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Part II

ERT 216 HEAT & MASS TRANSFER

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1. Introduction to Mass Transfer and Diffusion

2. Molecular Diffusion in Gasses3. Molecular Diffusion in Liquids

4. Molecular Diffusion in Biological Solutions and Gels

5. Molecular Diffusion in Solids6. Unsteady State Diffusion7. Convection Mass Transfer

Coefficients

Part I

Part II

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4. Molecular Diffusion in Biological Solutions and Gels.

4.1 Diffusion of biological solutes in liquids

4.2 Diffusion in biological gels5. Molecular Diffusion in Solids

5.1 Introduction and types of diffusion in solids.

5.2 Diffusion in solids following Fick‟s law6. Unsteady state Diffusion

6.1 Derivation of basic equation

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7. Convection Mass Transfer Coefficients7.1 Introduction to convective mass

transfer7.2 Types of mass transfer coefficients

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4.1 Diffusion of biological solutes in liquidsDiffusion of small solute molecules

[especially macromolecules (e.g. proteins)] in aqueous solutions are important in the processing & storing of biological systems & in life processes of microorganisms, animals, plants.

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Food processing: Drying of liquid solutions of fruit juice,

coffee & tea. Water & volatile flavor or aroma

constituents are removed. These constituents diffuse through the

liquid during evaporation.

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In fermentation processes:Nutrients, sugars, oxygen etc diffuse to

the microorganisms, and waste products & at times enzymes diffuse away.

Macromolecules in solution having molecular weights of tens of thousands or more were often called colloids. The diffusion behavior of protein

macromolecules is affected by size & shape.

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Interaction and “binding” in diffusion.Protein macromolecules are very large

compared to small solute molecules (e.g. urea, KCl, sodium coprylate) and often have a number of sites for interactions or “binding” of the solute or ligandmolecules.Human serum albumin protein binds most

of the free fatty acids in the blood & increases their apparent solubility.

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Experimental data for biological solutes. Table 6.4-1 shows a tabulation of

diffusivities of a few proteins and small solutes often present in biological systems.

The diffusion coefficients for the large protein molecules are on the order of magnitude of 5 x10-11 m2/s compared to the values of about 1x10-9 m2/s for the small solutes.

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Macromolecules diffuse at a rate about 20 times as slow as small solute molecules for the same concentration differences.The diffusivity of macromolecules is

strongly dependent on the surface charges on the molecules.

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Prediction of diffusivities for biological solutesFor predicting the diffusivity of small solutes alone in aqueous solution with molecular weights < 1000 or solute molar volumes < 0.500 m3/kg mol:

60

2116101731.

AB

/

BABVμ

TMφx.D

Viscosity

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For larger solutes, an approximation the Stokes-Einstein equation:

For a better approximation, Polson equation can be used. It is recommended for a molecular

weight above 1000.

31

1610969/

A

ABVμ

Tx.D

Viscosity

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When the shape of the molecule deviates greatly from a sphere, this equation should be used with caution.

31

1510409/

A

ABMμ

Tx.D

Molecular weight of the

large molecule AViscosity

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EXAMPLE 6.4-1

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Prediction of diffusivity of small solutes in protein solutions When a small solute (A) diffuses

through a macromolecule protein (P) solution, blockage to diffusion by the large molecules happens.

The data needed to predict these effects are diffusivity DAB of solute A in water alone, the water if hydration on the protein, & an obstruction factor.

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A semitheoretical equation that can be used to approximate the diffusivity DAP of A in globular-type protein P solutions (where only the blockage effect is considered and no binding is present):

The diffusion equation:

pABAP cx.DD 3108111

kg P/m3

12

21

zz

ccDN AAAP

AConcentration of A (kg mol / m3)

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Prediction of diffusivity with binding present When A is in a protein solution P and

binds to P, the diffusion flux A is equal to the flux of unbound solute A in the solution plus the flux of the protein-solute complex.

