tx69299 ch3

© 2003 by CRC Press LLC

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3

Introduction to Transport Phenomena

3.1 Historic Introduction

Unit operations of the agricultural and food industries constitute the physical,chemical, or biochemical stages that integrate industrial processes by meansof which agricultural products are handled or transformed. That these stagesare the same as those found in the processes of chemical industries makesthe knowledge and advances of unit operations of chemical engineeringapplicable to the agricultural and food industries when they are adapted tocharacteristics of the raw material (particularly natural products, which aregenerally perishable) and particular conditions (hygiene, cleanliness, etc.)normally required by agricultural–industrial processes. The importance ofthe concepts of process engineering is that they unify the techniques of indus-tries normally considered separate. Thus, the basic principles common to allfood industries are unified in a logical way despite their apparent diversity.

Food engineering is a relatively new branch of engineering based on thechemical industry, where process engineering was developed. Chemicalengineers have built process technology with dimensions not frequentlyfound, which has been very advantageous. Most of this technology can beapplied to the food industry.

The processes can be divided into so-called unit or basic operations com-mon to many processes; thus, the individual study of each unit operation ismore efficient than the study of each process. The knowledge of unit oper-ations is vast; the food engineer should take this knowledge and apply it tothe development of the food industries.

It should be recalled that the term “unit operation” was established in 1915by Professor Little of the Massachusetts Institute of Technology (MIT). Becauseof its historic and conceptual value, it is interesting to recall its definition:

… every chemical process conducted at any scale can be decomposed inan ordered series of what can be called unit operations, as pulverization,drying, crystallization, filtration, evaporation, distillation, etc. The amountof these unit operations is not very large and, generally, only few of themtake place in a determined process.

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Unit Operations in Food Engineering

This simplification reduced the complexity of the study of processes fromthe almost infinite set of processes that could be imagined to the set ofexisting unit operations. A given process would be formed by a combinationof unit operations.

A new development and growth stage began with the systematic study ofthese unit operations, adding new operations and generalizing, for many ofthem, their didactic presentation through dimensional analysis and experi-mental study. This was a phase of reasoned empiricism in which theory andpractice were skillfully combined, and during which the theoretical funda-mentals of the different operations were gradually established. This tradi-tional concept of unit operations has been one of the main factors of theextraordinary success of this branch of engineering in the past.

Continuing with the systematization effort, a new generalization periodbegan, grouping unit operations according to the general principles on whichthey were based. In this way, they were classified within the followingsections: fluid treatment, mass transfer by multiple contact, and energy andmass transfer by continuous contact.

Later, because of better knowledge of the fundamentals of unit operations,it was noticed that all of them were based on three phenomena: momentumtransport, energy transfer, and mass transfer.

In all three cases, the flow of the transferred property is directly propor-tional to an impelling force (velocity, temperature, or concentration gradient)and inversely proportional to a resistance that depends on the properties ofthe system and on operation conditions. Therefore, it is possible to developan abstract doctrine body from which these three transport phenomena canbe taken as particular cases.

3.2 Transport Phenomena: Definition

It can be deduced that all the physical stages constituting the differentindustrial processes of industrial elaboration are based on momentum trans-port, energy transfer, and mass transfer. In all processes in which a systemis not in equilibrium, the system evolves in such a way that it tends toequilibrium by transferring one or more of the cited properties.

Transport phenomena can be defined as physical phenomena revealedwhen a system evolves towards an equilibrium state. Some examples toexplain this definition may be:

• When, in a fluid stream, there are two points whose velocities, takennormal to its movement, are different, the system will evolve to coun-teract this velocity difference by means of a momentum transport.




45

• If, in a solid, there are zones of different temperature, there will bea heat transport from the hottest zone to the coldest one, so thesystem tends to thermal equilibrium.

• When, within the same phase, there is a concentration differencebetween two points, a mass transfer will tend to equilibrate thisconcentration difference.

