CP302 Separation Process Principles
Mass Transfer - Set 6
Course content of
Mass transfer section
Diffusion
Theory of interface mass transfer
Mass transfer coefficients, overall coefficients
and transfer units
Application of absorption, extraction and
adsorption
Concept of continuous contacting
equipment
Simultaneous heat and mass transfer in gasliquid contacting, and solids drying
20 Oct 2011
Prof. R. Shanthini
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Plate columns are used mainly in distillation
You probably know how
to calculate the number
of plates required for the
desired separation.
20 Oct 2011
Prof. R. Shanthini
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Packed columns are used in absorption, stripping
and adsorption
There are no plates here.
We need to calculate the
height of packing in a packed
column where the separation
process takes place.
How to do that?
20 Oct 2011
Prof. R. Shanthini
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Equations for Packed Columns
Treated gas
Gs,out, Gout, Yout, yout
Counter-current packed
column used for absorption
Inlet gas
Gs,in, Gin, Yin, yin
20 Oct 2011
Prof. R. Shanthini
Inlet solvent
Ls,in, Lin, Xin, xin
Gs
G
Y
y
Ls
L
X
x
Spent solvent
Ls,out, Lout, Xout, xout
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Notations
Gs
Ls
G
L
Y
y
X
x
- inert gas molar flow rate (constant)
- solvent molar flow rate (constant)
- total gas molar flow rate (varies as it looses the solute)
- total liquid molar flow rate (varies as it absorbs the solute)
- mole ratio of solute A in gas
= moles of A / moles of inert gas
- mole fraction of solute A in gas
= moles of A / (moles of A + moles of inert gas)
- mole ratio of solute A in liquid
= moles of A / moles of solvent
- mole fraction of solute A in liquid
= moles of A / (moles of A + moles of solvent)
Solute in the gas phase = Gs Y = G y
Solute in the liquid phase = Ls X = L x
20 Oct 2011
Prof. R. Shanthini
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Relating Y to y and X to x:
moles of solute A
y = moles of solute A + moles of inert gas
1
=
1 + (moles of inert gas / moles of solute A )
=
1
1 + (1/Y)
=
Y
Y+1
Therefore y = Y / (Y+1)
Similarly, it can be shown x = X / (X+1)
20 Oct 2011
Prof. R. Shanthini
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For dilute mixers
Y
y
X
x
- mole ratio of solute A in gas
= moles o fA / moles of inert gas
- mole fraction of solute A in gas
= moles of A / (moles of A + moles of inert gas)
- mole ratio of solute A in liquid
= moles of A / moles of solvent
- mole fraction of solute A in liquid
= moles of A / (moles of A + moles of solvent)
For dilute mixtures, y ≈ Y
x≈X
20 Oct 2011
Prof. R. Shanthini
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For dilute mixers
Gs
Ls
G
L
- inert gas molar flow rate (constant)
- solvent molar flow rate (constant)
- total gas molar flow rate (varies as it looses the solute)
- total liquid molar flow rate (varies as it absorbs the solute)
For dilute mixtures, Gs ≈ G
Ls ≈ L
20 Oct 2011
Prof. R. Shanthini
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Equations for Packed Columns for dilute solutions
The operating equation for the Treated gas
packed column is obtained by Gout, yout
writing the mass balance for
solute over the control volume:
(74)
Lin xin + G y = L x + Gout yout
If dilute solution is assumed, then
Lin = L = Lout and Gin = G = Gout.
Therefore, the above becomes
L xin + G y = L x + G yout
G
y
Inlet solvent
Lin, xin
L
x
Control
volume
(75)
The operating line therefore becomes
y = (L / G) x + yout - (L / G) xin
20 Oct 2011
(76)Inlet gas
Gin, yin
Prof. R. Shanthini
Spent solvent
Lout, xout
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Equations for Packed Columns for dilute solutions
Mass transfer of solute from gas phase to liquid phase is
assumed to be explained by the Two-film Theory:
20 Oct 2011
Prof. R. Shanthini
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Equations for Packed Columns for dilute solutions
Rate of mass transfer from vapour phase to liquid phase is
therefore given by the following:
ky a (y - yi) = kx a (xi - x)
(77)
ky and kx are film mass transfer coefficients based on unit area.
