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1 Chapter 7 Equilibrium-Stage Operations

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

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3 Ideal Stage 理想级 /Equilibrium Stage 平衡级 /theoretical Stage 理论级 y2 is in equilibrium with x2. y2 is in equilibrium with x2.

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4 Ideal Plate/Equilibrium Plate/Theoretical Plate/Perfect plate y2 is in equilibrium with x2. y2 is in equilibrium with x2.

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5 1. Equipment for stage contacts 1) Typical distillation equipment 1) Typical distillation equipment Fig.20.1. Equipment for continuous distillation. Fig.20.1. Equipment for continuous distillation.

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6 Equipment for continuous distillation. Equipment for continuous distillation. Rectifying section Stripping /Enriching section Feed plate

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8 2) Typical leaching equipment （自学） 2) Typical leaching equipment （自学） Fig.20.2. Fig.20.2.

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9 2. PRINCIPLES OF STAGE PROCESSES 1) Terminology for stage-contact plants 1) Terminology for stage-contact plants Fig.20.3 Fig.20.3

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10 Plate 1 Plate n-1 Plate n Plate n+1 Plate N Fig.20.3 Material- balance diagram for plate column (Two- component system)

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11 1) Terminology for stage-contact plants 1) Terminology for stage-contact plants In this book,the stages are numbered in the direction of flow of the L phase, and the last stage is that discharging the L phase. In this book,the stages are numbered in the direction of flow of the L phase, and the last stage is that discharging the L phase. 2)Material balances 2)Material balances Under steady flow, there is neither accumulation nor depletion, the input and the output are equal and Under steady flow, there is neither accumulation nor depletion, the input and the output are equal and Total material balance:

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12 Material balance for component A: Entire cascade: Total material balance: component A material balance :

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13 3) Enthalpy balances 3) Enthalpy balances The general energy balance can be simplified by neglecting mechanical potential energy and kinetic energy. If in addition, the process is worklessand adiabatic, a simple enthalpy balance applies. The general energy balance can be simplified by neglecting mechanical potential energy and kinetic energy. If in addition, the process is worklessand adiabatic, a simple enthalpy balance applies. For two-component system: For two-component system: Where H L and H V are the enthalpies per mole of L phase and V phase, respectively. overall cascade:

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14 4) Graphical methods for two-component system to find stage numbers 4) Graphical methods for two-component system to find stage numbers The methods are based on material balances and equilibrium relationships; some more complex methods require enthalpy balances as well. The methods are based on material balances and equilibrium relationships; some more complex methods require enthalpy balances as well.

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15 5)Operating line diagram 5)Operating line diagram From eq.(7.2-2): From eq.(7.2-2): Operating-line equation 操作线方程 : Operating-line equation 操作线方程 :(7.2-7) When the flow rates are not constant in the column, the operating line on a simple arithmetic plot is not straight. When the flow rates are not constant in the column, the operating line on a simple arithmetic plot is not straight.

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16 Operating-line equation becomes: Operating-line equation becomes: Operating line: Operating line:WhenWhen If L n and V n+1 are constant through the column, the equation is that of a straight line with slope 斜率 L/V and intercept 截距 : If L n and V n+1 are constant through the column, the equation is that of a straight line with slope 斜率 L/V and intercept 截距 :

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17 Operating line Equilibrium curve Operating-line diagram for gas absorber （吸收）

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18 The position of the operating line relative to the equilibrium line: The position of the operating line relative to the equilibrium line: (1) For rectification （精馏） in a distillation column, the operating line must lie below the equilibrium line (Fig.7.2-4a, p.48), why? (1) For rectification （精馏） in a distillation column, the operating line must lie below the equilibrium line (Fig.7.2-4a, p.48), why? is in equilibrium with is in equilibrium with

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19 Plate 1 Plate n-1 Plate n Plate n+1 Plate N For rectification: y= mole fraction of more volatile component A

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20 Operating line Equilibrium curve Fig.7.2-4(a) for rectification Driving force: is in equilibrium with is in equilibrium with

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

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22 Plate 1 Plate n-1 Plate n Plate n+1 Plate N For absorption: y=conc. of soluble material in an inert gas 稀端 Lean terminal 浓端 Thick terminal

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23 Operating line Equilibrium curve Fig.7.2-4(b) for gas absorption Driving force: y n+1 - y n is in equilibrium with is in equilibrium with

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24 (3) Desorption/stripping: the reverse of gas absorption: recover valuable solute from the absorbing solution and regenerate the solvents. The operating line must lie below the equilibrium line (Fig.7.24c), why? (3) Desorption/stripping: the reverse of gas absorption: recover valuable solute from the absorbing solution and regenerate the solvents. The operating line must lie below the equilibrium line (Fig.7.24c), why? is in equilibrium with is in equilibrium with x=concentration of solute in absorbing solution

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25 Plate 1 Plate n-1 Plate n Plate n+1 Plate N For desorption or stripping: y=conc. of soluble material in an inert gas x=conc. of solute in absorbing solution

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26 Operating line Equilibrium curve Fig.7.2-4 (c) for stripping Driving force: is in equilibrium with is in equilibrium with

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27 6) Ideal contact stages 6) Ideal contact stages Ideal Stage /Equilibrium Stage /theoretical Stage Ideal Stage /Equilibrium Stage /theoretical Stage Ideal Plate/Equilibrium Plate /Theoretical Plate/ Perfect plate Ideal Plate/Equilibrium Plate /Theoretical Plate/ Perfect plate y2 is in equilibrium with x2. y2 is in equilibrium with x2.