Actually, in all operations, at least two out of the three phenomena aresimultaneously present, but there are some in which one phenomenon nor-mally predominates. Thus, momentum transport predominates in operationsof fluid transport, sedimentation, filtration, etc., heat transfer dominates inthe design of heat exchangers, condensers, etc., and mass transfer dominatesin operations such as absorption, solvent extraction, distillation, etc. In oper-ations such as air–water interaction, drying, crystallization, etc., mass andheat transfer phenomena are equally important.

3.3 Circulation Regimes: Reynolds’ Experiment

Before studying the mechanisms of transport phenomena, it is interesting toprove experimentally the existence of such mechanisms. When studyingcirculation regimes, the experiment carried out by Reynolds in 1833 isextremely important. This experiment consisted of circulating water throughtransparent tubing with constant cross section, varying the circulation veloc-ity of the liquid by means of a valve placed at the entrance of the pipes. Inthe middle of the pipes, and in the entrance section, a colored solution wasintroduced. The variation of the colored vein was observed along the pipesat different circulation velocities of the liquid.

In this type of experiment, the velocity of the liquid is low, the coloredvein does not lose its identity while circulating through the center of thepipes, and, although a low and progressive increase of the vein’s width canbe observed, mixture in a transversal sense is not seen, indicating that theflow takes place as parallel streams with no interference between them. Massexchange occurs only at the molecular level. As the velocity of the liquidincreases, oscillations in the colored filament appear until, at a certain velocity,it breaks into whirlpools and transversally inundates the conduction. Dif-ferent images of Reynolds’ experiment can be observed in Figure 3.1.

Based on these observations, it can be stated that there are two types ofwell-differentiated circulation regimes in which mass transfer mechanismsare different. For low velocities, liquid moves in a horizontal way withconcentric parallel layers and without transversal movement. This regime iscalled laminar and is characterized by the absence of a global movementperpendicular to the main direction of the stream.



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For high velocities, there is movement of the liquid in a transversal way ofmacroscopic proportions. This type of circulation regime is called turbulent,and is characterized by rapid movement of the liquid in the form of whirl-pools with no pre-established direction in the transversal section of the pipe.

The only variable modified in these experiments was velocity, but theremay be other variables that could alter the regime, such as diameter of thepipe and the nature of the liquid. Hence, for a better study of circulationregimes, a dimensionless module that relates the magnitudes that character-ize the circulation phenomenon is defined, setting up the boundaries of thedifferent regimes. Such a module is called the Reynolds number, whichrepresents the quotient between the inertia and viscosity forces of the movingfluid. In the case of a cylindrical conduit and a Newtonian fluid, it is:

FIGURE 3.1

Reynolds’ experiment.

Flow

Flow

Flow

Increase of fluid velocity

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47

(3.1)

where:Re = Reynolds number (dimensionless)

ρ

= density of the fluid (kg/m

3

)

v

= mean velocity of the fluid (m/s)

d

= diameter of the conduit (m)

η

= viscosity of the fluid (Pa ·s)

The numerical value of the Reynolds number (Re) is used to define thetype of circulation regime of a fluid stream. It has been observed that, forvalues lower than a given Re, called critical, the oscillations of the fluid areunstable and any perturbation disappears quickly. For values higher thanthis critical Re, the oscillations become stable and of greater amplitude,giving place to a high radial mixture. For Newtonian fluids, the critical Revalue is 2100. Values lower than 2100 indicate a laminar regime, while, forgreater values, there is a range of Re number values called transition, inwhich metastable phenomena could appear. The regime is completely tur-bulent for values higher than 10,000. This study can also be applied tomomentum transport and heat transfer.

If the pressure drop (

∆

P) in a fluid between the inlet and outlet sections ofa conduit is measured in the Reynolds experiment, an increase in the volu-metric flow of the fluid will be observed. Figure 3.2 shows the variation of thepressure drop as a function of the Reynolds number. The loss of charge of the

FIGURE 3.2

Variation of the pressure drop in relation to the Reynolds number.

