O VERALL M ASS -T RANSFER C OEFFICIENTS

11.5 DESIGN TECHNIQUES FOR COUNTERCURRENT ABSORPTION COLUMNSABSORPTION COLUMNS

11.5.6 O VERALL M ASS -T RANSFER C OEFFICIENTS

N^A=k yy

(

A−yAi

)

⁼k xx

(

Ai⁻xA

)

^(11.35)

NA=^{K y}y

(

A−yAi+^yAi−^yA^*

)

^(11.38)

Then combining Equations 11.37 and 11.38,

N K N

k

mN

A y k

A y

A x

=  +

 

 (11.39)

Then dividing both sides of the Equation 11.39 by NA and rearranging,

1 1

K k

m k

y y x

= + (11.40)

11.5.7 VOLUME-BASED MASS-TRANSFER COEFFICIENTS

Absorption towers are packed with plastic, metal, or ceramic pieces to provide a greater area for contact between the vapor and liquid phases. Each of these packings has a char- acteristic area per unit volume of packing, which can be denoted by the quantity a, where

a = Interfacial area Unit volume

Flow rate of both phases, viscosity, density, surface tension, and size and shape of the packing determine the value of a. These same factors affect the value of the mass-transfer coefficients K_y and K_x. Therefore, it is expedient to include a in the mass-transfer equation and define two new quantities K_ya and K_xa. These quanti- ties would then be correlated with the solution parameters as functions of various chemical systems. If A is the absorption tower cross-sectional area and z the packing height, then Az is the tower packing volume. Defining A_i as the total interfacial area:

Ai= aAz (11.41)

or in differential form

dAi=aAdz (11.42)

We can now rewrite Equation 11.36 in terms of a differential rate of mass transfer dNA, with units (moles/unit time) = (moles/unit time − area) × (area),

dNA =N dAA i=K yy

(

A−y aAdzA^*

)

^(11.43)

or

dNA=K a yy

(

A−y Adz K a xA^*

)

⁼ x

(

A⁻x AdzA^*

)

^(11.44)

Equation 11.44 defines the volumetric mass-transfer coefficients K_ya and K_xa, which have the typical units moles/(h-m³-mole fraction). Note also that the term Adz repre- sents the differential packed tower volume.

11.5.8 DETERMINING HEIGHTOF PACKINGINTHE TOWER: HEIGHTOFA TRANSFER UNIT METHOD

The differential rates of mass transfer given by Equation 11.44 are equal to the differ- ential rate of change of mass Equation 11.23 within a phase. Then, for the gas phase, the differential rate of mass transfer of component A is equal to the differential rate of change of the mass of A in the incoming gas stream.

dNA=d Gy

(

A

)

^(11.45)

Rewriting Equation 11.25, where G_B is a constant,

GB=

(

1− y GA

)

(11.46)

then

d Gy d G y

A y

B A A

( )

⁼

(

₋

)













1 (11.47)

It can be shown that

d G y y

Gdy y

B A

1− A 1

( )











=

(

− ^AA

)

^(11.48)

Therefore,

Gdy

y^A K a y y Adz

A

y A a

*

(

1−

)

⁼

(

⁻

)

^(11.49)

becomes the basic design equation. A similar equation based on the liquid phase can be written, which would be useful in stripping calculations.

We now define a mass-transfer coefficient Ky

o, which is independent of concentration:

K K

y y

y o A LM

=

(

1−

)

^(11.50)

where

(

1−^y_{A LM}

)

is the log mean concentration defined by

1 1 1

1 1

(

−

)

⁼

(

⁻

)

^{− −}

( )

(

−

) (

⁻

)

 



y y y

y y

A LM

A A

A A

*

ln *

(11.51)

Define set of flow rates based on tower cross section G G

A L L

= and = A (11.52)

G dy y

Ky a

y y y dz

A A

A o A LM

1− 1 A



 

 = −

( ) (

^{− *}

)

^(11.53)

Assuming constant flow rate, G, through the column, this equation can now be inte- grated from z = 0 to z = z down the column to determine tower height.

z G

K a

y

y y y dy

y y

=

y o

A LM

A A A

A A1

 A0

 



(

−

) (

−

) (

⁻

)

∫

1 ^{1 *}

(11.54)

The ratio of flow rate to mass transfer has been designated as the height of a transfer unit, or, for the gas phase, HOG.

