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Inengineering,physics, andchemistry, the study oftransport phenomena concerns the exchange ofmass,energy,charge,momentum andangular momentum between observed and studiedsystems. While it draws from fields as diverse ascontinuum mechanics andthermodynamics, 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 thevelocity 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,[1] and mechanical engineering, but the subject is a fundamental component of the curriculum in all disciplines involved in any way withfluid mechanics,heat transfer, andmass transfer. It is now considered to be a part of the engineering discipline as much asthermodynamics,mechanics, andelectromagnetism.
Transport phenomena encompass all agents ofphysical change in theuniverse. Moreover, they are considered to be fundamental building blocks which developed the universe, and which are responsible for the success of all life onEarth. However, the scope here is limited to the relationship of transport phenomena to artificialengineered systems.[2]
Inphysics,transport phenomena are allirreversible processes ofstatistical nature stemming from the random continuous motion ofmolecules, mostly observed influids. Every aspect of transport phenomena is grounded in two primary concepts : theconservation laws, and theconstitutive equations. The conservation laws, which in the context of transport phenomena are formulated ascontinuity equations, describe how the quantity being studied must be conserved. Theconstitutive equations describe how the quantity in question responds to various stimuli via transport. Prominent examples includeFourier's law ofheat conduction and theNavier–Stokes equations, which describe, respectively, the response ofheat flux totemperature gradients and the relationship betweenfluid flux and theforces applied to the fluid. These equations also demonstrate the deep connection between transport phenomena andthermodynamics, a connection that explains why transport phenomena are irreversible. Almost all of these physical phenomena ultimately involve systems seeking theirlowest energy state in keeping with theprinciple of minimum energy. As they approach this state, they tend to achieve truethermodynamic 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 is the system's attempt to achieve thermal equilibrium with its environment, just as mass andmomentum transport move the system towards chemical andmechanical equilibrium.[citation needed]
Examples of transport processes includeheat conduction (energy transfer),fluid flow (momentum transfer),molecular diffusion (mass transfer),radiation andelectric charge transfer insemiconductors.[3][4][5][6]
Transport phenomena have wide application. For example, insolid state physics, the motion and interaction of electrons, holes andphonons are studied under "transport phenomena". Another example is inbiomedical engineering, where some transport phenomena of interest arethermoregulation,perfusion, andmicrofluidics. Inchemical engineering, transport phenomena are studied inreactor design, analysis of molecular or diffusive transport mechanisms, andmetallurgy.
The transport of mass, energy, and momentum can be affected by the presence of external sources:
An important principle in the study of transport phenomena is analogy betweenphenomena.
There are some notable similarities in equations for momentum, energy, and mass transfer[7] which can all be transported bydiffusion, as illustrated by the following examples:
The molecular transfer equations ofNewton's law for fluid momentum,Fourier's law for heat, andFick's law for mass are very similar. One can convert from onetransport coefficient to another in order to compare all three different transport phenomena.[8]
| Transported quantity | Physical phenomenon | Equation |
|---|---|---|
| Momentum | Viscosity (Newtonian fluid) | |
| Energy | Heat conduction (Fourier's law) | |
| Mass | Molecular diffusion (Fick's law) | |
| Electric charge | Electric current (Ohm's law) |
A great deal of effort has been devoted in the literature to developing analogies among these three transport processes forturbulent transfer so as to allow prediction of one from any of the others. TheReynolds analogy assumes that the turbulent diffusivities are all equal and that the molecular diffusivities of momentum (μ/ρ) and mass (DAB) are negligible compared to the turbulent diffusivities. When liquids are present and/or drag is present, the analogy is not valid. Other analogies, such asvon Karman's andPrandtl's, usually result in poor relations.
The most successful and most widely used analogy is theChilton and Colburn J-factor analogy.[9] This analogy is based on experimental data for gases and liquids in both thelaminar 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.
In fluid systems described in terms oftemperature,matter density, andpressure, it is known thattemperature differences lead toheat flows from the warmer to the colder parts of the system; similarly, pressure differences will lead tomatter 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 inconvection) 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 byLars Onsager usingstatistical mechanics as a consequence of thetime reversibility 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.[10]
In momentum transfer, the fluid is treated as a continuous distribution of matter. The study of momentum transfer, orfluid mechanics can be divided into two branches:fluid statics (fluids at rest), andfluid dynamics (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 isυxρ. By random diffusion of molecules there is an exchange of molecules in thez-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 isNewton's law of viscosity written as follows:
whereτzx is the flux of x-directed momentum in the z-direction,ν isμ/ρ, the momentum diffusivity,vx is the velocity of the fluid in the x-direction,z is the distance of transport or diffusion,ρ is the density, andμ 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 symbolτzx; the indices are reversed as compared with standard usage in solid mechanics, and the sign is reversed.[11]
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 byFick'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:[12]
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.
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 theFourier's law which is expressed as follows for a static system:
The net flux of heat through a system equals the conductivity times therate 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:
Heat transfer is analyzed inpacked beds,nuclear reactors andheat exchangers.
The heat and mass analogy allows solutions formass transfer problems to be obtained from known solutions toheat transfer problems. Its arises from similar non-dimensional governing equations between heat and mass transfer.
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 theReynolds number, is thePrandtl 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 theSchmidt 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 andNewton's law of cooling:
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 [kg/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.
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, andLewis number, yielding:
For fully developed turbulent flow, with n=1/3, this becomes the Chilton–Colburn J-factor analogy.[13] Said analogy also relates viscous forces and heat transfer, like theReynolds analogy.
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 thechemical potential 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.
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.[14] 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,[15] and HVAC dehumidification equipment that combine heat transfer and selective membranes.[16]
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 fromurban runoff, and policies intended to reduce thecopper content of vehicle brake pads in the U.S.[17][18]
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