Re =

ρηvd

2100

∆P

R e



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fluid is a result of the energy consumed by the fluid because a momentumtransport took place. It can be seen in this figure that, starting from Re = 2100,the loss of charge increases rapidly, which favors the momentum transportprocess since radial components appear in the velocity of the fluid particles.

In order to study heat transfer, one can consider a tubing in which waterwith mass flow rate (

w

) at a temperature

t

1

enters, and the same volume ofwater at temperature

t

2

, due to the heat gained by the fluid through thetubing wall, exits. The total heat gained will be:

(3.2)

When the mass flow rate (

w

) varies, the exit temperature (

t

2

) will also vary.A graphical representation of the variation of gained heat as a function ofthe Reynolds number will yield a graphic similar to that in Figure 3.2. It canbe observed that, for a laminar fluid, the exchanged heat increases in directproportion to the volume of the fluid; however, starting at the critical valueof the Reynolds number, a sharp increase is observed. In a laminar regime,heat transfer occurs in radial form, molecule by molecule, but, in a turbulentregime, there are streams or whirlpools that favor radial heat transport.

3.4 Mechanisms of Transport Phenomena

The mechanism of energy transfer by means of electromagnetic waves iscalled radiation and it can be performed through vacuum, not needing amaterial media for transmission. However, other forms of energy transferand of momentum transport are associated with material movement,although there is not a net transfer. Thus, in heat transfer by conduction in acontinuous material media, there is no material movement at the macroscopiclevel, although there is movement at the molecular level due to the motionof free electrons (in metals) or to the vibration of molecules or ions in solids.

Because these different transport phenomena are associated, it is interest-ing to study them as a whole. Transport of the three previously mentionedproperties may take place by means of two well-differentiated mechanisms:molecular transport and turbulent transport.

In molecular transport the transfer of the property is carried out, moleculeby molecule, by movement of the individual molecules or by interactionsamong them. Turbulent transport is produced when large groups of mole-cules move as aggregates or whirlpools, transporting with them momentum,mass, or energy. These aggregates serve as transportation media and theytransfer the property to other groups of molecules that interact with them.Molecular transport can happen alone; turbulent transport never occurs inisolation but is always accompanied by molecular transport.

q w Cp t t= −( ) 1 2




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3.4.1 Mass Transfer

To study the mechanism of mass transfer, suppose a given component of theconsidered material is transferred from one point to another of the studiedsystem. This mass transfer may take place according to two mechanisms:molecular flow or forced convective flow. When there is a concentrationgradient of the considered component between two points of the system,mass transfer is produced by molecular flow. However, when the entire massmoves from one point to another, the transfer is produced by forced convec-tive flow.

According to the physical nature of the media, different situations canoccur and the mass transfer is carried out by one or by both of the transportmechanisms considered:

• When there is no concentration gradient of the considered compo-nent and if the media is a fluid, there can only be convectivetransport. Usually, this type of problem is studied as momentumand not as mass transport.

• When there is a concentration gradient of the component and themedia is a fluid in repose, the mass transfer is carried out bymolecular flow, due only to molecular diffusion. Thus, if one con-siders a beaker filled with water with a crystal of colorant placedon the bottom, the crystal dissolves, gradually diffusing through-out all the beaker, since the concentration around the crystal isgreater than in other zones. This diffusion will take place untilequilibrium is reached.

• When there is a concentration gradient and the media is a fluidmoving in a laminar regime, mass transfer is carried out by thetwo mechanisms. Recalling the Reynolds experiment, the colorantis injected into an entrance point

P

of the tubing (Figure 3.3). Atthe exit at point

Q

, the colorant is transferred from the entrance tothe exit by forced convection flow, and from the center of the pipeto point

Q

by molecular flow.

FIGURE 3.3

Simultaneous molecular and forced convective flows.