H G

OG K a

y

= o

 

 (11.55)

Therefore, H_OG has been defined in such a way that it remains constant through the absorption column. The integral portion of Equation 11.54 is designated as the number of overall mass-transfer units, or, for the gas phase, N_OG. Thus,

N y

y y y dy

y y

OG A LM

A A A

= A A1

A0 1

1

(

−

)

(

−

) (

⁻

)

∫

^{* (11.56)}

The integral may be evaluated by graphical or numerical techniques. The height of the column may now be calculated from

z H N= OG OG (11.57) 11.5.9 DILUTE SOLUTION CASE

For dilute solutions, Henry’s law is usually a good choice for an equilibrium rela- tionship. In this case, yA mx

*

= A, which is associated with the overall mass-transfer coefficient, as defined in Equation 11.36. For a dilute solution,

1− ≈ −1 ≈

(

^yA

) (

^y_{A LM}

)

^{1.0 (11.58)}

and Equation 11.56 reduces to

N dy

y y

y y OG

A

A A

=

A1 A0

(

−

)

∫

^{* (11.59)}

The operating line, Equation 11.24, may now be rewritten as

y L

G x x y

A=  A A A1

 



(

− ¹

)

^{+ (11.60)}

Rewrite Equation 11.60, multiplying and dividing and multiplying by Henry’s law constant, m,

yA=  A A A1

 



(

−

)

⁺

L

mG mx mx 1 y (11.61)

Note that the ratio L G/ = /L G. Then an absorption factor, Ab, can be defined as Ab = L

mG L

=mG (11.62)

Then rewrite Equation 11.61 using Henry’s law and the absorption factor and solve for yA

*

yA y y mx

* A A1

Ab A1

= −

 

 (11.63)

Substitute Equation 11.63 in Equation 11.59 and integrate

N mx y mx

OG

A0 A1 / A1 A1 /Ab /Ab

=

{

_

(

−

) (

⁻ /Ab

)

_{ −}

( )

⁺

( ) }

(

−

)

ln y 1 1 1

1 1

(11.64)

When a pure solvent such as water is used, xA1 = 0.0, and Equation 11.64 reduces to

NOG yA yA

Ab ln Ab Ab

= −



 

 _

( ) (

⁻

)

⁺ _

1

1 1 0 1 1 1 1 (11.65)

11.5.10 USING MASS EXCHANGE NETWORK CONCEPTSTO SIMULTANEOUSLY

EVALUATE MULTIPLE MASS SEPARATING AGENT (ABSORBENT) OPTIONS

As indicated previously in Chapter 8, systems-based methodologies exist that allow the simultaneous evaluation of multiple absorbents that could be used for the removal of contaminants from gaseous emission streams. Examples of contaminants in gas- eous emissions that have been removed by liquid absorbents are VOCs (solvents) and sulfur-based contaminants such as hydrogen sulfide. Numerous absorbents can be employed for the separation task. One type of absorbents, physical absorbents, facilitates the direct physical transfer of the contaminants from the gaseous emis- sion stream to the absorbent but the contaminant transferred is not altered. This approach can be typically employed by using heavy oils as adsorbents to remove VOCs from gaseous emission streams. A second type of absorbents, reactive absor- bents, facilitates the direct physical transfer of the contaminants from the gaseous emission stream to the absorbent, where a chemical reaction with the absorbent occurs that converts the contaminant transferred to a chemical compound that can be more readily recycled, regenerated, discharged, or sold. A specific example of this approach reported in literature is the use of sodium hydroxide solutions to absorb hydrogen sulfide from gaseous emission streams from pulp and paper plants.

sulfide, a raw material that can be recycled for reuse in the pulping manufacturing process.¹² Both physical and reactive absorbents are collectively classified as mass separating agents, as has been previously discussed in Chapter 8. It has previously been shown in the literature that both physical and chemical mass separating agents can be modeled using an operating line and equilibrium line similar to the depiction previously provided in Figure 11.9.¹³

As previously shown, the composition driving force between the operating and equilibrium line drives the transfer of a contaminant from a gaseous emission stream to the absorbent. A visual representation of this concept showing the oper- ating and equilibrium lines along with the composition driving forces is provided as Figure  11.10. The minimum horizontal distance between the operating and the equilibrium line is called the minimum composition driving force, ε, as shown in this figure. This minimum composition driving force is used to trade off the capital ver- sus operating costs for an absorption column.¹⁴ As the minimum composition driving force increases, more absorbent is required to achieve the separation while the capital cost of the absorption column decreases and vice versa as the composition driving force decreases. This is analogous to the effect of the minimum temperature driving force on the operating cost and capital cost of a heat exchanger. Ultimately, an optimal minimum composition driving force will result in an absorption column that realizes the minimum total annualized cost as illustrated in Figure 11.11. This concept is par- ticularly helpful when designing a network of multiple absorption columns.

