In
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
,
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, and
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
, the study of transport phenomena concerns the exchange of
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
,
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
,
charge,
momentum and
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
between observed and studied
systems. While it draws from fields as diverse as
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles.
Continuum mec ...
and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, it places a heavy emphasis on the commonalities between the topics covered. Mass, momentum, and heat transport all share a very similar mathematical framework, and the parallels between them are exploited in the study of transport phenomena to draw deep mathematical connections that often provide very useful tools in the analysis of one field that are directly derived from the others.
The fundamental analysis in all three subfields of mass, heat, and momentum transfer are often grounded in the simple principle that the total sum of the quantities being studied must be conserved by the system and its environment. Thus, the different phenomena that lead to transport are each considered individually with the knowledge that the sum of their contributions must equal zero. This principle is useful for calculating many relevant quantities. For example, in fluid mechanics, a common use of transport analysis is to determine the
velocity profile of a fluid flowing through a rigid volume.
Transport phenomena are ubiquitous throughout the engineering disciplines. Some of the most common examples of transport analysis in engineering are seen in the fields of process, chemical, biological, and mechanical engineering, but the subject is a fundamental component of the curriculum in all disciplines involved in any way with
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
, and
mass transfer. It is now considered to be a part of the engineering discipline as much as
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
,
mechanics
Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
, and
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
.
Transport phenomena encompass all agents of
physical change
Physical changes are changes affecting the form of a chemical substance, but not its chemical composition. Physical changes are used to separate mixtures into their component compounds, but can not usually be used to separate compounds into c ...
in the
universe
The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
. Moreover, they are considered to be fundamental building blocks which developed the universe, and which are responsible for the success of all life on
Earth
Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
. However, the scope here is limited to the relationship of transport phenomena to artificial
engineered systems.
Overview
In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, transport phenomena are all
irreversible processes of
statistical
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
nature stemming from the random continuous motion of
molecules
A molecule is a group of two or more atoms that are held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemistry ...
, mostly observed in
fluids. Every aspect of transport phenomena is grounded in two primary concepts : the
conservation laws, and the
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
s. The conservation laws, which in the context of transport phenomena are formulated as
continuity equations
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantit ...
, describe how the quantity being studied must be conserved. The
constitutive equations
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
describe how the quantity in question responds to various stimuli via transport. Prominent examples include
Fourier's law
Thermal conduction is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy ...
of
heat conduction and the
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
, which describe, respectively, the response of
heat flux to
temperature gradients and the relationship between
fluid flux and the
forces applied to the fluid. These equations also demonstrate the deep connection between transport phenomena and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, a connection that explains why transport phenomena are irreversible. Almost all of these physical phenomena ultimately involve systems seeking their
lowest energy state in keeping with the
principle of minimum energy. As they approach this state, they tend to achieve true
thermodynamic equilibrium, at which point there are no longer any driving forces in the system and transport ceases. The various aspects of such equilibrium are directly connected to a specific transport:
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
is the system's attempt to achieve thermal equilibrium with its environment, just as mass and
momentum transport move the system towards chemical and
mechanical equilibrium.
Examples of transport processes include
heat conduction (energy transfer),
fluid flow
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
(momentum transfer),
molecular diffusion
Molecular diffusion is the motion of atoms, molecules, or other particles of a gas or liquid at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid, size and density (or their product, ...
(mass transfer),
radiation
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or a material medium. This includes:
* ''electromagnetic radiation'' consisting of photons, such as radio waves, microwaves, infr ...
and
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
transfer in
semiconductor
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
s.
Transport phenomena have wide application. For example, in
solid state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state p ...
, the motion and interaction of electrons, holes and
phonons are studied under "transport phenomena". Another example is in
biomedical engineering
Biomedical engineering (BME) or medical engineering is the application of engineering principles and design concepts to medicine and biology for healthcare applications (e.g., diagnostic or therapeutic purposes). BME also integrates the logica ...
, where some transport phenomena of interest are
thermoregulation,
perfusion, and
microfluidics. In
chemical engineering
Chemical engineering is an engineering field which deals with the study of the operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials ...
