223VOLUME V / INSTRUMENTS

4.2.1 Introduction

More than half of the chemical engineering literatureconcerns transport phenomena. This topic deals with all thoseproblems where a physical property, like mass, energy ormomentum, is transferred from one point to another in space.For example, the study of these phenomena allow thefollowing calculations: a) the loss of pressure in a fluidduring its flow inside a pipe; b) the velocity profile of a fluidmoving inside a channel; c) the length of pipe needed to heatthe fluid flowing inside it to the desired temperature; d ) theamount of heat dissipated from a surface in contact with afluid stream; e) the contact area needed between two phasesto transfer the desired amount of matter in a defined time.

In this chapter, only the problems of momentum, energyand matter transfer without chemical reactions will be analyzed.The interactions existing between transport phenomena andchemical reactions are analyzed in chapters 5.1 and 6.3.

The topics examined by classic thermodynamics aresystems in equilibrium, while transport phenomena deal withthose systems far from equilibrium conditions, wheregradients of intensive properties, like velocity, temperatureand concentrations, are present. From a microscopic point ofview, transport phenomena are caused by the chaoticmovement of molecules and their aggregates in turbulentmotion, and which cause systems to evolve towardsequilibrium conditions. Dissipative processes are associatedwith these evolutions that, from a phenomenological point ofview, are identified through the resistance caused by thedissipation itself. Consequently, the phenomenologicalcorrelations developed during the years to describe thedifferent fluxes (e.g. the amount of the considered propertytransferred per unit of area and time) are, in fact, correlated tothese gradients with more or less complex functional forms.

The study of transport phenomena began towards the endof the Nineteenth century, with the beginning of engineeringapplications related to the construction of thermodynamicmachines and to the building of industrial plants. Initially, thecorrelations developed were the macroscopic balances that dealwith a finite portion of space. The origin of these macroscopicbalance equations is difficult to trace; however, they are widelyused, together with the assumptions and the impliedapproximations in their formulations, not only in chemical

engineering but also in civil, mechanical and aeronauticengineering. Indeed, most of the work was done in theframework of these last three disciplines and only later werethey used in a chemical context (for example, the theories onturbulence or on the velocity boundary layer, both developedLudwig Prandtl). Once the importance of the macroscopicbalance equations was ascertained, the necessity to understandthe involved mechanisms emerged, together with the need toformulate operational schemes to make the application of suchequations easier. Consequently, charts that describe the desiredbehaviour as a function of the main physical variables, groupedas dimensionless numbers, were developed. For example, it ispossible to refer to the graphs for the drag coefficient as afunction of the Reynolds number (adopted for the calculationof the pressure drop inside pipes) or to those equating theSherwood number as a function of the product between theReynolds and Schmidt numbers, each elevated to a suitableexponent (used for the estimation of the mass transfercoefficient). This treatment was developed under the physics ofcontinuum systems and the properties of the involved fluidswere identified with suitable variables introduced from aphenomenological point of view such as viscosity, thermalconductivity and diffusivity, all experimentally measurable.

At the same time, the development of kinetic moleculartheories made it possible to justify the differentphenomenological coefficients previously introduced, withconsequent great impact on their theorical estimation startingfrom the known properties of the involved molecules. For thispurpose it is possible to mention the work of Chapman andEnskog on the theory of monoatomic gases and that ofChapman and Cowling (1939) on the binary gaseous mixtures,up through the extension to the multicomponent mixtures byHirschfelder et al. (1954), whose contribution represents amilestone in the examination of molecular aspects.

Although the study of the transport of mass, of energy andof momentum were historically developed independently, todayit is increasingly important to address transport phenomena asa whole for two reasons: the mechanism of transport of thedifferent properties is often the same and, consequently, themathematic formalism for their description is the same.

In conclusion, the fundamentals of transport phenomenawere already considered well-defined in the mid Twentiethcentury, as is testified by the publication by Bird et al. (1960)

4.2

Transport phenomena

of the textbook that become the ‘reference’ in the field. In thesame period the translations from the Russian of thetextbooks by Landau and Lifshitz (1959) and by Levich(1962) also appeared. Today, dozens of texts dedicated totransport phenomena are available, from the introductory tothose dealing with the most innovative aspects. Often theexamples developed are chosen according to who the finaluser is and then, for example, texts specifically dedicated tochemical, metallurgical, biomedical, and so on, becomeavailable.

4.2.2 Macroscopic and molecular vision

From the above it emerges that, in a broad sense, theexpression transport phenomena means the transfer ofphysical properties within a system or through its border. Theproperties under consideration for chemical and engineeringsystems are mass, momentum and energy, while the systemscan evidence a greatly differentiated degree of complexity, asillustrated by some examples reported in Fig. 1. Hence,systems run from homogeneous, where a single phase ispresent, to heterogeneous, where multiple phases are present.Moreover, each phase can be monocomponent ormulticomponent, depending on the number of chemicalspecies present in it. Often, one of the phases present is afluid in motion, thus the complexity of the problem isincreased by the discontinuous changes in its behaviour, as inthe case of the transition from laminar to turbulent motion orto that existing from the subsonic to the supersonic motion.Other peculiar examples of motion transitions are present, forexample, in two-phase fluids, where a dozen different flowconfigurations are known, depending on the relative velocitybetween the two phases and their volumetric ratio.

A system can be analyzed at different scales, eachidentified by its own characteristic size. In general terms, a

system, or more properly the portion being examined, isconsidered as a continuous medium and consequently thisproperty is also extended to all the intensive and extensivequantities used for its description.

At the macroscopic level, the study is performed on finitedimensions, where the control volume includes the entiresystem and where the change in value of its properties isobtained by writing balance equations containing the input andthe output quantities per unit time. As a first approximation,uniform values can be assigned to the intensive variables in thedifferent regions of the system. This is defined as the ‘lumped’parameters approach. From a mathematical point of view, theformulation of balances leads to the writing of algebraicequations if the system is in steady state conditions, or toordinary differential equations if the system is transient. Thecharacteristic length of the system coincides with one of itsdimensions and thus it can span from centimetres to meters. Aswill be explained below, in such an approach, the transportproperties can be expressed by the transfer coefficients, thatstate in an averaged form the contributions of the matterproperties and the transport regime.

The study of the same system at an intermediate scaleimplies the analysis and the description of phenomenaoccurring at a characteristic length between a micrometre anda centimetre. Mathematically, it is appropriate to describe theinvolved phenomena by considering a significant elementaryvolume, definitely assimilated to an infinitesimal one, by anapproach to the limits. Because the dimensions considered aresignificantly greater than those of the molecules contained inthis volume, it is often correct to consider the system as acontinuum. The writing of the balance equations leads todifferential equations with partial derivatives with respect tothe three spatial coordinates and the temporal one asindependent variables. The equations obtained in this way, ifintegrated on the whole system, provide the fluid velocity, thetemperature and the composition fields. As will be described

FLUID MOTION

224 ENCYCLOPAEDIA OF HYDROCARBONS

A

B

C Dbubbly slug churn annular

Fig. 1. Some examples of systems typicallyencountered during thestudies of transportphenomena in a channel. A, motion of a single-phase fluid in laminar regime; B, motion of a single-phasefluid in turbulent regime; C, two-phase flow (gas-liquid or liquid-liquid);D, two-phase flowinstability (bubble regime,slug regime, churn regime,annular regime).

below, writing the balance equations implies the knowledge ofsome parameters, the so-called phenomenological properties,which include the thermal conductivity, the diffusivity, theviscosity and the surface tension of the material making up theexamined system. From a mathematical point of view, theseproperties actually reflect the use of a linear approximation inthe correlations linking the local flux values of the transportedproperties to the corresponding gradients of the intensivevariables. In a strictly phenomenological vision, the gradientsare identified with the driving forces for the transport, whilethe fluxes in the direction opposed to them represent theeffects. These phenomenological properties must beexperimentally measured.

At the molecular level, it is necessary to consider themechanisms that cause the transport on the basis of theproperties of the molecules making up the examinedsubstance. In other words, by analyzing the problem of thetransport at this scale the aim is to estimate the value of theabove phenomenological properties from the ultimateproperties of the matter. In this case, the characteristicdimension is identified by a reasonable interval around themolecular dimensions and thus it ranges in the intervalbetween a nanometre and a micrometre.

In the following, transport phenomena will be analyzedabove all by considering a continuum system and thus derivingthe expression of the differential equations at distributedparameters, the so-called microscopic balance equations (ormicroscopic equation of change). Until the last decade, theirintegration represented a tricky problem; consequently, textsdealing with transport phenomena examined a great number ofsimplified cases where it was possible to obtain analyticalsolutions, sometimes approximated. The reconstruction of thebehaviour of the system was then obtained by the combinationof the asymptotic trends obtained as shown above. Today,improvements in methods of calculation and especiallyelectronic computers have permitted the use of reliablesoftware that allow the integration of the conservationequations on complex geometric domains, also taking intoconsideration multiphase systems, with highly non-linearphenomenological transport laws, both for laminar andturbulent flow regimes.

Successively, the theories at the molecular level that allowthe estimation of phenomenological properties, like viscosity,thermal conductivity and diffusivity will be examined.

Finally, some cases of particular interest will beexamined, such as the determination of velocity profiles inchannels, the study of boundary layer development forvelocity, temperature and concentration for a fluidapproaching a wall, temperature distribution across acomposite wall, heating dynamics of a solid and diffusionthrough a solid or a stagnant fluid, mass transport throughinterfaces and some aspects of natural convection.

4.2.3 Phenomenological correlations

Physical entities like the ones examined here (mass, energy andmomentum) can be transferred from one part of the system toanother in two completely different ways. An example of thesemechanisms is illustrated in Fig. 2. The former is linked to themacroscopic fluid movement and is called convection. Themolecules of the fluid move within the system and they conveythe mass, as well as all the other associated properties.

Convection can be externally induced using machines to movethe fluid (stirrers, pumps, compressors); in this case themechanism is identified as forced convection. Alternatively, asalready seen, the overall fluid motion can be due to thepresence of intensive variable gradients within the system.When these gradients exceed a critical value, circular motionphenomena within the fluid start and this mechanism is callednatural convection. The second transport mode identifies atransfer modality to which is not associated any overall massmovement. In the energy transport this mechanism is indicatedas conduction and it is associated with the presence oftemperature gradients, while in the cases of mass transport, i.e.diffusion, and of momentum, it is associated with theconcentration and velocity gradients, respectively.

To complete the description of the different transport modesit is also necessary to examine the way in which the motion in afluid occurs. Using as an example, for sake of simplicity, asingle-phase fluid, there can be a case where each single fluidelement moves following well-defined trajectories representedby regular lines, esentially steady within the core of the fluid inmotion (the so-called streamlines). If, for example, a fluidmoves inside a channel, its elements move in a direction parallelto the channel walls and the velocity components at right anglesto the channel axis are absent. In this case, the motion is calledin laminar regime. The motion can also occur followingirregular, tortuous and unsteady trajectories. In this second case,the perturbation of the fluid movement is called turbulence andindicates the motion component that is superimposed on themain flow along the channel axis. Thus, at a certain instant,motion components exist both parallel to and at right angleswith respect to the channel axis that identifies the main flowdirection. If the system is observed for enough time the velocitycomponents along the right angles are seen to have a zero localaverage value. The parallel component, on the other hand,presents an average value different from zero and is responsiblefor the overall fluid transport. In this case the motion is called inturbulent regime.

Therefore, in the case of laminar regime motion, thetransport in the direction at right angles to the main flow canhappen uniquely by molecular collision and thus this lastmechanism is responsible for the transports of conductive anddiffusive nature. In case of turbulent motion, although the longtime average of transversal velocities is zero, a very effective

TRANSPORT PHENOMENA

225VOLUME V / INSTRUMENTS

diffusion(chaotic molecular

motion with velocity c)

convection(overall motionwith velocity v)

velocityprofile

uvc

Fig. 2. Illustration of the different transport mechanisms:convection (long-range aggregate motion), diffusion (short-range motion).

transport mechanism, superimposed on the molecular one, isactivated by the presence of the instantaneous fluctuations.Such a mechanism is called turbulent transport.

Convective fluxesBy considering a fluid element that moves uniformly with

a velocity, u, the convective flux, J, indicates the amount of agiven property that in the unit time flows through thereference unit surface by the effect of the transport of theentire fluid ensemble. Mathematically, this corresponds to theproduct of the velocity of the fluid element and an intensivevariable identified by the amount of the given propertycontained in the unit volume. Thus, for mass, momentum andenergy, the three expressions are, respectively:

[1]

[2]

[3]

where r, wi, U, F and u22 are the density, the mass fractionand the specific contributions of the unit mass for theinternal, potential and kinetic energies, respectively. Althoughthe result shown in equation [3] rigorously correspond to thetotal energy flux, in chemical systems the last twocontributions are usually some order of magnitude lower thanthat of the internal energy and consequently they can beignored. Accordingly, in the following only the contributiondue to the internal energy, conveniently expressed asUU°rCV T, will be considered, where °, CV and ∆Tindicate the reference value, the specific heat at constantvolume and the temperature difference existing between thelocal value and that of the reference state, respectively.

