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Internal Column Balances*

External column balances allow the calculation

of only a few of the process variables (B, D, QC,QR)

Column design, however, requires knowledge of

additional parameters, e.g., number of stages,

optimum feed location (stage), column diameter, etc

Therefore, it becomes necessary to write down and

solve the internal column balances as well

For convenience, we will separate the column intothree sections:

A. The enriching section, which includes the colum

stages above the feed and the condenser.

B. The stripping section, which includes the column

stages below the feed and the reboiler.

C. The feed stage

We will then write the internal balances around

each stage for all three sections* We restrict our discussion to binary mixtures

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Enriching Section: Stage 1

(By convention, stage 1 is the topmost stage)

Definitions

V1, L1: vapour and liquid streams leaving stage 1

(They are considered to be at thermo. equilibrium) V2 : vapour stream rising from stage 2

Lo : reflux stream (entering stage 1)

D: distillate

Qc: Heat removed in the condenser

Fig. 1: Balance envelope for stage 1 and condenser

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Enriching Section: Stage 1

Degrees of freedom analysis

Known design variables:

(These will be given in the design problem, or

will be calculated from the external balances)

Distillate flow rate (D)

Reflux ratio (Lo/D); hence, V1 is also known Mole fraction: xD (=y1=xLo; why?)

Amount of heat removed in the condenser (Qc)

Also, hD, HV1, hLo (not actual design variables!!)

Unknown variables:

L1, V2 Mole fractions: y2, x1(Also unknown: H2, h1)

Total: 6 => We need 6 independent equations

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Enriching Section: Stage 1

Mass and energy balances

Mass balances:

DLV 12 +=Overall: (1)

Most volatile component: (2)D1122 DxxLyV +=

Equilibrium relationship: x1=x1(y1, P) (4)

Energy balance: D11C22 DhhLQHV +=+ (3)

Finally, the molar enthalpies are calculated from:

=

=C

1iref1i,PL1,i1 )TT(Cxh

])TT(C[yHC

1iiref2i,PV2,i2

=

+=

(5)

(6)

Where, is the latent heat of vaporisation for

component i at Tref .

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Enriching Section: Stage j

For the general case of stage j the same procedure

must be followed (also notice the symmetry!).

Fig. 2: Balance envelope for stage j

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Enriching Section: Stage j

Approach #1: Use of h vs. x,y data

Mass balances:

DLV j1j +=+Overall: (1)

Most volatile component: (2)Djj1j1j DxxLyV +=++

Equilibrium relationship: xj=xj(yj, P) (4)

Energy balance: DjjC1j1j DhhLQHV +=+++ (3)

The enthalpies are calculated from:

)P,x(hh jjj = (5)

)P,y(HH 1j1j1j +++ = (6)

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Use of Enthalpy-composition data

Enthalpy-composition diagram for ethanol-water

Graphical estimation of enthalpies and temperature

Concentration of Alcohol, weight fraction

En

thalpy,Kcal/kg

Example: Ethanol-water mixture

H=H(y,P)

h=h(x,P)

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Enriching Section: Stage j

Alternative way: Tj+1 as a direct unknown

Mass balances:

DLV j1j +=+Overall: (1)

Most volatile component: (2)Djj1j1j DxxLyV +=++

Equilibrium relationship: xj=xj(yj, P) (4)

Tj+1= Tj+1(yj+1, P) (4)

Energy balance: DjjC1j1j DhhLQHV +=+++ (3)

Finally, the enthalpies are calculated from:

=

=C

1irefji,PLj,ij )TT(Cxh

=

+++ +=C

1iiref1ji,PV1j,i1j ])TT(C[yH

(5)

(6)

Where, i is the latent heat of vaporisation

of component i at Tref

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Forms of Tj+1 vs. yj+1 data

T

x, y

Graphical representation

Analytical equations of the type: Tj+1=Tj+1(yj+1, P(e.g., polynomial fit to graphical data)

Indirect relations:

Note that V j+1 is a saturated vapour =>Tj+1 =Tj+1,dp

For a binary mixture we can write:

1K

)y1(

K

y1

)P,T(K

y1x

2

1i 1j,2

1j,1

1j,1

1j,1

1j1j,i

1j,i2

1i1j,i =

+==

= +

+

+

+

++

+

=+

dp: dew-point of mixture