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**Chapter 7 Equilibrium-Stage Operations**

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Cascades 逐级接触设备 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.

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**Ideal Stage理想级/Equilibrium Stage平衡级/theoretical Stage理论级**

y2 is in equilibrium with x2.

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**Ideal Plate/Equilibrium Plate/Theoretical Plate/Perfect plate**

y2 is in equilibrium with x2.

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**1. Equipment for stage contacts**

1) Typical distillation equipment Fig Equipment for continuous distillation.

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**Equipment for continuous distillation.**

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

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**2) Typical leaching equipment（自学）**

Fig.20.2.

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**2. PRINCIPLES OF STAGE PROCESSES**

1) Terminology for stage-contact plants Fig.20.3

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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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**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. 2)Material balances Under steady flow, there is neither accumulation nor depletion, the input and the output are equal and Total material balance:

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**Material balance for component A:**

Entire cascade: Total material balance: component A material balance :

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**For two-component system:**

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. For two-component system: Where HL and HV are the enthalpies per mole of L phase and V phase, respectively. overall cascade:

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**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.

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**5)Operating line diagram**

From eq.(7.2-2): 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.

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

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

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**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? is in equilibrium with

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

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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**is in equilibrium with Driving force: Equilibrium curve Operating line**

Fig.7.2-4(a) for rectification Driving force: is in equilibrium with

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(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 y=concentration of soluble component in an inert gas

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

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

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**Driving force: yn+1 - yn is in equilibrium with Operating line**

Equilibrium curve Fig.7.2-4(b) for gas absorption is in equilibrium with Driving force: yn+1 - yn

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(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 x=concentration of solute in absorbing solution

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**For desorption or stripping: **

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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**is in equilibrium with Driving force: Equilibrium curve Operating line**

Fig (c) for stripping is in equilibrium with Driving force:

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

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**Plate (Murfree) efficiency:**

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: Plate (Murfree) efficiency: （默弗里效率）

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**7) Determining the number of ideal stages**

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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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:

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

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

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**Utilize alternately the operating and equilibrium lines **

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

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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. Fractional stage? (See example 7.4)

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**When the operating and equilibrium lines are both straight: **

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: Let the equation of the equilibrium line be Where, by definition, m and B are constant. If stage n is ideal,

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**Substitution for xn into Eq. (20. 7)[p**

Substitution for xn into Eq.(20.7)[p.628] gives, for ideal stages and constant L/V, Define 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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Therefore Because , the total number of stages,and

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Then 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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**(Kremser equation克列姆塞尔方程)**

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

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**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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**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.] For case of N=NTP=1,

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

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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.

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**For absorption, using Eq.(20.22) or (22.24) or eq.(20.21); **

Summary: 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 stripping, using eq.(20.28) or (20.30). [Because equations in x are more common.] [EXAMPLE 20.2.]

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

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