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Part III

ERT 216 HEAT & MASS TRANSFER

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7. Convection Mass Transfer Coefficients

8. Mass Transfer Coefficients for various geometries

9. Mass Transfer to Suspensions of Small Particles

Part III

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7. Convection Mass Transfer Coefficients7.3 Mass transfer coefficients for

general case of A and B diffusing and convective flow using film theory

7.4 Mass transfer coefficients under high flux conditions.

8. Mass Transfer Coefficients for various geometries

8.1 Dimensionless numbers used to correlate data

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8.2 Derivation of Mass-transfer coefficients in laminar flow

8.3 Mass transfer for flow inside pipes8.4 Mass transfer for flow outside solid

surfaces9. Mass Transfer to Suspensions of Small

Particles9.1 Introduction9.2 Equations for mass transfer to small

particles

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Assuming a simplified film theory where the mass transfer is assumed to occur through a thin film next to the wall of thickness f and by molecular diffusion.

The experimental value of kc for dilute solutions is used to determine the film thickness f :

f

AB'

c

Dk

bblee@UniMAP6

Rewriting:

The convective term is

Rearranging and integrating:

BAAAAB

A NNxdz

dxDcN

BAA NNx

2

10

1 A

A

f x

xBAAA

Az

zAB NNxN

dxdz

cD

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1

2

ABAA

ABAA'

c

BA

AA

xNNN

xNNNlnck

NN

NN

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In section 7.3, it is assumed that the film thickness is unaffected by high fluxes and bulk or convective flow (diffusion-induced convection). In the case of A diffusing through

stagnant, nondiffusing B where diffusion-induced convection is present.For the flux NA at the surface z = 0

where xA=xA1,01

0

AA

z

AABA Nx

dz

dxcDN

Not average value

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Defining a mass-transfer coefficient in terms of the diffusion flux,

Solving for NA,

At low concentrations & fluxes, kc0

approaches kc for no bulk flow:

21

0

0

AAc

z

AAB xxck

dz

dxcD

1

21

0

1 A

AAcA

x

xxckN

21 AA

'

cA xxckN

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A coefficient kc may be defined without regard to convective flow:

Combining:

The relationship between kc0 or kc for high flux and kc for low fluid will be derived using the film theory.

21 AAcA xxckN

cAc kxk 10 1

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Based on film theory, for transferring of A by molecular diffusion and convection flow with B being stagnant & nondiffusing.Setting NB=0,

For the film theory,

2121 AAcAA

BM

'

cA cckcc

x

kN

'

x

x

BM

'

c

c

k

k

xk

k 1

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Combining,

'

x

x

BM

A

'

c

c

k

k

x

x

k

k 010 1

EXAMPLE 7.2-2

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EXAMPLE 7.2-2

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8.1 Dimensionless numbers used to correlate data

Schmidt number (NSc) The ratio of the shear component for

diffusivity (/) to the diffusivity for mass transfer DAB.

It physically relates the relative thickness of the hydrodynamic layer and mass transfer boundary layer.

AB

ScD

N Viscosity

Density

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Schmidt number (NSc): the ratio of momentum diffusivity to

molecular diffusivity.

ABABAB

ScD

D

D

N

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Reynolds number (NRe) the most important dimensionless no. it indicates degree of turbulence.

LvNRe

Diameter (Dp) for a sphere or Diameter for a pipe or Length for a flat plate.

Mass average velocity (in a pipe) or Superficial velocity (v) in the empty cross section of a packed bed or Interstitial velocity (v/); =void fraction of bed.

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Sherwood number (Nsh): It is also known as mass transfer

Nusselt number. It represents the ratio of convective to

diffusive mass transport.

AB

'

x

AB

BMc

AB

'

cshD

L

c

k

D

Lyk

D

LkN

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Take the case of gas phase mass transfer for flow past a sphere, 1cm in diameter, at low partial pressure of the solute.

s/m

ms/m~y

D

DkN BM

AB

cSh 25

22

10

1010= 10

sm

sm

D

N

AB

Sc 25

25

10

10= 1

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For liquid phase mass transport in a similar geometry,

s/m

ms/m~y

D

DkN BM

AB

cSh 29

25

10

1010= 100

sm

sm

D

N

AB

Sc 29

26

10

10= 1000

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8.2.1 IntroductionWhen a fluid is flowing in laminar flow and

mass transfer by molecular diffusion is occurring.The equations are very similar to those for

heat transfer by conduction in laminar flow(But not always true).In mass transfer several components may

be diffusing, the flux of mass perpendicular to the direction of the flow must be small (not distort the laminar velocity profile).

