CHG 3111
Unit Operation
Chapter 10
Gas-Liquid Separation and Operations
4.3 Integrated Approach
CHG 3111/B. Kruczek 2
Design of Packed Towers Integrated Approach Absorption and Stripping
Thus far the height of the packed tower (Z) was estimated using:
where: N = the number of theoretical equilibrium contacts and HEPT = the height of the theoretical stage
Equilibrium Relationship
Mass Transfer
Operating Conditions
Operating Conditions
Mass Balance
Equilibrium Relationship N HEPT x
Z
In the integrated approach
Equilibrium Relationship
Mass Balance
Mass Transfer
Operating Conditions
Z For dilute solutions it will simplify to:
CHG 3111/B. Kruczek 3
Absorption/Stripping in Packed Towers
moles of A leaving gas = moles of A entering liquid
Application of integrated approach for determination of Z:
Material balance on solute in a differential height (dz) of the tower:
or: A AG ALn d Vy d Lx
Both V and yAG as well as L and xAL change along the tower, but if liquid is not volatile and inert gas not soluble
in liquid, then:
1 1' constant and ' constantAG ALV V y L L x
Thus:
21 1 11
' ''AG AG AG AGA
AG AG AGAG
V y y V dy Vdyn d V d
y y yy
Similarly:
21 1 11
' ''AL AL AL ALA
AL AL ALAL
L x x L dx Ldxn d L d
x x xx
CHG 3111/B. Kruczek 4
Absorption/Stripping in Packed Towers
1
y
A A AG Ai
A iM
k an N dA y y Sdz
y
Application of integrated approach for determination of Z:
Combining mass balance with the rate equations:
Application of rate equation for a differential height (dz) of the tower:
where: dA aSdz
Tower Height
1
xA A Ai AL
A iM
k an N dA x x Sdz
x
or:
1 1
yAGAG Ai
AG A iM
k aVdyy y Sdz
y y
1 1
xALAi AL
AL A iM
k aLdxx x Sdz
x x
1
0 2 11
yZ
y y
i
iM
Vdydz Z
k aSy y y
y
yAG = y
Tower Height
1
0 2 11
xZ
x xi
iM
Ldxdz Z
k aSx x x
x
xAL = x
CHG 3111/B. Kruczek 5
Absorption/Stripping in Packed Towers Application of integrated approach for determination of Z:
Dropping the subscripts, i.e. xAL = x and yAG = y
Integrated approach can be used with the overall mass transfer coefficients
1
0 2 11
*
'*
yZ
y y
M
Vdydz Z
K aSy y y
y
1
0 2 11
*
'*
xZ
x xi
M
Ldxdz Z
K aSx x x
x
x*A
y*A
Slope = m''
E
D
Slope = m'
1 1
1 1 1*y A y A x A iMM iM
m
K a y k a y k a x
AGA
AGA
MAyy
yyy
11ln
111
*
*
*
1 1 1
1 1 1*x A y A x AM iMiM
K a x m k a y k a x
*
*
*11ln
111
AAL
AAL
MAxx
xxx
where:
and
NB1: The integrated approach allows the design of the tower in the case when the mass transfer coefficients and the total flow of phases vary along the column, which occurs for the concentrated solution(s)
NB2: Evaluation of the height of tower requires graphical or numerical integration of derived equations
CHG 3111/B. Kruczek 6
Absorption/Stripping in Packed Towers Simplifications for dilute solutions
Consider application of the integrated approach for the determination of Z based on ky
NB 1: The above analysis provides justification from the HEPT approach
1
2 11
y
G Gy y
i
iM
VdyZ H N
k aSy y y
y
1
2
1
1av
yiM
G Gyy i
y dyVZ H N
k aS y y y
where: is the arithmetic average at the top and the bottom of the tower
1
1av
iMG
y
yVH
k aS y
11 2
2
y
Gy i i M
dy y yN
y y y y
and :
Furthermore, for dilute solutions:
11 0
1.iM
y
y
av av
G
y y
V VH
k aS k aS
thus
NB 2: The above expression provides an alternative to graphical and analytical methods for the
estimation of theoretical equilibrium contacts
1 11 1 2 22 2
where: ln ii i iMi
y yy y y y y y
y y
CHG 3111/B. Kruczek 7
Absorption/Stripping in Packed Towers Simplifications for dilute solutions
Summary of other expressions for height of the transfer towers for dilute solutions
Liquid-phase film coefficient
1
2 11
x
L Lx x
i
iM
LdxZ H N
k aSx x x
x
av av
L
x x
L LH
k aS k aS
1 2
L
i M
x xN
x x
where and
Gas-phase overall coefficient
1
2 11
*
'*
y
OG OGy y
M
VdyZ H N
K aSy y y
y
1 2
*OG
M
y yN
y y
where
av av'
OG
y y
V VH
K aS K aS
and
Liquid-phase overall coefficient
where
av av'
OL
x x
L LH
K aS K aS
and
1
2 11
*
'*
x
OL OLx x
M
LdxZ H N
K aSx x x
x
1 2
*OL
M
x xN
x x
CHG 3111/B. Kruczek 8
Absorption/Stripping in Packed Towers
A gas stream contains 4.0 mol % NH3 and its ammonia content is reduced to 0.5 mol % in
a packed absorption tower at 293 K and 1.013x105 Pa. The inlet pure water flow is 68.0 kg
mol/h and the total inlet gas flow is 57.8 kg mol/h. The tower diameter is 0.747 m. The film
mass-transfer coefficients are kya = 0.0739 kg mol/s m3 mol frac. and kxa = 0.169 kg mol/s
m3 mol frac.
