Fundamentals of Heat and Mass Transfer

Introduction

Heat and mass transfer are central to mechanical engineering. Engines, turbines, refrigeration systems, heat exchangers, boilers, electronic cooling systems, drying equipment, combustion devices, and environmental-control systems all depend on the movement of energy and matter. Thermodynamics determines whether a process is possible and quantifies the energy changes between equilibrium states. Heat and mass transfer determine how rapidly those processes occur and how equipment must be designed to achieve them.

Heat transfer is the transport of thermal energy caused by a temperature difference. Mass transfer is the movement of a chemical species caused by a difference in concentration, chemical potential, or related driving force. Energy conversion concerns the transformation of energy from one form to another, such as chemical energy into mechanical work and heat in an internal-combustion engine. These subjects are related but not identical.

This chapter develops the foundation required for later study of conduction, convection, radiation, diffusion, phase change, heat exchangers, and combined heat-and-mass-transfer systems.

Learning Objectives

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

1. Distinguish temperature, heat, internal energy, and work.
2. Apply conservation of mass and the first law of thermodynamics to control volumes.
3. Explain conduction, convection, radiation, and molecular diffusion.
4. Define important thermophysical and transport properties.
5. Use thermal resistance networks for simple one-dimensional systems.
6. Select consistent SI units and interpret boundary conditions.
7. Distinguish heat transfer, mass transfer, and energy conversion.
8. Identify assumptions, limitations, and engineering implications in transfer problems.

1. Thermodynamic Foundations

1.1 System, surroundings, and control volume

A system is the matter selected for analysis. Everything outside it is the surroundings. A closed system contains a fixed amount of matter, whereas an open system, or control volume, permits mass to cross its boundary. A control surface is the boundary of a control volume.

Thermodynamics describes macroscopic states using properties such as pressure, temperature, density, and specific internal energy. A state is an equilibrium condition in which these properties have well-defined values. Heat transfer analysis often concerns systems that are not in complete equilibrium; therefore, local properties and spatial temperature gradients become important.

1.2 Temperature, heat, and internal energy

Temperature measures the thermal state of matter and determines the direction of heat transfer: in ordinary situations, heat flows spontaneously from higher temperature to lower temperature. Temperature is an intensive property, while total internal energy is extensive.

Heat is energy transferred across a system boundary solely because of a temperature difference. It is not a property stored in a body. Once transferred, it changes the energy of the receiving system or is converted into another form. The symbol Q commonly denotes total heat transfer and Q̇ denotes heat-transfer rate.

Internal energy, U, is the microscopic energy associated with molecular translation, rotation, vibration, bonding, and interactions. For a closed system, the first law is:

where ∆E is the change in total energy, Q is heat added to the system, and W is work done by the system. The convention is positive Q into the system and positive W out of the system. Total energy may be written as:

where U is internal energy, KE is kinetic energy, and PE is potential energy.

For a stationary closed system with negligible changes in kinetic and potential energy:

These equations express conservation of energy; they do not specify the rate or spatial distribution of heat transfer.

1.3 Control-volume energy balance

For a general control volume, conservation of energy may be expressed as:

Here E_cv is energy inside the control volume, Q̇_cv is heat transfer into it, Ẇ_cv is work done by it, ṁ is mass-flow rate, h is specific enthalpy, V is velocity, g is gravitational acceleration, and z is elevation. The equation assumes negligible electrical, magnetic, and nuclear energy contributions unless they are added explicitly. Enthalpy, h = u + pv, conveniently includes flow work.

For steady operation with one inlet and one outlet, negligible changes in stored energy and potential energy, the balance becomes:

The signs and terms must be adapted to the selected control-volume convention.

1.4 Conservation of mass

Mass cannot be created or destroyed in ordinary heat-and-mass-transfer equipment. For a control volume:

At steady state, total mass flow in equals total mass flow out. For a single stream, ṁ = ρAV, where ρ is density, A is cross-sectional area, and V is average velocity normal to the area. The units are kg/s when ρ is in kg/m³, A in m², and V in m/s.

