Coker A.K. Fortran Programs for Chemical Process Design, Analysis, and Simulation

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507

508 Fortran Programs for Chemical Process Design

The abscissa on the generalized AP curve is expressed as"

10.5

X -- ~ (PL -- PG)

(7-62)

where 0.01 < X < 2.0 and the ordinate is expressed as:

0.1

= (G~FplRL)

(7-63)

[OG (OL -- PG)g]

where 0.001 < Y < 0.15

Y - exp[C o + Cj In X

+ C 2

(In X) 2 + C 3 (In X) 3

+ C 4 (ln X) 4 ] (7-64)

and C o - C4 are constants derived for each AR The values of these con-

stants for each AP correlation are shown in Table 7-14 [13].

The mass vapor rate, Gi, is expressed as:

ft2.s

(7-65)

The tower area is

A - (7-66)

(3600)(G,)

71;D 2

= (7-67)

The tower diameter is defined by

D- (4A) ~

ft (7-68)

71;

At flooding condition, Eckert's flooding curve is represented as"

log Ps

versus log(FEB ) (7-69)

Mass Transfer 509

Table 7-14

Constants for Each Pressure Drop Correlation

Pre s sure

Drop in.

HzO/ft. of

Packed Depth

0.05

0.10

0.25

Source: Blackwell [13]

where

I 105

L PG

constants

Co= -6.30253

C]= -o. 6o8o9

C2=

-0. I1932

C3= -0.00685

C4=

0.00032

Co= -5.50093

CI= -0.78508

C2=

-0. 13496

C3=

0.00134

C4=

0.00174

Co--

-5.00319

CI=

-0.95299

C2= -0.13930

C3=

0.01264

C4=

0.00334

Pressure

Drop in.

H20/ft. of

Packed Depth

0.50

1.0

1.5

Constants

Co=

-4.39918

CI= -0.99404

C2= -0.16983

C3=

0.00873

C4=

0.00343

Co= -4.09505

C]=

-1.00120

C2= -0.15871

C3= 0.00797

C4= 0.00318

Co= -4. 02555

CI= -0. 98945

C2--

-0. 08291

C3=

0.03237

C4= 0.00532

(7-70)

Using Kessler and Wankat's model to represent the flooding condition,

X l - FLG

(7-71)

P g'/) (7-72)

YI = ( Gz2F ~P 02

(PG'PL'g)

510

Fortran Programs for Chemical Process Design

At flooding condition,

log(Yl) = -1.6678 - 1.085 log(Xl) - 0.29655(log

Xl) 2

(7-73)

The mass vapor rate lb/(sft 2) at 70% flooding condition can be

expressed as:

/0.5

G 2 -0 7 YI"PG'Po'2 g

9 Up i ~}/~ ~-~L-

(7-74)

The vapor velocity (ft/sec) at 70% flooding condition is expressed as:

6 2

VGF = (7-75)

Guidelines for Selecting Packed Towers

Use the following rules of thumb for selecting packed towers:

1. Structured and random packings are suited to packed towers

under 3 ft diameter and when low AP is desirable. In such cases,

volumetric efficiences, given adequate liquid distribution through-

out the column, can exceed trayed towers.

2. Replacing trays with packing allows greater throughput or sepa-

ration in existing tower shells.

3. For gas rates of 500 ft3/min, use 1-inch packing. For gas rates of

200 ft3/min, or more, use 2-inch packing.

4. The ratio of the diameter of the tower to the packing should be at

least 15.

5. Because of deformability, plastic packing is limited to an unsup-

ported depth of 10-15 ft, metal to 20-25 ft.

6. Liquid redistributors are required every 5-10 tower diameters with

pall rings, and at least every 20ft for other types of dumped

packings. The number of liquid streams should be 3-5/ft 2 in tow-

ers larger than 3 ft diameter. Some experts advocate 9-12/ft 2 [27].

7. The height equivalent to a theoretical plate (HETP) for vapor-

liquid contacting is 1.3-1.8 ft for 1-inch rings, 2.5-3.0 ft for

2-inch rings.

8. Packed towers should operate near 70% of the flooding rate given

by the correlation of Lobo

et al.

[19] and Sherwood.

Mass Transfer 511

9. Limit the tower height to about 175 ft maximum because of wind

load and foundation considerations. Also, L/D should be less

than 30.