, transport phenomena are studied in
reactor design, analysis of molecular or diffusive transport mechanisms, and
metallurgy.
The transport of mass, energy, and momentum can be affected by the presence of external sources:
* An odor dissipates more slowly (and may intensify) when the source of the odor remains present.
* The rate of cooling of a solid that is conducting heat depends on whether a heat source is applied.
* The
gravitational force acting on a rain drop counteracts the resistance or
drag imparted by the surrounding air.
Commonalities among phenomena
An important principle in the study of transport phenomena is analogy between
phenomena
A phenomenon ( phenomena), sometimes spelled phaenomenon, is an observable Event (philosophy), event. The term came into its modern Philosophy, philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be ...
.
Diffusion
There are some notable similarities in equations for momentum, energy, and mass transfer which can all be transported by
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
, as illustrated by the following examples:
* Mass: the spreading and
dissipation of odors in air is an example of mass diffusion.
* Energy: the conduction of heat in a solid material is an example of
heat diffusion.
* Momentum: the
drag experienced by a rain drop as it falls in the atmosphere is an example of
momentum diffusion (the rain drop loses momentum to the surrounding air through viscous stresses and decelerates).
The molecular transfer equations of
Newton's law for fluid momentum,
Fourier's law
Thermal conduction is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy ...
for heat, and
Fick's law for mass are very similar. One can convert from one
transport coefficient to another in order to compare all three different transport phenomena.
A great deal of effort has been devoted in the literature to developing analogies among these three transport processes for
turbulent transfer so as to allow prediction of one from any of the others. The
Reynolds analogy assumes that the turbulent diffusivities are all equal and that the molecular diffusivities of momentum (μ/ρ) and mass (D
AB) are negligible compared to the turbulent diffusivities. When liquids are present and/or drag is present, the analogy is not valid. Other analogies, such as
von Karman's and
Prandtl
Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German Fluid mechanics, fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlyin ...
's, usually result in poor relations.
The most successful and most widely used analogy is the
Chilton and Colburn J-factor analogy. This analogy is based on experimental data for gases and liquids in both the
laminar and turbulent regimes. Although it is based on experimental data, it can be shown to satisfy the exact solution derived from laminar flow over a flat plate. All of this information is used to predict transfer of mass.
Onsager reciprocal relations
In fluid systems described in terms of
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
,
matter density, and
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
, it is known that
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
differences lead to
heat
In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
flows from the warmer to the colder parts of the system; similarly, pressure differences will lead to
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
flow from high-pressure to low-pressure regions (a "reciprocal relation"). What is remarkable is the observation that, when both pressure and temperature vary, temperature differences at constant pressure can cause matter flow (as in
convection
Convection is single or Multiphase flow, multiphase fluid flow that occurs Spontaneous process, spontaneously through the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoy ...
) and pressure differences at constant temperature can cause heat flow. The heat flow per unit of pressure difference and the density (matter) flow per unit of temperature difference are equal.
This equality was shown to be necessary by
Lars Onsager using
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
as a consequence of the
time reversibility
In mathematics and physics, time-reversibility is the property (mathematics), property of a process whose governing rules remain unchanged when the direction of its sequence of actions is reversed.
A deterministic process is time-reversible if th ...
of microscopic dynamics. The theory developed by Onsager is much more general than this example and capable of treating more than two thermodynamic forces at once.
Momentum transfer
In momentum transfer, the fluid is treated as a continuous distribution of matter. The study of momentum transfer, or
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
can be divided into two branches:
fluid statics
In physics, a fluid is a liquid, gas, or other material that may continuously move and deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot r ...
(fluids at rest), and
fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
(fluids in motion).
When a fluid is flowing in the x-direction parallel to a solid surface, the fluid has x-directed momentum, and its concentration is ''υ''
x''ρ''. By random diffusion of molecules there is an exchange of molecules in the ''z''-direction. Hence the x-directed momentum has been transferred in the z-direction from the faster- to the slower-moving layer.
