mass transfer summary

Lecture 2

Mass Transfer Coefficients

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Mass Transfer Coefficients and other rate

coefficients

Effect Basic Equation Coefficient

Mass transfer mass transfer coefficent

k([=]L/t) is a function of

flow

Diffusion diffusion coefficient

D([=]L2/t) is a property

independent of flow

Dispersion dispersion coefficient

E([=]L2/t) depends on flow

Homogeneous chemical

reaction

rate constant k1([=]1/t) is a

physical property

independent of flow

Heterogeneous chemical

reaction

rate constant k1([=]L/t) is a

surface property often

defined in terms of a bulk

concentration

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1 1N k c

1i D c j

1 1' 'ic E c v

1 1 1r c

1 1 1r c

Analysis

mass transfer coefficient is a combination of diffusion

and dispersion

however, the diffusion and dispersion coefficients

have dimensions of length squared per time, because

their driving force is a gradient in concentration while,

mass transfer coefficient has dimensions of length per

time.

There are still inconsistencies in the accepted dimensions

of the mass transfer coefficient, unlike the case for the

others.

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Other definitions of mass transfer coefficients

Basic Equation Typical units of k+ Remarks

cm/sec Common in earlier literature, used for its

simple physical significance (Treybal,

1980)

mol/cm2-sec-atm Common for gas absorption; equivalent

forms occur in biological problems

(McCabe Smith, and Harriot,

1985;Sherwood, Pigford and Wilke,

1975)

mol/cm2-sec Preferred for practical calculations,

especially in gases (Bennett and Myers,

1974)

cm/sec Used in an effort to include diffusion-

induced convection (Bird, Stewart and

Lightfoot, 1960)

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1 1N k c

1 1pN k p

1 1xN k x

1 1 1

oN k c c


Usually use the simplest definition k

For design of gas adsorption, distillation and extraction

equipment, use alternative forms such as kx and kp

The given definitions are ambiguous since (take the

example of a gas absorption column)

concentration difference (local? average?) must be defined

interfacial area may be unkown ( use lumped k)

there may be complications introduced by diffusion-induced

convection normal to the interface in concentrated solutions

(use the last definition in the table)

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In the above unit NH3 is separated from a gas stream by

washing the gas with water


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1. The concentration difference between bulk and interface is different along the column. A local MTC should be used

Local concentration difference Local mass transfer coefficient

The local k does not change much compared to other variables.

In case there is no sufficient information to determine the local MTC, use the average MTC

Example: packed bed

Consider 0.2-cm diameter spheres of benzoic acid packed into a bed. The packed bed of spheres has a surface area: volume ratio of 23 cm2 per 1cm3. Pure water flows into the bed at a superficial velocity of 5 cm/s. The water becomes 62% saturated with benzoic acid after passing through 100 cm of bed length.

What is the mass transfer coefficient?

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Example: packed bed

select the appropriate ∆c

Always the difference between the

concentration ON the sphere and

that IN the solution is selected.

However, this ∆c is different

along the bed axis.

At the bed’s entrance:

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1 1( )0

satN k c

Example: packed bed

The flux N1 can be calculated by

mass balance:

Benzoic acid left the spheres:

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1N area time

_ sec __

areaarea bed length cross tional area

sphere volume

2

323 100

cmarea cm A

cm

_ 100

5 /

bed length cmtime

velocity cm s

Example: packed bed

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Example: packed bed

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1 1( ) 1satN k c c Divide by A∆z with ∆z0

Example: packed bed

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2. The interfacial area between water

and gas is unknown

Hard to define the flux per unit

areaLumping the area with k

3. Diffusion induced convection will also

affect k.

Example: Averaging a mass transfer coefficient

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Imagine we have a porous solid containing a solution of concentration c1while outside the concentration is c1∞

Diffusion will take place (unsteady state) and the flux is

As a result the mass transfer coefficient is:

• After a long time t0 the average flux can be defined as:

1 1 1( ) N D t c c

k D t

1 1N k c

Example: Averaging a mass transfer coefficient

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Howis k related to k ?

0 0

0

1 10 0

1

00

/t t

t

N dt D t c dtN

tdt

1/2

0 1

1 0 1

0

2 /2 /

D t cN D t c

t

2k k

Correlations of Mass Transfer

For prediction of how the value of the mass transfer

coefficient changes with some process variables.