Diffusive fluxes and constitutive equationsTo analyze the origin of the diffusive fluxes it is necessary

to consider a direction at right angles to the direction of themain flow. By its intimate nature and definition, the transportof diffusive nature in one direction should not be associatedwith any overall transport (i.e. convective) in the samedirection. In principle, any of the different gradients present inthe system, like those of composition, temperature, pressure,potential of an external force field and momentum, provide acontribution to the diffusive transport of the consideredproperty. The simplest functional form to consider all of thesecontributions is a linear combination. Not all of the mentionedgradients provide a numerically significant contribution andthus, in practice, it is common to consider only the mostimportant ones.

The expressions for the mass, energy and momentumfluxes are called ‘laws’ or more correctly, in modern terms,constitutive equations because they express the existing linkbetween the driving force of the phenomenon and theresulting action.

For sake of simplicity, in the following the system will beassumed isotropic in order to identify through a single valueany of its properties independently of the considereddirection. Obviously it is easy to remove this hypothesis whennon-isotropic systems are examined.

Mass diffusive fluxWhen only molecular diffusion is present, the mass

transport is induced by the contributions of the ordinary

diffusion Ni(ord) (due to the composition gradients), of the

thermal diffusion Ni(T ) (due to the temperature gradients), of

the diffusion by pressure Ni(P) (due to the pressure gradients)

and of the diffusion by the effect of external force fields Ni(F),

electromagnetic, for example, that act selectively on somespecies. As a first approximation, each of these isproportional to the force inducing such a flux, which can beidentified through the opposite of the gradient of anexamined variable by means of a coefficient that is called thediffusion coefficient. In conclusion, the mass diffusive flux isexpressed by the sum of all the above contributions:Ni

(tot)Ni(ord)Ni

(T)Ni(P)Ni

(F).For the ordinary diffusion contribution, by far the most

important, the Fick law is valid:

[4]

where Di,m is the ordinary diffusion coefficient or diffusivityof the ith species in the mixture, expressed in m2/s. Values forthe diffusion coefficients are in the range of 0.5-2105 m2/sfor gases at atmospheric pressure and ordinary temperatures,to about 108-109 m2/s for liquids at ambient temperatureand to 1011-1013 m2/s for the diffusion through solids.Obviously, for a concentration gradient to exist, the systemmust contains at least two chemical species and thus morecorrectly, it should be considered a binary diffusioncoefficient. Alternatively, it is possible to refer to systemswhere two isotopes of the same specie are present. In thatcase, it is correct to refer to a self-diffusion coefficient. It isimportant to note that in the case of two chemical species Aand B, seeing that diffusion should not give origin to anoverall fluid motion, the equimolar counterdiffusionphenomena takes place. In other words, the flux of the firstspecies through the second one is equal and opposite to thatof the second species through the first one:

[5]

It is easy to verify that, since wB1wA, this leads to theimportant consequence of the equality between the twobinary diffusion coefficients (DABDBA). Considering thedependence of diffusivity on composition in multicomponentsystems complicates matters further, given that the massfluxes of the different species are all mutually interconnected(see Section 4.2.5).

The thermal diffusion contribution, known as the Soreteffect, is related to the gradient of the temperature logarithm.The temperature difference induces significant mass transportonly when there are large gradients and significant molecularweight differences between the species present in the system.Its effect is to transfer the ‘heavy’ species to the colderregions and, instead, the ‘light’ ones to the hotter regions. Athermal diffusion coefficient Di

(T ) is also defined in this case:

[6]

The thermal diffusion contribution is generally negligiblein common chemical systems, where the temperaturegradients are modest. However, in reactors used inmicroelectronics for thin-film deposition technologies, it isnot rare to find temperature gradients of about 30,000 K/mand thus this contribution becomes numerically important.

The pressure diffusion contribution is due to the fact thatit is possible to induce the ith species displacement if apressure gradient is present inside the system; it can beconcisely expressed as follows:

N iT

iTD T( ) ( ) ln= − ∇r

N NAord

Bord( ) ( )+ = 0

N iord

i m iD( )

,= − ∇r ω

J uE U u= + +

⋅Φ2

2r

J up = ⋅ruJ ui i= rω

FLUID MOTION

226 ENCYCLOPAEDIA OF HYDROCARBONS

[7]

where Di(P) indicates the diffusion coefficient by pressure and

R is the universal constant of gases. However, the tendency ofa mixture to separate under these conditions is indeed verysmall and usually this contribution is completely negligible,with the exception of the case of centrifuging where it ispossible to obtain very high pressure gradients.

The diffusion contribution induced by external forcesdepends on the properties of the examined forces. Inchemical systems, the most important contribution is thatinduced by the action of an electric field on the ions presentin a solution:

[8]

where zi, zi, NA and E are the electric charge of the ion,expressed in units of the electron electric charge, e, its ionicmobility, Avogadro’s number and the electric potentialgradient, respectively. The product eNA corresponds toFaraday’s constant, ℑ. Ionic mobility, zi, is related to the iondiffusivity through the Nernst-Einstein equation:

[9]

Energy diffusive fluxAs in the case of mass transport, all the previously

examined contributions for the energy flux should also betaken into consideration, but actually, when, in a single-component system, selective external field forces are notpresent, the relevant contribution is only the one due to thetemperature differences existing within the system. In amulticomponent system, the contribution induced by thepresence of mass diffusion must also be considered, becauseevery molecule is indissolubly linked to its energetic content.In the event that the mass transport is induced by atemperature gradient (thermal diffusion), the consequentenergy transport is known as the Dufour effect. The energytransport by a diffusive mechanism is called conduction.

The expression for the conductive flux is thusproportional to the temperature gradient through a coefficientkT called thermal conductivity, expressed in W/mK; thecorresponding constitutive law is known as Fourier’s law:

[10]

Typical values for the thermal conductivity are in theorder of 10-300 W/mK for metals, 0.1-0.5 W/mK for liquids,0.05-2 W/mK for solids and about 102 W/mK for gases. Theorigins of such diversities will be understood when thethermal conductivity is correlated to the molecular properties.It is useful to define a new quantity, related to the previousone, that assumes the same dimension as the mass diffusivity,and is analogously called thermal diffusivity:

[11]

where CP is the specific heat at constant pressure.In multicomponent systems, the contribution of the

energy transport induced by the presence of diffusive massfluxes is given by:

[12]

where Hi° indicates the massive enthalpy of formation of theith species.

Radiative energy fluxA material object, as a consequence of its temperature,

emits electromagnetic radiation. This transmission mode doesnot need any medium to take place, and thus its propagationcan even occur in a vacuum. By its own nature, it is acontribution relevant only at elevated temperatures. Forengineering purposes, to know the flux it is sufficient toknow the temperature difference existing between the twosurfaces involved in the energy exchange and then to applythe Stefan-Boltzmann law. With reference to two grey bodies,the radiant flux leaving a grey body is equal to the sum of theradiant flux emitted by the body and of the reflected radiantflux. Considering that the emissivity of a body, ei, is equal toits absorbance, ai, it is possible to demonstrate the followingexpression for the radiative flux between two surfaces of areaAi and temperature Ti, respectively:

[13]

were sSB and F12 indicate the Stefan-Boltzmann constant andthe view factor between the two surfaces, respectively. Thelatter is a geometric factor that, as illustrated in Fig. 3,expresses the projection of the first surface on the second.

It is worth noticing that equation [13] does not containany matter properties, except surface emissivity. Therefore,this contribution does not participate directly in the writing ofthe energy balance equation, where only volumecontributions are present. However, always in principle, it ispresent in the boundary conditions, even if in practice itbecome relevant only at the higher temperatures.

Momentum diffusive fluxWhile for the definition of diffusive transports for mass

and energy the systems can be considered indifferently inmotion or at rest; for the definition of molecular momentumtransport it is obviously necessary to examine a system inmotion. It is to be considered, therefore, a fluid in laminarregime, where its motion in a channel develops in parallelstreamlines. The momentum flux of diffusive nature isidentified with the tangential stress component (shear stress).

q( )

( )

rad SB T Te

e FA ee A

=−( )

−+ +

−σ

1

4

2

4

1

1 12

1 2

2 2

1 1 1

q Nxii iH( ) = ∑ °

α =kCT

Pr

q = − ∇k TT

ζiiD

RT=

N iF

i i i Az e N E( ) = − ⋅( ) ∇ζ ωr

N iP i

PDRT

P( )

( )

= − ∇

TRANSPORT PHENOMENA

227VOLUME V / INSTRUMENTS

A1

A1A2A1 A2 pr2

T1

θ1

A2

T2

θ2

F12F21 dA1dA21

cos1 cos2

Fig. 3. View factor betweentwo surfaces (q anglebetween the normal directionof the surface and the joiningdirection with the othersurface).

The tangential stress component is related to the velocitygradient (shear rate) by the fundamental law of rheologywhich, in the simplest case where only the velocitycomponent along the x direction exists and the momentumdiffusive flux is directed along the y direction, becomes:

[14]

where the proportionality coefficient is a property of the fluidthat assumes the name of dynamic viscosity (or simplyviscosity) and it is expressed in Pa⋅s. This equation is calledNewton’s law of viscosity. Although originally introduced asthe simplest connection between stress and the velocitygradient, it was found to be valid for a large class of fluids, inparticular for all gases and liquids with molecular weightlower than about 5,000. Accordingly, this class of fluids iscalled Newtonian fluids. Conversely, the fluids not obeyingthis simple law are called non-Newtonian fluids. Examples ofnon-Newtonian fluids are liquid polymers, solid suspensions,pastes, muds and other complex fluids. Typical values forviscosity are in the order of 105 Pas for gases and 103-10Pas for liquids. In this case as well it is convenient tointroduce a quantity homogeneous with the mass diffusivity,called kinematic viscosity (or diffusivity for the molecularmomentum), defined as follows:

[15]

The expression of the momentum flux derived in this wayis evidently too simple to be adopted for any flowconfiguration, even laminar. In a generic fluid system inmotion all three of the velocity components are present, eachof which is a function of the three spatial coordinates. Thisgeneralization is not direct and about 150 years were neededto pass from the simple formulation suggested by Newton tothe more general equation. The details of this demonstrationwill not be developed here, instead only the main hypothesiswill be mentioned. Because each of the three velocitycomponents depends on the three coordinates, it is evident thatin total nine stress components, tij, will be present. Moreover,in addition to the tangential stress components induced by theviscous forces, the perpendicular components, associated withpressure, P, will also be present. In general terms, a molecularstress component that includes both the above mentionedcontributions can be introduced, whose definition is:

[16]

where dij is Kronecker’s symbol, which assumes a nil value ifij and unitary value if ij. The stresses with identicalindexes are indicated as normal stresses, while the others arecalled tangential or shear stresses. Physically, pij can beassociated with two different meanings, fully equivalent toeach other. In the first case, it represents the force in directionj acting on an area at right angles to its direction. In thesecond, it represents the flux of the jth momentum componentin the ith direction. The first meaning is usually adopted whenthe forces applied to a surface by a fluid are being analyzed,while the second is more suitable when the attention isfocused on aspects of fluid motion. Mathematically, pp and ttare second order tensors, called tensor of molecular stressesand tensor of viscous stresses, respectively. In general, byassuming that any viscous stress component will be a linearfunction of all the velocity gradients, the resulting tensor has

81 components, which in principle give rise to 81 differentviscosity coefficients. If, however, the symmetry propertiesare assumed valid and the fluid is considered isotropic, theexpression of the tensor of viscous stresses in compacttensorial notation reduces to:

[17]

where dd is the unitary tensor, ∇u is the tensor of the velocitygradient, (∇u)t is its transposed counterpart and ∇u is thedivergence of the velocity vector. In detail, the individualcomponents of the tensor assume the following structure:

[18]

The generalization reported here involves two differentcoefficients to characterize the fluid properties. The former, m,is the viscosity, while the latter, κ, is the second viscosity(dilatational viscosity). Commonly, it is not strictly necessary toknow this second coefficient. For ideal gases κ is zero, while forincompressible fluids ∇u0, and thus the entire second termvanishes. This coefficient is important when the transmission ofsound in polyatomic gases is to be described or when the fluiddynamics of liquids containing gases is to be analyzed.

Non-Newtonian fluids For non-Newtonian fluids the viscosity concept, as a

chemico-physical fluid property, loses its meaning because itsvalue is not dependent only on the considered fluid and onthe external conditions such as temperature and pressure, butalso on fluid motion. To maintain the formalism adopted sofar, an apparent viscosity h is then introduced which is afunction also of the local stress status:

[19].gg being the so-called rate of the strain tensor (or rate of thedeformation tensor) that identifies the dissipation. Asillustrated in Fig. 4, the dependence of the apparent viscosityon

.gg identifies the non-Newtonian fluids. If the apparentviscosity decreases when the viscous dissipation increases thefluid is called a pseudo-plastic fluid, while in the oppositecase it is called a dilatant fluid. For example, liquid polymerstend to behave prevalently as pseudo-plastic fluids becausewhen the stress status (or equivalently the velocity gradient)increases the macromolecules tend to extend and to place

tt tt tt= − ∇ + ∇ = −h h( ) ( ) ( )u u t γγ

τ µ µ κijj

i

i

j

xux

ux

ux

=− +

+ −

+

23

uy

uz

y zij+

d

tt dd= − ∇ + ∇ + −

∇ ⋅( )µ µ κu u u( )t

2

3

π τij ij ijP= +d

υ µ=r

Np z yxxdu

dy,= = −tt µ

FLUID MOTION

228 ENCYCLOPAEDIA OF HYDROCARBONS

t

g.

t

g.

t

g.

h

g.

h

g.

h

g.

