Tx69299 ch4

© 2003 by CRC Press LLC

53

4

Molecular Transport of Momentum, Energy,

and Mass

4.1 Introduction

In many cases, it is difficult to describe in a complete and quantitative waythe problems set up in engineering. The velocity laws that govern the pro-cesses studied may not be exactly known when the behavior of static fluidsis studied. However, the behavior of moving fluids sets up complicated prob-lems to be quantified. The work required to pump a fluid through a conduitdepends on the momentum losses experienced by the fluid rubbing along thewalls, on the circulation regime, and on the type and nature of the fluid.

For transport in one direction under steady state and laminar flow, i.e.,molecular transport, the last chapter showed that the velocity expressionsor laws may be expressed as: (flux density) = (transport property) (potentialgradient).

4.2 Momentum Transport: Newton’s Law of Viscosity

Consider a static fluid, between two parallel slabs, of area

A

separated by adistance

y

. If, at a given time (t = 0), the lower slab begins to move at avelocity

v

, a time will come when the velocity profile is stable, as shown inFigure 4.1. Once the steady state is reached, a force

F

must continue to beapplied to maintain the motion of the lower slab. Assuming that the circu-lation regime is laminar, the force per unit area that should be applied isproportional to the velocity and distance ratio, according to the followingequation:

(4.1)

FA

vy

= η

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54

Unit Operations in Food Engineering

The proportionality constant

η

is called the viscosity of the fluid.For certain applications it is convenient to express the latter equation in a

more explicit way. The shearing stress exerted in the direction

x

on the fluid’ssurface, located at a distance

y

, by the fluid in the region where

y

is smalleris designated by

τ

YX

. If the velocity component in the direction

x

has a value

v

x

, the last equation may be expressed as:

(4.2)

FIGURE 4.1

Velocity profile.

y t < 0

v

t = 0

v

v

Formation ofvelocity profileUnsteady

x ( y, t )tsmall

v

v

y

x

Velocity profileSteady flow

x ( y ) t big

Movinginferior slab

a)

b)

c)

d)

τ ηYXXd v

d y= −



Molecular Transport of Momentum, Energy, and Mass

55

That is, the shear stress or force per unit area is proportional to the localvelocity gradient. This equation is the expression of Newton’s law of viscos-ity. All those fluids that follow this law are termed Newtonian fluids.

The shearing stress shear stress

τ

YX

in a Newtonian fluid, for a distance

y

from the limit surface, is a measure of the velocity of momentum transportper unit area in a direction perpendicular to the surface. The momentumflux is the amount of momentum per unit area and unit time, which corre-sponds to a force per unit area. For this reason,

τ

YX

can be interpreted as fluxdensity of momentum

x

in the direction

y

.According to Equation 4.2, the momentum flux goes in the direction of the

negative velocity gradient. That is, the momentum is transferred from thefastest moving fluid to the slowest one. Also, shear stress acts in such adirection that it opposes the motion of the fluid.

Another way to express Newton’s law of viscosity is:

(4.3)

where

v

is the kinematic viscosity, which is the viscosity divided by thedensity:

(4.4)

4.3 Energy Transmission: Fourier ’s Law of Heat Conduction

To study heat transfer, a solid material with the form of a parallele pipedwith surface A and thickness

y

is considered. Initially, the temperature ofthe slab is

T

o

. At a given time (t = 0), the lower part of the slab suddenlyreaches a temperature

T

1

, higher than

T

o

, and remains constant over time.The temperature will vary along the slab until reaching the steady state, inwhich a temperature profile such as that shown in Figure 4.2 is reached. Tomaintain this profile, a heat flow

Q

should be sent through the slab.Values of the temperature difference (

∆

T

=

T

1

–

T

o

) small enough complywith the following relation:

(4.5)

This equation indicates that the heat flow per unit area is proportional tothe increase of temperature with distance

y

. This proportionality constant

k

is called thermal conductivity of the solid slab. This equation also applies

τ ν

ρYX

Xd v

d y= −

( )

v = η ρ

QA

kT

y=

∆



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for liquids and gases placed between slabs, provided that there are no con-vection or radiation processes. Therefore, this equation is applicable to allheat conduction processes in solids, liquids, and gases.

