model: Build a thermal-resistance network for a multilayer wall and calculate heat rate, interface temperatures, and the effect of adding insulation.
Convection experiment: Design a safe experiment comparing natural and forced convection from the same warm surface and estimate how the effective heat-transfer coefficient changes.
Fin investigation: Model a straight fin, vary its material or length, and explain the trade-off between added area and diminishing fin efficiency.
Heat exchanger study: Use the interactive simulation to compare parallel and counterflow arrangements, then produce a short engineering report with temperature profiles and interpretation.

Advanced

Transient conduction simulation: Solve a one-dimensional transient conduction problem numerically and compare the result with a lumped-capacitance prediction over the same time interval.
Radiation network: Construct a radiative exchange model for three diffuse-gray surfaces and evaluate how emissivity and geometry influence net heat rates.
Conjugate heat transfer project: Develop a coupled conduction-convection model for cooling an electronic component and justify all boundary conditions and property choices.
Engineering interview: Interview a thermal engineer, energy manager, laboratory researcher, or HVAC professional about a real heat-transfer design problem and connect their decisions to theory, uncertainty, safety, and validation.

Learning Assessment

Model formulation: Given an unfamiliar thermal system, define the control volume, identify dominant mechanisms, state assumptions, and derive a solvable governing model.
Correlation selection: Compare two published convection correlations for the same geometry and justify which is valid for a specified Reynolds and Prandtl number range.
Transient reasoning: Decide whether a lumped-capacitance approximation is defensible for a cooling object and quantify the consequences if the Biot criterion is violated.
Heat exchanger design: Determine the area required for a specified duty, then explain how fouling, uncertainty in the overall coefficient, and flow arrangement affect the design margin.
Radiative transfer: Analyze how changing surface emissivity or adding a radiation shield alters the energy balance of a high-temperature enclosure.
Verification and validation: Evaluate a numerical temperature field using energy conservation, grid refinement, limiting cases, and comparison with an analytical or experimental benchmark.

Evidence of Learning

Important evidence includes accurate use of heat-transfer terminology and units; derivation and interpretation of governing equations; correct selection of boundary and initial conditions; defensible use of property data and correlations; calculation of conduction, convection, radiation, and exchanger performance; construction of thermal-resistance and radiation networks; transient and steady-state modeling; uncertainty and sensitivity analysis; numerical verification; experimental comparison; and communication of engineering assumptions, limitations, and design consequences.

Products may include a validated calculation, laboratory notebook, plotted dataset, simulation, thermal image or annotated diagram, technical report, design recommendation, presentation, or reproducible computational model. Strong transfer is demonstrated when you can apply the same principles to a new device or geometry and explain which assumptions must change.

OERs on the Topic

The English Wikipedia article provides a broad overview of heat-transfer mechanisms and links to related concepts.

LearnChemE offers university-level screencasts and interactive simulations covering conduction, convection, radiation, boiling, condensation, fins, and heat exchangers.

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