Newtonian Bingham

pseudo-plastic

pseudo-plastic

Newtonian Bingham dilatant

dilatant

Fig. 4. Diagrams of the rheologic behaviour of Newtonian and non-Newtonian fluids.

themselves along the flow direction to offer as littleresistance as possible to the motion. An analogous behaviouris shown by the colloidal solutions, where micelles tend toorient themselves to favour the motion. Instead, fats andstarches behave like dilatant fluids because the stress to beapplied to keep them in motion increases together with theirvelocity. For completeness, it is important to mentionanother class of non-Newtonian fluids, Bingham’s fluids. Inthese systems, to induce motion it is necessary that theapplied stress exceed a critical threshold value, below whichthe fluid behaves as a rigid body.

To describe many of the complex behaviour patterns ofnon-Newtonian fluids, different models have been proposed,for example, the Bingham, the Ostwald-De Waële, theEyring and the Reiner-Philippof models. For the sake ofsimplicity, following the trend in engineering, the simpleOstwald-De Waële model, better known as the power law, iscommonly used to describe with satisfactory approximationtheir rheologic behaviour:

[20]

where m and n are the parameters of the fluid. It is easy toverify that if n1 the fluid is Newtonian and the mcoefficient is identified with the viscosity m, if n1 it is adilatant fluid, and finally if n1 it is a pseudo-plastic fluid.

Analogy between the diffusive transportsObserving the three constitutive laws described so far, it

can be noted that they show the same mathematicalstructure. The analogy among the diffusive moleculartransports of mass, energy and momentum should not besurprising because these relationships find their origin in thesame physical principles. To highlight this, it is useful togroup the three homogeneous coefficients, diffusivity,kinematic viscosity and thermal diffusivity, into twodimensionless groups called the Prandtl and the Schmidtnumbers, respectively:

[21]

[22]

It is easy to verify that these equations represent the ratiobetween two characteristic times and thus identify therelative importance of the different transport mechanisms.The Schmidt number defines the relative importance of massdiffusion with respect to the momentum diffusion. ThePrandtl number represents instead the relative importance ofheat conduction with respect to the momentum diffusion. Forgases, the long distance transport of energy, matter andmomentum occurs by the same mechanism. The moleculemoving within space, between one collision and the next,carries its momentum, its energy and its mass. Thus, it isreasonable to suppose that the three diffusivities assumemore or less the same value (uDa), and consequentlyScPr1. For liquids, it is instead important to distinguishbetween the ordinary liquids and the liquid metals. In fact,for the latter, the transport by conduction is particularlyfavoured because of the activation of the electronicmechanism, much more efficient for energy transport thancollisional and the vibrational mechanisms. And so for thesePr1. For the ordinary liquids, instead, the most effective

transport is momentum, while mass transport is the mostinhibited. Therefore, since u a D, this means that Pr 1(with values ranging from a few dozen to 106 for liquidpolymers) and Sc 1. Evidently, for the solid systems, thetwo dimensionless numbers defined above have no physicalmeaning because it does not make sense to refer tokinematic viscosity (which assumes an infinite value whenapproaching the limit). It is, however, important to noticethat for the solid systems a D, since evidently it is easier totransfer heat instead of mass through them.

Transfer coefficientsIn the presence of a fluid in motion, generally speaking,

both transport mechanisms, diffusive and convective, arepresent at the same time. Moreover, the presence of themotion alters the shape of the gradient for the examinedvariable with respect to that of the stagnant system.Therefore, in practice, it is advisable to introducecoefficients that take into account both contributions, byexpressing the driving force in linear terms in the frameworkof the diffusive fluxes expressions. These coefficients, calledtransfer coefficients, are defined as follows:

[23]

[24]

[25]

where kc, h and (fru) are the mass, the heat and themomentum transfer coefficients, respectively. The definitionof the last, compatibly with its historically development, usesthe f coefficient, known as the Fanning friction factor. Thesuperscripts and 0 indicate, respectively, the function valuein an unperturbed region of the fluid and the value incorrespondence with the surface through which the flux hasto be calculated. It is to be noted that, under ordinaryconditions, the fluid velocity in correspondence with a wallis always zero (no-slip conditions), and so the classicexpression tyxf ru2/2 is obtained.

4.2.4 Microscopic conservationequations

All equations of conservation or balance show the samemathematical structure. Once the volume being examined isidentified, which in the considered case has infinitesimaldimensions, the change in the amount contained in such avolume of the considered variable is given by the differencebetween the amounts of the variable entering and leavingthrough the boundary surfaces in the unit time, besides anyamount that could be generated inside the volume:

[26]

where the symbols X, RX, JX and NX are the value referred tothe unit volume of the variable in question (i.e. the‘concentration’ of the variable), the source term specific tothe volume, and the convective flux and the diffusive flux ofthe same variable through the boundary surfaces,

Xt

RX X X= −∇ ⋅ +( ) +J N

ttyxxuy

f u u u= − =

−( )∞ ∞µ

0

0

2r

q = − = −( )∞k Ty

h T TT

0

0

Ni i mi

c i i iDy

k= − = −( )∞, ,r r

ωω ω

0

0

Pr = =υα

µCk

P

T

ScD Di m i m

= =υ µ, ,r

h=−

mn

γ1

2

TRANSPORT PHENOMENA

229VOLUME V / INSTRUMENTS

respectively. The formulation reported here examines avolume in fixed position with respect to an external referencesystem (i.e. the Eulerian reference system).

To write the conservation equation it is useful to identifythe more suitable intensive variables. To identify the amountof matter contained within a volume, the mass concentration(or the product between the mass fraction of the ith chemicalspecies and the density, rwi) is usually adopted. Thecorresponding ‘concentration’ for momentum is identified bythe product between the density and fluid velocity, ru.Finally, for energy, by ignoring the contribution ofmechanical energy, and thus considering only internal energy,this quantity is identified with the product rU, where U is theinternal energy for unit mass. The term of change representsthe variation in time of the amount of the variable containedwithin the volume, which is, obviously, zero in steady-stateconditions. The source term, RX, identifies the amount of thevariable, in algebraic sense, in the unit time and volume andit depends on the nature of the analyzed variable. In the caseof the overall mass contained in the system, obviously thesource term is absent. If a single species is considered, by theconsequence of the chemical reactions such a contributioncan be present if these reactions lead to the production or theconsumption of the examined species. For the energy balancethe source terms can originate from the dissipative effects(like the Joule effect in conductors affected by electriccurrent or to the work performed onto a fluid by themechanical forces acting on it). The mechanical forces actingon the system play a role in the momentum balance.

To obtain compact formulations, in the following thesubstantial derivative operator, defined as the sum of the timederivative and of the scalar product between the fluid velocityand the gradient of the examined variable, will be adopted:

[27]

This derivative includes both the transient effects andthose due to the convective transport. Thus, the termsremaining explicitly in the balance equation are all inherentto the contributions of diffusive nature.

Each of the microscopic balance equations needs to becompleted with initial and boundary conditions, both typicalof the system and of the problem in question. Generally, theinitial condition, necessary in transient problems, coincideswith the initial instant value of the examined variable withinthe whole integration domain. The boundary conditions canbe of two different kinds. The first one assigns the value ofthe function on the system boundary (the Cauchy-Dirichletcondition) while the second assigns the value of the fluxthrough the wall (the von Neumann condition). Usually,boundary conditions of this second kind are closer to physicalreality.

Microscopic balance equation for massThe microscopic equation for mass balance of the

individual chemical species is presented in the followingform:

[28]

where Ri is the production rate of the examined ith species(mol/m3s) due to the chemical reactions occurring in thesystem and where Mi and wi are the molecular weight and the

mass fraction of the same species, respectively. Theproduction rate of the species is linked to the rate of theindividual chemical reactions, through the relationship (seeChapter 5.1):

[29]

where νij and rj are the stoichiometric coefficient of the ith

species in the jth chemical reaction and the rate of the latter,respectively. If, as often occurs, the relevant contribution tothe diffusive flux is the ordinary equation [4], then equation[28] becomes:

[30]

Equation [28] can be written for all the species that arepresent. Instead of taking into account each of the massbalance equations for the individual species, in many cases, itis convenient to use the overall mass balance equation, whichcan be obtained by summing the balance equations for all thespecies that are present in the system. Considering that theensemble of the chemical reactions does not alter the overallmass in the system (iMiRi0), just as the ensemble of thediffusive fluxes does not produce a net mass transport(iNi0), the following equation is then obtained, usuallycalled a continuity equation:

[31]

Microscopic balance equation for momentumThe microscopic balance equation describing the motion

of a fluid is well known in fluid mechanics. Together with thecontinuity equation it provides a system of equations that, inthe case of Newtonian fluids, is commonly indicated asNavier-Stokes equations. By ignoring the stresses producedby the molecular fluxes of diffusive nature, inmulticomponent systems it reduces to:

[32]

where the pressure is indicated by P, while equation [17] isvalid for the stress tensor, tt.

Microscopic balance equation for energyAs already pointed out, the balance equation for energy

will be written here by ignoring the mechanical energycontributions to better highlight the internal energycontribution, which is, numerically, the most important one inchemical systems:

[33]

where the product between the pressure and the divergence ofvelocity represents the energy increase due to the fluidcompression, while the double scalar product (:) between theviscous stress tensor and the velocity gradient indicates theinternal energy generation due to the viscous dissipations(irreversible phenomenon). Finally, the last contribution is ofinterest only when the possible external forces fields actdifferently on the species that are present, as in the case of theelectrochemical systems where the electric field selectivelyinfluences the motion of the present ions. Obviously, if theonly external field present is gravity, this last term vanishes.

rDUDt

Pxii i= −∇ ⋅ +( ) − ∇ ⋅ − ∇ + ⋅∑q q u u N g( ) :tt

r rDDt

Pu g= − ∇ ⋅ − ∇tt

DDtr

r= − ∇ ⋅u

r rDDt

D M Rii m i i i

ω ω=∇⋅ ∇( )+,

R ri ij jj=∑ ν

rDDt

M Rii i i

ω= −∇ ⋅ +N

DDt t

= + ⋅∇

u

FLUID MOTION

230 ENCYCLOPAEDIA OF HYDROCARBONS

In practice, it is more convenient to consider the statefunction enthalpy (HUP/r) so that, when no otherexternal field forces besides gravity are present, equation[33] becomes:

[34]

If the thermal dilatations associated with the pressurechanges and the heat dissipations due to viscous flow areignored as well, it is possible to obtain an even moresimplified expression, which is in any case valid for a widerange of common chemical systems. Usually, this equation isexpressed directly in term of temperature and is expressed as:

[35]

and contains the contributions due to the chemical reactionsand those due to the conductive energy transport caused bythe temperature gradients.

Dimensionless numbers deductible from microscopicbalance equations

Microscopic balance equations contain terms whoserelative importance vary depending on the examinedconditions. To quantify this point, it is useful to consider asuitable group of variables, known as dimensionless numbers.

By examining the momentum balance equation, it ispossible to define the following dimensionless numbers:

[36]

[37]

[38]

[39]

The Reynolds number, Re, expresses the relativeimportance between the inertial forces and the viscous forces,while the ratio between the pressure originated forces and theinertial forces is expressed by the resistance number, NF. theFroude number, Fr, and the Grashof number, Gr, indicate therelative importance of the inertial forces with respect to thegravitational forces and that of Archimede’s forces(buoyancy) with respect to the viscous forces, respectively.

Conversely, by analyzing the microscopic balance equationsfor mass and energy, it is possible to single out two additionaldimensionless numbers, both defined as Péclet numbers (massand thermal, respectively), that express the ratio between theconvective and the diffusive transport mechanisms:

[40]

[41]

Finally, the contribution of the chemical reactions relativeto the microscopic mass balance equation is quantified usingthe Damkhöler number, Da, which expresses the relativeimportance between these reactions and diffusive masstransport:

[42]

In the event that the chemical reaction proceeds through asimple kinetics of the first order, with rate constant, k, theprevious equation becomes DakL2/Di,m.

Dimensionless numbers linked to transfer coefficientsThe functional form of dimensionless numbers

containing the various transfer coefficients, in other wordsthe contributions originated by the linearization of thediffusive transport laws, always identifies the ratio betweenthe transfer coefficient and the transport variable of interest.Because the exchange coefficient expresses a suitable meanvalue over the characteristic length where the phenomenon isexamined, this last contribution also appears in the expressionof the dimensionless number. Therefore, the Sherwoodnumber, Sh, and the Nusselt number, Nu, are defined, whichare applied to the study of the mass and the heat transport,respectively:

[43]

[44]

Functional links between dimensionless numbersObviously, dimensionless numbers are not all

independent from each other given that they simply expressratios between variables contained in the different terms ofthe same balance equation. Usually, some dimensionlessnumbers are considered dependent on other umbers, whichare independent. Among these, Re, Sc, Pr, Gr, Fr and Da canbe included. Substantially, they are those numbers identifyingthe fluid (by means of its physical properties), the kind ofmotion present in the system (viscous laminar, inertiallaminar, turbulent, and so on) and they are those directlydefinable from the terms present in the microscopic balanceequations. Consequently, typical dependent numbers are Shand Nu which express the value of the transfer coefficients.

The links between the dimensionless numbers are usuallyexpressed with monomial forms like the following:

[45]

[46]

which are strictly deductible only within a well-definedinterval the parameters. Usually, these expressions, or a linearcombination, are used as empirical expressions to state thefunctional links among the dimensionless numbers and theyare then applied in practical engineering to estimate the massand heat transfer coefficient values, which are necessarywhen the transport phenomena are analyzed at themacroscopic level.