For certain applications it is convenient to express this equation in a dif-ferential form. That is, if the thickness

y

of the solid tends to zero, the limitform of Equation 4.5 will be:

FIGURE 4.2

Development of the temperature profile.

T0

T0

T0

T( y, t )

T( y )

T0

T0

T1

T1

T1

t < 0

t = 0 Lower slab attemperature

T1

tsmall

t bigy

x

a)

b)

c)

d)




57

(4.6)

where

q

Y

represents the heat flux in the direction

y

. This equation expressesthat heat flow density is proportional to the negative gradient of temperatureand is the unidimensional form of Fourier’s law

of heat conduction.For other directions, the equations are analogous to that described for

direction

y

. Therefore, Fourier’s law is expressed as:

(4.7)

that is, the heat flux vector is proportional to the temperature gradient, withopposite direction. It is assumed that the medium is isotropic; i.e., the thermalconductivity has the same value in all the directions of the material.

The units in which heat conductivity is normally expressed are W/(m ·K)or kcal/(h ·m ·°C), and the dimensions are: [

k

] = MLT

–3

θ

–1

.Besides the thermal conductivity, in the latter equations thermal diffusivity

can be used, defined according to the equation:

(4.8)

where ˆ

C

P

is the specific heat of the material.Taking into account the definition of thermal diffusivity, the expression of

Fourier’s law for one direction is:

(4.9)

for an isotropic material in which

ρ

and ˆ

C

P

are constants.

4.4 Mass Transfer: Fick’s Law of Diffusion

Fick’s law of diffusion refers to the movement of a substance through abinary mixture due to the existence of a concentration gradient. The move-ment of a substance within a binary mixture from high concentration pointsto points with lower concentrations may be easily deduced by recalling thedissolution of a color crystal in water, as described in Chapter 3. The diffu-sion of a component due to the existence of a concentration gradient receivesthe name of ordinary diffusion. There are also other types of diffusion,

q k

d Td yY = −

r rq k T= − ∇

αρ

=k

CP ˆ

q

d C T

d yYP= −

( )α

ρ

ˆ

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58


according to the property that confers the movement to the component ofthe mixture; thus, if movement is due to a pressure gradient, it is calledpressure diffusion; if it is due to a thermal gradient, the diffusion is thermal.When there is an inequality in the external forces that cause such movement,it is called forced diffusion.

The study of diffusion is more complicated than the cases of momentumand energy transport because diffusion involves the movement of a specieswithin a mixture. In a diffusive mixture, the velocities of the individualcomponents are different and should be averaged to obtain the local velocityof the mixture that is required to define diffusion velocities. To get theexpression of Fick’s law, it is convenient to define the different forms forexpressing concentrations, velocities, and fluxes:

• Mass concentration

ρ

i

: the mass of species

i

per unit of mixturevolume.

• Molar concentration

C

i

: the number of moles of the species

i

perunit of mixture volume.

C

i

=

ρ

i

/

M

i

in which

M

i

is the molecularweight of species

i

.• Mass fraction

w

i

: the mass concentration of the species

i

dividedby the total molar density of the mixture;

w

i

=

ρ

i

/

ρ

.• Molar fraction

X

i

: the molar concentration of the species

i

dividedby the total molar density (global concentration) of the mixture:

X

i

=

C

i

/

C

.