Species conservation includes generation and consumption by chemical reaction. For species i:

where Ṙ_i is the net rate of production of species i. In a nonreacting system, Ṙ_i = 0.

2. Transport Mechanisms

2.1 Conduction

Conduction is energy transfer through a stationary medium or between neighboring particles caused by microscopic interactions. It occurs in solids, liquids, and gases. In metals, mobile electrons contribute strongly; in nonmetals, molecular vibrations and interactions are more important.

Fourier’s law for one-dimensional conduction is:

where q̇_x'' is heat flux in W/m², k is thermal conductivity in W/(m·K), and dT/dx is the temperature gradient. The negative sign indicates heat flows toward decreasing temperature. For constant k through a plane wall of thickness L and area A:

This relation assumes steady one-dimensional conduction, constant thermal conductivity, no internal heat generation, and uniform area.

2.2 Convection

Convection is heat transfer between a surface and a moving fluid. It includes conduction within the fluid adjacent to the surface and advection caused by fluid motion. Newton’s law of cooling is an engineering boundary relation:

where h is the convection heat-transfer coefficient in W/(m²·K), A_s is surface area, T_s is surface temperature, and T_∞ is the bulk fluid temperature away from the surface. The coefficient h depends on fluid properties, velocity, geometry, surface condition, and flow regime. It is not generally a pure material property.

Natural convection is driven by buoyancy caused by density differences. Forced convection is driven by fans, pumps, or other external devices. The value of h must normally be obtained from correlations or experiments, often using dimensionless groups such as Reynolds, Prandtl, and Nusselt numbers.

2.3 Thermal radiation

Radiation is energy transfer by electromagnetic waves. Unlike conduction and convection, it does not require matter and can occur through a vacuum. The ideal blackbody emission law is:

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

For a real opaque surface exchanging radiation with a large surrounding enclosure at temperature T_sur:

where ε is surface emissivity, with 0 ≤ ε ≤ 1. This simplified equation assumes diffuse-gray behavior, a large enclosure, and negligible participating-medium effects. Radiation is especially important at high temperatures.

2.4 Molecular diffusion and mass transfer

Mass transfer is driven by differences in species concentration or chemical potential. For dilute, stationary, one-dimensional diffusion, Fick’s first law is:

where J_A,x is diffusive molar flux of species A relative to the molar-average velocity, D_AB is the binary diffusion coefficient in m²/s, and c_A is molar concentration in mol/m³. The negative sign indicates diffusion toward lower concentration.

Convection can also transport species. The total species flux is therefore the sum of diffusive and convective contributions. In flowing mixtures, careful definition of the reference velocity is essential.

2.5 Coupled heat and mass transfer

Evaporation, condensation, drying, humidification, combustion, sweating, and cooling towers involve simultaneous heat and mass transfer. For example, evaporation removes latent energy from a liquid surface while vapor diffuses into the surrounding gas. Temperature and concentration fields influence one another, so treating the processes as entirely independent may be inaccurate.

Figure: Heat and mass transfer are driven by gradients in temperature and composition, respectively.

3. Thermophysical and Transport Properties

Properties determine how a material responds to thermal or mass-transfer driving forces.

Specific heat is not identical to thermal conductivity. A material may store substantial thermal energy while conducting it poorly. Thermal diffusivity describes the relative rates of conduction and energy storage:

where α has units m²/s. High α means temperature disturbances penetrate relatively rapidly, subject to the assumptions of constant properties and local thermodynamic equilibrium.

Dimensionless groups help organize transfer behavior. Important examples include:

where Re is the Reynolds number and compares inertia with viscous effects;

where Pr is the Prandtl number and compares momentum diffusivity with thermal diffusivity; and

where Nu is the Nusselt number and compares convection with conduction across a characteristic length L.

4. Thermal Resistance and Overall Heat Transfer

The resistance method is analogous to electrical circuits. For a plane wall, the conduction resistance is:

and a convection resistance is:

Each resistance has units K/W. For resistances in series:

For a wall with convection on both sides:

This model assumes steady one-dimensional heat transfer, constant properties, no contact resistance, no heat generation, and equal heat rate through each series element. Contact resistance, fouling, cylindrical geometry, and radiation may require additional terms or a different model.