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28 To use ideal stages in design, it is necessary to apply a correction factor, called the stage efficiency 级效率 or plate efficiency 板 效率, which relates the ideal stage to an actual one. (See Chapter 9 and 12) To use ideal stages in design, it is necessary to apply a correction factor, called the stage efficiency 级效率 or plate efficiency 板 效率, which relates the ideal stage to an actual one. (See Chapter 9 and 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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29 7) Determining the number of ideal stages 7) Determining the number of ideal stages The usual method of designing cascades: The usual method of designing cascades: Determining the number of ideal stages Finding the stage efficiencies Calculating the number of actual stages

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30 A simple method of determining the number of ideal stages when there are only two components in each phase is a graphical construction using the operating-line diagram. A simple method of determining the number of ideal stages when there are only two components in each phase is a graphical construction using the operating-line diagram. E.g.: Gas absorption: E.g.: Gas absorption:

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31 Plate N For absorption: y=conc. of soluble material in an inert gas. From How many ideal stages are needed?How many ideal stages are needed?

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32 Fig.20.5 Operating-line diagram for gas absorber. Operating-line Equilibrium curve Points (x1,y1), (x2,y2), (x3,y3) must lie on equilibrium curve. Every step, or triangle represents one ideal stage. Utilize alternately the operating and equilibrium lines

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33 The same construction can be used for determining the number of ideal stages needed in any cascade, whether it is used for gas absorption, distillation, leaching, or liquid extraction. The same construction can be used for determining the number of ideal stages needed in any cascade, whether it is used for gas absorption, distillation, leaching, or liquid extraction. The graphical step-by-step construction can be started at either end of the column. The graphical step-by-step construction can be started at either end of the column. Fractional stage? (See example 7.4) Fractional stage? (See example 7.4)

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34 8) Absorption factor method 吸收因数法 for calculating the number of ideal stages 8) Absorption factor method 吸收因数法 for calculating the number of ideal stages When the operating and equilibrium lines are both straight: When the operating and equilibrium lines are both straight: Let the equation of the equilibrium line be Let the equation of the equilibrium line be Where, by definition, m and B are constant. If stage n is ideal, Where, by definition, m and B are constant. If stage n is ideal,

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35 Substitution for x n into Eq.(20.7)[p.628] gives, for ideal stages and constant L/V, Substitution for x n into Eq.(20.7)[p.628] gives, for ideal stages and constant L/V, Define Define Where A=absorption factor, ratio of the slope of the operating line L/V to that of the equilibrium line m. Where A=absorption factor, ratio of the slope of the operating line L/V to that of the equilibrium line m.

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36 Therefore Therefore Because Because, the total number of stages,and

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37 Then Then The sum of geometric series is The sum of geometric series is =sum of first n terms of series =first term =constant ratio of each term to preceding term （公比） There

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38 Equation(20.16) can then be written Equation(20.16) can then be written (Kremser equation 克列姆塞尔方程 ) Other Forms of Kremser equation[For absorption]:

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39 When A=1, (the operating line and the equilibrium line are parallel): When A=1, (the operating line and the equilibrium line are parallel):

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40 Question: If the operating line and equilibrium line are straight and parallel, A=1,Question: If the operating line and equilibrium line are straight and parallel, A=1, Where, N=NTP=Number of theoretical plates Why?

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41 When the operating line is straight but steeper than the equilibrium line, as in Fig.8.2-2b,When the operating line is straight but steeper than the equilibrium line, as in Fig.8.2-2b, NTP=N=Number of theoretical plates Why? [Refer to chapter 8.]Why? [Refer to chapter 8.] For case of N=NTP=1,

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42 9)L-phase form of Eq.(20.24): 9)L-phase form of Eq.(20.24): Where x*=equilibrium concentration corresponding to y Where x*=equilibrium concentration corresponding to y S=stripping factor S=stripping factor Eq.(20.28) mainly for stripping. Eq.(20.28) mainly for stripping.

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43 The stripping factor is the ratio of the slope of the equilibrium line to that of the operating line. The stripping factor is the ratio of the slope of the equilibrium line to that of the operating line. It is not assumed that the linear extension of the equilibrium line passes through the origin. It is only necessary that the line be linear in the range where the steps representing the stages touch the line. It is not assumed that the linear extension of the equilibrium line passes through the origin. It is only necessary that the line be linear in the range where the steps representing the stages touch the line.

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44 Summary: Summary: In the design of a plant, N is calculated from the proposed terminal concentrations and a selected value of A or S. In the design of a plant, N is calculated from the proposed terminal concentrations and a selected value of A or S. For absorption, using Eq.(20.22) or (22.24) or eq.(20.21); For absorption, using Eq.(20.22) or (22.24) or eq.(20.21); For stripping, using eq.(20.28) or (20.30). [Because equations in x are more common.] For stripping, using eq.(20.28) or (20.30). [Because equations in x are more common.] [EXAMPLE 20.2.]

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45 *10)Equilibrium-Stage Calculations for Multicomponent system( 自学 ) *10)Equilibrium-Stage Calculations for Multicomponent system( 自学 )

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