Heat and Mass Transfer

Introduction

Heat and mass transfer are central to mechanical engineering because temperature and species concentration differences drive many industrial and natural processes. Engines, refrigerators, boilers, heat exchangers, electronic cooling systems, drying equipment, batteries, and building envelopes all depend on controlled transfer of thermal energy and matter.

Heat transfer is the movement of thermal energy caused by a temperature difference. Mass transfer is the movement of a chemical species caused by a concentration, partial-pressure, or chemical-potential difference. The two subjects are closely related: both involve transport through a medium, both may occur by diffusion and bulk motion, and both are described using conservation laws and constitutive relations.

This chapter develops the foundations required to analyze conduction, convection, radiation, transient thermal response, mass diffusion, and heat exchangers. The treatment emphasizes physical interpretation, assumptions, engineering models, and design limitations.

Learning Objectives

By the end of this chapter, a student should be able to:

1. Distinguish conduction, convection, radiation, and mass diffusion.
2. Apply Fourier’s, Newton’s cooling, and Stefan–Boltzmann laws.
3. Analyze one-dimensional steady conduction using thermal resistance networks.
4. Explain transient conduction and use the lumped-capacitance model when justified.
5. Interpret the Nusselt, Reynolds, Prandtl, Biot, and Fourier numbers.
6. Analyze parallel-flow and counter-flow heat exchangers using energy balances and the effectiveness–NTU method.
7. Apply Fick’s law to simple steady and transient mass-diffusion problems.
8. Identify assumptions, limitations, safety issues, and design trade-offs in thermal systems.

1. Conservation Principles and Modes of Transfer

A thermal system is governed first by conservation of energy. For a control volume,

where Ė denotes an energy rate and E_cv is energy stored in the control volume. At steady state, storage is zero. A complete energy balance must include sensible energy, latent energy, shaft work, electrical power, and chemical energy where relevant.

Heat transfer occurs by three mechanisms:

• Conduction: molecular and electronic energy transfer through a stationary or moving medium.
• Convection: energy transfer between a surface and a moving fluid; it includes conduction within the fluid immediately adjacent to the surface.
• Radiation: electromagnetic energy transfer, which can occur through a vacuum.

Mass transfer commonly occurs by:

• Molecular diffusion, caused by random molecular motion and concentration gradients.
• Convection, caused by bulk fluid motion carrying a species.
• Migration, such as electrically driven ion transport, in specialized systems.

In real equipment, mechanisms often act simultaneously. For example, heat crosses a furnace wall by convection from hot gas to the wall, conduction through the wall, and radiation from the wall to its surroundings.

2. Conduction

2.1 Fourier’s law

For one-dimensional conduction in the x-direction,

Here q̇_x is heat-transfer rate in watts, k is thermal conductivity in W/(m·K), A is area normal to heat flow in m², and dT/dx is the temperature gradient in K/m. The negative sign indicates that heat flows toward decreasing temperature.

For a plane wall of thickness L, constant conductivity, area A, and surface temperatures T₁ and T₂,

This expression assumes steady, one-dimensional conduction, constant k, no internal heat generation, and a uniform cross-sectional area.

The heat flux is

where q'' has units W/m².

Thermal conductivity depends on material and temperature. Metals generally have high conductivity; insulation, gases, and porous materials have low conductivity. Contact resistance may be important when two solid surfaces meet because microscopic roughness traps air and reduces the actual conducting area.

2.2 Cylindrical and spherical conduction

For radial conduction through a cylindrical wall,

where r₁ and r₂ are inner and outer radii and L is cylinder length. For a spherical shell,

These formulas assume steady radial conduction, constant conductivity, no heat generation, and negligible end effects.

2.3 Internal heat generation

Electrical resistance heating, nuclear fuel, chemical reaction, and viscous dissipation can generate heat inside a solid. For a plane wall with uniform volumetric generation q̇''' and symmetry at the mid-plane,

The temperature distribution is parabolic. If the wall has half-thickness L, its maximum temperature at the center exceeds the surface temperature by

This result applies to a slab with uniform generation, constant conductivity, and equal surface temperatures.

