EXPERIMENTAL MEASUREMENT OF

MASS TRANSFER COEFFICIENTS

The mass transfer coefficient, kc, can be studied in experimental devices in which the area of the contact between phases is known.

The wetted-wall tower is one of the instruments used in practice.

It can give valuable information on mass transfer to and from fluids in turbulent flow.

2 EKC 217: Mass Transfer Coefficients

Wetted-wall tower

3 EKC 217: Mass Transfer Coefficients

Generally, the gas enters the bottom of the tower and flows countercurrent to the liquid, but parallel flow (concurrent) can also be used.

Experiments were conducted for obtaining the numerical values of cA or xA. These values can then be used to calculate the mass transfer coefficient.

Measurement of the rate of evaporation of the liquid into the gas stream over the known surface permits calculation of the mass transfer for the gas phase.

The use of different gases and liquids provides variation of Schmidt number, Sc.

4 EKC 217: Mass Transfer Coefficients

Mass transfer coefficient, k would depend on: 1) Diffusivity, DAB 2) Velocity, 3) Viscosity, 4) Density, 5) Linear dimension, D

),,,,( DDk ABor:

Dimensional analysis gives:

Shnumber, Sherwood,1

ABAB

c

D

DG

D

Dk

where: G = ------ (1.84)

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Mass transfer with flow inside pipes

Correlations for mass transfer to the inside wall of a pipe is given by Sherwood number, Sh: a) For laminar flow (Re < 2100):

3/1'Gz76.1Sh ------ (1.85)

where:

L

D

LD

m

AB

ScRe4

'Gz number, Graetz

------ (1.86)

* Recall: ABD

Sc

DRe

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b) For turbulent flow (Re > 2100):

14.0

3/18.0 ScRe023.0Sh

w

------ (1.87)

The term /w is usually about 1.0 for mass transfer, hence can be omitted from Eq. (1.87). c) For flow inside wetted-wall towers: Data for evaporation of several liquids in wetted-wall towers were correlated with slightly higher exponents for both Re and Sc numbers:

44.081.0 ScRe023.0Sh ------ (1.88)

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A correlation for mass transfer at high Schmidt number (430 to 100000) was obtained by measuring the rate of solution of tubes of benzoic acid in water and viscous liquid:

346.0913.0 ScRe0096.0Sh ------ (1.89)

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Example 9: a) What is the effective thickness of the gas film for the

evaporation of water into air in a 2-in.-diameter wetted-wall column at a Reynold’s number of 10,000 and a temperature of 40C?

b) Repeat the calculation for the evaporation of ethanol under the same conditions. At 1 atm, the diffusivities are 0.288 cm2/s for water in air and 0.145 cm2/s for ethanol in air.

Solution: For air at 40C, 33

3g/cm10x129.1

313

273x

/molecm 22410

g/mole29

Recall: Molar volume at STP = 22.41 m3/kmole

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From Appendix 8 (McCabe, Smith and Harriott):

cP0186.0

/scm165.0g/cm10x1.129

sg/cm10x86.1 2

33-

-4

a) For the air-water system:

573.0288.0

165.0Sc

ABD

From Eq. (1.88):

3.31)573.0()10000(023.0Sh 44.081.0

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In the film theory, from Eq. (1.67):

T

ABc

z

Dk and since

AB

c

D

DkSh

Therefore:

in.064.03.31

0.2or Sh T

T

zz

D

b) For the air-ethanol system:

14.1145.0

165.0Sc

3.42).141()10000(023.0Sh 44.081.0

Similar to a), determine the Sc and Sh numbers:

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in.047.03.42

0.2Tz

Finally, calculate the effective thickness of the gas film:

Note that thickness, zT become smaller as the diffusivity decreases with mass transfer coefficient, kc.

Additional info: Gas density is the mass of one mole of the gas divided by its volume (22.4 L) at STP. Molar volume of ideal gases at STP (0C , 1 std atm) = 22.4 L/gmole

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Exercise: a) Calculate the steady-state mass transfer coefficient for

diffusion from the wall of a smooth 3-cm tube when the average fluid velocity = 1.8 m/s, = 1050 kg/m3, = 5 cP and DAB = 2.0 x 10-6 cm2/s.

b) Calculate the effective film thickness, zT.

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INTERPHASE MASS TRANSFER

In many separation processes, material must diffuse from one phase to another phase (e.g.: gas absorption operation which occurs when ammonia is dissolved from an ammonia-air mixture by liquid water).

The rates of diffusion in both phases affect the overall rate of mass transfer.

Equilibrium is assumed at the interface, and the resistances to mass transfer in the two phases are added to get the overall resistance.

The overall coefficient is actually the reciprocal of the overall resistance.

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Consider a situation at a particular level along the tower, e.g.: at a point midway between top and bottom. Since the solute is diffusing from the gas phase into the liquid, there must be a concentration gradient in the direction of mass transfer within each phase.

Figure 4: Concentration gradients near a gas-liquid interface

Figure 4 shows graphically in terms of the distance through the phases. The concentration of A in main body of the gas is yA,G

and it falls to yAi at the interface. In the liquid, the concentration falls from xAi at the interface to xA,L in the bulk liquid.

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The rate of transfer to the interface is set equal to the rate of the transfer from the interface:

)( ,LAAixA xxkN

)( , AiGAyA yykN

------ (1.90)

------ (1.91)

The rate of transfer is also set equal to an overall coefficient Ky times an overall driving force yA,G – y*A :

)( *

, AyyKN GAyA ------ (1.92)

where y*A is the composition of the vapor that would be in equilibrium with the bulk liquid of composition xA,L.

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To get Ky in terms of ky and kx, Eq. (1.92) is rearranged and the term yA,G – y*A is replaced by (yA,G – yAi) + (yAi – y*A):

A

AAi

A

AiGA

A

AGA

y N

yy

N

yy

N

yy

K

*,

*

,1

------ (1.93)

Eq. (1.91) and (1.92) are used to replace NA in the last two terms of Eq. (1.93):

)()(

1

,

*

,

,

LAAix

AAi

AiGAy

AiGA

y xxk

yy

yyk

yy

K

------ (1.94)

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Figure 5: Overall concentration differences.

Figure 5 shows the typical values of composition at the interface, and it is apparent that (yAi – y

*A)(xAi – xA,L) is actually the local slope of the equilibrium curve, which is denoted by m’.

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Then, Eq. (1.94) can be written as:

xyy k

m

kK

'11 ------ (1.95)

Overall resistance to mass transfer

Resistance in the liquid film Resistance in the

gas film

Similarly, the overall coefficient Kx can be defined as follows:

)( ,

*

LAAxA xxKN ------ (1.96)

yxx kmkK "

111 ------ (1.97)

where m” is the slope of the chord MD in Figure 5.

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Eq. (1.95) and (1.97) lead to the following relationship between the mass transfer resistance:

y

y

K

k

/1

/1

phasesboth ,resistance Total

phase gasin Resistance

x

x

K

k

/1

/1

phasesboth ,resistance Total

phase liquidin Resistance

------ (1.98)

------ (1.99)

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