Packed Towers Versus Trayed Towers

Both trays and packings are used to provide intimate contact between

the ascending vapor and descending liquid without a great reduction in

the throughput or capacity of a fractionating column. The most striking

difference between trays and packings is the percentage opening of each

phase-contacting device. A tray has an opening of 8-15% of the tower

cross-sectional area, while the projected opening of a typical packing

design is usually more than 50% of tower cross section. The void frac-

tion of a packed column is even higher, and is usually more than 90% of

the tower volume. In addition, with packing, contact is readily achieved

between the vapor and the liquid phases throughout the column, rather

than at specific points as with trays.

The following further illustrates the advantages of using packed towers

over trays.

High capacity with high liquid rates or high viscosity.

Trayed col-

umns employ the energy of the vapor to create a mass-transfer surface

area by bubbling through the liquid. Packed towers, with the aid of grav-

ity, create a mass-transfer surface area by the action of the liquid falling

over the packing. Thus, there are no downcomers in a packed tower,

and 100% of the tower cross section is used for mass transfer.

High capacity/efficiency combinations.

Because the capacity of a

packed tower is greater than a comparable sized trayed one, a smaller,

more efficient packing can be used to handle the same capacity. The

range of packing sizes and types allows the combination of efficiency

and capacity to be optimized.

High capacity in foaming system.

Trayed columns use the continuous

liquid phase to create a froth that is difficult to separate. Packed towers

make the vapor phase continuous, and the liquid phase discontinuous.

Low pressure drop.

Packed towers have a low AP per theoretical

stage or transfer unit, which is beneficial in low-pressure and vacuum

applications.

Low residence time.

Packed towers offer low liquid holdup. This gives

lower residence time for materials that are sensitive to high processing

512

Fortran Programs for Chemical Process Design

temperature. In contrast, trayed columns typically impose a holdup vol-

ume of 20-30%.

Economic Trade-Offs

In the design of industrial-scale packed towers, an economic balance

is important between the capital cost of the column, its ancillary equip-

ment, and the operating cost. If the diameter of the tower is reduced, the

capacity cost will also be reduced, but the cost of pumping the gas through

the column will increase because of the higher AR

For columns operating at high pressures, the capital cost of the col-

umn shell is more significant, and it may be more economic to operate

at gas velocities above the loading conditions. Morris and Jackson [28]

suggest that a gas velocity 75-80% of the flooding velocity for normal

systems, and less than 40% of the flooding rate if foaming is present.

In general, the actual operating vapor rate is some percentage of G2,,,,o ~ .

The usual range is 65-90% of flooding, with 70-80% being most com-

mon. Because the flooding correlation is not always accurate, some con-

fidence limit is used. For a 95% confidence, Bolles and Fair [29,30]

recommend a safety factor of 1.32 for the calculated cross-sectional

area. The tower diameter is calculated once the tower area is known.

Because the liquid and vapor properties and gas and liquid flow rates all

vary, the designer must calculate the diameter at several locations, and

then use the largest value. Sizing the column diameter has often been

influenced by changes in the vapor flow rate.

Eckert [21] develops an economic characterization number for various

types of packing. Usually, the most suitable choice of packing is depen-

dent on the material of construction of the equipment. In distillation

towers when low-carbon steel is used, 2-inch metal tings are often favored

because they give a low AR high capacity, and efficiency. Alternatively,

saddles are the most frequently selected plastic and ceramic packing.

Troubleshooting packed towers is well illustrated in the literature [31 ].

Nomenclature

A = tower area,

ft 2

C o -C 4 = packed drop correlation constants

D = packed tower diameter, ft

FEB = flow parameter (Equation 7-67)

Fp- packing factor for packing material

Mass Transfer 513

G = vapor rate, lb/h

G~ = vapor mass velocity, lb/(ft 2 sec tower cross section)

G 2 =

vapor mass velocity,

lb/(ft 2 sec

tower cross section)

g = gravitational constant, 32.2 ft/s 2

L = liquid rate, lb/h

L~ = liquid mass velocity, lb/(ft 2 sec tower cross section)

X = abscissa in the generalized plot

Y = ordinate in the generalized plot

Y~ = ordinate in Eckert's flooding curve

VGF = vapor velocity at 70% of flooding conditions, ft/s

Greek Letters

= constant in Table 7-11

[3 = constant in Table 7-11

gL - liquid viscosity, cP

PL -- liquid density, lb/ft 3

PG = vapor density, lb/ft 3

- ratio of water density to liquid density

DETERMINATION OF PLATES IN FRACTIONATING

COLUMNS BY THE SMOKER

EQUATIONS

Introduction

Smoker [32] developed an analytical equation that determines the

number of stages when the relative volatility is constant (that is c~ is

constant). Smoker's method can be used in preference to the McCabe-

Thiele [33] graphical method for calculating the number of theoretical

plates. This is because the accuracy of the McCabe-Thiele depends on

the care given to the construction of the plots. The graphical construc-

tion for the number of stages becomes more difficult at very low con-

centrations. Also, if the relative volatility is close to one, the number of

stages required will be very large, and the McCabe-Thiele diagram will

be difficult to construct.