3. Thermal Resistance Networks

The resistance form of a heat-transfer relation is

where ΔT is the temperature difference driving heat transfer and R_total is total thermal resistance in K/W.

For a plane wall,

For convection at a surface,

where h is the convection coefficient in W/(m²·K). Contact resistance is commonly represented as

where h_c is a contact conductance.

For resistances in series,

For parallel paths, conductances add:

Worked Example 1: Composite wall

A wall has area 10 m². It consists of a 0.10 m brick layer with k = 0.70 W/(m·K), a 0.08 m insulation layer with k = 0.04 W/(m·K), and convection coefficients of 8 and 25 W/(m²·K) on the two sides. The fluid temperatures are 20°C and −5°C. Find the heat-transfer rate, neglecting contact resistance.

The resistances are

Thus,

and

The insulation dominates the resistance. Increasing its thickness would reduce heat loss more effectively than a small change in either convection coefficient.

4. Convection

Newton’s law of cooling is

where T_s is surface temperature and T_∞ is the bulk-fluid temperature. The coefficient h is not a pure material property. It depends on geometry, fluid properties, velocity, flow regime, and whether the flow is forced or natural.

Forced convection results from fans, pumps, or external flow. Natural convection results from buoyancy caused by density variations. The coefficient is usually obtained from correlations or experiments rather than from a simple universal formula.

4.1 Boundary layers and dimensionless groups

A velocity boundary layer develops near a wall because fluid velocity satisfies the no-slip condition at the surface. A thermal boundary layer develops when the wall and fluid temperatures differ. Their relative thickness depends strongly on the Prandtl number.

The Reynolds number is

where ρ is density, V velocity, L characteristic length, μ dynamic viscosity, and ν kinematic viscosity. It compares inertial and viscous effects and helps identify laminar or turbulent flow.

The Prandtl number is

where c_p is specific heat, k thermal conductivity, and α = k/(ρc_p) thermal diffusivity.

The Nusselt number is

It expresses the relative importance of convection and conduction across a characteristic fluid layer. A correlation may have the form Nu = C Re^m Prⁿ, but its constants and valid range must be taken from an appropriate experimental or analytical source.

For natural convection, the Grashof number is

where g is gravitational acceleration and β is the volumetric expansion coefficient. The product GrPr is the Rayleigh number.

A correlation must match geometry, boundary condition, flow regime, property-evaluation temperature, and range of dimensionless parameters. Extrapolating outside its stated range can produce serious design errors.

5. Radiation

All bodies above absolute zero emit electromagnetic radiation. The ideal blackbody relation is

where E_b is emissive power in W/m², σ = 5.670×10⁻⁸ W/(m²·K⁴) is the Stefan–Boltzmann constant, and T is absolute temperature in kelvin.

A real opaque gray surface emits

where emissivity ε lies between 0 and 1. Net radiation between a small gray surface and large surroundings at temperature T_sur is approximately

This approximation assumes the surroundings behave as a large isothermal enclosure and the surface is gray and diffuse.

Radiation depends on absolute temperature, not temperature in degrees Celsius. Surface finish, oxidation, wavelength, view factor, and enclosure geometry influence exchange. Radiation can be reduced by low-emissivity coatings, radiation shields, or reduced view factors.

6. Transient Conduction

Transient conduction occurs when temperature changes with time. The thermal diffusivity is

A high thermal diffusivity means that temperature disturbances propagate rapidly through a material.

The Biot number is

where the characteristic length is often L_c = V/A_s, with V solid volume and A_s exposed surface area. If Bi ≤ 0.1, internal temperature gradients are usually small enough for the lumped-capacitance model.

