Transport phenomena

In physics, transport phenomena are all irreversible processes of statistical nature stemming from the random continuous motion of molecules, mostly observed in fluids. Every aspect of transport phenomena is grounded in two primary concepts : the conservation laws, and the constitutive equations. The conservation laws, which in the context of transport phenomena are formulated as continuity equations, describe how the quantity being studied must be conserved. The constitutive equations describe how the quantity in question responds to various stimuli via transport. Prominent examples include Fourier's law of heat conduction and the Navier–Stokes equations, 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, 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 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 (momentum transfer), molecular diffusion (mass transfer), radiation and electric charge transfer in semiconductors.[3][4][5][6]

Transport phenomena have wide application. For example, in solid state physics, the motion and interaction of electrons, holes and phonons are studied under "transport phenomena". Another example is in biomedical engineering, where some transport phenomena of interest are thermoregulation, perfusion, and microfluidics. In chemical engineering, 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.

An important principle in the study of transport phenomena is analogy between phenomena.

Diffusion

There are some notable similarities in equations for momentum, energy, and mass transfer[7] which can all be transported by diffusion, 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 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.[8]

Comparison of diffusion phenomena

Transported quantity	Physical phenomenon	Equation
Momentum	Viscosity (Newtonian fluid)	${\displaystyle \tau =-\nu {\frac {\partial \rho \upsilon }{\partial x}}}$
Energy	Heat conduction (Fourier's law)	${\displaystyle {\frac {q}{A}}=-k{\frac {dT}{dx}}}$
Mass	Molecular diffusion (Fick's law)	${\displaystyle J=-D{\frac {\partial C}{\partial x}}}$

(Definitions of these formulas are given below).

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's, usually result in poor relations.

The most successful and most widely used analogy is the Chilton and Colburn J-factor analogy.[9] 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.

In momentum transfer, the fluid is treated as a continuous distribution of matter. The study of momentum transfer, or fluid mechanics can be divided into two branches: fluid statics (fluids at rest), and fluid 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 is υ_xρ. 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 transport is Newton's Law of Viscosity written as follows:

${\displaystyle \tau _{zx}=-\nu {\frac {\partial \rho \upsilon _{x}}{\partial z}}}$

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 is the simplest relationship between the flux of momentum and the velocity gradient.

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:[11]

• 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

This can be compared to Fick's Law of Diffusion, for a species A in a binary mixture consisting of A and B:

${\displaystyle J_{Ay}=-D_{AB}{\frac {\partial Ca}{\partial y}}}$

where D is the diffusivity constant.

All processes in engineering involve the transfer of energy. Some examples are the heating and cooling of process streams, phase changes, distillations, etc. The basic principle is the first law of thermodynamics which is expressed as follows for a static system:

${\displaystyle q=-k{\frac {dT}{dx}}}$

The net flux of energy through a system equals the conductivity times the rate of change of temperature with respect to position.

For other systems that involve either turbulent flow, complex geometries or difficult boundary conditions another equation would be easier to use:

${\displaystyle Q=h\cdot A\cdot {\Delta T}}$

where A is the surface area, :${\displaystyle {\Delta T}}$ is the temperature driving force, Q is the heat flow per unit time, and h is the heat transfer coefficient.

Within heat transfer, two 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, and Prandtl number. The commonly used equation is ${\displaystyle Nu_{a}={\frac {h_{a}D}{k}}}$.

• 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.[11]

Heat transfer is analyzed in packed beds, nuclear reactors and heat exchangers.

• Constitutive equation
• Continuity equation
• Wave propagation
• Pulse
• Action potential
• Bioheat transfer

• Transport Phenomena Archive in the Teaching Archives of the Materials Digital Library Pathway
• "Some Classical Transport Phenomena Problems with Solutions - Fluid Mechanics".
• "Some Classical Transport Phenomena Problems with Solutions - Heat Transfer".
• "Some Classical Transport Phenomena Problems with Solutions - Mass Transfer".