1 Chapter 7 Equilibrium-Stage Operations. 2 One class of mass-transfer devices consists of assembliesof individual units, or stages,interconnected so.

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Chapter 7 Chapter 7 Equilibrium-Stage Equilibrium-Stage

OperationsOperations

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• One class of mass-transfer devices consists of One class of mass-transfer devices consists of assembliesof individual units, or assembliesof individual units, or stages,interconnected so that the materials stages,interconnected so that the materials being processed pass through each stage in being processed pass through each stage in turn. The two streams move turn. The two streams move countercounter currently currently through the assembly; in each stage they are through the assembly; in each stage they are brought into contact, mixed, and then brought into contact, mixed, and then separated. Such multistage systems are called separated. Such multistage systems are called cascades.cascades.

Cascades Cascades 逐级接触设备逐级接触设备

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Ideal StageIdeal Stage 理想级理想级 /Equilibrium /Equilibrium StageStage 平衡级平衡级 /theoretical Stage/theoretical Stage 理论理论

级级

• y2 is in equilibrium y2 is in equilibrium with x2.with x2.

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Ideal Plate/Equilibrium Ideal Plate/Equilibrium Plate/Theoretical Plate/Plate/Theoretical Plate/Perfect Perfect

plateplate


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1. Equipment for stage contacts1. Equipment for stage contacts

• 1) Typical distillation equipment1) Typical distillation equipment• Fig.20.1. Equipment for continuous Fig.20.1. Equipment for continuous

distillation.distillation.

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• Equipment Equipment for for continuous continuous distillation.distillation.

Rectifying section

Stripping /Enriching section

Feed plate

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• 2) Typical leaching equipment2) Typical leaching equipment （（自学）自学）• Fig.20.2. Fig.20.2.

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2. PRINCIPLES OF STAGE 2. PRINCIPLES OF STAGE PROCESSESPROCESSES

• 1) Terminology1) Terminology for stage-contact plantsfor stage-contact plants• Fig.20.3Fig.20.3

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Plate 1

Plate n-1

Plate n

Plate n+1

Plate N

Fig.20.3 Fig.20.3 Material-Material-balance balance diagram for diagram for plate column plate column ((Two-Two-component component systemsystem))

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• 1) Terminology for stage-contact plants1) Terminology for stage-contact plants• In this book,the stages are numbered in the In this book,the stages are numbered in the

direction of flow of the L phase, and the last direction of flow of the L phase, and the last stage is that discharging the L phase.stage is that discharging the L phase.

• 2)Material balances2)Material balances• Under steady flow, there is Under steady flow, there is neither neither

accumulation nor depletionaccumulation nor depletion, , the input and the input and the output are equal andthe output are equal and

Total material Total material balance:balance:

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Material balance for component A:Material balance for component A:

Entire cascade:Entire cascade:

Total material balance:Total material balance:

component A material balance :component A material balance :

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• 3) Enthalpy balances3) Enthalpy balances• The general The general energy balance energy balance can be simplified by can be simplified by

neglecting mechanical neglecting mechanical potential energy potential energy and and kinetic kinetic energy energy . If in addition, the process is . If in addition, the process is worklessworklessand and adiabaticadiabatic, a simple enthalpy balance applies., a simple enthalpy balance applies.

• For two-component system:For two-component system:

aVanLnnVnaLa HVHLHVHL ,,1,1,

Where HWhere HLL and H and HVV are the are the enthalpiesenthalpies per mole of L per mole of L

phase and V phase, respectivelyphase and V phase, respectively..

aVabLbbVbaLa HVHLHVHL ,,,, overall cascade:overall cascade:

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• 4) 4) Graphical methods Graphical methods for two-component for two-component system to find stage numberssystem to find stage numbers

• The methods are based on The methods are based on material balancesmaterial balances and and equilibrium relationshipsequilibrium relationships; some more ; some more complex methods require complex methods require enthalpy balancesenthalpy balances as well.as well.

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• 5)Operating line diagram5)Operating line diagram• From eq.(7.2-2):From eq.(7.2-2):

• Operating-line equationOperating-line equation 操作线方程操作线方程 ::

111

n

aaaan

n

nn V

xLyVx

V

Ly (7.2-(7.2-

7)7)• When the When the flow rates flow rates are not are not constantconstant in the in the

column, the operating line on a simple column, the operating line on a simple arithmetic plot is arithmetic plot is not straightnot straight..