For a composite wall, the overall heat-transfer coefficient U is often defined by:

where U includes all relevant resistances and ∆T_overall is the selected temperature difference. The area associated with U must be stated because inner and outer areas may differ in cylindrical systems.

Worked Example: Composite wall

A plane wall has area A = 10 m², thickness L = 0.20 m, and conductivity k = 1.5 W/(m·K). The convection coefficients on the hot and cold sides are h₁ = 12 W/(m²·K) and h₂ = 8 W/(m²·K). The fluid temperatures are 120°C and 20°C. Determine the steady heat-transfer rate.

The three resistances are:

Thus:

Because temperature differences in Celsius and kelvins are numerically equal:

The heat-transfer rate is approximately 2.93 kW from the hot fluid to the cold fluid. The corresponding heat flux is Q̇/A ≈ 293 W/m². The calculation is valid only if the stated steady one-dimensional assumptions are reasonable.

5. Initial and Boundary Conditions

A transfer problem requires governing equations, material properties, geometry, initial conditions, and boundary conditions.

An initial condition specifies the state at the starting time, such as T(x,0) = T_i. A boundary condition specifies behavior at a spatial boundary. Common thermal conditions include:

1. Specified temperature: T at the boundary is known.
2. Specified heat flux: −k∂T/∂n = q̇'' is prescribed.
3. Convection boundary: −k∂T/∂n = h(T_s − T_∞).
4. Radiation boundary: surface heat flux is related to fourth-power absolute temperatures.
5. Insulated or symmetry boundary: ∂T/∂n = 0, implying zero normal heat flux.

Mass-transfer boundaries may specify concentration, species flux, convective mass-transfer coefficient, or an equilibrium relation at a phase interface. A boundary condition must be physically compatible with the system. For example, temperature in a radiation equation must be in kelvins, whereas a temperature difference may be expressed in kelvins or degrees Celsius.

6. Units, Dimensions, and Engineering Modeling

The SI base units relevant to this subject include metre (m), kilogram (kg), second (s), kelvin (K), and mole (mol). Common derived units are watt (W = J/s), pascal (Pa = N/m²), and joule (J = N·m).

Dimensional checking is a powerful error-detection method. In Fourier’s law, kA∆T/L has units:

In the resistance equation, ∆T/R has units K/(K/W) = W. Numerical consistency does not guarantee physical correctness; assumptions and boundary conditions must also be examined.

Good engineering practice includes stating the system boundary, sketching the geometry, identifying known and unknown quantities, selecting a sign convention, estimating the expected magnitude, and checking limiting cases. Computer-aided analysis and computational fluid dynamics can resolve complex geometries, but numerical results remain dependent on governing equations, property data, mesh quality, and boundary conditions.

7. Heat Transfer, Mass Transfer, and Energy Conversion Compared

Heat transfer can accompany energy conversion, but it is not itself a conversion device. A heat exchanger transfers energy between streams without necessarily converting it into shaft work. A turbine converts part of fluid energy into mechanical work while heat transfer may be negligible or significant. Mass transfer may change composition without changing the total mass of a control volume.

The second law adds directionality. Heat naturally flows from hot to cold, and real processes generate entropy. The first law accounts for quantity of energy; the second law constrains quality and irreversibility. Consequently, a thermodynamically possible process may still be impractical because its transfer rate is too low or its required area is too large.

8. Applied Case: Cooling an Electronic Enclosure

Background

Consider an illustrative design case involving a sealed electronic enclosure containing a power converter that dissipates 180 W. The enclosure is mounted in an indoor environment at 25°C. Excessive component temperature would reduce reliability.

Problem

The designer must determine whether a simple wall and external natural-convection surface can maintain the internal air below 60°C. The analysis requires a heat-transfer path from the component to the internal air, through the enclosure wall, and from the external wall to ambient air. Contact resistances and radiation may also matter.