Some relationships of this kind will be examined,together with some important typical cases, in Section 4.2.7.The use of these functional relationships to predict thetransfer coefficient values, and thus of the inherent fluxes,represents the core of the practical application of the resultsof transport phenomena studies in engineering. Substantially,their availability makes the study of even complex systemspossible at a greater scale, i.e. at the macroscopic level, thus

Sh a Scb c= Re

Nu a b c= Re Pr

Nu hLkT

=

Sh k LD

c

i m

=,

DaM R LDi i

i m=

2

r,

Pe uLT = = ⋅

αRe Pr

Pe uLD

Scmi m

= = ⋅,

Re

Gr g TLT= r2 3

2

βµ∆

Fr ugL

=2

N PuF =∆r 2

Re = ruLµ

rC DTDt

H M R k TP ii i i T= − + ∇ ⋅ ∇( )∑ ( )°

rDHDt

DPDt

x= −∇ ⋅ +( ) − ∇ +q q u( ) :ττ

TRANSPORT PHENOMENA

231VOLUME V / INSTRUMENTS

making all the design aspects easier. It is important to notethat the exponent values do not assume a generic free value.Characteristic values typical of the existing flow regime areidentified. Evidently, this is due to the existence of a well-defined chemical-physical connection between the variablescontained in the dimensionless numbers. This connection isexpressed by the microscopic balance equations.

4.2.5 Molecular aspects

In the following, the most reliable theories for the estimationof the phenomenological coefficients from molecularproperties will be examined. The examination of transportphenomena at this level allows their most fundamentalaspects to be understood. Molecular transport propertiesdepend on the local state of the materials, liquids or solids,and thus on temperature, pressure and composition, as well ason the molecular properties like mass, molecular dimensionsand their interactions (see Chapter 2.3).

As illustrated in Fig. 2, by observing systems at themolecular level, it is necessary to highlight the molecularmotions of chaotic nature that superimpose themselves ontothe convective ones. In these terms, the fluid velocity is givenby the sum of these two contributions. Although the chaoticvelocity component is not important to the convectivetransport, it is the one on which attention must be focused todetermine the transport coefficients at the molecular level.

For gaseous systems, as illustrated in Table 1, the basictool for approaching these problems is the kinetic theory ofgases. Being, however, in its first formulation, based on theconcept of mean free path and on the use of rigid elasticspheres, leads to correlations that are not fully accurate. Inthis framework, the mean free path, l, identifies the spacecovered by a molecule between two successive collisions,whose value can be estimated by the product of the mean

velocity of the molecular motion, c, with the relaxation time,t (ltc), while the flux of the generic function, Y, is simplyexpressed by NYnclY/z (n is the molecular density).To improve the model, it was then necessary to introduce thepotentials of interaction between molecules, like the Lennard-Jones potential (see Chapter 2.3). In this approach, the pathof a molecule is no longer described through a succession oflinear segments because of the presence of repulsiveinteractions. Consequently, the concept of mean free path alsobecome less clear. Thus, the description of the system mustbe conducted using a distribution function f(u,x,t) whichgives the fraction of molecules that hold a defined energyvalue and whose shape can be estimated by integrating theBoltzmann equation. Using this approach, the estimation ofviscosity, thermal conductivity and of binary diffusivity forthe gases is largely improved. Accordingly, the availability ofa unifying theory (the kinetic theory of gases) demonstratesthe interconnection existing between the transport of differentproperties in the framework of the same mechanism.

The theory of liquids, based on the vacancy modelallowed similar developments even if less accurate than thoseobtained for gases. The nature of liquids is intrinsically morecomplex than the nature of gases and, therefore, theirtheorical description is indeed less precise. Moreover, there isan additional complication brought about by the kind ofliquid under examination (ordinary liquid, liquid metal, liquidpolymer) so that, in practice, different theories are formulateddepending on the kind of liquid handled.

ViscosityThe simplest system to be considered is the monoatomic

ideal gas, for which it is possible to adopt the hard spheresmodel for a rough calculation. In this context, it is possible todemonstrate that the viscosity, m, depends on the density, r,the mean free path, l, and the mean kinetic velocity ofmolecules, 33c:

FLUID MOTION

232 ENCYCLOPAEDIA OF HYDROCARBONS

Table 1. Examples of transport molecular models for diluted gases( f ° is the distribution function of the system in equilibrium conditions)

Non-interactingmassive points

Non-interactingrigid spheres

Interactingobjects

Molecules do not have volumeand do not interact

with each other

Molecules occupy a volumeand do not interact

with each other

Molecules occupy a volume,have a generic shape, andinteract with each other

Representation of motion

Equation of state(bcovolume, ainteraction parameter)

Transport model –

Transport equation(of the variable Yin direction z)

–

PV

RT

= 1

PV

RT

V

V b

=

−

λ τ= c

N ncz

zΨΨ

, = − λ

N f t u duz z zΨ Ψ

,( , , )= ∫ u x

f

tf

f fcoll+ ⋅∇ = ≈

− °u Γ

τ

PV

RT

V

V b T

a T

RTV

=

−−

( )

( )

[47]

where 33c8RT/pM and lM/rNA2pd2, and R, NA, M and dare the universal constant of gases, Avogadro’s number, themolecular weight and the molecular diameter of theexamined species, respectively. In conclusion, the followingequation is obtained:

[48]

from which it is possible to observe that the viscosity of amonoatomic ideal gas is independent from pressure, while itdepends on the square root of temperature. The group pd2 iscalled the collisional cross section of the molecule. Thisresult, found by James Clerk Maxwell in 1860, is stillessentially valid also for polyatomic gases in supercriticalconditions up to a pressure of about 10 bar. Actually, thepresence of intermolecular forces makes the collisionsbetween the molecules inelastic, with the consequence thatthe exponent of temperature approaches a value of about 0.7.

A more rigorous kinetic theory, based on the Boltzmannequation for low density monoatomic gases, was developedby S. Chapman and D. Enskog, introducing anintermolecular potential. Today, for convenience, theLennard-Jones expression is adopted, which assumes thefollowing form as a function of the intermolecular distance r:

[49]

where s and e are the collisional diameter of the moleculeand its characteristic energy, respectively. They can beestimated in a semi-empirical way from the knowledge ofthe fluid properties (temperature and molar volume) incritical conditions, or at the normal boiling point or atmelting point:

[50a] critical conditions

[50b] boiling point

[50c] melting point

where the subscripts c, b, and m indicate the criticalconditions, the normal boiling point and the normal meltingpoint, respectively. In these terms, the expression forviscosity becomes:

[51]

where Wm indicates a dimensionless function, known ascollisional integral for viscosity, which expresses thedeviation of behaviour from that of rigid spheres:

[52]

given that T*kBT/e. The calculation of the viscosity for gaseous mixtures, in

a rigorous interpretation, is based on the extension of theChapman-Enskog theory; for convenience a semi-empiriccorrelation giving a so-called mixing rule can still be used:

[53]

where

[54]

The performed analysis is valid for low-density fluidswhere dl. When the mean free path decreases, it loses itsvalidity and thus it cannot be extended to liquids, because inthat case dl. In liquids, the viscosity decreases, notincreases, with a temperature increase.

To calculate liquid viscosity a simple theory wasdeveloped by Henry Eyring. The key hypothesis of thetheory is to assume that the motion of the molecules of theliquid, because the value of the mean free path is small, issubstantially limited to vibration within a volume confinedby the vicinal molecules that thus represent a sort of ‘cage’.In parallel, the liquid structure presents a series of ‘reticularholes’ that are continuously in motion, which can host amolecule. The transfer of one molecule from its cage to thenearest hole means that an activation barrier must becrossed. The frequency, n, of these ‘cage-hole’ transfers canbe estimated through the following equation:

[55]

where h and DG0 are Planck’s constant and the activation

free energy of the process, respectively. This last value islinked to the internal energy of vaporization incorrespondence with the normal boiling point (DG0

0.408DUvap), whose value can be estimated through the Troutonrule (DUvapDHvapRTb9.4 RTb). In a fluid in motion,and thus in a state of stress, this free energy value increaseswith respect to the value of a fluid at rest so that:

[56]

where V, a and d are the molar volume, the distance to be

traveled during the jump toward a hole, and the distancebetween two molecular planes, respectively. Usually, theapproximation da is adopted. A positive sign indicates thatthe molecular jump is coherent with the stress direction andvice versa for a negative sign value. The viscosity value canbe estimated by supposing that a linear variation of the fluidvelocity exists between two adjacent molecular layers; byapproximating the second exponential with a Taylor seriestruncated at the first term, the following expression isobtained because tyx

V/2RT1:

[57]

This relation agrees closely with the empirical relationusually adopted to define the dependence of the liquidviscosity on temperature (mAeB/T). Obviously, over theyears many other empirical relationships that introducecorrective parameters to better fit the experimental data havebeen developed.

Unfortunately, to calculate the viscosity for liquidmixtures, the best approach is still that of performing a seriesof experimental measures for the mixture viscosity at differenttemperatures and then describing the results through anequation like [57]. An often-used mixing rule is the following:

[58] ln lnµ µmix i iix=∑

µ =N hVA T Tb

e3 8. /

ν τ±

− ±=±k T

hB G RT a V RTyxe e

∆0

2/ / d

ν = − ±k ThB G RT

e∆

0/

Ψiji

j

i

j

i

j

MM

MM

= +

+

1 8 1

0 252

µµ

.

µ µmix

i i

j ijji

xx

= ∑∑ Ψ

Ωµ = + +1 16145 0 52487 2 161780 77320 2 43

.

T T* . .

. .*

e e 7787T *

µσ µ

= 5

16 2

πMRTNAΩ

ε σ/ . .,

/k T VB m m sol= =1 92 1 222 1 3

ε σ/ . .,

/k T VB b b liq= =1 15 1 166 1 3

ε σ/ . . /k T VB c c= =0 77 0 841 1 3

ϕ ε σ σ( )rr r

=

−

4

12 6

µ = 2

3 1 5 2

MRTN dAπ ,

µ λ= 13

rc

TRANSPORT PHENOMENA

233VOLUME V / INSTRUMENTS

In the framework of liquid systems, polymers and theirmixtures obviously require more complex treatment than thepreceding one because their molecular structure that cannotbe approximated as a sphere and this leads to non-Newtonianrheological behaviour. The final goal is to estimate thedifferent coefficients contained in the expression of the stresstensor. The kinetic theories for polymers are substantiallydivided into two classes, networks and single molecule. Thenetwork theories were originally developed to describe themechanical properties of rubber and were then extended tomelted polymers and to their concentrated solutions. Thesingle-molecule theories were originally developed todescribe diluted polymeric solutions where each molecule isrepresented by a ‘bead-spring’ model where a system ofsprings are connected to some spheres, which are then free tomove within a solution, where friction forces act between thefluid solvent and the spheres. The theory was then extendedto concentrated solutions and to melted polymers, studyingthe single molecule behaviour through a mean forceapproximation able to represent in an effective way itssurroundings. In both cases, 4 to 6 parameters are obtained,whose values must be estimated by matching the rheologicmeasurements.

The last cases to be examined are the heterogeneoussystems, i.e. suspensions and emulsions. The easiestapproach is to approximate the heterogeneous fluid as apseudo-homogeneous system, whose viscosity depends onthat of the continuum media and on the properties and thevolumetric fraction of the dispersed phase. The earliestequation is the one derived in 1906 by Albert Einstein forsuspensions containing rigid spheres of the samediameter:

[59]

where meff, m0 and / are the pseudo-homogeneous fluidviscosity, the solvent viscosity and the volumetric fraction ofthe suspended solid, respectively. This equation was modifiedrepeatedly to be extended to non-spherical particles and toconcentrated suspensions (i.e. with /0.05). In particular,the Mooney equation can be used for concentratedsuspensions of spheres:

[60]

where /0 is a constant whose value is between 0.74 and 0.52depending on how the system of spheres is packed. In allthese relationships, as well as in others not summarized here,the viscosity deviation from the solvent depends on thevolumetric fraction of the solid and not on the diameter of thesolid particles. For diluted emulsions, the Taylor equationimplies a combination of the viscosity values for the twoliquids; the subscript 1 adopted here is used to identify thedispersed phase:

[61]

Thermal conductivityThe kinetic theory for gases supplies a valid tool for the

estimation of the thermal conductivity values for monoatomicideal gases in the same way as viscosity does. In this case it ispossible to demonstrate that:

[62]

CV1.5 R/M being the specific heat. The simple substitutionof the expressions for the mean free path and for themolecular mean velocity leads to the equation:

[63]

from which it can be seen that the thermal conductivity ofgases is independent from pressure while it depends on thesquare root of temperature. Even in this case the temperaturedependence is underestimated because in a real system thecollisions between molecules are inelastic. TheChapman-Enskog theory takes this last effect into account,leading to:

[64]

where the expression for the collisional integral coincideswith that for viscosity [52]. Equation [64] is very similar toequation [51] for viscosity; comparing them it is possible toobserve that for a monoatomic gas kT2.5 CVm. Forpolyatomic systems, besides the translational component, therotational and vibrational components of the internal energyare also present. The theory, obviously more complex, leadsto an equation that was originally empirically deduced byArnold Thomas Eucken in 1913:

[65]