But the area for mass transfer in a packed column in difficult to
determine.
Therefore we use a factor ‘a’ which gives the area of mass
transfer per unit volume of packed bed.
In packed columns, we use ‘kya’ and ‘kxa’ which are the film
mass transfer coefficients based on unit volume of packed bed.
Molar fractions yi and xi are the interface properties that are
related by the equilibrium ratio K (= yi / xi). (78)
20 Oct 2011
Prof. R. Shanthini
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Equations for Packed Columns for dilute solutions
Operating line:
y = (L / G) x + yout - (L / G) xin
Equilibrium line:
K = yi / xi
(76)
(78)
Relating equilibrium mole fractions to operating mole fractions:
y - yi
kx a
Equation (77) gives
=x - xi
ky a
y
Operating line
kx a
Slope = - k a
y
(x,y)
(slope = L/G)
yout - (L / G) xin
20 Oct 2011
(xi,yi)
Prof. R. Shanthini
Equilibrium line
(slope = K)
x
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Equations for Packed Columns for dilute solutions
y* is the gas-phase mole fraction that would have been in
equilibrium with the liquid-phase mole fraction x.
y* = K x
x* is the liquid-phase mole fraction that would have been in
equilibrium with the gas-phase mole fraction y.
y = K x*
yi – y*
= x -x
i
y - yi
K = x* - x
i
y
(x,y)
(x*,y)
yout - (L / G) xin
(xi,yi)
(x,y*)
20 Oct 2011
Prof. R. Shanthini
x
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Equations for Packed Columns for dilute solutions
1
1
=
+
Kxa
kxa
1
kya
x* - xi
y - yi
1
=
+
kxa
1/K
kya
(77)
1
1
=
+
Kya
kya
1
kxa
yi – y*
xi - x
1
=
+
kya
K
kxa
(78)
Since the liquid usually has strong affinity for the
solute, mass transfer resistance is mostly in the gas.
Therefore, determination of the packed height of a
column most commonly involves the overall gasphase mass transfer coefficient based on unit volume
of packed base, which is Kya.
20 Oct 2011
Prof. R. Shanthini
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Equations for Packed Columns for dilute solutions
Treated gas
Gout, yout
Let us now obtain equations
required to determine the
height of packing required for
a specified separation in a
packed column operated with
dilute solution.
Inlet solvent
Lin, xin
G
y
dz
G
y+dy
Prof. R. Shanthini
L
x+dx
Z
z
Inlet gas
Gin, yin
20 Oct 2011
L
x
Spent solvent
Lout, xout
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Equations for Packed Columns for dilute solutions
Mass of solute lost from the
gas over the differential height
of packing dz
= G y - G (y + dy) = - G dy
Treated gas
Gout, yout
Mass of solute transferred from
the gas to the liquid
= Kya (y – y*) S dz
where S is the inside crosssectional area of the tower.
Inlet solvent
Lin, xin
G
y
dz
G
y+dy
Therefore, mass balance for solute
gives the following:
-G dy = Kya (y – y*) S dz
(79) Inlet gas
Gin, yin
20 Oct 2011
Prof. R. Shanthini
L
x
L
x+dx
Z
z
Spent solvent
Lout, xout
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Equations for Packed Columns for dilute solutions
Rearranging and integrating
(79) gives the following:
Z
KyaS
G
0
Treated gas
Gout, yout
Inlet solvent
Lin, xin
yin
∫ ∫
dz
dy
y – y*
=
G
y
yout
dz
The packed height is therefore given
by:
y
G
y+dy
L
x+dx
Z
z
in
G
=
Z
KyaS
L
x
∫
dy
y – y*
yout
HOG
20 Oct 2011
NOG
Inlet gas
Gin, yin
Prof. R. Shanthini
Spent solvent
Lout, xout
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Equations for Packed Columns for dilute solutions
Definition of HOG and NOG:
G
KyaS
HOG ≡
yin
∫
dy
y – y*
NOG ≡
yout
(80) which is termed as the overall
height of a transfer unit (HTU)
based on the gas phase which
has the dimension of length
(81) which is termed as the overall
number of transfer units
(NTU) based on the has phase
which is dimensionless
Height of the packed column is written in terms of transfer
units as follows:
Z = HOG NOG
(82)
20 Oct 2011
Prof. R. Shanthini
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Equations for Packed Columns for dilute solutions
HTU: HOG =
G
KyaS
The smaller the HTU, the more efficient is
the contacting.