For a body initially at T_i exposed to fluid at constant T_∞,

This assumes constant properties, uniform internal temperature, constant h, negligible radiation, and a constant surrounding-fluid temperature.

The Fourier number is

and measures the degree of thermal diffusion. When Bi > 0.1, spatial temperature gradients matter and charts, series solutions, numerical methods, or finite-element analysis may be required.

7. Heat Exchangers

A heat exchanger transfers energy between fluids, usually without allowing them to mix. A steady energy balance, neglecting heat loss to surroundings, is

where subscripts h and c denote hot and cold streams.

The heat-capacity rate is

and the smaller and larger values are C_min and C_max. The capacity ratio is

7.1 Log-mean temperature difference

For a heat exchanger with approximately constant overall coefficient,

where U is overall heat-transfer coefficient and A is heat-transfer area. The log-mean temperature difference is

The temperature differences must be defined consistently at the two ends. For cross-flow and multipass arrangements, a correction factor F is often used: q̇ = UAFΔT_lm.

7.2 Overall heat-transfer coefficient

For a plane composite exchanger wall, a simplified resistance relation is

Fouling resistances represent deposits such as scale, corrosion products, or biological growth. In design, a clean exchanger may perform adequately initially but fail to meet duty after fouling.

7.3 Effectiveness–NTU method

Effectiveness is

where

The number of transfer units is

For a parallel-flow exchanger,

For a counter-flow exchanger with C_r ≠ 1,

For C_r = 1, the counter-flow relation becomes

These relations assume steady operation, negligible heat loss, constant properties, and a specified exchanger flow arrangement.

8. Mass Transfer

Mass transfer is often described using mass concentration C_A, mass fraction w_A, mole fraction, or partial pressure. The chosen concentration basis must be stated clearly.

For dilute species A diffusing through a stationary medium, Fick’s first law is

where J_A is diffusive molar flux in mol/(m²·s), D_AB is binary diffusion coefficient in m²/s, and C_A is molar concentration in mol/m³.

For steady diffusion through a plane layer of thickness L with constant D_AB,

The total flux may include convection of the mixture. In gas absorption, evaporation, drying, and membrane systems, distinguishing diffusive flux from total species flux is essential.

The transient diffusion equation for constant diffusivity is

This is mathematically analogous to transient heat conduction, with concentration replacing temperature and diffusivity replacing thermal diffusivity.

A mass-transfer coefficient is commonly defined by

where k_c is a mass-transfer coefficient. The dimensionless Sherwood number is

The Schmidt number is

Forced-convection mass-transfer correlations often relate Sh to Re and Sc, just as heat-transfer correlations relate Nu to Re and Pr. This analogy is useful but does not remove the need to verify physical properties and boundary conditions.

9. Applied Case: Cooling of an Electronic Module

Background

Consider an illustrative design case involving a 60 W electronic module mounted on an aluminum heat sink. The module must remain below 85°C in room air at 25°C. The heat sink is cooled by a fan, and the designer must assess conduction, convection, and transient start-up.

Problem and analysis

The module-to-sink contact resistance is 0.08 K/W, while conduction through the mounting plate contributes 0.12 K/W. The required maximum sink-to-air resistance is therefore estimated from

The allowable sink-to-air resistance is

Thus the heat sink must provide at least

If the selected sink has area 0.20 m², the required average convection coefficient is

A fan and fin geometry are then selected using a validated forced-convection correlation or manufacturer data. Fin efficiency must be considered: the entire geometric fin area is not necessarily at the base temperature. Surface fouling and fan failure should also be assessed.

Transient check

Suppose the heat sink has mass 0.8 kg, specific heat 900 J/(kg·K), area 0.20 m², and effective convection coefficient 8 W/(m²·K). With a characteristic length based on volume-to-area ratio giving Bi < 0.1, lumped capacitance is acceptable. The time constant is

After one time constant, the temperature excess above room air has fallen to approximately 36.8% of its initial value during cooling, or has reached 63.2% of its final value during heating toward a constant load condition.

