IDEAL STAGEFeedF, xF

DistillateD, xD

Bottom ProductB, xB

SINGLE-STAGE (FLASH) DISTILLATION

UNECONOMICAL

3

2

4

1

5

F, xF

V1

V2

V3

V5

V4

V6L5

L4

L3

L2

L1

L0

Countercurrent multistage contact

Simple counter-current flow cannot give as complete a separation as required

N = More concentrated L0

uneconomical x0 is fixed by other consideration

Where does L0 come from?

Multistage cascade with reflux at both ends for distillation

V1

-qC

m

p

LC

D

F

Enrichingsection

Strippingsection

F-1

F+1

N

1

B

C

S qS

L0

Condenser

Reboiler

NL

1NV

SL

F

ZERO REFLUX

• No liquid returned to stage 1

• No condensation of V2 to supply liquid leaving stage 1

• The vapor leaving stage 1 would be the same quantity and composition of the vapor leaving stage 2.

• The vapor leaving stage 2 would be the same quantity and composition of the vapor leaving stage 3.

• Etc.

V1

-qCLC

D

F F

1

C

2

V2

V3

3

Multistage cascade with no liquid reflux

F

N-2

N-1

N

F

B

S qS

NL

1NV

0LS

1NL

2NL

If the vapor reflux were eliminated:

• No vapor returned to stage N

• No vaporization of to supply vapor leaving stage N

• The liquid leaving stage N would be the same quantity and composition of the liquid leaving stage N-1.

• The liquid leaving stage N-1 would be the same quantity and composition of the liquid leaving stage N-2.

• Etc.

1NL

Multistage cascade with no vapor reflux

A fractionating column by its inherent nature has two limits of operation based upon reflux ratio:

• Minimum reflux

• Total reflux

MINIMUM REFLUX

D

B

F

L0 • There is insufficient liquid returned

to the column

• There is only an infinitesimal change in vapor and liquid compositions through the plates.

• Infinite number of plates would be needed.

• Actual operation of a column below or at minimum reflux is impossible.

Schematic representation of minimum reflux operation

TOTAL REFLUX

• All condensate is returned to the column

• It requires the least number of stages.

• Practically no overhead product and no bottom product can be made and no feed is introduced.

• It is possible to operate experimentally a fractionating column at total reflux when the system inventory is large and only very small samples of distillate and bottoms are removed.

D = 0

B = 0

F = 0

Schematic representation of total reflux operation

MINIMUM REFLUX VS TOTAL REFLUX

Large reflux ratio

Small reflux ratio

• More coolant• More heating medium• Greater operating cost

• Greater number of plates

• Greater investment cost

$/u

nit

pro

du

ct

Number of stages

Nmin

Total cost

Operating cost

Equipment cost

Optimum designat minimum cost

Schematic relationship between reflux ratio and number of stages

V1, y1, H1

L1, x1, h1

L0

x0

h0

DxD

hD

qD

Total condenser

1

2

Over-all:

Component i:

(1)

(2)

(3)

(4)

(5)

V1 = L0 + D

D01 ii0i1 xDxLyV

D01 iii xxy

11

0

VD

1VL

1DV

DL 10

MATERIAL BALANCE AROUND TOTAL CONDENSER

V1 H1 + qD = L0 h0 + D hD (6)

ENTHALPY BALANCE AROUND TOTAL CONDENSER

The total heat removed in the condenser can be expressed in terms of heat per unit mass of distillate stream times the mass of stream.

qD = D QD

V1 H1 + D QD = L0 h0 + D hD

V1 H1 + (V1 – L0) QD = L0 h0 + (V1 – L0) hD

(7)

(8)

(9)

Introducing eq. (1) into eq. (8) to eliminate D yields:

D0D100D0D111 hLhVhLQLQVHV

DD0

DD1

1

0

QhhQhH

VL

01

1DD0

hHHQh

DL

(10)

(11)

V1 H1 + (V1 – L0) QD = L0 h0 + (V1 – L0) hD

DD11DD00 QhHVQhhL

Introducing eq. (1) into eq. (8) to eliminate V1 yields:

MATERIAL BALANCE IN ENRICHING SECTION

V1, y1, H1

L1, x1, h1 L0

x0

h0

DxD

hD

qD

Vm+1

ym+1

Hm+1

Lm

xm

hm

F

Over-all:

Component i:

(12)

(13)

(14)

Vm+1 = Lm + D

Dm1m iimi1m xDxLyV

Dm1m im1mimi1m xLVxLyV

Dm

D1m

ii

ii

1m

m

xx

xy

VL

Introducing eq. (12) into eq. (13) to eliminate D results in

D1mDm ii1miim xyVxxL

(15)m1m

1mD

ii

iim

xy

yx

DL

Introducing eq. (12) into eq. (13) to eliminate Vm+1 results in:

Dm1m iimim xDxLyDL

1mDm1m iiiim yxDxyL

ENTHALPY BALANCE IN ENRICHING SECTION

(16)DmmD1m1m hDhLQDHV

Dm1mmmDm1m1m1m hLVhLQLVHV

Introducing eq. (12) into eq. (16) to eliminate D results in

DmD1mmmDmD1m1m1m hLhVhLQLQVHV

DD1m1mDDmm QhHVQhhL

DDm

DD1m

1m

m

QhhQhH

VL

(17)

m1m

D1mDm

hHQHh

DL

(18)

Introducing eq. (12) into eq. (16) to eliminate Vm+1 results in:

DmmD1mm hDhLQDHDL

DmmD1m1mm DhhLDQDHHL

D1mDm1mm QHhDhHL

Partial condenser

MATERIAL AND ENTHALPY BALANCES AROUND PARTIAL CONDENSER

V1, y1, H1

L1, x1, h1

L0

x0

h0

DyD

HD

qD

Over-all:

Component i:

(19)

(20)

(21)

V1 = L0 + D

D01 ii0i1 yDxLyV

0

D

D01i

iDiii x

yK;xxy

D01 i01i0i1 yLVxLyV

(22) D0

D01

D0

D1

ii

iii

ii

ii

1

0

K1x

Kxy

yx

yy

VL

D1D0 ii1ii0 yyVyxL

(23)

D01 ii0i0 yDxLyDL

01

1D0

01

1D

ii

iii

ii

ii0

xy

yKx

xy

yy

DL

1D01 iiii0 yyDxyL

ENTHALPY BALANCE:

V1 H1 + qD = L0 h0 + D HD (24)

The total heat removed in the condenser (qD) can be expressed in terms of heat per unit mass of distillate stream times the mass of stream.

qD = D QD

V1 H1 + D QD = L0 h0 + D HD

(25)

(26)

V1 H1 + (V1 – L0) QD = L0 h0 + (V1 – L0) HD

DD11DD00 QHHVQHhL

DD0

DD1

1

0

QHhQHH

VL

(27)

Replacing D in equation (26) with (V1 – L0):

01

1DD0

hHHQH

DL

(28)

(L0 + D) H1 + D QD = L0 h0 + D HD

1DD010 HQHDhHL

Replacing V1 in equation (26) by (L0 + D):

MATERIAL BALANCE IN ENRICHING SECTION WITH PARTIAL CONDENSER

V1, y1, H1

L1, x1, h1

L0

x0

h0

DyD

hD

qD

Vm+1

ym+1

Hm+1

Lm

xm

hm

F

m

Over-all:

Component i:

(29)

(30)

(31)

Vm+1 = Lm + D

Dm1m iimi1m yDxLyV

Dm1m im1mimi1m yLVxLyV

Dm

D1m

ii

ii

1m

m

yx

yy

VL

D is eliminated from equation (30) by substituting D with (Vm+1 – Lm):

D1mDm ii1miim yyVyxL

(32)m1m

1mD

ii

iim

xy

yy

DL

Dm1m iimim yDxLyDL

Vm+1 is eliminated from equation (30) by substituting Vm+1 with Lm + D

1mDm1m iiiim yyDxyL

ENTHALPY BALANCE IN ENRICHING SECTION

V1, y1, H1

L1, x1, h1

L0

x0

h0

DxD

hD

qD

Vm+1

ym+1

Hm+1

Lm

xm

hm

F

(33)