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The mass transfer coefficients for laminar flow for certain geometries (e.g. flow past a cylinder or in a packed bed) are difficult to be described using mathematical approach. Experimental mass-transfer

coefficients are often obtained and correlated.

A simplified theoretical derivation will be given for two cases of laminar flow.

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8.2.2 Mass transfer in laminar flow in a tube

In the case of mass transfer from a tubewall to a fluid inside in laminar flow, the wall is made of solid benzoic acid which is dissolving in water.This is similar to heat transfer from a

wall to the flowing fluid where natural convection is negligible.

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For fully developed flow, the parabolic velocity derived as:

For steady state diffusion in a cylinder, a mass balance can be made on a differential element where the rate in by convection plus diffusion equals the rate out radiallyby diffusion:

2

2

12

1

R

rv

R

rvv

av

maxxVelocity in the x direction at the distance r from the center.

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2

2

2

21

x

c

r

c

r

c

rD

x

cv AAAAB

Ax

02

2

x

cA

If the diffusion in the x direction is

negligible compared to that by convection,

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Mass transfer of solute A into a laminar falling film, which is important in wetted wall columns, in developing theories to explain mass transfer in stagnant pockets of fluids, and in turbulent mass transfer.

Figure 7.3-1: Diffusion of solute A in a laminar falling film: (a) velocity profile & concentration profile.

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Figure 7.3-1: Diffusion of solute A in a laminar falling film: (b) small element for mass balance.

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The solute A in the gas is absorbed at the interface and then diffuses a distance into the liquid so that it has not penetrated the whole distance x= at the wall.At steady state the inlet concentration

cA=0.The concentration profile of cA at a point

z distance from the inlet is shown in Fig 7.3-1a.A mass balance on the element is shown in

Fig. 7.3-1b.

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We obtain:

To determine the local molar flux at the surface x=0 at position z from the top entrance,

2

2

x

cD

z

cv AAB

Az

z

vDc

x

cDzN maxABA

x

AABxAx 0

00

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The total moles of A transferred per second to the liquid over the entire length z=0 to z=L, where the vertical surface is unit width:

dzN.LNL

AxlxA

0

011

dzz

vDc

/

/

maxAB

L

A 21

21

0

0

11

L

vDc.L maxABA

41 0

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The rate of mass transfer is proportional to and

The basis for the penetration theory in turbulent mass transfer where pockets of liquid are exposed to unsteady-state diffusion (penetration) for short contact times.

L

max

tv

LTime of exposure of the liquid to the solute A in the gas.

50.

ABD501 .Lt

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8.3.1 Mass transfer for laminar flow inside pipes.

When a liquid or gas is flowing inside a pipe & NRe (Dv/) is below 2000, laminar flow occurs.

Figure 7.3-2: Data for

diffusion in a fluid in

streamline flow inside a

pipe.

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The dimensionless coordinates:or

The velocity profile is assumed fully developed to parabolic form at the entrance.

LD

W

AB

4LDNN ScReFlow (kg/s)

Length of mass transfer section (m)

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For liquids that have small values of DABData follow the parabolic flow lineFor W/DAB L > 400:

32

0

0 55

/

ABAAi

AA

LD

W.

cc

cc

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For turbulent flow when NRe (Dv/) is above 2100 for gases or liquids flowing inside a pipe,

The equation holds for NSc of 0.6 to 3000.Note: The NSc for gases is in the range 0.5-

3 and for liquids is above 100 in general.

AB

BMc

AB

'

cShD

D

P

k

D

DkN

330830

0230

.

AB

.