Outcome and learning points

The resistance model identifies the dominant design constraints before detailed simulation. Improving the contact interface, increasing fin area, enhancing airflow, and implementing fan-failure protection can all improve reliability. A design based only on steady-state temperature may be unsafe because blocked airflow, dust accumulation, variable loads, and transient start-up can cause overheating.

10. Design Practice, Limitations, and Professional Responsibility

Good thermal design begins with a clearly stated control volume, reliable material properties, realistic boundary conditions, and an energy balance. Engineers should use SI units consistently, evaluate fluid properties at an appropriate film or bulk temperature, and state whether radiation is included.

Common sources of error include using Celsius in a radiation equation, confusing heat flux with heat rate, applying a laminar correlation to turbulent flow, ignoring contact resistance, assuming a constant convection coefficient when airflow varies, and using a correlation outside its valid range.

Thermal systems also present safety and environmental responsibilities. Excess temperature can cause fire, pressure rise, insulation failure, battery thermal runaway, or release of hazardous substances. Refrigerants and process gases may have environmental or toxicity implications. Designs should comply with applicable pressure-vessel, electrical, fire, building, and environmental regulations. Safety factors, alarms, thermal cutoffs, containment, and maintainability are engineering requirements rather than optional additions.

Computational fluid dynamics, finite-element analysis, digital twins, and machine-learning models can support thermal design. However, AI or numerical software does not replace conservation laws, mesh and time-step verification, experimental validation, or engineering judgment. A model is credible only when its assumptions, input data, uncertainty, and validation evidence are documented.

Chapter Summary

Heat transfer occurs by conduction, convection, and radiation; mass transfer occurs primarily by diffusion and convection. Fourier’s law describes conduction, Newton’s law describes surface convection, and the Stefan–Boltzmann law describes thermal radiation. Thermal resistance networks simplify multilayer steady systems and reveal dominant resistances.

Transient conduction is governed by thermal diffusivity and can be approximated by lumped capacitance when the Biot number is no greater than about 0.1. Reynolds, Prandtl, Nusselt, Grashof, Rayleigh, Fourier, Sherwood, and Schmidt numbers organize transport behavior and support experimentally based correlations.

Heat exchangers are analyzed using energy balances, the log-mean temperature difference method, or the effectiveness–NTU method. Mass diffusion is described by Fick’s laws and has strong mathematical similarity to heat diffusion. Reliable engineering analysis requires stated assumptions, appropriate correlations, property accuracy, validation, safety consideration, and awareness of fouling, contact resistance, transient behavior, and model limitations.

Key Terms

Heat transfer: Movement of thermal energy caused by a temperature difference.

Mass transfer: Movement of a chemical species caused by a concentration or chemical-potential difference.

Conduction: Energy transfer through molecular or electronic interactions.

Convection: Heat transfer between a surface and a moving fluid.

Radiation: Electromagnetic energy transfer emitted by matter.

Thermal conductivity: Material property measuring ability to conduct heat.

Thermal resistance: Opposition to heat flow, expressed as temperature difference divided by heat rate.

Convection coefficient: Proportionality coefficient relating surface heat flux to temperature difference.

Emissivity: Ratio of real-surface emission to blackbody emission.

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

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

Fourier number: Dimensionless measure of transient diffusion time.

Nusselt number: Dimensionless convection-to-conduction heat-transfer ratio.

Reynolds number: Dimensionless ratio of inertial to viscous effects.

Heat exchanger: Device that transfers heat between fluids.

Effectiveness: Actual heat transfer divided by maximum possible heat transfer.

Fouling resistance: Resistance caused by deposits on a heat-transfer surface.

Fick’s law: Constitutive relation for mass diffusion.

Sherwood number: Dimensionless mass-transfer coefficient.

Diffusion coefficient: Property describing molecular species mobility.