Finally, the thermal conductivity for a gaseous mixturecan be estimated through the following mixing rule, wherethe y coefficients have the same expression of thoseintroduced for viscosity [54]:

[66]

To describe the mechanism for thermal conductivity inordinary liquids it is useful to refer to Bridgman’s simpletheory, which assumes that the molecules are arranged rigidlyin a cubic lattice characterized by a reticular parameter equalto the cubic root of the molecular volume (

V/NA)1/3 and that

energy is transferred from one reticular plane to the other atthe sonic velocity, us. Starting from the expression of thethermal conductivity of gases derived through the hardspheres model [62], it is possible to obtain:

[67]

The specific heat for monoatomic liquids almostcoincides with that of a solid at high temperature, thus it ispossible to estimate its value through the Dulong and Petitequation, CV3 R/M, obtaining:

[68]

The coefficient 3 was substituted by 2.80 to increase theagreement with the experimental data and the sonic velocitywas calculated as a function of the CP/CV ration and of theisothermal compressibility coefficient (lnr/P)T. Toestimate the thermal conductivity of mixtures manycorrelations, of about the same accuracy, have been proposed.

k RM

CC

P VNT

P

V T A

=

2 80 3. rr

k C c C u VNT V V s

A

= =13

3r rλ

kx k

xT mixi T i

j ijji,

,= ∑∑ Ψ

k C RMT P= +

µ 5

4

k RMTN

CTA k

V= 25

32 2

/ πσ Ω

kMd

R TN

MRTN d

CTA A

V=

=2 2

32

1 5

2 2π π

.

k C cT V= 13

r λ

µµ

µ µµ µ

φeff

0

0 1

0 1

12 5

= +++

,

µµ

φφ φ

eff

0 0

2 5

1=

−

exp

,

/

µµ

φeff

0

1 52

= +

FLUID MOTION

234 ENCYCLOPAEDIA OF HYDROCARBONS

Consequently, it is convenient to refer to the simplest one:

[69]

In liquid metals the energy is essentially transported bythe motion of free electrons. Because the electron motiontransfers both the electric charge and the heat, there is a strictanalogy between electric and thermal conductivities. Inparticular, these two properties are proportional to each otherthrough the Lorentz constant, L, whose value can beassumed, with fair approximation, to be the same for allmetals:

[70]

where se indicates the electric conductivity. Equation [70] isvalid for solid metals as well. It is important to note that themetals with the greatest electric conductivity (Al, Cu, Ag) arealso those that present the greatest thermal conductivity.Instead, metal alloys have an electric conductivity lower thanthat of their constituting elements.

The thermal conductivity of solids is difficult to predictbecause it depends on many factors (dimension of the crystalgrains and their orientation, porosity, volumetric fraction ofamorphous, etc.); it is, therefore, necessary to use experimentalmeasurements. It should be noted that the nature of the solidgreatly influences the conductivity value, which is very low fordry, inorganic, porous solids (which are very good insulatingmaterials) while it is high for metals. In general, the thermalconductivity of amorphous materials is lower than that ofcrystalline ones. Generally, for a rough estimate, it is possible toassume a relation between electric and thermal conductivities,analogously to what was observed for metals (refractorymaterials are poor electric conductors too).

It is important to obtain the relationships that allow theestimation of thermal conductivity for heterogeneous systems,formed by a mixture of two different solids or by a poroussystem. The fundamental equation, thanks to Maxwell, is validfor systems where the inclusion of one phase into the otheroccupies a small volumetric fraction, /, of the solid:

[71]

where the subscripts 0 and 1 indicate the solid constitutingthe matrix and the solid representing the inclusion,respectively. For solids containing gas inclusions (poroussolids) the radiative effects can be important, especially whenthe solid must be used at high temperatures as an insulatingmaterial. In this case, the effective thermal conductivity canbe estimated through the following equation:

[72]

where kT,1, L and sSB are the thermal conductivity of the gas,the thickness of the material in the direction of the heatconduction and the Stefan-Boltzmann constant, respectively.

DiffusivityThe kinetic theory of gases allows estimations of the

binary diffusivity of a gas phase within a 5% approximation

when the more accurate Chapman-Enskog approach isadopted. To illustrate the main results of the theory, it is againconvenient to start from the theory developed for hard-spheresystems, considering only self-diffusion phenomenon, that isthe diffusion of species of the same type, such as isotopes. Inthis framework, the diffusion coefficient is given by:

[73]

The simple substitution of the expressions for the meanfree path and for the molecular mean velocity leads to therelation:

[74]

From this equation it is possible to obtain the formula forthe binary coefficient by replacing the molecular weight ofthe species with the reduced weight 2/(1/MA1/MB) andsubstituting the molecular diameter with the mean arithmeticone 0.5(dAdB):

[75]

From the above equation it is clear that the diffusivitydepends linearly on the inverse of pressure (the densitydependence of an ideal gas, rMmixP/RT) and ontemperature with exponent 1,5. However, while thedependence on pressure is correct, the dependence ontemperature is underestimated because of the inelasticity ofcollisions between real molecules, and in practice the correctexponent is about 1.75. Using the Chapman-Enskogapproach, it is possible to obtain accurate values for DAB byintroducing the collisional integrals WD in the previousequation:

[76]

where T*kBT/e. Averaged values, such as sAB0.5(sAsB)and eAB(eAeB)0.5 must be used in the calculations. Hence,the relationships for the binary diffusivity become:

[77]

Although this equation was derived for monoatomic idealgases it can also be applied to polyatomic ones. Bycomparing the previous equation with the corresponding onefor viscosity [51], it is possible to observe that a link betweenthe kinematic viscosity and the self-diffusion coefficientexists for gaseous systems:

[78]

Because the ratio between the collisional integrals isalmost constant (Wm1.1 WD), it derives that DAA1.32 u,confirming that the Sc number for gases is close to 1.

The theories for diffusion in liquids do not reach aquantitative level analogous to that obtained for gaseoussystems. Two models are indeed available and they can betaken as references to derive semi-empirical relationships

υ

µDAA

D= 5

6

ΩΩ

D RTM M PAB

A B AB D

= +

3

16

2 1 1 13

2

( )

π σ Ω

ΩD TT= + +1 06036 0 19300 1 035

0 15610 0 47635

. . .* . . *

e

887 1 764741 52996 3 89411e e. .* *

.T T+

DM M

RT

d dABA B

A B

=+

+

−

21 1

2

3 0 25

1

1 5 2π . . ( ) rNA

DM RTd NAA

A

A A

=2

3 1 5 2π . r

D cAA =13

λ

kk k

kT L

k

T eff

T T

T

SB

T

,

, ,

, ,

0 1

0

3

0

1

1 4=

− + +

φ

φσ

−−1

kk k k

k k

T eff

T T T

T T

,

, , ,

, ,

0 1 0

1 0

1 32

= ++−

−

φ

φ

k TT e= Λ σ

12 2k kT

i

T ii

=∑ ω,

TRANSPORT PHENOMENA

235VOLUME V / INSTRUMENTS

correlating diffusivity to easily measured properties likeviscosity or molar volumes.

The first available theory to estimate the diffusivity of abinary mixture is the hydrodynamic theory, based on theNernst-Einstein equation [9], originally developed for themotion of particles in a stagnant fluid. This equation relatesthe diffusivity to the mobility, zA, the latter being the particlevelocity, uA, in steady-state regime when it is subject to theaction of a constant force, FA, i.e. zAuA/FA. If the relativemovement between the particle (whose diameter is dA) andthe fluid (whose viscosity is mB) is only creep movement(Re1), it is possible to demonstrate that:

[79]

where bAB is the sliding friction factor between particle andfluid. Two limiting solutions are possible. In the former, thefluid is supposed to adhere perfectly to the particle (non-slipconditions) and consequently bAB. On the contrary, in thelatter, free-slip conditions are adopted and thus bAB0.Inserting equation [79] into equation [9] two limitingexpressions for diffusivity are thus obtained:

[80]

[81]

The first is known as the Stokes-Einstein equation andits use is recommended for the estimation of diffusivityfor molecules of large dimensions in low molecular weightsolvents. If in the second the molecular dimensions areestimated from the molar volume, dA(

VA/NA)1/3, it is

possible to obtain an expression for self-diffusivity thatwas found to be reliable (uncertainty below 12%) both forordinary (polar and non-polar) liquids and for liquidmetals:

[81a]

An analogous expression can be obtained through thesecond model, developed by Eyring which, similarly to theestimation of the viscosity of liquids, assimilates diffusion toa monomolecular activated process:

[82]

where x is a packing parameter that defines the number ofmolecules of the ‘nearest-neighbour’ solvent to the diffusingmolecule. When examining self-diffusion, x2p, then theEyring formula substantially coincides with equation [81],despite the conceptual difference existing between theexamined models used to derive the two equations.

Because of the limits of the two previous approaches, theestimation of diffusivity in liquid phase is usually performedby means of the Wilke and Chang empirical equation whichis able to estimate the diffusion coefficients in dilutedsystems with an uncertainty below 10%:

[83]

where YB is the association parameter for the solvent, whosevalues are 2.6 for water, 1.9 for methanol, 1.5 for ethanol

and 1.0 for benzene, ether, heptane and all the othernon-polar unassociated solvents.

For solutions of electrolytes it is necessary to estimatethe diffusivity of ions in a different way. It is evident that thecharged species experience the effect of the electric field intheir motion and thus their diffusivity is related to theelectric conductivity. Therefore, by replacing the equationlinking the ionic mobility to the equivalent ionicconductance of the ion li (liziℑ2zi) the followingrelationship is obtained:

[84]

Moreover, the electro-neutrality constraint imposes thecoupled migration of anions and cations and thus thediffusivity of interest is that of the ionic couple inside thesolvent. The property directly measurable is the equivalentconductance of the electrolyte Le, which is the sum of theconductance of the constituting ions (Lell), linked inturn to the electric conductivity, se, of the solution throughthe relationship:

[85]

where z, n and C are the charge of the positive ion, itsstoichiometric coefficient in the reaction of electrolyticdissociation, and the molar concentration of the salt in thesolution, respectively.

An even more complex case is represented by the solutionof polymers in low molecular weight solvents. For thesesystems a detailed theory is available which describes thepolymer as an ensemble of N spheres connected through N1elastic springs in order to form a chain. Each sphere interactswith the solvent through a viscous interaction, thus alsoaffecting the solvent in the environment of the nearest spheres.In terms of order of magnitude, this theory predicts that thediffusivity of polymer A in solvent B is proportional to theinverse of the square root of the polymer molecular weight:

[86]

For solutions of melted polymers, the theory is evenmore complex and the results are approximated. In general,for self-diffusion, a link with the molecular weight of thepolymer is present:

[87]

with quadratic dependence (n2) in the theoricaldevelopment, while in practice the exponent can assumeeven a value of 3 for some polymers.

In a liquid mixture, molecules interact in groups, andthus all the expressions reported above, valid for very dilutedconditions, need to be modified to take into account theinteractions present at the higher concentrations. One of themost adopted expressions is the following:

[88]

where gA and DAA are the activity coefficient of the speciesin solution and its self-diffusion coefficient, respectively.

For diffusion through solids, because of the packing ofthe lattice to be crossed, it is normal to find very low

D DAB AAA

A

= +

1

lnγω

DMAA

An≈ 1

DMAB

A

≈ 1

Λee

z C=

+ +

σν

D RTzi

i

i

=ℑλ

2

DM TVAB

B B

B A

= ⋅ −2 95 10 8

0 6.

.

Ψµ

N DRT

NV

A AB B A

A

µξ

=

11 3

/

DRT N

NV

AA A

A

A

A

µ=

1

2

1 3

π

/

DRT N dAB B

A A

µ= 1

2π

DRT N dAB B

A A

µ= 1

3π

uF d

dd

A

AA

B A

B A AB

B A AB

= =++

ζ

µµ βµ β

1

3

6

4π

FLUID MOTION

236 ENCYCLOPAEDIA OF HYDROCARBONS

diffusivity values. The reference model is similar to that ofliquids, where the solute displacement from one position toanother is possible only if a hole is present nearby thatcreates the space available for the transfer. Vacancies arecommon defects of crystalline solids, whose numeral densitydepends on thermodynamic and kinetic factors. Moreover,vacancies are mobile and they can even aggregate and giverise to defects of lager dimensions (microcavities).Analogously to the theories already seen for liquids, theresistance of the solute to moving towards a nearby vacancyis great, and so that this can happen the solute moleculemust overcome a potential energy barrier. In other words, thediffusion into a solid matrix is an activated phenomena andthus the expression to be used is the following:

[89]

where the AD term is connected to the vibration frequency ofthe atoms in the lattice (which identifies the mobility of thevacancies), while the activation energy is related to the bondenergy.

To complete the analysis of molecular aspects it is alsonecessary to derive the equation for the estimation ofdiffusivity for the processes induced by temperature andpressure gradients. Because these phenomena are usuallyrelevant only for gaseous systems, only expressions directlyderived from the Chapman and Enskog theory will beaddressed here. The thermal diffusion coefficient is relatedto the ordinary coefficient through a thermal diffusionfactor, ai

(T ), which takes into account the differences inmolecular weight and in dimension between the species. In abinary system this link assumes the following form:

[90]

where the thermal diffusion factor can be estimated throughthe Liley and Gambill equation:

[91]

even if its rigorous expression is much more complex,because it includes the dependence on the Lennard-Jonesparameters through the collisional integrals (Hirschfelder etal., 1954).