yin
NTU: NOG =
∫
dy
y – y*
yout
20 Oct 2011
It represents the overall change in
solute mole fraction divided by the
average mole-fraction driving force.
The larger the NTU, the greater is
the extent of contacting required.
Prof. R. Shanthini
19
Equations for Packed Columns for dilute solutions
yin
Integration of NOG =
∫
dy
y – y*
(81)
yout
Equilibrium relationship gives:
Operating line gives:
y* = K x
x = (G/L) y - (G/L) yout + xin
Using these the denominator in (81) can be written as follows:
y - y* = y - K x
= y - K(G/L) y + K(G/L) yout - K xin
= (1 - KG/L) y + (KG/L) yout - K xin
20 Oct 2011
Prof. R. Shanthini
(82)
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Equations for Packed Columns for dilute solutions
Combining (81) and (82), we get
yin
dy
NOG =
(1 - KG/L) y + (KG/L) yout - K xin
yout
∫
yin
=
ln[(1 - KG/L) y + (KG/L) yout - K xin]
(1 - KG/L)
yout
1
=
(1 - KG/L)
20 Oct 2011
(1 - KG/L) yin + (KG/L) yout - K xin
ln
yout - K xin
Prof. R. Shanthini
21
Equations for Packed Columns for dilute solutions
1
NOG =
(1 - KG/L)
=
1
(1 - KG/L)
1
=
(1 - KG/L)
20 Oct 2011
yin - K xin + KG/L (yout - yin)
ln
yout - K xin
ln
yin - K xin + KG/L (yout - K xin + K xin - yin)
yout - K xin
(1 - KG/L) (yin - K xin)
ln
+ KG/L
yout - K xin
Prof. R. Shanthini
(83)
22
Example 6.10 of Ref 2
Experimental data have been obtained for air containing
1.6% by volume of SO2 being scrubbed with pure water in a
packed column of 1.5 m2 in cross-sectional area and 3.5 m
in packed height. Entering gas and liquid flow rates are
0.062 and 2.2 kmol/s, respectively. If the outlet mole fraction
of SO2 in the gas is 0.004 and column temperature is near
ambient with KSO2 = 40, calculate the following:
a) The NOG for absorption of SO2
b) The HOG in meters
c) The volumetric, overall mass-transfer coefficient, Kya for
SO2 in kmol/m3.s
20 Oct 2011
Prof. R. Shanthini
23
Example 6.11 of Ref 2 (modified)
A gaseous reactor effluent consisting of 2 mol% ethylene
oxide in an inert gas is scrubbed with water at 30oC and 20
atm. The total gas feed rate is 2500 lbmol/h, and the water
rate entering the scrubber is 3500 lbmol/h. The column, with
a diameter of 4 ft, is packed in two 12-ft-high sections with
1.5 in metal Pall rings. A liquid redistributer is located
between the two packed sections. Under the operating
conditions for the scrubber, the K-value for ethylene oxide is
0.85 and estimated values of kya and kxa are 200 lbmol/h.ft3
and 2643 lbmol/h.ft3 , respectively. Calculate the following:
a) Kya
b) HOG and NOG
c) Yout and xout
20 Oct 2011
Prof. R. Shanthini
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