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• Operating-line equation becomes:Operating-line equation becomes:

)(1 aann xV

Lyx

V

Ly

• Operating line:Operating line:

WhenWhen

WhenWhenbNnbNn

anan

yyyxxx

yyxx

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1

,

,

• If LIf Lnn and V and Vn+1n+1 are are constantconstant through the through the

column, the equation is that of a straight line column, the equation is that of a straight line with with slopeslope 斜率斜率 L/V and L/V and interceptintercept 截距截距 : :

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Operating lineOperating line

Equilibrium curveEquilibrium curve

Operating-line diagram for gas absorberOperating-line diagram for gas absorber （（吸收）吸收）

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• The positionThe position of the operating line relative to of the operating line relative to the equilibrium line:the equilibrium line:

• (1) For (1) For rectificationrectification （（精馏）精馏） in a distillation in a distillation columncolumn, , the operating line must lie below the the operating line must lie below the equilibrium lineequilibrium line (Fig.7.2-4a, p.48), why? (Fig.7.2-4a, p.48), why?

is in equilibrium with is in equilibrium with

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Plate 1

Plate n-1

Plate n

Plate n+1

Plate N

For rectification:For rectification:

y= mole fraction y= mole fraction of more volatile of more volatile component Acomponent A

ab

ab

xx

yy

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Fig.7.2-4(a) for rectificationFig.7.2-4(a) for rectification

Driving force:Driving force:

is inis in equilibrium with equilibrium with

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• (2) (2) AbsorptionAbsorption:: When one component is to When one component is to be transferred from the V phase to L phase, be transferred from the V phase to L phase, as in the absorption of soluble material from as in the absorption of soluble material from an an inert gasinert gas, , the operating line must lie the operating line must lie above the equilibrium lineabove the equilibrium line (Fig.7.2-4b), why? (Fig.7.2-4b), why?


y=concentration of soluble y=concentration of soluble component in an inert gascomponent in an inert gas

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Plate 1

Plate n-1

Plate n

Plate n+1

Plate N

For For absorptionabsorption::

y=conc. of soluble y=conc. of soluble material in an material in an inert gasinert gas

ab

ab

xx

yy

稀端稀端 Lean Lean terminalterminal

浓端浓端 Thick terminalThick terminal

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Fig.7.2-4(b) for gas absorptionFig.7.2-4(b) for gas absorption


yyn+1n+1 - y - ynn

is in equilibrium withis in equilibrium with

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• (3) (3) Desorption/strippingDesorption/stripping:: the reverse of gas the reverse of gas absorption: recover absorption: recover valuable solute valuable solute from the from the absorbing solution and absorbing solution and regenerate the regenerate the solventssolvents. . The operating line must lie below the The operating line must lie below the equilibrium lineequilibrium line (Fig.7.24c), why? (Fig.7.24c), why?


x=concentration of solute in x=concentration of solute in absorbing solutionabsorbing solution

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Plate 1

Plate n-1

Plate n

Plate n+1

Plate N

For desorption or For desorption or stripping:stripping:

y=conc. of soluble y=conc. of soluble material in an material in an inert gasinert gas

x=conc. of solute x=conc. of solute in absorbing in absorbing solutionsolution

ab

ab

xx

yy

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Fig.7.2-4 (c) for strippingFig.7.2-4 (c) for stripping


is in equilibrium withis in equilibrium with

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• 6) Ideal contact stages6) Ideal contact stages• Ideal Stage /Equilibrium Stage Ideal Stage /Equilibrium Stage

/theoretical Stage/theoretical Stage• Ideal Plate/Equilibrium Plate /Theoretical Ideal Plate/Equilibrium Plate /Theoretical

Plate/ Plate/ Perfect platePerfect plate


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• To use ideal stages in design, it is To use ideal stages in design, it is necessary to apply a correction factor, necessary to apply a correction factor, called the called the stage efficiencystage efficiency 级效率级效率 or or plate plate efficiencyefficiency 板效率板效率 , which relates the , which relates the ideal ideal stagestage to an to an actual oneactual one. (See Chapter 9 and . (See Chapter 9 and 12)12)

• Overall efficiency:Overall efficiency:

• Plate (Murfree) efficiency:Plate (Murfree) efficiency:• （（默弗里效率）默弗里效率）

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• 7) Determining the number of ideal 7) Determining the number of ideal stagesstages

• The usual method of designing cascades: The usual method of designing cascades:

Determining the number of ideal stagesDetermining the number of ideal stages

Finding the stage efficienciesFinding the stage efficiencies

Calculating the number of actual stagesCalculating the number of actual stages

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• A simple methodA simple method of determining the number of determining the number of ideal stages when there are only two of ideal stages when there are only two components in each phase is components in each phase is a graphical a graphical constructionconstruction using the operating-line using the operating-line diagram.diagram.

• E.g.: Gas absorption:E.g.: Gas absorption:

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Plate N

For absorption:For absorption:

y=conc. of soluble y=conc. of soluble material in an material in an inert gas.inert gas.

FromFrom

)( ba

ab

xx

yy

•How many ideal How many ideal stages are stages are needed?needed?

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Fig.20.5 Operating-line diagram for gas Fig.20.5 Operating-line diagram for gas absorber.absorber.

Operating-lineOperating-line


Points (x1,y1), Points (x1,y1), (x2,y2), (x3,y3) (x2,y2), (x3,y3) must lie on must lie on equilibrium equilibrium curve.curve.

Every step, or Every step, or triangle triangle represents one represents one ideal stage.ideal stage.

Utilize alternately the Utilize alternately the operating and equilibrium operating and equilibrium lineslines

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• The same construction can be used for The same construction can be used for determining the number of ideal stages determining the number of ideal stages needed in any cascade, whether it is used for needed in any cascade, whether it is used for gas absorptiongas absorption, , distillation,distillation, leachingleaching, or , or liquid extractionliquid extraction..