For diffusion induced by pressure in a binary system thediffusivity is correlated to the difference between thevolumetric fraction, /A, and the mass fraction, wA, of thespecies by means of the following equation:

[92]

As highlighted above for diluted systems, the volumetricfraction can be estimated through the product between themass fraction and the solution and solute density ratio,/AwAr/rA.

As a final aspect, it is necessary to consider that, inmicroscopic balance equations, the symbol Di,m was adoptedas the effective diffusion coefficient for the ith species toexpress the ordinary mass flux as a function only of thecomposition gradient for the examined species. It is indeedevident that in a binary system the flux of a species in onedirection corresponds with the flux of the other species inthe opposite direction. To account for this point in generalmathematical terms the Stefan-Maxwell equations can beused for reference, through which the driving force, (wi),

or directly the effective diffusion coefficient, Di,m, dependson the diffusive fluxes of all the species that are present inthe system:

[93a]

[93b]

Equation [93b] can be simplified for some cases ofpractical interest. For example, because one of the diffusioncoefficients can be arbitrarily chosen, by assuming thatDi,mDi,i, the Blank equation can be obtained:

[94]

which is often used to estimate the effective diffusioncoefficient instead of the more precise, but also morecomputationally complex, Stefan-Maxwell equations [93].Another relevant case is the one where a species, typicallythe solvent, is present in large quantities and so can bereasonably considered stagnant. In this case the diffusivityreduces to the binary coefficient:

[95]

4.2.6 Role of turbulence

In turbulent motion, unlike laminar motion, the streamlinesappear irregular and with abrupt direction changes. This facthas a strong influence on transport phenomena. Turbulenceis a phenomenon analogous to phase transitions: in a fluid inlaminar motion, once it has reached a critical velocity, asharp transition towards turbulence occurs, analogously towhat is observed when a liquid reaches the bubble point. Afluid in turbulent motion dissipates more energy than wouldbe dissipated in a hypothetical laminar regime with the sameReynolds’ number.

Although in turbulent motion the local instantaneousvelocity and pressure values are subject to chaotic variations,it is possible to identify some properties of time-averagedvalues that are well defined and significant. In other words,while the reproducibility of instantaneous values is notassured because of the chaotic nature of the motion, a

D Di m i B, ,=

D

D

i mi

j

i jj i

,

,

=−

≠∑

1 ωω

1 1

r

r

D

D

i m

i j i j j ij

i i mm,

,( / )

=−( )

−∑

∑ω ω

ω

N N

N N

∇ = −( )∑ω ω ωii j

j i j j iD1

,

N N

D D DAP

A A ABA

A AB( ) = −( ) ≈ −

φ ω ωr

r1

αAT B A

A B

M MM M

( ) = −+

105118

D DAT

AT

A B AB( ) ( )=α ω ω

D AAB DE RT= − ±e

/

TRANSPORT PHENOMENA

237VOLUME V / INSTRUMENTS

u'uu

u

u

time

velo

city

Fig. 5. Graph of the time fluctuations of velocity of a fluid in turbulent regime.

reproducibility in term of time-averaged values (long timeaverage) is possible instead:

[96]

where A indicates the function of interest (like velocity,concentrations, temperature and pressure) in the generic xposition within the system.

Considering a fluctuating function like the one illustratedin Fig. 5, it can be observed that its instantaneous value can beexpressed by adding a fluctuating contribution to the time-averaged contribution:

[97]

where the averaged value over a sufficiently long time of thefluctuating contribution is zero (

_A0). By applying this

definition to the microscopic balance equations, and by usingthe linearity properties of the mean operator, it is possible todemonstrate that the local transport equations, expressedthrough the averaged terms, keep the same structure of theoriginal equations for laminar motion, provided that theinstantaneous functions (wi, T, P, u) are substituted with thecorresponding averaged functions (

33

wi, 33

T,33

P,33

u) and acontribution related to the turbulent velocity fluctuation isadded, presenting the common structure (

3

u3 3333

3

A33

). This lastcontribution represents the diffusive flux of turbulent origins forthe examined function. To write these terms, it is generallyaccepted to refer to expressions with the same structure as thelaminar ones. However, while in molecular transports ofNewtonian fluids, viscosity, thermal conductivity and diffusivityare properties of the fluid and do not depend on its motion, inturbulent regime the corresponding ‘turbulent’ propertiesdepend substantially on the motion. They cannot, therefore, becompared to scalar properties, but should be strictly consideredthrough all their tensorial components. Actually, considering allthe uncertainties present in the theory of turbulent motion,generalized expressions to deal with this question are notavailable and thus it is opportune to apply Boussinesq’shypothesis, in which scalar properties are introduced (even ifstill dependent on the motion and the kind of turbulencepresent) such as the following:

[98]

[99]

[100]

It is natural, therefore, to refer to overall transportproperties, which are the sum of the laminar and turbulentcontributions: m(tot)mm(t), D(tot)Di,mD(t), kT

(tot)kTkT(t).

In turbulent regime, incidentally, all chemical species havethe same coefficient of turbulent diffusion which,consequently, is indicated omitting the index inherent to theconsidered compound. Even more significant are theproperties homogeneous with turbulent ‘diffusivity’, such asturbulent kinematic viscosity, u(t)m(t)r, and turbulentthermal diffusivity, a(t)kT

(t)rCp. Because any turbulenttransport mechanism is associated to the time fluctuation ofthe variables induced by the fluctuations of fluid velocity, as aclose approximation it is valid to consider that the sameelementary mechanism governs all the transport properties:

[101]

Some additional comments are needed for the functionsof instantaneous generation of mass and heat, which in a firstapproximation can be substituted by functions of theaveraged variables. It is important to understand the limits ofsuch an approximation, which in the case of chemicalreactions can lead to errors reaching up to several order ofmagnitude because of the strongly non-linear nature of therate expression. By applying the mean operator [96] even to asimple first order reaction rate expression, rk0e

E/RTCi, thefollowing is obtained:

[102]

from which it appears that as a consequence of turbulencehomogeneous chemical reactions can be faster than thevalues that can be calculated through the simple introductionof the averaged variables in the rate expression,(RK0e

E/R33

T33

Ci). By managing the derivative it is possible toverify that the correction is important only if E

33

T. A casewhere this is verified is represented by the combustionreactions that occur by means of a free radical mechanism. Inthis case E/R

33

T20-30, and reaction rate enhancements ofabout 400-1.000 times the value estimated through theaveraged values can be observed.

In conclusion, with the introduction of the abovedefinitions, the microscopic balance equations assume ashape analogous to those valid in laminar regime, so oftenthat the superbar identifying the averaged values is notreported. Accordingly, equations [28], [32] and [35] become:

[103]

[104]

[105]

Dimensional analysis and scaling effectsExperimentally, it has been observed that the intensity of

turbulence, expressed as the absolute value of the fluctuatingcomponent of velocity, is proportional to the mean value ofthe fluid velocity itself. In particular, if the motion of a fluidin a pipe is examined, the following relationship is assumed:

[106]

where f is the friction factor. This dependence can beunderstood through the analysis of the characteristic time andlength of the turbulence. As already mentioned, beyond theRe threshold value, fluid motion destabilizes and vortices(eddies) start to form which are responsible for energydissipation. In a vortex, because the Re number is very high(Re105), inertial contributions are prevalent compared toviscous ones and thus the correspondent energy dissipation isnegligible. Accordingly, an energy transfer with a very lowdissipation from the higher dimensions vortices to the smallerones can be hypothesized which, however, increases when thevortex dimension becomes so small that the viscous termsbecome important.

Based on Andrej N. Kolmogorov’s classic theory, it isuseful to identify different turbulence scales, l, eachcorresponding to the average vortex dimension. Each scale is

u f u' ≈

rC DTDt

H M Rii i itot

P q= − − ∇ ⋅∑ ( )°

r rDDt

Ptotu g= − ∇ ⋅ − ∇tt

rDDt

M Rii i

ω= −∇ ⋅ +N i

tot

r C T r C T rT

Ti i( , ) ( , ) '≈ + 1

2

2

2

2

υ α( ) ( ) ( )t t tD≈ ≈

q( ) ( )tTtk T≈ − ∇

N it

it

iD( ) ( )≈ − ∇r ω

tt ijt t

iju( ) ( )≈ − ∇µ

A t A A t( , ) ( ) ( , )x x x= + '

At

A t dtt

( ) ( , )x x= ∫1

0∆

∆

FLUID MOTION

238 ENCYCLOPAEDIA OF HYDROCARBONS

associated with a well-defined turbulence intensitycorresponding to the vortex velocity, ul. On the basis ofthese two variables it is then possible to define a Reynoldsnumber typical of turbulent vortices, Rell ul/u. To each ofthese vortices corresponds a dissipated power per unitvolume, e, which, by means of a dimensional analysis, canbe related to the Rel value:

[107]

where el is the energy transferred per vortex. If only theasymptotic behaviour is examined, two limiting situationscan be seen. In the first, where Re 1, are contained themacrovortices , where inertial aspects are prevalent and thusthe energy is transferred to the lower level vortices withoutdissipation. In the second, where Re1, microvortices arepresent, where viscous aspects are prevalent, and thus wherea dissipative process takes place. Thus, in the former case,the dissipated energy must be level independent, and then:

[108]

where L is the maximum possible dimension for the vorticesthat are present in the system and thus it coincides with thedimension of the apparatus generating the turbulence. It ispossible to estimate the characteristic velocity of vortices atsome scale beginning from the velocity at the maximumscale, uL:

[109]

As already observed, the energy transfer betweenvortices stops when the viscous phenomena becomeprevalent, i.e. when Re assumes values close to 1.Substantially, the energy transfer proceeds unaltered up tothe microvortex level where it sharply decays. Thecharacteristic dimension of microvortices, l0, can beestimated through the following relationship:

[110]

Thus the aggregates whose dimensions are lower than l0are not able to transfer energy but only to dissipate it. Byexamining the above concepts inherent to turbulent motioninside a pipe, it is possible to see that macrovortices have acharacteristic dimension of the order of the pipe diameter(Ld), while the turbulence intensity will correspond toabout 10% of the fluid mean velocity in the channel.Numerically, for Re105, ReL0.1Re, the maximumdimension of microvertexes will be l0103d. If the pipehas a diameter of 10 cm, the microscale for turbulence is onthe order of 100 mm with a characteristic time of about 1 ms.It is to be noted that this value is bigger than the mean freepath of molecules in a gas at ambient temperature (about 0.1mm) and thus, in principle, the simulation of turbulentmotion, DNS (Direct Numerical Simulation), can still beperformed assuming a continuous system and by applyingthe Navier-Stokes equations, being careful to partition theintegration domain at microvortex dimension and to use atime step of about one ms. It is clear that even with today’scomputers this simulation is a difficult task.

The considerations in terms of order of magnitude likethe above can in any case be used to estimate the values ofturbulent diffusivities. Remembering equation [73], and byidentifying the mean free path of the vortices with theirdimensions, the turbulent diffusivity associated with thevortices with dimension l is:

[111]

and so the average value that takes into account thecontributions of all vortices whose dimensions are betweenl0 and L will be:

[112]

The direct simulation of turbulence is a problem whosesolution is still highly difficult; thus alternative approacheshave been developed. One of the most frequently used todayis called K-e, which adds the equations for turbulent kineticenergy, K0.5u2, and turbulent dissipation, e, to theaveraged equations [103-105]:

[113]

[114]

where the three constants assume the values C00.09,C11.44 and C21.92, while the turbulent diffusivity iscalculated directly as a function of the variables reportedabove:

[115]

4.2.7 Some typical problems

The theories that have been developed so far set the stage toapproach in principle any problem dealing with transportphenomena. As already mentioned, it is sufficient to integratethe microscopic balance equations on have been defined, toobtain, in the most general case, the velocity, temperature andcomposition within the examined domain. In the frameworkof the mentioned approximations, by means of the computersavailable today, such a procedure can be adopted, both inlaminar and in turbulent regimes, for any geometrical domainand even for an elevated number of chemical species. Thisstage has been reached gradually by the solution of a highnumber of simple cases for which it was possible to obtainanalytical solutions, even if simplified. Substantially, aconsistent fraction of the typical problems consists in the useof microscopic balance equations to obtain monomialrelationships between the characteristic dimensionlessnumbers for the examined problem. These equations can thenbe applied to the solution of similar problems in physicalterms even if with geometries that are not strictly identical. Itis possible to rely on the extended similitude principle, bywhich the functional dependences between the dimensionlessnumbers are seldom influenced by the system geometry, withthe exception of small adjustments of their proportionalitycoefficient, whose value is indeed close to one. In otherwords, it means that the physical phenomena underlying the

D C Kt t t( ) ( ) ( )= = =υ αε0

2

DDt

C K C C K Cε υε

ε= ∇ +

∇

+ ∇( ) +

0

2

1 0

2

2u εε 2

K

DKDt

C K K C K= ∇ +

∇

+ ∇( ) −υ

ε εε

0

2

0

22

u

D u Lt t tL

( ) ( ) ( ) .= = ≈υ α 0 14

D utλ λλ( ) = 1

3

λ0

3 4

L= ( )−ReL

/

u uLLλλ=

1 3/

λε ε

ε ε

λ

λ

λ

uLu

L

L

L

3 3= =

=

constant

λελλ

λuf

3= ( )Re

TRANSPORT PHENOMENA

239VOLUME V / INSTRUMENTS

examined case are only slightly affected by geometry. Thus,for the estimation of mass and energy transfer coefficients, aswell as for the friction factor for the fluid motion, it is possibleto refer to equations derived for simplified geometricalconfigurations, or to the combination of some asymptoticbehaviour. For that reason, a compilation of the most adoptedrelationships for the estimation of the mass transfercoefficients is summarized in Table 2.