Analysis and approach

A first model treats the three dominant elements as series resistances: internal convection, wall conduction, and external convection. The predicted temperature rise is:

If the predicted internal temperature is too high, possible design changes include increasing wall conductivity, adding fins to increase external area, improving internal circulation, reducing component power, or adding forced ventilation. However, adding vents changes the problem: air exchange may introduce dust, moisture, electromagnetic-interference concerns, and a new mass-flow path.

The designer should also assess radiation from the external surface, nonuniform component heating, transient startup, material-property variation, and the validity of natural-convection correlations. A conservative design includes safety margin and verifies the result experimentally with calibrated sensors.

Outcome and learning points

The resistance model provides a rapid preliminary estimate and identifies the dominant bottleneck. It should not be treated as a substitute for detailed analysis when temperature limits are strict or geometry is complex. The case illustrates the engineering sequence: define the boundary, identify transfer modes, select properties and correlations, calculate, test, and revise.

9. Limitations, Safety, and Contemporary Practice

Heat- and mass-transfer predictions can be limited by uncertain properties, fouling, surface oxidation, turbulent fluctuations, phase-change nonequilibrium, and poorly known boundary conditions. Correlations are usually valid only over stated ranges of Reynolds number, geometry, fluid properties, and surface conditions.

Safety considerations include burn hazards, pressure containment, flammable vapor release, toxic species, cryogenic exposure, and thermal runaway in batteries. Engineers must follow applicable codes, material compatibility requirements, ventilation standards, and environmental regulations. Accurate species accounting is particularly important when emissions or hazardous refrigerants are involved.

Modern practice increasingly combines experiments, reduced-order models, computational fluid dynamics, sensor networks, and data-driven tools. Artificial intelligence can assist with property estimation, fault detection, surrogate modeling, and design optimization, but it does not remove the need for conservation laws, dimensional checks, uncertainty analysis, and human engineering judgment. Training data may not represent unusual operating conditions, and an apparently accurate model may violate physical constraints if not properly supervised.

Chapter Summary

Heat transfer is energy transport caused by a temperature difference, while mass transfer is species transport caused by concentration or chemical-potential differences. Conduction is described by Fourier’s law, convection by an empirical surface relation, radiation by electromagnetic emission and exchange, and diffusion by Fick’s law. Conservation of mass and energy establishes the balances that every transfer analysis must satisfy.

Thermal conductivity, specific heat, density, viscosity, diffusivity, emissivity, and latent heat govern material behavior. Thermal resistance networks provide useful steady one-dimensional approximations, but their assumptions must be stated. Initial and boundary conditions are essential for obtaining a unique physical solution. Consistent SI units and dimensional checks reduce errors.

Heat and mass transfer often occur together in evaporation, drying, combustion, and environmental systems. Energy conversion is related but distinct: it concerns transformation among energy forms, whereas transfer concerns movement across space or boundaries. Effective engineering design combines conservation laws, property data, correlations, experimental validation, safety requirements, and awareness of model limitations.

Key Terms

Temperature: Intensive property indicating thermal state and heat-transfer direction.

Heat: Energy transferred solely because of a temperature difference.

Internal energy: Microscopic energy stored within a substance.

Work: Energy transfer associated with generalized force and displacement or other nonthermal interactions.

Control volume: Defined region through which mass and energy may pass.

Heat flux: Heat-transfer rate per unit area.

Conduction: Heat transfer through microscopic interactions in matter.

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

Radiation: Energy transfer by electromagnetic waves.

Diffusion: Molecular transport caused by composition or chemical-potential gradients.

Thermal conductivity: Material ability to conduct heat.

Specific heat: Energy required per unit mass for a unit temperature change under specified conditions.

Thermal diffusivity: Ratio k/(ρc_p), indicating relative thermal response speed.

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

Thermal resistance: Temperature difference divided by steady heat-transfer rate.

Boundary condition: Mathematical or physical specification at a system boundary.

Nusselt number: Dimensionless ratio characterizing convective enhancement over conduction.

Reynolds number: Dimensionless ratio of inertial to viscous effects.