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The flux can be calculated:

100100108111 3 boundA%

DfreeA%

cx.DD PpABAP

Diffusivity of the protein

alone in solution (m2/s)

12

21

zz

ccDN AAAP

A

Total concentration of A in the solution

(kg mol / m3)

A not bound to the protein (determined from experimental binding coefficient)

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Gels:Semisolid materials, which are pores.They are composed of macromolecules

which are usually in dilute aqueous solution with the gel comprising a few %wt of the water solution.The pores (open spaces) in the gel

structure are filled with water.

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The rates of diffusion of small solutes in the gels are somewhat less than in aqueous solution.

Some typical gels: agarose, agar & gelatin.A number of organic polymers exist as

gels in various types of solutions.The unsteady-state methods are used

to measure the diffusivity of solutes in gels.The gel is melted & poured into a

narrow tube open at one end.

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After solidification, the tube is placed in an agitated bath containing the solute for diffusion.The solute leaves the solution at the

gel boundary and diffuses through the gel itself.After a period of time, the amount

diffusing in the gel is determined to give the diffusion coefficient of the solute in the gel.

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Table 6.4-2:Typical values of diffusivities of some

solutes in various gels.The diffusivity of the solute in pure

water is given so that the decrease in diffusivity due to the gel can be seen. at 278K urea in water has a diffusivity

of 0.880x10-9 m2/s; in 2.9 wt% gelatin, has a value of 0.640

x 10-9 m2/s; Decrease of 27%.

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Table 6.4-2:Typical

diffusivities of solutes in

dilute biological

gels in aqueous solution.

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Example 6.4-2

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5.1 Introduction & types of diffusion in solids

Even though rates of diffusion of gases, liquids, & solids in solids are generally slower than rates in liquids and gases, mass transfer in solids is quite important in chemical & biological processing.Leaching of soybeans and metal ores Drying of timber, salts, foods Diffusion of gasses through polymer

film used in packaging

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The transport in solids can be broadly classified:1. Diffusion that can be considered to

follow Fick’s law and does not dependprimarily on the actual structure of the solid.

2. Diffusion in porous solids where the actual structure and void channels are important.

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5.2.1 Derivation of equationsThe diffusion occurs when the fluid or

solute diffusing is actually dissolved in the solid to form a more or less homogenous solution. The diffusion in solids does not depend

on the actual structure of the solid. Example, leachingThe solid contains a large amount of

water and a solute is diffusing through this solution, or

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The diffusion of zinc through copper, (solid solutions are present).The diffusion of nitrogen or hydrogen

through rubberThe diffusion of water in foodstuffs.

For binary diffusion,

BAAA

ABA NNc

c

dz

dxcDN

Bulk-flow term

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Since cA/c or xA is quite small, it is neglected.

Also c is assumed constant, giving for diffusion in solids:

dz

dcDN AAB

A

Diffusivity (m2/s) of A though B

[Assumed constant]Note: DAB ≠ DBA

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Integration for a solid slab at steady state:

For the case of diffusion radially through a cylinder wall of inner radius, r1 and outer r2

and length L:

12

21

zz

ccDN AAAB

A

dr

dcD

rLπ

N AAB

A

2

12

21

2

rrln

LπccDN AAABA

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The diffusion coefficient DAB in the solid is not dependent upon the pressure of the gas or liquid on the outside of the solid.The solubility of a solute gas (A) in a solid

is usually expressed as S m3 solute (at STP of 00C and 1atm) / m3 solid per atm partial pressure of A.

solidm

molAkg

.

pSatmp

Amolkg/)STP(m.

atm.solidm/STPmSc A

AA 33

33

4142241422

solidcm

molAg

.

pS A

341422

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EXAMPLE 6.5-1

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5.2.2 Permeability equations for diffusion in solids

Diffusion of gases in solids are not given as diffusivities and solubilities but as permeabilities (PM) PM, m3 of solute gas (A) at STP (00C, 1

atm) diffusing per second per m2 cross sectional area through a solid 1 m thick under a pressure difference of 1 atmpressure.