The equation for momentum transfer is
Newton's law of viscosity written as follows:
:
where ''τ''
zx is the flux of x-directed momentum in the z-direction, ''ν'' is ''μ''/''ρ'', the momentum diffusivity, ''z'' is the distance of transport or diffusion, ''ρ'' is the density, and ''μ'' is the dynamic viscosity. Newton's law of viscosity is the simplest relationship between the flux of momentum and the velocity gradient. It may be useful to note that this is an unconventional use of the symbol ''τ''
zx; the indices are reversed as compared with standard usage in solid mechanics, and the sign is reversed.
Mass transfer
When a system contains two or more components whose concentration vary from point to point, there is a natural tendency for mass to be transferred, minimizing any concentration difference within the system. Mass transfer in a system is governed by
Fick's first law: 'Diffusion flux from higher concentration to lower concentration is proportional to the gradient of the concentration of the substance and the diffusivity of the substance in the medium.' Mass transfer can take place due to different driving forces. Some of them are:
["Griskey, Richard G. "Transport Phenomena and Unit Operations." Wiley & Sons: Hoboken, 2006. 228-248.]
* Mass can be transferred by the action of a pressure gradient (pressure diffusion)
* Forced diffusion occurs because of the action of some external force
* Diffusion can be caused by temperature gradients (thermal diffusion)
* Diffusion can be caused by differences in
chemical potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
This can be compared to Fick's law of diffusion, for a species A in a binary mixture consisting of A and B:
:
where D is the diffusivity constant.
Heat transfer
Many important engineered systems involve heat transfer. Some examples are the heating and cooling of process streams, phase changes, distillation, etc. The basic principle is the
Fourier's law
Thermal conduction is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy ...
which is expressed as follows for a static system:
:
The net flux of heat through a system equals the conductivity times the
rate of change of temperature with respect to position.
For convective transport involving turbulent flow, complex geometries, or difficult boundary conditions, the heat transfer may be represented by a heat transfer coefficient.
:
where A is the surface area,
is the temperature driving force, Q is the heat flow per unit time, and h is the heat transfer coefficient.
Within heat transfer, two principal types of convection can occur:
*
Forced convection can occur in both laminar and turbulent flow. In the situation of laminar flow in circular tubes, several dimensionless numbers are used such as
Nusselt number,
Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
, and
Prandtl number. The commonly used equation is
.
* Natural or
free convection is a function of
Grashof and
Prandtl numbers. The complexities of free convection heat transfer make it necessary to mainly use empirical relations from experimental data.
Heat transfer is analyzed in
packed beds,
nuclear reactor
A nuclear reactor is a device used to initiate and control a Nuclear fission, fission nuclear chain reaction. They are used for Nuclear power, commercial electricity, nuclear marine propulsion, marine propulsion, Weapons-grade plutonium, weapons ...
s and
heat exchangers.
Heat and mass transfer analogy
The heat and mass analogy allows solutions for
mass transfer problems to be obtained from known solutions to
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
problems. Its arises from similar non-dimensional governing equations between heat and mass transfer.
Derivation
The non-dimensional energy equation for fluid flow in a boundary layer can simplify to the following, when heating from viscous dissipation and heat generation can be neglected:
Where
and
are the velocities in the x and y directions respectively normalized by the free stream velocity,
and
are the x and y coordinates non-dimensionalized by a relevant length scale,
is the
Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
,
is the
Prandtl number, and
is the non-dimensional temperature, which is defined by the local, minimum, and maximum temperatures:
The non-dimensional species transport equation for fluid flow in a boundary layer can be given as the following, assuming no bulk species generation:
Where
is the non-dimensional concentration, and
is the
Schmidt number.
Transport of heat is driven by temperature differences, while transport of species is due to concentration differences. They differ by the relative diffusion of their transport compared to the diffusion of momentum. For heat, the comparison is between viscous diffusivity (
) and thermal diffusion (
), given by the Prandtl number. Meanwhile, for mass transfer, the comparison is between viscous diffusivity (
) and mass Diffusivity (
), given by the Schmidt number.
In some cases direct analytic solutions can be found from these equations for the Nusselt and Sherwood numbers. In cases where experimental results are used, one can assume these equations underlie the observed transport.