(a) Calculate the tower height using kya
(b) Calculate the tower height using Kya
Use equilibrium data from Appendix A.3.
Example 5: Design of Packed Tower using Integrated Approach
CHG 3111/B. Kruczek 9
Continuous
Humidification/Dehumidification General information
Parameter Absorption Humidification
Number of phases two liquid and gas two liquid and gas
Solute component soluble in both phases
water vapor and heat
Heat effects negligible purpose of the process
Contact of phases packed, tray, spray, bubble towers
packed and spray towers
Operation countercurrent, liquid on top countercurrent, liquid on top
Method: direct contact of dry (not necessarily) cold air and water
Purpose: cooling of hot water by water evaporation, and thus air humidification, but also removal of water vapor from moist cold air (dehumidification)
Comparison of absorption and humidification processes
CHG 3111/B. Kruczek
Continuous Humidification - Approach
10
Heat and Mass Transfer
Rate Equations Thermodynamics
(Equilibrium Relation)
Mass and Energy
Balances
Integrated Approach Complete Design
Height of Tower = Height of Transfer Unit x Number of Equilibrium Contacts
CHG 3111/B. Kruczek
Continuous Humidification
11
Rate Equations for Heat and Mass Transfer
Consider a single point close to the top of a humidification tower
Liquid phase:
Heat transfer: " L L iq q A h T T Mass transfer: there is no mass transfer in the liquid phase because liquid is pure water
Gas phase:
Mass transfer: A A c G B i GN M R k PM H H
Heat transfer: " G i G c o G i G G B i G oq q A h T T R h T T k PM H H
Combining on the basis of heat transfer: L L i G i G G B i G ih T T h T T k PM H H
Rc in terms of DH was derived in Chapter 9
Why we do not express Rc in terms of DT?
CHG 3111/B. Kruczek
Continuous Humidification
12
Rate Equations for Heat and Mass Transfer
Consider a single point close to the bottom of a humidification tower
Question: What is the effect of humidification on air temperature?
At the bottom of the tower, liquid is already cooled down, but dry air is still at relatively
high temperature
Therefore, it is possible that close to the bottom of the tower TG > TL
Regardless of the relationship between TG and TL in humidification: TL > Ti = interface
temperature
Recall Humidity Chart:
Since TL > Ti , water loses heat, humidification is always associated with cooling of water
The net effect of humidification is cooling of air In adiabatic process, TG decreases along the
adiabatic saturation line.
TG1 TG2
NB: Humidification is a unique heat transfer process
in which temperature of both streams decrease
CHG 3111/B. Kruczek
Continuous Humidification
13
Equilibrium Considerations
Conditions at the interface
Hyi and Hy are the total enthalpies of air at interface and bulk
Bulk air phase and bulk water phase are not in equilibrium, but at the interface the air and water are in equilibrium
The air is in equilibrium with water when for a given temperature the partial pressure of water in air (pA) equals to the vapor pressure of water (pAs).
yi iH f T
Criterion for the humidification: yi yH H
1 005 1 88 2501 4kJ kg dry air . . .y s oH c T H H T Recall from Chapter 9, that taking
reference temperature, To = 0oC:
0
100
200
300
400
500
15 25 35 45 55 65
Hy [
kJ/
kg
dry] a
ir]
Water Temperature, TL [oC]
Equilibrium Relationship:
Generation of equilibrium data:
NB: Evaluation of Hyi requires Hi = Hs = saturation humidity at Ti
1. Vapor pressure (pAs ) at given T from Steam Tables or:
20 386 5132mmHg exp . KASp T
2. Saturated humidity:
18 02
28 97
.