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Fick‟s equation:

Figure 6.5-1

12

21

zz

ccDN AAAB

A

41422

11

.

spc A

A41422

22

.

spc A

A

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Substituting:

The permeability:

When there are several solids 1,2,3… in series & L1, L2, L3,… represent the thickness of each:

2

12

21

12

21

4142241422m.s/molkg

zz.

ppP

zz.

ppSDN AAMAAAB

A

m/atm.S.Cm.s

STPmSDP ABM 2

3

Cross section

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where pA1-pA2 is the overall partial pressure difference.

2211

21 1

41422 MM

AAA

P/LP/L.

ppN

in series

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EXAMPLE 6.5-2

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5.2.3 Experimental diffusivities, solubilities, & permeabilities

Accurate prediction of diffusivities in solids is generally not possible. Lack of knowledge of the theory of the

solid state. Experimental values are needed (see

Table 6.5-1).

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Table 6.5-1:Diffusivities & Permeabilities in solids

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For the simple gasses such as He, H2, O2, N2, CO2 with gas pressures up to 1 or 2 atm, the solubility in solids such as polymers and glasses generally follows Henry’s law.

S α PA1

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For these gasses, the diffusivity & permeability are independent of concentration & pressure.

Temperature effect: ln PM is approximately a linear function of 1/T.

The diffusion of one gas is approximately independent of the other gases present, such as O2 and N2.

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For metals (e.g. Ni, Cd, Pt) where gases such as H2, O2 are diffusing, it has been found experimentally that the flux is approximately proportional to (√pA1 -√pA2).

When water is diffusing through polymers, unlike the simple gases, PM may depend on the relative pressure difference.

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5.3.1 Diffusion of liquids in porous solidsThe effect of porous solids that have

pores or interconnected voids in the solid on diffusion.

Figure 6.5-2: A cross section of a typical porous solid

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In the situation where the voids are filled completely with liquid water, the concentration of salt in water in boundary 1 is cA1 & at point 2 is cA2.The salt, in diffusing through the water

in the void volume, takes a tortuous path which is unknown and > z2-z1 by a factor (τ) called tortuosity.Diffusion does not occur in the inert

solid.

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For a dilute solution / for diffusion of salt in water at steady state:

12

21

zzτ

ccDεN AAAB

A

Open void fraction

Diffusivity of salt in

waterA factor [corrects for the path longer than (z2-z1)]

Note: For inert-type solids, τ can vary from 1.5 to 5.

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An effective diffusivity:

s/mDτ

εD ABeffA

2

12

21

zz

ccDN AAeff,A

A

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EXAMPLE 6.5-3

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If the voids are filled with gases, then a somewhat similar situation exists [same equation can be used].

If the pores are very large so that diffusion occurs only by Fickian-type diffusion, then the equation becomes, for gases,

12

21

12

21

zzRTτ

ppDε

zzτ

ccDεN AAABAAAB

A

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τ must be determined experimentally. Diffusion is assumed to occur only through

the voids or pores and not through the actual solid particles.

A correlation of tortuosity (τ) versus the void fraction (ε) of various unconsolidated porous media of beds of glass spheres, sand, salt, talc, etc gives the following approximate values of τ for different values of ε.

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ε τ

0.2 2.00

0.4 1.75

0.6 1.65

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6.1 Derivation of basic equationIn previous sections, we considered

various mass transfer systemsThe concentration or partial pressure

at any point and diffusion flux were constant with time, hence at steady state.

Before steady state can be reached, time must elapse after the mass transfer process is initiated for the unsteady-state conditions to disappear.

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Refer to Figure 7.1-1:

Mass is diffusing in the x direction in a cube composed of a solid, stagnant gas, or stagnant liquid.