Review Questions

Short Answer Questions

1. What physical mechanism distinguishes conduction from convection?
2. Why must absolute temperature be used in the Stefan–Boltzmann equation?
3. What does a thermal resistance network represent?
4. State the criterion commonly used to justify lumped-capacitance analysis.
5. Define heat-exchanger effectiveness.

Long Answer Questions

1. Derive the thermal resistance of a plane wall and explain the assumptions behind it.
2. Explain the physical meaning of Reynolds, Prandtl, and Nusselt numbers and their role in convection correlations.
3. Compare parallel-flow and counter-flow heat exchangers.
4. Explain steady and transient mass diffusion using Fick’s first and second laws.
5. Discuss how contact resistance, fouling, and radiation influence practical thermal-system design.

Analytical / Application Questions

1. A 2 m² wall has thickness 0.15 m and conductivity 0.50 W/(m·K). The two surface temperatures are 80°C and 20°C. Calculate the steady heat-transfer rate and state the assumptions used.
2. A solid initially at 150°C is exposed to air at 25°C. Explain what information is needed to determine whether lumped capacitance is valid and how its temperature changes if the model is valid.
3. A counter-flow heat exchanger has hot-fluid inlet temperature 150°C and cold-fluid inlet temperature 25°C. Explain how you would determine outlet temperatures using the effectiveness–NTU method when both mass flow rates, specific heats, and UA are known.

Multiple Choice Questions with Answers

Question 1

Which mechanism can transfer heat through a vacuum?

A. Conduction

B. Radiation

C. Forced convection

D. Natural convection

Correct Answer: B. Radiation

Question 2

For a plane wall with constant conductivity and no heat generation, which relation gives the steady heat-transfer rate?

A. q̇ = kA(T₁ − T₂)/L

B. q̇ = kL(T₁ − T₂)/A

C. q̇ = hA(T₁ + T₂)

D. q̇ = σA(T₁ − T₂)

Correct Answer: A. q̇ = kA(T₁ − T₂)/L

Question 3

A wall has total thermal resistance 0.50 K/W and a temperature difference of 30 K. What is the heat-transfer rate?

A. 15 W

B. 30 W

C. 60 W

D. 75 W

Correct Answer: C. 60 W

Question 4

The Biot number primarily compares which two effects?

A. Inertial and viscous effects

B. Radiation and convection

C. Internal conduction and surface convection

D. Diffusion and chemical reaction

Correct Answer: C. Internal conduction and surface convection

Question 5

For lumped-capacitance analysis, the usual validity criterion is

A. Bi ≥ 10

B. Bi ≤ 0.1

C. Re = 0

D. Pr ≤ 0.001

Correct Answer: B. Bi ≤ 0.1

Question 6

Which dimensionless number is defined as Nu = hL/k?

A. Prandtl number

B. Nusselt number

C. Schmidt number

D. Fourier number

Correct Answer: B. Nusselt number

Question 7

For a gray surface exchanging radiation with large surroundings, the net radiation rate is proportional to

A. T_s − T_sur

B. T_s² − T_sur²

C. T_s³ − T_sur³

D. T_s⁴ − T_sur⁴

Correct Answer: D. T_s⁴ − T_sur⁴

Question 8

In a heat exchanger, C_min is

A. The larger heat-capacity rate

B. The smaller heat-capacity rate

C. The wall heat capacity

D. The radiation capacity

Correct Answer: B. The smaller heat-capacity rate

Question 9

Which law describes molecular mass diffusion in a dilute system?

A. Fourier’s law

B. Newton’s law of cooling

C. Fick’s law

D. Stefan–Boltzmann law

Correct Answer: C. Fick’s law

Question 10

A surface has area 0.20 m² and convection coefficient 10 W/(m²·K). Its convection resistance is

A. 0.02 K/W

B. 0.5 K/W

C. 2 K/W

D. 50 K/W

Correct Answer: B. 0.5 K/W