ShD

D.N

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When a gas is flowing inside the core of a wetted-wall tower, the same correlations that are used for mass transfer of a gas in laminar or turbulent flow in a pipe are applicable. The equations can be used for NRe up to

about 1200.

u

N

avz

Re

44

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EXAMPLE 7.3-1

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EXAMPLE 7.3-1

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The mass transfer and vaporization of liquids from a plate or flat surface to a flowing stream is of interest in the dryingof inorganic & biological materials, in the evaporation of solvents from paints, for plates in wind tunnels & in flow channels in chemical process equipment.When the fluid flows past a plate in a free

stream in an open space the boundary layer is not fully developed.

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8.4.1 Mass transfer in flow parallel to flat plates

For gases or evaporation of liquids in the gas phase & for the laminar region of NRe,L=Lv/ < 15 000, the data can be represented within 25% by:

Writing in terms of Sherwood number:

506640 .LRe,D N.J

31506640 /Sc.

LRe,Sh

AB

'

c NN.ND

LkLength of the plate in the direction of flow

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For gases & NRe,L of 15 000 30 000, the data are represented within 30% by JD=JH=f/2:

Experimental data for liquids are correlated within about 40% for a NRe,Lof 600 50 000 by:

200360 .LRe,D N.J

50990 .LRe,D N.J

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For flow past single spheres and for very low NRe=Dpv/ (v average velocity in the empty test section before the sphere), The NSc = kcDp/DAB should approach 2.0.

The mass transfer coefficient kc which is kc for a dilute solution:

2121

2AAcAA

p

ABA cckcc

D

DN

p

AB'

cD

Dk

2

Diameter

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Rearranging,

Note: Natural convection effects could increase kc

For gases, for NSc of 0.6 2.7 and NRe of 1 48 000:

02.ND

DkSh

AB

p

'

c

3153055202 /Sc.

ReSh NN.N

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For liquids, NRe of 2 to about 2000,

For liquids, NRe of 2000 17 000,

315009502 /Sc.

ReSh NN.N

3162034702 /Sc.

ReSh NN.N

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EXAMPLE 7.3-3

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EXAMPLE 7.3-3

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EXAMPLE 7.3-3

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EXAMPLE 7.3-3

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Mass transfer to and from packed beds occurs often in processing operations, including drying operations, adsorption or desorption of gases or liquids by solid particles such as charcoal, & mass transfer of gases & liquids to catalyst particles.By using a packed bed a large amount of

mass-transfer area can be contained in a relatively small volume.

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The void fraction () in a bed is m3 volume void space divided by the m3 total volume of void space plus solid.The values range from 0.3 to 0.5.Because of flow channeling, nonuniform

packing, accurate experimental data are difficult to obtain and data from different investigators can deviate considerably.

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For NRe range of 10 10 000 for gases in a packed bed of spheres, the recommended correlation with an average deviation of about 20% and a maximum of about 50%:

4069045480 .ReHD N

.JJ

'vDN

p

Re

Diameter of the spheres

Superficial mass average velocity in the empty tube

without packing.

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For NRe range of 0.0016 55 & NSc range of 165 70 000,

For liquids, NRe range of 55 - 1500 & NScrange of 165 10 690,

For liquids, NRe range of 10 - 1500,

32091 /ReD N

.J

310250 .ReD N

.J

4069045480 .ReD N

.J

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For fluidized beds of spheres, for gases and liquids, NRe range of 10 4000,

For liquids in a fluidized bed, NRe range of 1 10,

72010681 .ReD N.J

4069045480 .ReD N

.J

Nota: If packed beds of solids other than spheres are used, approximate correction factors can be used. Particle diameter = surface area of the solid particle.

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The total flux in a packed bed:1. JD is obtained 2. kc (m/s) is determined from JD3. Total volume Vb (m

3) of the bed [voids + solids]

4. Total external surface area A (m2) of the solids for mass transfer:

pD

a

16Surface area (m2) / total volume of bed (m3) [when the solids are spheres]

baVA

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5. To calculate the mass transfer rate, the log mean driving force at the inlet & outlet of the bed:

2

1

21

AAi

AAi

AAiAAicA

cc

ccln

ccccAkAN

Concentration at the surface of the solid

(kg mol / m3)

Inlet bulk fluid concentration (kg mol / m3)

Outlet bulk fluid

concentration (kg mol / m3)

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6. The material-balance equation on the bulk stream:

The equations can be used for a fluid flowing in a pipe or past a flat pipe, where A = the pipe wall area or plate area.