The cube having dimensions Δx, Δy, Δz

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For diffusion in the x direction:

∂cA/∂x means the partial of cA with respect to x / the rate of change of cAwith x when the other variable, time t is kept constant.

x

cDN A

ABAx

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A mass balance on component A in terms of moles for no generation:

Rate of input

Rate of output

Rate of accumulation

x

AABAxlx

x

cDN

xx

AABxAxlx

x

cDN

Δ

Δ

t

czyx AΔΔΔ

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Solving the equations:

The above holds for a constant diffusivity DAB.If DAB is a variable,

2

2

dx

cD

t

c AAB

A

x

xcD

dt

c AABA

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For diffusion in all three directions a similar derivation gives:

2

2

2

2

2

2

z

c

y

c

x

cD

t

c AAAAB

A

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For diffusion in one direction:

Dropping the subscripts A & B for convenience,

2

2

x

cD

t

c AAB

A

2

2

x

cD

t

c

Figure 7.1-2: Unsteady-state diffusion in flat plate

with negligible surface resistance

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6.3.1 Convection and boundary conditions at the surface

As shown in Figure 7.1-2, there was no convective resistance at the surface.A convective mass-transfer coefficient,

kc (similar to that of heat transfer):

LiLcA cckN 1

Mass transfer coefficient (m/s)

Bulk fluid concentration (kg mol A/m3)

Concentration in the fluid

just adjacent to the surface

of the solid (kg mol A/m3)

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In Fig. 7.1-3a, the case where a mass transfer coefficient is present at the boundary. The concentration drop across the fluid

is cL1-cLi. The concentration in the solid ci at the

surface is in equilibrium with cLi. The concentration cLi in the liquid

adjacent to the solid & ci in the solid at the surface are in equilibrium & are equal.

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Figure 7.1-3: Interface conditions for convective mass transfer & an equilibrium distribution coefficient K=cLi/ci

(a) K=1, (b) K>1, (c) K<1, (d) K>1 and kc=α.

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The concentrations are in equilibrium & are related by:

K value in Fig. 7.1-3a is 1.0.The distribution coefficient K>1 & cLi>ci,

even though they are in equilibrium (see Fig. 7.1-3b).Other cases are shown in Fig.7.1-3 c & d.

i

Li

c

cKEquilibrium distribution

coefficient (Henry‟s law coefficient)

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6.3.2 Relation between mass & heat transfer parameters

Table 7.1-1:The relations between these variables

are tabulated.For K≠1.0, whenever kc appears, it is

given Kkc, and whenever c1 appears, it is given as c1/K.

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Table 7.1-1: Relation between mass & heat transfer parameters for unsteady-state diffusion.

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6.3.3 Charts for diffusion in various geometries.

The various heat-transfer charts for unsteady-state conduction can be used for unsteady-state diffusion:

No Geometries Chart

1. Semi-infinite solid Fig. 5.3-3

2. Flat plate Fig. 5.3-5 &-6

3. Long cylinder Fig. 5.3-7 &-8

4. Sphere Fig. 5.3-9 &-10

5. Average concentration,zero convective resistance

Fig. 5.3-13

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EXAMPLE 7.1-1

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EXAMPLE 7.1-1

(b) Half the thickness, X = 0.658 / (0.5)2

= 2.632, n=0, m=0, Y=0.0020, c = 2.0 x 10-4 kg mol/m3.

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EXAMPLE 7.1-2

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EXAMPLE 7.1-2

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Diffusion in a rectangular block in the x, y, z directions:

01

1

cKc

cKcYYYY

z,y,x

zyxz,y,x

Concentration at the point x, y, z from the center of the block

Obtained from Fig, 5.3-5 or -6

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7.1 Introduction to convective mass transfer

In previous sections, molecular diffusion in stagnant fluids or fluids in laminar flow was emphasized.

In many cases the rate of diffusion is slow & more rapid transfer is desired. The fluid velocity is

increaseduntil turbulentmass transfer occurs.