P Q

Flow



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• When the media is a fluid in which there are turbulence and con-centration gradients, the mechanisms of molecular and forced con-vection mass transport occur simultaneously. Although thephenomenon is complex, it is similar to the previous case, usingan analogous model. In this case, consider an effective diffusionthat brings together the molecular diffusion due to the gradientand the denominated turbulent diffusion, passing from

P

to

Q

bymeans of turbulent transport by whirlpools.

3.4.2 Energy Transfer

As mentioned at the beginning of this section, energy transfer by radiationoccurs by a different mechanism than do those of conduction and convection.It is interesting to mention some aspects of energy transfer by these twomechanisms.

Conduction supposes a molecule-to-molecule flow of energy, due to tem-perature gradients, by mechanisms that depend on the physical nature ofthe medium. By analogy with mass diffusion, the principle of these mecha-nisms can be explained at an atomic-molecular level; however, these mech-anisms differ because, in conduction, there is no net mass flux.

If the considered medium is a fluid and there is a temperature gradient, inmany cases a density difference will be noticeable. Therefore, there will be amass flow due to flotation forces associated with an energy flow of the naturalconvection type. Forced convection exists as well and, as in natural convec-tion, is due to the associated energy of moving fluids; however, in this casethe energy applied to move the fluid comes from mechanical devices. Besidesconvection, energy transfer by conduction will also occur but is less impor-tant. In a general way, energy transfer is studied as a convection phenomenonin fluid media, bringing together convection and conduction.

3.4.3 Momentum Transport

To study the mechanisms of momentum transport, an analogous study tothe mass transfer study can be carried out. Molecular and forced convectionmomentum flows can also be considered.

3.4.4 Velocity Laws

In the mechanisms of molecular transport, property transfer occurs due toa potential gradient, which can be a concentration, temperature, or velocitygradient, depending on whether the transferred property is mass, energy, ormomentum, respectively.

In molecular transport, the flux density of the property is proportional tothe potential gradient. The proportionality constant is an intensive propertyof the media. Depending on the nature of the property, the proportionality




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constant receives different names, as well as the laws of each of the transportphenomena.

Conductivity:Fick’s Law: (flux density of mass) = (diffusivity) (concentration

gradient)Fourier’s Law: (flux density of thermal energy) = (thermal conduc-

tivity) (temperature gradient)Newton’s Law: (flux density of momentum) = (viscosity) (velocity

gradient)

When the transport regime is turbulent, a laminar regime subsists in theinner part of the whirlpools involving molecular transport in this region;normally, the parameters include both phenomena.

3.4.5 Coupled Phenomena

The velocity laws for mass, thermal energy, and momentum transfer areexpressed in a similar manner, in which the flux density of the consideredproperty is proportional to the gradient of the impelling force:

(3.3)

where

→

J

= flux density (quantity of property/m

2

·s)

k

= proportionality constant

→

X

= potential gradient

This equation is applicable to all systems not in equilibrium. The flow ofan equal property may be caused by various simultaneous potential gradi-ents, or a given gradient can generate diverse flows. Coupled phenomenaor processes are those that occur in systems in which different flows andgradients take place simultaneously. Thus, for example, a temperature gra-dient, in addition to causing an energy flow, can cause a mass flow (Soreteffect of thermodiffusion).

Onsegar generalized the latter expression to a system with

R

flows and

S

gradients as:

(3.4)

This equation indicates that each flow

J

j

depends not only on its combinedgradient, but also on other gradients acting on the system (

X

j

≠

i

).

r rJ k X=

r rJ k X

i R

i ij jj

S

= ∑

For = , , ...,

=1

1 2



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TABLE 3.1

Coupled Phenomena

Potential

Flow DensityMass Energy Momentum

Concentration gradient Diffusion(Fick’s law)

Thermodiffusion(Dufour’s effect)

Temperature gradient Thermodiffusion(Soret’s effect)

Thermal conductivity(Fourier’s law)

Velocity gradient Molecular transport of momentum(Newton’s law)