In the considered mixture, each component moves at a different velocity.If a component

i

has a velocity

v

i

with respect to steady coordinate axes, thedifferent types of velocity are defined as follows:

• Mean mass velocity

v

:

(4.10)

• Mean molar velocity

v

*:

(4.11)

r

r r

rv

v v

w vi i

i

n

ii

n

i ii

n

i ii

n

= = ==

=

=

=

∑

∑

∑∑

ρ

ρ

ρ

ρ1

1

1

1

r

r r

rv

C v

C

C v

CX v

i ii

n

ii

n

i ii

n

i ii

n* = = ==

=

=

=

∑

∑

∑∑1

1

1

1




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When considering flow systems, it is convenient to refer to the velocity ofthe component

i

with respect to

v

or

v

*, rather than referring to steadycoordinate axes. In this way, one obtains the so-called diffusion velocitiesthat represent the movement of the species

i

with respect to the movementof the fluid stream:

• Diffusion velocity of component

i

with respect to

→

v

: the velocity ofcomponent

i

with respect to an axis system that moves at velocity

v

; it is given by the difference (

v

i

–

v

).• Diffusion velocity of the component

i

with respect to

→

v

*: the veloc-ity of component

i

with respect to an axis system that moves atvelocity

v

*; it is given by the difference (

v

i

–

v

*).

The flux density can be of mass or molar and is a vectorial magnitudedefined by the mass or moles that cross through a unit area per unit time.The motion can be referred to as steady axis or as axis moving at velocity

v

or

v

*. In this way, the different forms for expressing the flux density of acomponent

i

in a mixture made of

n

components will be:

• With respect to steady axes:Mass flux:

(4.12)

Molar flux:

(4.13)

• For moving axes:Diffusion mass flux in relation to velocity

→

v

:

(4.14)

Diffusion mass flux in relation to velocity

→

v

*:

(4.15)

Diffusion molar flux in relation to velocity

→

v

:

(4.16)

Diffusion molar flux in relation to velocity

→

v

*:

(4.17)

r rm vi i i= ρ

r r rN C vi i i=

r r rj v vi i i= −( )ρ

r r rj v vi i i

* *= −( )ρ

r r rJ C v vi i i= −( )

r r rJ C v vi i i

* *= −( )



60


Some of these forms are rarely applied, such as

J

i

and

j

i

*. Most often usedin engineering is the molar flux, referred to as steady axis

N

i

.Once the different forms in which concentrations, velocities, and flux can

be expressed have been reviewed, mass transfer is studied. Consider a binarymixture with components A and B in which the diffusion of one of thecomponents occurs due to a concentration gradient of the considered com-ponent. As for momentum and energy transfer, viscosity and thermal con-ductivities are defined as proportionality factors between momentum fluxand the velocity gradient for viscosity (Newton’s law of viscosity), andbetween heat flux and temperature gradient for thermal conductivity (Fou-rier’s law of heat conduction). In an analogous way the diffusivity DAB =DBA in a binary mixture is defined as the proportionality factor between themass flux and the concentration gradient, according to the equation:

(4.18)

that is, Fick’s first law of diffusion for molar flux. In addition to the concen-tration gradient, the temperature, pressure, and external forces contribute tothe diffusion flux, although their effect is small compared to that of theconcentration gradient. This indicates that the diffusion molar flux relatedto the velocity v* is proportional to the gradient of the molar fraction. Thenegative sign expresses that this diffusion takes place from higher to lowerconcentration zones.

It is easy to deduce that the diffusion sum of a binary mixture is zero, i.e.:

since, if a component diffuses to one side, the other component diffuses inthe opposite way.

When the global concentration is constant, a chemical reaction does notexist, or there is a chemical reaction, but the number of moles does not vary,Equation 4.18 can be transformed into:

(4.19)

There are other ways to express Fick’s first law according to the flux andthe correspondent concentration gradient considered. Thus, for the mass fluxin reference to the steady axes, the gradient considered is that of the massfraction, so this law is expressed by the equation:

(4.20)

r rJ C D XA AB A

* = − ∇

r rJ JA B

* *+ = 0

r rJ D CA AB A

* = − ∇

r r r rm w m m D wA A A B AB A= +( ) − ∇ρ



Molecular Transport of Momentum, Energy, and Mass 61

Of all possible expressions for Fick’s first law of diffusion, the one withthe greatest importance is that related to the molar flux in steady or fixed axis:

(4.21)