Mass-transfer coefficient: Parameter relating species flux to a concentration driving force.

Energy conversion: Transformation of energy from one form to another.

Review Questions

Short Answer Questions

1. Why is heat not considered a thermodynamic property?
2. State the three principal modes of heat transfer.
3. What physical meaning does the negative sign in Fourier’s law have?
4. Distinguish a closed system from a control volume.
5. Why must absolute temperature be used in the Stefan–Boltzmann relation?

Long Answer Questions

1. Explain the first-law energy balance for a control volume and define each major term.
2. Compare conduction, convection, and radiation in terms of mechanism, medium, and governing relation.
3. Derive the total thermal resistance for a plane wall with convection on both sides.
4. Explain the role of thermal conductivity, specific heat, density, and thermal diffusivity in transient heating.
5. Discuss why heat transfer, mass transfer, and energy conversion should not be treated as interchangeable concepts.

Analytical / Application Questions

1. A wall has known area, thickness, conductivity, fluid temperatures, and convection coefficients. Formulate a resistance-network solution and identify the assumptions required for its validity.
2. A wet porous product is dried by warm air. Identify the simultaneous heat- and mass-transfer processes and explain how surface temperature and vapor concentration interact.
3. A numerical model predicts a physically impossible negative absolute temperature. Describe the checks required before accepting the result, including units, boundary conditions, material properties, and numerical implementation.

Multiple Choice Questions with Answers

Question 1

Which statement best defines heat transfer?

A. Energy stored as microscopic molecular motion

B. Energy transferred solely because of a temperature difference

C. Conversion of chemical energy into mechanical work

D. Mass transported by a concentration gradient

Correct Answer: B. Energy transferred solely because of a temperature difference

Question 2

For steady one-dimensional conduction through a plane wall with constant properties, which relation applies?

A. Q̇ = hA(T_s−T_∞)

B. Q̇ = σAT⁴

C. Q̇ = kA(T₁−T₂)/L

D. Q̇ = ṁc_pT

Correct Answer: C. Q̇ = kA(T₁−T₂)/L

Question 3

What is the SI unit of thermal conductivity?

A. W/(m·K)

B. W/m²

C. J/(kg·K)

D. m²/s

Correct Answer: A. W/(m·K)

Question 4

In Newton’s law of cooling, the coefficient h primarily represents:

A. A universal fluid property independent of flow

B. A convection heat-transfer coefficient dependent on flow and geometry

C. The wall’s thermal conductivity

D. The surface emissivity

Correct Answer: B. A convection heat-transfer coefficient dependent on flow and geometry

Question 5

Which mechanism can transfer energy through a vacuum?

A. Conduction only

B. Convection only

C. Radiation

D. Molecular diffusion only

Correct Answer: C. Radiation

Question 6

A wall has total thermal resistance 0.05 K/W and a temperature difference of 100 K. The heat-transfer rate is:

A. 0.5 W

B. 2 W

C. 50 W

D. 2,000 W

Correct Answer: D. 2,000 W

Question 7

Fick’s first law describes:

A. Molecular mass diffusion

B. Blackbody radiation

C. Conservation of total energy

D. Fluid momentum storage

Correct Answer: A. Molecular mass diffusion

Question 8

Which expression defines thermal diffusivity?

A. α = ρc_p/k

B. α = k/(ρc_p)

C. α = hA/∆T

D. α = μ/ρ

Correct Answer: B. α = k/(ρc_p)

Question 9

At steady state, a control volume with one inlet and one outlet and no accumulation must satisfy:

A. Inlet mass flow equals outlet mass flow

B. Inlet temperature equals outlet temperature always

C. Heat transfer must be zero

D. Work transfer must equal zero

Correct Answer: A. Inlet mass flow equals outlet mass flow

Question 10

Which condition is represented by zero normal temperature gradient at an insulated or symmetry boundary?

A. Specified surface temperature

B. Maximum heat flux

C. Zero normal heat flux

D. Infinite convection coefficient

Correct Answer: C. Zero normal heat flux