Velocity profile for a fluid in a pipeBy considering an incompressible fluid in steady state

laminar regime flowing through a circular-section pipe, it iseasy to verify that the equations in [32] reduce to the singleequation dealing with the uniquely present velocitycomponent, uz, that is independent from the axial coordinatebecause of the continuity equation [31] and from theazimuthal coordinate because of cylindrical symmetry:

[116]

[117]

By assigning nil velocity values in correspondence withthe walls and by using the symmetry conditions incorrespondence with the pipe axis, the integration of equation[116] provides the velocity profile as a function of the

pressure gradient (P/L), of the fluid viscosity and of thechannel radius R:

[118]

which show the classic parabolic trend. It is just as easy toverify that, if instead of a Newtonian fluid a non-Newtonianfluid is considered and where the Ostwald-De Waëlerheological model [20] is adopted, the integration leads to theexpression (with s1/n):

[119]

whose mean velocity value is given by:

[120]

Obviously, if n1/s1, the previous equation becomesNewtonian. It is just as important to note that the stresswithin the fluid, maximum at the wall and nil in the centre, isindependent from the fluid rheology:

[121]

A comparison between the velocity profiles, at constantmean velocity, as a function of the rheologic behaviour of thefluid is illustrated in Fig. 6, from which it can be seen thatpseudoplastic fluids (n1) tend to a flat front profile, while

τrzPL

r= −

∆2

u Rs

PLz med

s s

, =+

+1

31

2∆

µ

u u ss

rRz z med

s

= ++

+

+

,31

11

u PL

R rRz = −

−

∆ 2 2

41

µ

uzz =0

ruuz

Pz r r

rur

urz

z z z

=− +

+

µ 1 2

2

FLUID MOTION

240 ENCYCLOPAEDIA OF HYDROCARBONS

Table 2. Examples of some common monomial expressions between dimensionless numbersfor the estimation of the mass transfer coefficient

Flow configuration Relationship

Flat horizontal wall,forced convection

Laminar:

Turbulent:

Flat vertical wall,natural convection

Laminar:

Turbulent:

Steady state flat disk

Laminar:

Turbulent:

Flat rotating disk

Laminar:

Turbulent:

Flow around spheres

Laminar:

Turbulent:

L

u

x

R

R

w

d

Sh ScL L= 0 646 1 2 1 3. / /Re

ShL L= 0 365 0 8. .Re

Sh Gr Sc Scx = + −0 508 0 9521 4 1 2 1 4. ( . )/ / /

Sh Gr Sc Scx = + −0 249 1 0 4941 4 7 15 2 3 2 5. ( . )/ / / /

ShR = 8 / π

ShR R= 0 562 0 558. .Re

Sh ScR R= 0 879 0 5 1 3. . /Re

ShR R= 0 365 0 8. .Re

Sh Scd d= +2 0 60 1 3 1 3. / /Re

Sh Scd d= 0 347 0 62 1 3. . /Re

dilatant fluids (n1) tend to a linear profile. By consideringequation [98], it is easy to verify that the velocity profile inturbulent regime approaches that of a pseudoplastic fluid. If auniversal turbulent velocity profile is assigned, an exponent inthe order of 1/7 is observed, which corresponds in the abovedevelopments to exponents n in the order of about 0.1-0.2. Inparticular, the velocity gradient in correspondence with thewall is even steeper, so that in practice the maximum velocityis identified by the averaged velocity (plug-like flow).

Velocity, temperature and mass boundary layersIf a Newtonian fluid approaches a wall whose linear

dimension is L with a velocity, ux, so that Re 1, the

inertial forces are significantly greater than the viscous

ones and the momentum convective flux is larger than thediffusive one. This situation is not found in proximity of thewall. As is shown in Fig. 7 A, it is possible to identify aregion of space, whose thickness is d, called the boundarylayer and introduced for the first time by Prandtl in 1904,where the viscous forces are more prevalent than theinertial forces. Because the viscous dissipation isresponsible for the resistance encountered by the fluidduring its motion, the knowledge of the boundary layerfeatures plays a paramount technological importance.Substantially, the boundary layer thickness identifies thepoint where viscous and inertial forces balance. Within theboundary layer the velocity profile can be assumed linearwith good approximation.

When referring to the case where the fluid approaches aflat wall placed along the direction of its movement (seeagain Fig. 7 A), with unperturbed velocity, ux

, the equationsfor the motion reduce to two equations for the velocitycomponents (ux, uy) and to the continuity equation forpressure:

[122]

[123]

[124]

The boundary conditions for the velocity componentsare: zero velocity in correspondence with the wall (y0),velocity ux

, unaffected and moving parallel to the wall(uy0) at an infinite distance from the wall; analogouslyfor pressure, P. In correspondence with the boundarylayer, by definition, the velocity value along the xdirection coincides with the unaffected value (ux(d)ux

).In terms of order of magnitude, the continuity equation[124] states that uyux

d/x, i.e. the transverse velocity ismuch smaller than the longitudinal velocity, because ofthe small value of d, and that the order of magnitude ofthe longitudinal coordinate approaches the wall length.Comparing equations [122] and [123] it is possible toobserve that:

ux

uy

x y+ =0

r uux

uuy

Py

ux

ux

yy

y y y

+

=− + +µ

2

2

2

yy2

r u ux

u uy

Px

ux

ux

xy

x x x

+

=− + +µ

2

2

2

yy2

TRANSPORT PHENOMENA

241VOLUME V / INSTRUMENTS

norm

aliz

ed v

eloc

ity

0

0.5

0.2

0.5

1.0

1.0

1.5

2.5

2.0

2.0

3.0

5.0

n

r/R0 0.2 0.4 0.6 0.8 1.0

Fig. 6. Velocity profile for a fluid in a circular pipe in steady-stateconditions and as a functionof the rheologic fluidbehaviour.

y

dux(y)

ux

x

y

ddT

ux(y) T(y)

T

ux

x

y

A

B

C

ddC

ux(y) C (y)

C

ux

x

Fig. 7. Graphs of the boundary layer developments during a fluid-wall interaction: A, velocity boundary layer; B, thermal boundary layer; C, composition boundary layer.

[125]

and thus it is possible to ignore the pressure variationalong the transversal coordinate and thus the pressurevalue in any point within the boundary layer is equal tothat in the unperturbed zone outside the layer. Byexamining again equation [122] it is easy to identify thatthe two left hand side convective terms are of the sameorder of magnitude, while the first one of the two viscousterms in the right hand side is much smaller than thesecond one. Thus, by imposing the balance of theconvective term with the viscous one (boundary layerdefinition) it follows that:

[126]

or in other words:

[127]

having established that Rexruxx/m. If the transverse velocity

gradient (ux/d) is estimated, it is possible to obtain the stress

txym(ux/y), and thus to derive the expression for thefriction factor:

[128]

Four years after the publication of Prandtl’s paper, theproblem was reconsidered in a more rigorous way byHeinrich Blasius, who determined more accuratelyf0.64 Rex

0.5.If the system is not isothermal, it is necessary to add the

energy balance to equations [122-124]:

[129]

with boundary values for temperature T° in correspondencewith the wall and T in the zone that is not disturbed. In thiscase it is also possible to identify a region in space whoseextension is small compared to the dimensions of the wallinside which all the temperature variations occur, theso-called thermal boundary layer as illustrated in Fig. 7 B. Tocalculate the order of magnitude of the thermal boundarylayer, dT, the approach developed above can be followed. Twocases can be distinguished as a function of the value of theReynolds number, Re 1 and Re1.

In the first case there is the development of twoindependent boundary layers, the velocity layer and thethermal layer, which can be either inside or outside velocitylayer depending on the value of the Prandtl number. Inparticular, dTd if Pr1 and vice versa if Pr1. For smallPr values, which is the case for liquid metals, it can be foundthat the phenomenon is controlled by the thermal Pécletnumber (PeTRePr):

[130]

[131]

If instead Pr1, which is the case for ordinary liquids,the exponents for Re and Pr are different:

[132]

[133]

In practice, these trends remain valid for gases as well,where Pr1. Thus, the Nu trend (and consequently that of theheat transfer coefficient) follows the combination of profilesshown in Fig. 8 A.

If Re1, the velocity boundary layer does not developand thus the only boundary layer present is the temperaturelayer. If Pe is small as well, which means that convection canbe ignored with respect conduction, Nu is constant and itsvalue is close to one. If instead Pe is great, a dependence withrespect to its value is found:

[134]

[135]

Thus, the Nu trend for Re1 follows the combination ofprofiles illustrated in Fig. 8 B.

Because the mass balance equation [30] assumes thesame shape of the energy balance equation [129], the onlydifference being the substitution of the thermal diffusivitywith mass diffusivity, the previous information can also be

Nu ≈ ⋅( )Re Pr1 3/

dT

L≈ ⋅( )−Re Pr

1 3/

Nu ≈ Re Pr1/2 1 3/

dT

L≈ − −Re Pr1 2 1 3/ /

Nu L

T

= ≈ ⋅( )d

Re Pr0 5.

dT

L≈ ⋅( )−Re Pr

0 5.

u Tx

u Ty

Tx

Tyx y

+ = +

α2

2

2

2

fu

xy

x

= ≈ ( )∞

−2

2

0 5τr( )

.

Rex

d

x≈ ( )−Rex

0 5.

rd

( )ux

ux x∞ ∞

≈2

2µ

Py

u u

xPx

ux

x y x≈ ≈∞ ∞

r r( )2

FLUID MOTION

242 ENCYCLOPAEDIA OF HYDROCARBONS

Re 1

Nu.

Re

1/2

PrA B

Pr1/2

Pr1/3

1

Re1

Nu

PeT

Pe1/3

1

1

Fig. 8. Nusselt number as a function of the Prandtl and Péclet numbers. A, in presence of the velocity boundary layer (Re 1); B, in absence of the velocity boundary layer (Re1).

velocity

temperature

composition

Fig. 9. Direct simulation of the velocity, temperature and composition fields for a fluid approaching a wall parallel to its motion (cold wall, mass transfer towards the wall).

applied to the examination of mass transport. In this case, amass boundary layer is defined, shown in Fig. 7 C, whosetrends are exactly those illustrated in Fig. 8, substituting Shfor Nu and Sc for Pr. Naturally, the same substitution isapplied to equations [130-135].

This shows that if Re 1 there is an analogy between heatand mass transport and transport of momentum. This fact isstated by the Chilton and Colburn analogy, which calculatesthe transfer coefficient using the knowledge of either of theother two:

[136]

[137]

The previous results can be verified through theintegration of the balance equations. The development of theboundary layers is in fact clearly shown in Fig. 9, where thedetails of the velocity, temperature and composition fields are

reported for the case of a fluid approaching a flat plane, inanalogy with the above examined cases.

The analysis above can also be repeated in turbulentregime because the structure of the balance equations, whenthe averaged variables are considered, is identical to that ofthe laminar case. Moreover, it must be remembered that‘diffusive’ transport is entirely due to turbulence and that thevalue of the three turbulent diffusivities is the same.Accordingly, the boundary layer analysis for velocity leads tothe following expression for the friction factor:

[138]

from which, by applying the Chilton and Colburn analogy,the expressions for Nu and Sh can be obtained:

[139]

[140]

From this it follows that in turbulent regime the exponentfor Pr and Sc is one, differently to what happens in laminarregime, where its value is 1/3.

Temperature distribution within solidsIn steady state conditions, as shown in Fig. 10, inside a

flat slab of finite thickness, a linear temperature profile isobserved and the heat flux through the slab by conductivemechanism is qkTT/s. In this case the divergence of theflux is nil in the energy balance equation [35], in other wordskT2T0. In the case of composite walls in contact with twofluids, in practice, it is common to define an overall heattransfer coefficient that accounts for all the ‘resistances’present in the heat transport from one fluid to the other:

[141]

[142]

where h1 and h2 are the heat transport coefficients of the twofluids, while (kT)j are the thermal conductivities of the solidmaterials making up the wall, each one of thickness sj. Tojustify the two previous equations it is sufficient to equate theexpressions for the heat flux through any phase and to imposethe temperature continuity for the walls in contact.

In a wall formed by a single material it is important toverify the relative importance between the external heattransfer by convection and the internal one by conduction.This ratio is given by the Biot number:

[143]

Another interesting case, illustrated in Fig. 11, deals withheat conduction in transient conditions. For a semi-infiniteslab, the analytical integration of the energy balance equationgives the following expression:

[144]

where TW and T0 indicate the ‘skin’ temperature of the slaband the initial temperature (constant at every point of theslab) and y is the internal coordinate along the slab. It is ofinterest to estimate the penetration thickness for heat, dT, thatidentifies the point where the temperature assumes 99% of itsasymptotic value. By inverting equation [144] dT4at is

T TT T

yt

W

W

−−

=

0 4

erfα

Bi hskT

=

1 1 1

1 2U h

s

k hj

T jj

= + ( ) +∑

q U T Thot cold= −( )

Sh f Sc= ⋅0 5. Re

Nu f= ⋅0 5. Re Pr

f = −0 791 0 25. .Re

Sh f Sc= ⋅0 5 1 3. /Re

Nu f R= ⋅0 5 1 3. /e Pr

TRANSPORT PHENOMENA

243VOLUME V / INSTRUMENTS

hotfluid

T1Thot

T2

T3

T4T5

T6Tcold

s1 s2 s3

coldfluid

Fig. 10. Temperature profile within a composite solid wall in contact with two fluids at different temperature.