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To have a fluid in convective flow usually requires the fluid to be flowing past another immiscible fluid or a solid surface. A fluid flowing in a pipe, where part of

the pipe wall is made by a slightly dissolving solid material like benzoic acid.

The benzoic acid dissolves & is transported perpendicular to the main stream from the wall.

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When a fluid is in turbulent flow & is flowing past a surface, the actual velocity of small particles of fluid cannotbe described clearly as in laminar flow.

In laminar flow the fluid flows in streamlines & its behavior can usually be described mathematically.

In turbulent motion there are no streamlines; instead there are large eddies or „chunks‟ of fluid moving rapidly in seemingly random fashion.

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When a solute A is dissolving from a solid surface, there is a high concentration of this solute in the fluid at the surface, and its concentration, in general, decreases as the distance from the wall increases.

Figure 7.2-1: Concentration profile in turbulent mass transfer from a surface to a fluid.

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Three regions of mass transfer:1. Adjacent to the surface: A thin, viscous sublayer film is present.Most of the mass transfer occurs by

molecular diffusion, since few or no eddies are present.A large concentration drop occurs

across this film as a result of the slow diffusion rate.

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2. Buffer region:Some eddies are present, and the mass

transfer is the sum of turbulent and molecular diffusion.

3. Turbulent region:Most of the transfer is by turbulent

diffusion, with a small amount by molecular diffusion.The concentration decrease is very

small since the eddies tend to keep the fluid concentration uniform.

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7.2.1 Definition of mass-transfer coefficient

For turbulent mass transfer for constant, c:

εM is variable & near zero at the interface or surface and increases as the distance from the wall thickness.

z

cεDJ A

MAB

*

A

Molecular diffusivity (m2/s)

Mass eddy diffusivity (m2/s)

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An average value εM since the variation of εM is not generally known. Integrating between points 1 & 2,

The flux JA1* is based on the surface area A1 since the cross-sectional area may vary.The distance of the path (z2-z1) is often

not known.

21

12

1 AAMAB*

A cczz

εDJ

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In simplified form:

Experimental mass-transfer coefficient:

211 AA

'

c

*

A cckJ

The flux of A from the surface A1

relative to the whole bulk phase

The concentration at point 2 (kg mol A / m3), or the average bulk concentration, cA2

12 zz

εDk MAB'

c

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The flux of A relative to stationary coordinates:

For the case of equimolar diffusion, NA=-NB, and integrating at steady state,

BAAA

MABA NNxdz

dxεDcN

21 AA

'

cA cckN

12 zzεDk MAB

'

c

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Often, the concentration is defined as mole fraction (if a liquid or gas); partial pressure (if a gas).If yA is mole fraction in a gas phase and

xA in a liquid phase. For equimolar counterdiffusion:Gases:

Liquids:

212121 AA

'

yAA

'

GAA

'

cA yykppkcckN

212121 AA

'

xAA

'

LAA

'

cA xxkcckcckN

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Substituting yA1=cA1/c; YA2=cA2/c:

2121

2121

AA

'

yAA'

y

AA

'

yAA

'

cA

ccc

k

c

c

c

ck

yykcckN

c

kk

'

y'

c

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Table 7.2-1: Flux equations & Mass-transfer coefficient.

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In the case, NB=0, for steady state:

2121 AAcAA

BM

'

cA cckcc

x

kN

2121 AAxAA

BM

'

xA xxkxx

x

kN

Mass transfer coefficient for A diffusing through stagnant B

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Rewrite:Gasses:

Liquids:

12

12

BB

BBBM

xxln

xxx

212121 AAyAAGAAcA yykppkcckN

212121 AAxAALAAcA xxkcckcckN

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All the mass-transfer coefficients can be related to each other:

Hence,

c

c

c

ckxxkcc

x

kN AA

xAAxAA

BM

'

cA

212121

c

k

x

k x

BM

'

c

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EXAMPLE 7.2-1

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EXAMPLE 7.2-1

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EXAMPLE 7.2-1