Many industrial design problems involve several gaseous emission streams that have been generated at different locations within a manufacturing process. In addi- tion, numerous liquid absorbents can potentially be used to remove the contaminants from the gaseous emission streams. A systems-based representation of the prob- lem of synthesizing a mass exchange network of absorption columns is provided in

Equilibrium line y = mx + b Operating line

y = m(x + ε) + b

Minimum composition

driving force Linearity exists for

many separations involving dilute gas streams Mole fraction of A in liquid

Mole fraction of A in vapor

ε

FIGURE 11.10 Minimum composition driving force for absorption mass transfer.

Figure 11.12. Any number of gaseous emission streams containing a single common contaminant can be simultaneously analyzed using this approach, and El-Halwagi has provided details of along with examples in his recent book.¹⁵

A useful representation for the simultaneous analysis of multiple absorbents is the composition-interval-diagram.¹⁶ This diagram contains a vertical line for each gaseous emission stream that extends on a rich composition scale from the ini- tial composition of the contaminant in each of the gaseous emission streams to the desired outlet contaminant composition. All of the gaseous emission streams are collectively referred to as rich streams. This diagram also has a vertical line for each liquid adsorbent that extends on its composition scale from its initial composition of contaminant to its maximum allowable outlet contaminant composition. All of the liquid absorbent streams are collectively referred to as lean streams. Each lean stream has its own composition scale that is based on the thermodynamic equilib- rium line separated by a unique composition driving force, ε, as previously shown on Figure  11.10. The rich stream composition scale and the lean stream compo- sition scales can be plotted on the same composition-interval-diagram since each respective lean stream scale can be related to a common rich stream composition scale. An example composition-interval-diagram with nomenclature is provided as Figure  11.13. El-Halwagi has shown that this composition-interval-diagram can be used with a graphical mass pinch diagram or via the use of a two-stage math- ematical optimization approach to identify feasibility level designs that represent the most cost-effective network of absorption columns.¹⁵  Details on these design methodologies can be found in El-Halwagi’s text.

Total annualized cost

Composition driving force (ε)

Capital cost Operating cost Total

annualized cost

ε^Optimum

FIGURE 11.11 Identifying the minimum total annualized cost for absorption-based VOC removal from gaseous emission streams.

Rich stream concentrations (C) Gaseous emission

streams (rich streams) Absorbents

(lean streams)

C₁ Intervals (k) k = 1 k = 2 k = 3 k = 4 k = 5 Rn

R₁

S₁

S_m x₁ (ppm) = y

Absorbent 1 C₂

C₃ C₅ C₄

C₆

Absorbent m

y1in x₁^out

x_n^out y_nⁱⁿ

x1in

x_nⁱⁿ y_n^out

m₁− ε₁ x₂ (ppm) = y m₂− ε₂

FIGURE 11.13 Composition-interval-diagram for the simultaneous analysis of multiple absorbent options for VOC removal from gaseous emission streams.

Absorption exchangemass

network

VOC laden gas stream 1 Clean gas recycled or discharged

Clean gas recycled or discharged

Absorbent 1 Absorbent m

Direct recycle, regenerated, discharge, or sale VOC Laden gas stream n

Flow1 (S1), concentration1in (x1in)

Flow1 (S1), concentration1out (x1out)

Flow₁ (R₁), concentration₁^out (y₁^out)

Flown (Rn), concentrationnout (ynout)

Flowm (Sm), concentrationmout (xmout) Flowm (Sm), concentrationmin (xmin)

Flow₁ (R₁), concentration₁ⁱⁿ (y₁ⁱⁿ)

Flown (Rn), concentrationnin (ynin)

Absorbent 1 containing VOC Absorbent m containing VOC VOC removal from gaseous emission streams

FIGURE 11.12 A schematic representation of the simultaneous analysis of multiple absor- bent options for VOC removal from gaseous emission streams.

11.6 COUNTERCURRENT FLOW PACKED