The relevant process variables may be numerous

and may have complicating interrelationships

It is convenient to reduce the number of variables

and interrelationships by the use of dimensionless

variables.

The mass transfer correlations are usually expressed

in terms of dimensionless variables.

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MTC Correlations

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Excellent for preliminary design of small pilot plants.

For design of full scale equipment you must

supplement them with data of the SPECIFIC

chemical system.

Fluid-Fluid interface Fluid-Solid interface

Dimensionless Numbers

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Mass Transfer Coefficient:

Sherwood No. (Sh = kl/D)

Stanton No. (St = k/υo) Different Kinds of Diffusion:

Schmidt No. (Sc = n/D)

Lewis No. (Le = a/D)

Prandtl No. (Pr = n/a)

Flow:

Reynolds No. (Re = lυor/m lυo/n)

Grashof No. (Gr = (l3gr/r)/v2)

Peclet No. (Pe = υol/D) Diffusion with Chemical

Reaction:

2nd Damköhler No. or

(Thiele Modulus) 2

(Da =l2/D)

Frequently Used Correlations (Fluid-Fluid Interfaces)

Physical situation Basic correlation Key variables

Liquid in a packed tower a , d

d

d

Gas in a packed tower a, d

d, e

Pure gas bubbles in a

stirred tank

d, P/V

Pure gas bubbles in an

unstirred tank

d, Dr

Large liquid drops rising in

unstirred solutions

d, Dr

Small liquid drops rising in

unstirred solution

d, υo

Falling films z, υo

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0.671 3 0.50 0.4

0.0051 ok l vg v av D v ad

0.45 0.5

25 okd D dv v v D

0.3 0.5o ok v dv v D va

0.70 1 3 2.0

3.6 ok aD v av v D ad

0.640.36 1 3

1.21 1 okd D dv v v D

1 4 1 34 30.13kd D P V d v v Dr

1 3 1 33 20.31 /kd D d g v v Dr r

1 3 0.53 20.42 /kd D d g v v Dr r

0.8

1.13 okd D d D

0.5

0.69 okz D z D

Frequently Used Correlations (Solid-Fluid Interfaces)

Physical situation Basic correlation Key variables

Membrane l

Laminar flow flat plat L, υº

Turbulent flow horiz slit d, υº

Turbulent flow circulr tube d, υº

Laminar flow circular tube d, υº, L

Flow outs // capillary bed de , υº

Flow outs capillary d, υº

Forced conv around sphere d, υº

Free conv around sphere d,g

Packed beds d, υº

Spinning disc d, ω

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1kl D

1 2 1 3

0.646 okL D L v v D

0.8 1 3

0.026 okd D d v D n

0.8 1 3

0.026 okd D d v v D

1 3

21.62 okd D d LD

0.93 1 321.25 o

ekd D d lv v D

1 4 1 33 22.0 0.6 /kd D d g v v Dr r

0.47 1 3

0.80 okd D d v v D

1 2 1 3

2.0 0.6 okd D d v v D

0.42 2 3

1.17o ok d v D n

1 2 1 320.62kd D d v v D

Note: mass transfer coefficient is the value averaged over the length

Dissolution rate of a spinning disc

A solid disc of benzoic acid 2.5 cm in diameter is

spinning at 20 rpm and 25oC. The diffusion

coefficients are 1.00 x 10-5 cm2/sec in water and

0.233 cm2 / sec in air. The solubility of benzoic

acid in water is 0.003 g/cm3; It’s equilibrium vapor

pressure is 0.30 mm Hg.

How fast will it dissolve in a large volume of water?

How vast will it dissolve in a large volume of air?

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Dissolution rate of a spinning disc

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Correlation for spinning

disc

Example: Gas scrubbing with a wetted-wall column

Air containing a water-soluble vapor is flowing up

and water is flowing down in the experimnental

column . The water flow iin te 0.07-cm-thick film is

3 cm/sec the column diameter is 10 cm, and the air

is essentially well-mixed right up to the interface.

The diffusion coefficient in water of the absorbed

vapor is 1.8 x 10-5 cm2/s.

How long a column is needed to reach a gas

concentration in water that is 10% of saturation?