At an interface, the boundary conditions for both equations are also similar. For heat transfer at an interface, the no-slip condition allows us to equate conduction with convection, thus equating Fourier's law and
Newton's law of cooling
In the study of heat transfer, Newton's law of cooling is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. The law is frequentl ...
:
Where q” is the heat flux,
is the thermal conductivity,
is the heat transfer coefficient, and the subscripts
and
compare the surface and bulk values respectively.
For mass transfer at an interface, we can equate Fick's law with Newton's law for convection, yielding:
Where
is the mass flux
">g/s is the diffusivity of species a in fluid b, and
is the mass transfer coefficient. As we can see,
and
are analogous,
and
are analogous, while
and
are analogous.
Implementing the Analogy
Heat-Mass Analogy:
Because the Nu and Sh equations are derived from these analogous governing equations, one can directly swap the Nu and Sh and the Pr and Sc numbers to convert these equations between mass and heat.
In many situations, such as flow over a flat plate, the Nu and Sh numbers are functions of the Pr and Sc numbers to some coefficient
. Therefore, one can directly calculate these numbers from one another using:
Where can be used in most cases, which comes from the analytical solution for the Nusselt Number for laminar flow over a flat plate. For best accuracy, n should be adjusted where correlations have a different exponent.
We can take this further by substituting into this equation the definitions of the heat transfer coefficient, mass transfer coefficient, and
Lewis number, yielding:
For fully developed turbulent flow, with n=1/3, this becomes the Chilton–Colburn J-factor analogy.
Said analogy also relates viscous forces and heat transfer, like the
Reynolds analogy.
Limitations
The analogy between heat transfer and mass transfer is strictly limited to binary diffusion in dilute (
ideal) solutions for which the mass transfer rates are low enough that mass transfer has no effect on the velocity field. The concentration of the diffusing species must be low enough that the
chemical potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
gradient is accurately represented by the concentration gradient (thus, the analogy has limited application to concentrated liquid solutions). When the rate of mass transfer is high or the concentration of the diffusing species is not low, corrections to the low-rate heat transfer coefficient can sometimes help. Further, in multicomponent mixtures, the transport of one species is affected by the chemical potential gradients of other species.
The heat and mass analogy may also break down in cases where the governing equations differ substantially. For instance, situations with substantial contributions from generation terms in the flow, such as bulk heat generation or bulk chemical reactions, may cause solutions to diverge.
Applications of the Heat-Mass Analogy
The analogy is useful for both using heat and mass transport to predict one another, or for understanding systems which experience simultaneous heat and mass transfer. For example, predicting heat transfer coefficients around turbine blades is challenging and is often done through measuring evaporating of a volatile compound and using the analogy.
Many systems also experience simultaneous mass and heat transfer, and particularly common examples occur in processes with phase change, as the enthalpy of phase change often substantially influences heat transfer. Such examples include: evaporation at a water surface, transport of vapor in the air gap above a membrane distillation desalination membrane,
and HVAC dehumidification equipment that combine heat transfer and selective membranes.
Applications
Pollution
The study of transport processes is relevant for understanding the release and distribution of pollutants into the environment. In particular, accurate modeling can inform mitigation strategies. Examples include the control of surface water pollution from
urban runoff
Urban runoff is surface runoff of rainwater, landscape irrigation, and car washing created by urbanization. Impervious surfaces (roads, parking lots and sidewalks) are constructed during land development. During rain, storms, and other Precipitati ...
, and policies intended to reduce the
copper
Copper is a chemical element; it has symbol Cu (from Latin ) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkish-orang ...
content of vehicle brake pads in the U.S.
See also
*
Constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
*
Continuity equation
*
Wave propagation
*
Pulse
*
Action potential
An action potential (also known as a nerve impulse or "spike" when in a neuron) is a series of quick changes in voltage across a cell membrane. An action potential occurs when the membrane potential of a specific Cell (biology), cell rapidly ri ...
*
Bioheat transfer
References
External links
Transport Phenomena Archive in the Teaching Archives of the Materials Digital Library Pathway
*
*
*
{{DEFAULTSORT:Transport Phenomena (Engineering and Physics)
Chemical engineering