TW

T0

t

dT

Fig. 11. Transient temperature evolution in a semi-infinite wall.

obtained, from which can be derived, for example, thethickness of a solid that can be assumed to be semi-infinite ifthe phenomenon is observed for times lower than t.

Mass transport through interphaseBecause the mass balance equation in absence of

chemical reaction is identical to that of energy balance,equation [144] also describes the penetration of a chemicalspecies through a solid or through a stagnant fluid, whentemperatures are substituted by concentrations (or by massfractions) and the thermal diffusivity is substituted by themass diffusivity:

[145]

In steady state conditions, as shown in Fig. 12, when themass transfer occurs through an interphase or through aninterface between two different phases, its presence induces adiscontinuity in the composition profile. It is usually assumedthat at the interface thermodynamic equilibrium conditionsare established. A typical example is the gas-liquid masstransfer.In steady-state conditions, the mass flux from the‘gas side’ must be equal to the corresponding flux on the‘liquid side’:

[146]

where the subscripts L and G indicate the liquid and gaseousphases, while the asterisk identifies the interfacial conditions.Moreover, by considering the equilibrium condition at theinterface, synthetically expressed by means of the partitionconstant Ki

[147]

it is possible to obtain the following expression for the massflux:

[148]

where, analogously to the heat transfer, an overall masstransport coefficient is defined:

[149]

Because the mass transport coefficient in liquid phase issignificantly smaller that that in gaseous phase (see the ratioof the two diffusivities Di,L/Di,G0.001), often kL,i

effkL,i isassumed.

Natural convectionIn natural convection, fluid motion is induced by the

presence of density gradients that can originate both fromtemperature and concentration gradients. In principle andwith the computational facilities available today, to solvethese problems only an equation of state expressing fluiddensity as a function of temperature and composition isnecessary. It is possible to obtain approximated analyticalsolutions of this problem through a linearization of theequation of state, as proposed by Boussinesq:

[150]

where bT is the thermal dilatation coefficient for the fluid andthe subscript 0 indicates the reference conditions. It isnecessary to indicate that in the presence of temperaturegradients, the first term is usually the dominant one. Itssubstitution within the balance equations allows thedefinition of the Grashof thermal number [39] and of thehomologous number for the concentration. It is important tonote that natural convection initiates only for values greaterthan a critical threshold. An example is evidenced by theheating of a fluid through a hot wall underneath. When theheat flux is small (and consequently the density andtemperature gradients are limited) the fluid remains at rest.Once the threshold value is surpassed it can be observed thatfluid motion starts with the formation of natural convection

r rr= + − + −

∑0 0 0

1 βω

ω ωTi

i iiT T( ) ( )

,

1 1k k K kL ieff

L i

L

i G G i, , ,

= +r

r

N k Ki Leff

G i i L iL i= −( )r

, , ,/ω ω

ω ωG i i L iK,*

,*=

N k ki G i G G i G i L i L L i L i= −( )= −( ), , ,*

, ,*

,r rω ω ω ω

ω ωω ω

i i W

i i W i

yD t

−−

=

,

, ,0 4erf

FLUID MOTION

244 ENCYCLOPAEDIA OF HYDROCARBONS

interface

w*G,i

w*G,iKiw*L,iwG,i

w*L,i

wL,i

mass transfer

mol

ar f

ract

ion

Fig. 12. Chemical composition profile during the mass transfer through an interface where thermodynamic equilibrium conditions are reached.

Fig. 13. Vortices caused by natural convection for different geometricconfigurations of the tankholding the fluid (fluid heated from below).

cells (Benard’s cells). The extension and the number of thesecells depends on the ratio between the vertical and horizontaldimensions for the tank holding the fluid. As illustrated inFig. 13, if the vertical dimension is prevalent, either a singlecell, whose extension covers the whole fluid, or two counter-rotating cells are present. Vice versa, if the horizontaldimension prevails, the formation of a great number of cellstakes place, because the maximum vortex extension isvertical.

Another interesting case, where it is possible to obtain ananalytical solution, is that of a fluid contained between twoparallel infinite vertical walls at different temperatures. Inthis case, the channel being infinite in length, only onevelocity component uz(x) exists, whose direction is parallelto the channel axis and whose value depends on thetransversal coordinate. Analogously, the fluid temperatureT(x) also depends uniquely on the same coordinate. It is easyto verify that the energy balance reduces to the Laplaceequation, in other words kT2T0, thus leading to a lineartemperature profile between the two walls. The integrationof the momentum balance equation for the uniquely presentvelocity component leads to the following expression for thevelocity profile, which is the sum of a zero average valuecubic profile (component due to natural convection) and of aPoiseuille’s parabolic profile (component due to forcedconvection):

[151]

where Re is calculated by means of the averaged fluidvelocity due to the forced convection and b is the semi-distance between the two walls. The fact that the velocityprofile is the sum of two profiles should not be surprising,seeing that the starting equations are linear. When forcedconvection is absent (Re0), the velocity profile becomes atzero mean value, because the rising flow rate in half of thechannel is exactly equal to the descending flow rate in theother half.

4.2.8 Conclusions and futuredevelopments

Research dealing with transport phenomena is at presentfocusing on molecular aspects, thanks to the availability ofmolecular dynamic software that simulates fluid behaviourin conditions closely approaching real conditions. Inparticular, the formulation of a reliable model for liquidpolymers surely represents one of the frontiers in the studyof transport phenomena, while considering that a moleculartheory for heat transport in polymers has not been yetformulated. In addition, fast computers are of greatimportance for the study of turbulent flow systems, becausethey allow the direct simulation of the flow (direct numericalsimulations) and thus permit, with time averaging techniques(long time average), to evidence the turbulence features thatlater can be adopted to formulate adequate turbulencemodels. The application of transport phenomena to smallsystems (micro-channels) produces surface phenomenawhich must be included in the balance equations. Althoughsuch a correction is known, fundamental theories on theinterface interaction (fluid-solid interaction, surfacewetting), are presently still incomplete. Finally, the

application to biological systems makes it necessary toinclude transport mechanisms reflecting the particular natureof the membranes present in the cellular organisms, forwhich the simple description through permeabilitycoefficients is greatly over simplified.

Bibliography

Beek W.J. et al. (1999) Transport phenomena, New York, John Wiley.

Bird R.B. (2004) Five decades of transport phenomena, «AmericanInstitute of Chemical Engineers Journal», 50, 273-287.

Dantzig J.A., Tucker C.L. (2001) Modeling in materials processing,Cambridge, Cambridge University Press.

Dente M. (1970) Principi di ingegneria chimica, Milano, Clup.

Mauri R. (2005) Elementi di fenomeni di trasporto, Pisa, PLUS.

Pedrocchi E., Silvestri M. (1980) Introduzione ai fenomeni ditrasporto, Milano, Clup.

Perry R.H., Green D.W. (edited by) (1997) Perry’s chemical engineers’handbook, New York, McGraw-Hill.

Reid R.C. et al. (1987) The properties of gases and liquids, New York,McGraw-Hill.

Rosner D.E. (1986) Transport processes in chemically reacting flowsystems, Boston (MA), Buttherworths.

References

Bird R.B. et al. (1960) Transport phenomena, New York, John Wiley.

Chapman S., Cowling T.G. (1939) The mathematical theory of non-uniform gases, Cambridge, Cambridge University Press.

Hirschfelder J.O. et al. (1954) Molecular theory of gases and liquids,New York, John Wiley.

Landau L.D., Lifshitz E.M. (1959) Fluid mechanics, London,Pergamon Press.

Levich V.G. (1962) Physicochemical hydrodynamics, EnglewoodCliffs (NJ), Prentice-Hall.

List of symbols

Ai area of the ith surfacea lattice parameter of the crystala molecular interaction parameterai absorbance of the ith surfaceb covolumeCi molar concentration of ith speciesCV specific heat at constant volumeCP specific heat at constant pressure33

c velocity module of molecular chaotic motionsDAB binary diffusion coefficientDi,m ordinary diffusion coefficient of ith species in the

mixtureDi

(T) thermal diffusion coefficientDi

(P) diffusion coefficient by pressured molecular diameterE electric potentialE activation energye absolute value of the electron electric charge,

1.6021019 Cei emissivity of ith surfaceF12 view factor between two surfaces

u x b Gr xb

xb

z ( )

υ=

−

+

12

2

3

3

Re xxb

−

2

1

TRANSPORT PHENOMENA

245VOLUME V / INSTRUMENTS

ℑ Faraday’s constant, 96.485 C/molf Fanning’s friction factorg gravity accelerationh heat transfer coefficienth Planck’s constant, 6.626 1034 m2kg/sJi mass convective flux of ith speciesJP momentum convective fluxJE energy convective fluxJX convective flux of variable X through the boundary

surfacesK kinetic energy by unit masskB Boltzmann’s constant, 1.38 1023 J/KkT thermal conductivity of the mediakc mass transport coefficientL characteristic lengthMi molecular mass of ith speciesm, n parameters for equation [20] – non-Newtonian fluid

rheologic modelNA Avogadro’s number, 6.023 1023 molecules/molNi

(tot) overall diffusive mass flux of ith speciesNi

(ord ) diffusive mass flux of ith species induced by theordinary diffusion

Ni(T ) diffusive mass flux of ith species induced by the

thermal diffusionNi

(P) diffusive mass flux of ith species induced by thepressure gradient

Ni(F) diffusive mass flux of ith species induced by the

external forces fieldNX diffusive flux for the X variable through the

boundary surfacesP pressureq diffusive energy fluxq(x) diffusive energy flux due to the presence of matter

diffusive fluxesq(rad) radiative energy fluxR universal gas constant, 8.313 J/molKRi production rate of ith species by the effect of

chemical reactionsRX source term for the generic X variable specific to

the volumer intermolecular distancerj rate of the jth chemical reactions wall thicknessT temperatureTb normal boiling point for a liquidU internal energy for unit massU overall heat transfer coefficientu fluid velocity vectoruj component of velocity in the jth directionus speed of soundul velocity of the turbulent vortexV molar volumeX specific value with respect volume for the

examined variablex, y, z Cartesian coordinatesxi mole fraction for ith specieszi electric charge for ith ion

Greek lettersa thermal diffusivityai

(T) thermal diffusion factor

bT volumetric dilatation coefficientbAB sliding friction coefficient between fluid and

particleGcoll collisional dissipation integralgI activity coefficient for ith species in solution.gg rate of strain tensor∆G0

free energy variation associated to the‘cage-vacancy’ transfer in liquids

∆Hvap enthalpy variation associated to the vaporization ofa liquid

∆T temperature difference∆Uvap internal energy variation associated to the

vaporization of a liquidd distance between two molecular planesd thickness of the velocity boundary layerdC Thickness of the concentration boundary layerdT thickness of the temperature boundary layerdij component of the Kroneker operatore characteristic energy for a molecule in the Lennard-

Jones potentialeL energy transferred by a turbulent vortexzi ionic mobilityh apparent viscosityκ dilatational viscosity or second viscosityL Lorentz’s constant (value dependent by the

considered metal within a limited interval)Le equivalent conductance for the electrolytel free mean path of moleculesl turbulence scalel,l equivalent conductance of the cation and anion

constituting the electrolytem dynamic viscosity of the fluidn vibration frequencyni stoichiometric coefficient of ith species x packing parameter, see equation [82] F potential energy per unit mass/ volumetric fraction of a solid in a suspensionf intermolecular potentialpp molecular stresses tensorpij molecular stresses tensor componentr mass densitys molecule collisional diameter in the Lennard-Jones

potentialse electric conductivitysSB Stefan-Boltzmann constant, 5.67 108 W/m2K4

t relaxation time of the systemtt viscous stresses tensortij viscous stresses tensor componentu kinematic viscosity of the fluidYB association parameter for the solvent, see equation

[83] Yij mixing rules parametersWk collisional integral for the kth variablewi mass fraction for ith species

Superscripts and supersigns° reference valuet transposed vector or matrix(t) turbulent value unperturbed value‘ fluctuant quantity- average quantity

FLUID MOTION

246 ENCYCLOPAEDIA OF HYDROCARBONS

* equilibrium conditionseff effective value

SubscriptsG gaseous phaseL liquid phasei ith speciesW value referred to the wall

Dimensionless numbers

Biot number

Damkholer number

Froude number

Grashof number

Nusselt number

mass Péclet number

thermal Péclet number

Prandtl number

Reynolds number

Schmidt number

Sherwood number

Maurizio Masi

Dipartimento di Chimica, Materialie Ingegneria Chimica ‘Giulio Natta’

Politecnico di MilanoMilano, Italy

Shk L

Dc

i m

=,

ScDi m

=µ

r ,

Re = ruLµ

Pr =µCk

P

T

Pe uLT = = ⋅

αRe Pr

Pe uLD

Scmi m

= = ⋅,

Re

NuhL

kT

=

Grg TLT=

r2 3

2

β

µ

∆

Fru

gL=

2

DaM R L

Di i

i m

=2

r ,

Bihs

kT

=

TRANSPORT PHENOMENA

247VOLUME V / INSTRUMENTS