The Equations

Smoker's equations start by considering Raoult's law for a binary

mixture, where a is a constant and represents the vapor-liquid equilib-

rium relationship. From Equation 7-25

514

Fortran ,Programs for Chemical Process Design

(XX

y = (7-25)

1 + (o:-

1)x

Any operating line that is either above or below the feed plate can be

represented by the general equation of a straight line.

y = mx + b (7-76)

Eliminating y from Equations 7-25 and 7-76 gives a quadratic equation in x.

m(~z- 1)X 2 + [m + (~- 1)b- ~]x + b = 0

(7-77)

For any particular distillation problem, Equation 7-77 will have only

one real root k, between 0 and 1. Representing this root k, and substitut-

ing in Equation 7-77 gives

m(cz- 1)k 2 +[m + (cz- 1)b-ct]k + b = 0

(7-78)

k is the value of the x-ordinate at the point where the extended operat-

ing lines intersect the vapor-liquid equilibrium curve.

Smoker shows that the number of stages required is given by

x o (1 - [3xo ) log 2

N-log x;(1-[3x;) mc

(7-79)

where

[3 - mc(~- 1)

2 (7-80)

0~- mc

N = number of stages required effecting the separation represented by

the concentration change from

x~tox

o,x*

-(x-k) andx

> x*" -

* n, C 1 + (r 1)k

(7-81)

k = composition of the liquid where the operating line intersects the

equilibrium line

m - slope of the operating line between x~ and x 0

~z- average relative volatility, assumed constant over x~ to x 0 (CZ~o p

+ (~bottom)/2

Application to a Distillation Column

Apply Smoker's equation to actual distillation conditions with a single

feed and no side streams.

Mass Transfer 515

Rectifying Section:

For a rectifying column, the number of plates required to enrich from

the feed composition, z v, to the distillate (product) composition, %, at

fixed reflux ratio, R, is required by

m = (7-82)

R+I

and

X o

R+I

(7-83)

Also

m(t~ - 1)k 2 +[m + (ct - 1)b - t~]k + b = 0

where k is the root between 0 and 1

x 0 = x D (7-84)

x 0- X D - k (7-85)

x. = z v (7-86)

x n - z v - k (7-87)

Stripping Section:

For the stripping section, the bottoms composition, x B, the feed, z v,

the overhead composition, x D and the reflux ratio, R, all fix the operat-

ing line. The values of m and b are

m - Rzv + xD -(R + 1)x B (7-88)

(R + 1)(z -

and

b ~.

- x )x.

(R + 1)(z F - XB)

(7-89)

*m m

x 0 z v k

(7-90)

X n X B --

(7-91)

516 Fortran Programs for Chemical Process Design

If the feed stream is not at its bubble point, z v is replaced by the value of

x at the intersection of operating lines given by

, b + zF/( q - 1) (7-92)

ZF --"

q/(q- 1)- m

For distillation at total reflux R = ~, m = 1, b = 0, k = 0, c = 1 and

Equation 7-79 becomes

N lo {X0 l Xn IfO x l

(7-93)

Equation 7-79 simplifies to the equation derived by Underwood [34].

Smoker's equation is not limited to the range of small concentra-

tions, but can be used for stage calculation over the entire range. In such

instances, the equation is written twice, one for each section of the tower.

The average relative volatilities for the rectifying section and for the

stripping section are determined.

Nomenclature

b = intercept of a straight line with the y-axis

c = constant defined by Equation 7-81

k = composition of the liquid where the operating line intersects the

equilibrium line

n = number of plates between any two points in the column under

consideration

R = reflux ratio, reflux per unit time divided by moles of distillate per

unit time

x B = mole fraction in the bottoms

x D = mole fraction in the distillate

z v = mole fraction in the feed

MULTICOMPONENT DISTRIBUTION AND MINIMUM

TRAYS IN DISTILLATION

COLUMNS

Determining the number of stages and reflux ratio for multicompo-

nent distillations is more complex than for binary mixtures. In multi-

component distillation with the feed containing more than two

components, it is often difficult to specify the complete composition of

the top and bottoms products. The separation between the top and