(34)

DmmD1m1m HDhLQDHV

Dm1mmmDm1m1m1m HLVhLQLVHV

D is eliminated from equation (33) by substituting D with Vm+1 – Lm

DDm

DD1m

1m

m

QHhQHH

VL

DD1m1mDDmm QHHVQHhL

Vm+1 is eliminated from equation (33) by substituting Vm+1 with Lm + D

DmmD1mm HDhLQDHDL

m1m

D1mDm

hHQHH

DL

D1mDm1mm QHHDhHL

(35)

MATERIAL BALANCE IN STRIPPING SECTION

qBBxB

hB

yp+1

Hp+1

xp

hp

F

1pV pL

Over-all: BVL 1pp

Component i:B1pp ii1pip xByVxL

B1pp i1ppi1pip xVLyVxL

Bp

B1p

ii

ii

1p

p

xx

xy

VL

(36)

(37)

(38)

Replacing B in equation (37) with 1pp VL

B1pBp ii1piip xyVxxL

ENTHALPY BALANCE:

B1p1pBpp hBHVqhL

qB = B QB

B1p1pBpp hBHVQBhL

B1pp1p1pB1pppp hVLHVQVLhL

BB1p1pBBpp QhHVhQhL

BBp

BB1p

1p

p

hQhQhH

VL

(39)

(40)

(41)

MATERIAL & ENTHALPY BALANCES ABOUT REBOILER

qB BxB

hB

yN+1

HN+1

yN

HN

1NV NL

Over-all: BVL 1NN

Component i:B1NN ii1NiN xByVxL

B1NN i1NNi1NiN xVLyVxL

BN

B1N

ii

ii

1N

N

xx

xy

VL

(42)

(43)

(44)

Replacing B in equation (43) with 1NN VL

B1pBp ii1piip xyVxxL

MATERIAL BALANCE AROUND THE FEED PLATE

F = FV + FL

LV FFF HHh

LV FFF xyx

1pV

Hp+1

yp+1

pL

hp

xp

Vm+1

Hm+1

ym+1

Lm

hm

xm

Over-all:

1mpm1pLV VLLVFF

Component i:

1mpm1pLV i1mipimi1pF,iLF,iV yVxLxLyVxFyF

1m1pV VVF

pmL LLF

(45)

(48)

(46)

(47)

1mp

m1pLV

i1mimL

imiV1mF,iLF,iV

yVxLF

xLyFVxFyF

(49)

V1pLp1p1mpm F,iiVF,iiLii1miim yyFxxFyyVxxL

pm

V1p

pm

Lp

pm

1p1m

ii1m

F,iiV

ii1m

F,iiL

ii

ii

1m

m

xxV

yyF

xxV

xxF

xx

yy

VL

If the feed is a saturated liquid, the last term in eq. (49) drops out.

If the feed is a saturated vapor, the middle term on the right side of eq. (49) drops out.

ENTHALPY BALANCE AROUND THE FEED PLATE

pp1m1mmm1p1pFLFV hLHVhLHVhFHFLV

pmL1m1mmm1pV1mFLFV hLFHVhLHFVhFHFLV

VL F1pVFpL1p1m1mpmm HHFhhFHHVhhL

pm

F1p

1m

V

pm

Fp

1m

L

pm

1p1m

1m

m

hh

HH

VF

hh

hh

VF

hhHH

VL VL

(50)

(51)

• If the feed is a saturated liquid, the last term in eq. (51) drops out.

• If the feed is a saturated vapor, the middle term on the right side of eq. (51) drops out.

In liquid mixture / solution the molal enthalpy of the mixture at a given T and P is the sum of the partial molal enthalpies of the components composing the mixture.

n

iim hh (52)

In “regular” / ideal mixtures: oiii hxh (53)

For gaseous / vapor mixtures at normal T and P:

n

iii

n

iim yhH (54)