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Example: Measuring stomach flow

We want to estimate the average flow in the stomach

by measuring the dissolution rate of a non-

absorbing solute present as a large spherical pill.

From in vitro experiments, we know that this pill’s

dissolution is accurately described with a mass

transfer coefficient.

How can we do this estimate.

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Example: Glucose uptake by red blood cells

The uptake of glucose across the red blood cell membrane has a maximum rate ranging from 0.1 to 5 mmole/cm2-hr. Apparently, these differences result from differences in experimental conditions. Assume that a typical experiment is made in a beaker containing 100 cm3 of red blood cells suspended in 1 liter of plasma. The beaker is stirred with a 1/50 hp motor. The cells originally contain little glucose. At time zero, radioactively tagged glucose is added and its uptake measured. The diffusion coefficient of glucose is about 6 x 10-6 cm2/s, and the plasma viscosity is approximately that of water.

Using the correlation for liquid drops, estimate the effect of mass transfer in the bulk to see when it could have affected these uptake rates.

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Dimensional Analysis

Method developed by Bridgeman, (1922) and

Becker (1976)

Done when existing correlations are inadequate

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Example: Aeration

Oxygen is injected into an aqueous solution and the steady–state oxygen concentration in the bulk is measured as a function of position in the bed using oxygen-selective electrodes. Different experiments are done by varying the bubble velocity υ, the solution density r, viscosity μ, the entering bubble diameter d, and the depth of the bed L.

Using dimensional analysis, derive the form of the expression to correlate the mass transfer coefficient with the variables mentioned above.

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Example: Artificial Kidney

An artificial kidney is basically a long tubular membrane where blood flowing through it is dialyzed against a well-stirred saline solution outside the tube. Toxins in the blood diffuse across the membrane into the saline solution, thus purifying the blood. This dialysis is often slow. To increase the rate of toxin removal, the agitation rate of the saline solution is increased and the membrane made as thin as possible.

After reducing these mass transfer effects to almost negligible, we can correlate the mass transfer coefficient as a function of blood flow υ, tube size d, density and viscosity.

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Mass Transfer Across Interfaces

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hot benzene

cool water

INITIAL CONDITIONS

FINAL CONDITIONS AFTER ALLOWING EQUILIBRATION

warm benzene

warm water

benzene w

bromine

water w

bromine

equal bromine

concentrations

higher conc Br in

benzene

lower conc Br in

water

low conc Br in

air

high conc Br in

water

higher conc Br in

air

lower conc Br in

water


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Concentration Profiles for Mass Transfer Across a Gas-

Liquid Interface

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The flux in the gas is:

N1=kp (p10-p1i)

Because the interfacial

region is thin it is at steady

state then the flux will be

equal to that in the liquid.

N1=kL (c1i-c10)

GAS LIQUID

p10 c10

c1i

p1i

Flux

The Overall Mass Transfer Coefficient

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So the flux N1 should be derived as:


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*

1 10 1

*

1 10

1

1

p

p

p L

N K p p

Kk H k

p Hc

*

1 1 10

* 101

1

1 1

L

L

L p

N K c c

Kk Hk

pc

H

Flux of specie 1

Overall mass transfer

coefficient

equilibrium relation

overall based on

Liquid

concentrations

overall based on

Gas

concentrations


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There are two limiting cases of interest:

a highly soluble gas (H<<1):

a sparingly soluble gas (H>>1):

p pK = k

L LK = k

Oxygen Mass Transfer

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5 2

3

2.1 10 /

0.01 0.01 18

LL

D x cm s molk

cm cm cm

4 21.2 10 /

Lk x mol cm s

Oxygen Mass Transfer

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Benzene mass transfer

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Estimate the overall liquid-side mass transfer

coefficient in the distillation of benzene and toluene.

At the concentrations used, you expect a temperature

of 90oC and at equilibrium

y*=0.70x1+0.39

The molar volume of the liquid is about 97cm3/mol.

Assume that the thickness in the liquid is 0.01cm and

0.1cm in the gas.

Perfume extraction

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Reading Assignment

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Overall mass transfer coefficients in packed tower

Theories of mass transfer

Film theory

Penetration theory

Surface-renewal theory

mass transfer summary

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