.AS
s
As
pH
P p
Dehumidification
Humidification
3 20 0035 0 2 7 2897 34 303o oAlternatively, for 15 C <
CHG 3111/B. Kruczek
Continuous Humidification
14
Material and Energy Balances
Consider material and energy balances on the entire tower
2 1 1 2 and L L L G G G
Material balance mass of liquid water decrease due to evaporation, but the loss of liquid water is typically less than 1.5%, thus:
where cL = 4.187 J/kg K
Energy balance
2 1 1 2y y L L LG H H Lc T T
Operating line Energy balance on the bottom (top) part of the
tower:
1 1L L
y L y L
Lc LcH T H T
G G
1 1y y L L LG H H Lc T T
0
100
200
300
400
500
15 25 35 45 55 65
Hy [
kJ/
kg
dry] a
ir]
Water Temperature, TL [oC]
Intercept Slope
Top
Bottom
Where on the Hy-TL diagram would be the top of a dehumidification tower?
CHG 3111/B. Kruczek
Continuous Humidification
15
Integrated Approach
Consider a differential element of tower of height dz Enthalpy balance: "y L LGdH Lc dT q
Focusing on the air-side of the interface: z
dz
At the same time, q can be expressed using heat balance at the interface:
" L L i G i G G B i G iq h a T T dz h a T T dz k aPM H H dz
NB: Since the contact area is unknown, hL, hG, and kG are replaced by hLa, hGa, and kGa
y G i G G B i G iGdH h a T T dz k aPM H H dz
Considering that, , the above equation becomes: G G sB y B G
h a h ac
M k a M Pk a
y B G s i i i s G G iGdH M k aPdz c T H c T H
Adding and subtracting csTG inside the brackets :
y B G s i o i o s G o G i B G yi yGdH M k aPdz c T T H c T T H M k aPdz H H
NB: Do not confuse Hy = enthalpy with H = humidity
CHG 3111/B. Kruczek
Continuous Humidification
16
Integrated Approach Height of Tower
The combined material and energy and heat balances along with application of rate equations on dz of the tower lead to:
Integrating above equation leads to the final design equation:
z
dz
y B G yi yGdH M k aPdz H H
2
0 1
HyZy
G GHyB G yi y
dHGdz Z H N
M k aP H H
Connection between operating and equilibrium lines
40
80
120
160
200
15 20 25 30 35 40 45
Hy [
kJ/
kg
dry] a
ir]
Water Temperature, TL [oC]
Consider different ways of expressing GdHy
y L L iGdH h a T T dz
y B G yi yGdH M k aPdz H H
yi yL
B G i L
H Hh a
M k aP T T
(Ti, Hyi)
(TL, Hy)
slope = L
B G
h a
M k aP
CHG 3111/B. Kruczek 17
Continuous Humidification Integrated Approach Overall Mass Transfer Coefficient
Design equation in terms of the overall mass transfer coefficient (KG) Analogy with the analysis of absorption/stripping processes
2
0 1*
HyZy
OG OGHyB G y y
dHGdz Z H N
M K aP H H
Where: H*y is the enthalpy air water vapor mixture
in contact with water at TL
Often only the overall coefficient KGa is known while hLa and kGa are not available,
but the resistance to heat transfer in the
liquid phase is negligible, i.e.:
large number, and P RL
B G
h a
M k aP
Design using HOG is limited to the cases when the equilibrium line is approximately
straight over the range used.
Question: What is the physical meaning of the triangle SMP ?
CHG 3111/B. Kruczek 18
Continuous Humidification Operation Constrains
Minimum air flow For a given water flow, what is the minimum air flow to cool water from TL2 to TL1? Constrain: the operating line cannot cross the
equilibrium line. Why?
Line MN has the maximum possible slope.
max min
min max
slope =slope
L LLc LcGG
The actual air flow rate should be 1.3 1.5 Gmin
NB: because of the shape of the equilibrium line,
the operating line may touch the equilibrium
line below the top of the tower
The slopemax can also be determined analytically. How?
Dehumidification operations Questions: 1) How a dehumidification operation is represented on Hy-TL diagram?
2) What is the constrain corresponding to the minimum air flow in dehumidification?
CHG 3111/B. Kruczek 19
Continuous Humidification
A forced draft counter-current water cooling tower is to cool water from 43.3C to 26.7 C. the air enters the bottom of the tower at 29.3 C with a wet bulb temperature of 21.1C. The value of
HG for the flow conditions is HOG = 0.533 m. The heat transfer resistance in the liquid phase will
be neglected, that is hL is very large. Hence, values of H*y should be used.
Calculate the tower height needed if 1.5 times the minimum air rate is used.
Example 6: Design of Packed Cooling Tower
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