It can be observed that NA is the result of two vectorial magnitudes,XA(

→NA +

→NB): the molar flux by convective transport, resulting in the global

motion of the fluid, and (–CDAB

→∇XA), due to the molecular transport, accord-

ing to the definition of JA*.The units of diffusivity can be expressed, for example, in cm2/s or m2/h.There is a lack of diffusivity data for most mixtures, so it is necessary to

use estimated values in many calculations in which diffusivities are required.When possible, experimental data should employed since, generally, theyare safer. The order of magnitude for diffusivities is given next for variouscases found in practice:

Viscosity and thermal conductivity of a pure fluid are only a function ofthe temperature and pressure, the diffusivity DAB for a binary mixture is afunction of temperature, pressure, and composition.

4.5 General Equation of Velocity

The dimensions of the diffusivity DAB are surface per unit time: DAB = L2T–1.If Newton’s law of viscosity is considered, the dimensions do not correspondto those of diffusivity: [η] = ML–1T–1. However, if the kinematic viscosity vis taken, i.e., the relationship between dynamic viscosity and density, it turnsout that its dimensions are surface per time unit: [ν] = L2T–1, that is, thedimensions coincide with those of diffusivity. So ν also receives the name ofmomentum diffusivity.

In an analogous way, in Fourier’s law of heat conduction, the dimensionsof thermal conductivity are: [k] = MLT–3T–1. However, if the relation α =k/(ρ CP) is considered, its dimensions coincide with those of diffusivity: [α] =L2T–1, where α is known as thermal diffusivity.

The analogy of these three magnitudes, DAB, v, and α, is could be observedfrom the velocity equations for the three transport phenomena in unidirectionalsystems:

Gas–gas diffusion: 0.776 to 0.096 cm2/sLiquid–liquid diffusion: 2 × 10–5 to 0.2 × 10–5 cm2/sDiffusion of a gas in a solid: 0.6 × 10–8 to 8.5 × 10–11 cm2/sDiffusion of a solid in another solid: 2.5 × 10–15 to 1.3 × 10–30 cm2/s

r r rN X N N CD Xa A A B AB A= +( ) − ∇



62 Unit Operations in Food Engineering

Fick’s law:

Newton’s law:

Fourier’s law:

where ρ and CP remain constant.In these equations the media is assumed to be isotropic, i.e., the material’s

viscosity, thermal conductivity, and diffusivity have the same value in anydirection. This is acceptable for fluids and most homogeneous solids.

Considering only one direction, the above equations can be grouped intoone expression:

(4.22)

where Φγ is the flux of any of the three properties in the direction y, ξ is theconcentration per unit volume of the considered property, and δ is the pro-portionality factor, called diffusivity. Similar equations will result for theother directions in such a way that the experimental equations of Fick,Newton, and Fourier can be grouped into three general equations, one foreach direction. These equations constitute the expressions of molecular trans-port of the three studied properties.

The analogies expressed by the last equations among momentum, energy,and mass cannot be applied to bi- and tridimensional problems, since shearstresses τ are grouped in a tensorial magnitude, while jA and q are vectorialmagnitudes of three components.

If vectorial notation is used to generalize these equations, for direction xone obtains:

(4.23a)

(4.24)

(4.25)

j D

dd yAB AB A= − ( )ρ

τ ν ρYX X

dd y

v= − ( ).

qd

d yCY P= − ( )α ρ θˆ

ΦY

dd y

= −δξ

τ ηX Xv= − ∇( )

q k TX= − ∇

j Dd

d yAB AB A= − ( )ρ



Molecular Transport of Momentum, Energy, and Mass 63

For the y and z directions, the only equation that changes is that relatedto Newton’s law; thus, the equations used for direction y and z will be:

(4.23b)

(4.23c)

Therefore, generalizing, it is obtained:

(4.26)

which expresses that the flux vector is proportional to the gradient of thedriving force or potential (for the experimental laws) and concentration ofthe properties (in the case of more strict laws), and of opposite direction.

τ ηY Yv= − ∇( )

τ ηZ Zv= − ∇( )

r rΦ = − ∇δ ξ


Tx69299 ch4

Food

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