• The graphical step-by-step construction can The graphical step-by-step construction can be be started at either end of the columnstarted at either end of the column..

• Fractional stage?Fractional stage? ( (See example 7.4)See example 7.4)

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• 8) Absorption factor method8) Absorption factor method 吸收因数法吸收因数法 for for calculating the number of ideal stagescalculating the number of ideal stages

• When the When the operating and equilibrium linesoperating and equilibrium lines are are both both straightstraight::

• Let the equation of the equilibrium line beLet the equation of the equilibrium line be

• Where, byWhere, by definition, definition, mm and and BB are constant. If are constant. If stage stage nn is ideal is ideal, ,

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)(1 aann xV

Lyx

V

Ly

• Substitution for xSubstitution for xnn into Eq.(20.7)[p.628] gives, for into Eq.(20.7)[p.628] gives, for

ideal stagesideal stages and and constant L/Vconstant L/V,,

)()(

1 aan

n xV

Ly

mV

ByLy

• DefineDefine

• Where Where A=A=absorption factorabsorption factor, ratio of the slope of , ratio of the slope of the operating line the operating line L/VL/V to that of the equilibrium to that of the equilibrium line line mm..

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• ThereforeTherefore

• BecauseBecause

, the total number of stages,and, the total number of stages,and

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• ThenThen

• The sum of geometric series is The sum of geometric series is

=sum of first n terms of series=sum of first n terms of series

=first term=first term

=constant ratio of each term to =constant ratio of each term to preceding termpreceding term （（公比）公比）

ThereThere

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• Equation(20.16) can then be writtenEquation(20.16) can then be written

)17.20(1

1

1

1 1

A

AAy

A

Ayy

N

a

N

ab

(Kremser equation(Kremser equation 克列姆塞尔方程克列姆塞尔方程 ))

Other Forms of Kremser equation[For Other Forms of Kremser equation[For absorptionabsorption]:]:

)24.20()]/()ln[(

)]/()ln[(

)22.20(ln

)]/()ln[(

)21.20()(

abab

aabb

aabb

bbaaN

yyyy

yyyyN

A

yyyyN

yyyyA

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• When When A=1A=1, (the operating line and the , (the operating line and the equilibrium line are equilibrium line are parallelparallel):):

)(

)25.20(

bbaa

bb

ab

aa

ab

yyyy

yy

yy

yy

yyN

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•Question:Question: If the operating line and equilibrium line If the operating line and equilibrium line are straight and parallel, are straight and parallel, A=1,A=1,

Where, Where, N=NTP=Number of N=NTP=Number of theoretical platestheoretical plates

Why? Why?

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•When the operating line is straight but steeper than When the operating line is straight but steeper than the equilibrium linethe equilibrium line, as in Fig.8.2-2b, , as in Fig.8.2-2b,

NTP=N=Number of theoretical platesNTP=N=Number of theoretical plates

•Why? Why? [Refer to chapter 8.][Refer to chapter 8.]

For case of N=NTP=1,

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• 9)L-phase form of Eq.9)L-phase form of Eq.(20.24):(20.24):

)29.20(1

)28.20(ln

)]/()ln[(

)]/()ln[(

)]/()ln[(

L

mV

AS

S

xxxx

xxxx

xxxxN

bbaa

baba

bbaa

• Where Where xx*=equilibrium concentration *=equilibrium concentration corresponding to corresponding to yy

• S=S=stripping factorstripping factor• Eq.(20.28) mainly for Eq.(20.28) mainly for strippingstripping..

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• The The stripping factorstripping factor is the is the ratioratio of the slope of of the slope of the the equilibrium lineequilibrium line to that of the to that of the operating operating line.line.

• It is not assumed that the linear extension of It is not assumed that the linear extension of the equilibrium line passes through the the equilibrium line passes through the origin. It is only necessary that the line be origin. It is only necessary that the line be linear in the range where the steps linear in the range where the steps representing the stages touch the line.representing the stages touch the line.

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• Summary:Summary:• In the design of a plant, N is calculated from In the design of a plant, N is calculated from

the proposed the proposed terminal concentrationsterminal concentrations and a and a selected value of A or S.selected value of A or S.

• For For absorptionabsorption, using Eq.(20.22) or (22.24) or , using Eq.(20.22) or (22.24) or eq.(20.21);eq.(20.21);

• For For strippingstripping, using eq.(20.28) or (20.30). , using eq.(20.28) or (20.30). [Because equations in x are more common.][Because equations in x are more common.]

)30.20(

bb

aaN

xx

xxS

[EXAMPLE 20.2.][EXAMPLE 20.2.]

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• *10)Equilibrium-Stage Calculations for *10)Equilibrium-Stage Calculations for Multicomponent systemMulticomponent system(( 自学自学 ))

1 Chapter 7 Equilibrium-Stage Operations. 2 One class of mass-transfer devices consists of assembliesof individual units, or stages,interconnected so.

Documents

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stage numbers

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plate n plate n