12 AAA ccVAN

Volumetric flow rate of fluid entering (m3/s)

bblee@UniMAP56

9.1 Introduction Mass transfer from/to small suspended

particles in an agitated solution occurs in a number of process applications. In liquid-phase hydrogenation, hydrogen

diffuses from gas bubbles, through an organic liquid, & then to small suspended catalyst particles.

In fermentation, oxygen diffuses from small gas bubbles, through the aqueous medium, & then to small suspended microorganisms.

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For liquid-solid suspension, increased agitation over & above that necessary to freely suspend very small particles has very little effect on the mass transfer coefficient, kL to the particle.When the particles in a mixing vessel are

just completely suspended, turbulence forces balance those due to gravity.

bblee@UniMAP58

With very small particles (m), which is the size of many microorganisms in fermentations and some catalyst particles, their size is smaller than eddies, which are about 100 m or so in size. increased agitation will have little

effect on mass transfer except at very high agitation.

bblee@UniMAP59

For a gas-liquid-solid dispersion (e.g. fermentation), increased agitation increases the number of gas bubbles and hence the interfacial area. The mass-transfer coefficients from

the gas bubble to the liquid and from the liquid to the solid are relatively unaffected.

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9.2.1 Mass transfer to small particles < 0.6mm

Equations for predicting mass transfer to small particles in suspension have been developed which cover 3 size ranges of particles:1) < 0.60mm2) >2.5 mm3) 0.60 - 2.5mm

bblee@UniMAP61

To predict mass-transfer coefficients from small gas bubbles (e.g. oxygen or air to the liquid phase or from the liquid phase to the surface of small catalyst particles, microorganisms, other solids, liquid drops):

31

2

323102

/

c

c/

Sc

p

AB'

L

gN.

D

Dk

Diffusivity

of the solute A in solution

(m2/s)

Diameter of the gas bubble / solid particle

(m)

Viscosity of the

solution (kg/m.s)

9.80665 m/s2

Density of continuous

phase (kg/m3)

Density of the gas or

solid particles (kg/m3)

bblee@UniMAP62

In aerated mixing vessels, the mass-transfer coefficients are essentially independent of the power input.

31

2

323102

/

c

c/

Sc

p

AB'

L

gN.

D

Dk

Molecular diffusion

Due to free fall / rise of the sphere due to gravitational

forces

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EXAMPLE 7.4-1

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EXAMPLE 7.4-1

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EXAMPLE 7.4-1

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As the power is increased, the bubble size decreases & the mass transfer coefficient continues to follow the above equation.The dispersions include those in which

the solid particles are just completely suspension of these small particles results in only small increase in kL.

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For large gas bubbles or liquid drops > 2.5mm,

Large gas bubbles are produced when pure liquid are aerated in mixing vessels & sieve plate columns.kL or kL is independent of the bubble size

and is constant for a given values of kLabout 3 4 times larger than that of small particles.

31

2

50420

/

c

c.

Sc

'

L

gN.k

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kL is essentially independent of agitation intensity in an agitated vessel and gas velocity in a sieve-tray tower.

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In mass transfer in the transition region between small & large bubbles in the size range of 0.6 - 2.5 mm, the mass transfer coefficient can be approximated by assuming that it increases linearly with bubble diameter.

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In the preceding 3 regions, the density difference between phases is sufficiently large to cause the force of gravity to primarily determine the mass-transfer coefficient.This includes solids just completely

suspended in mixing vessels.

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When agitation power is increased beyond that needed for suspension of solid or liquid particles & the turbulence forces becomes larger than the gravitational forces. Small increases in kL:

41

2

32 130

/

c

c/

Sc

'

L

VP.Nk

Power input per unit volume

bblee@UniMAP72

In the case of gas-liquid dispersions it is quite impractical for agitation systems to exceed gravitational forces.The experimental data are complicated by

the fact that very small particles are easily suspended, & if their size is on the order of the smallest eddies, the mass transfer coefficient will remain constant until a large increase in power input is added above that required for suspension.