Generalized Correlation of Flooding Rates - Industrial & Engineering

FLOW THROUGH POROUS MEDIA

Generalized Correlation of Flooding Rates B Y R O N C. SAKIADIS’, A. 1. J O H N S O N ,

STATE U N I V E R S I T Y ,

LOUISIANA

UNIVERSITY O F

EATON

ROUGE, L A .

TORONTO, TORONTO, CANADA

A generalized theoretical equation relating t h e flow rates of

t h e phases a t t h e flooding point has been derived, and this has been used i n the following forms t o correlate most published data on spray columns and packed towers: Spray Columns Phase combinations

Sol id- Iiq u id

1

+ 1.8 [(E)

I f 4

(E) uD ] 1/2

=

E:[

cs

P!“]

-114

Gas-liquid Packed Towers Phase combinations Liquid-liquid\ Gas-liquid

+

o.835

[(e) ‘I4 OD (,)1’2]

PC

=

cp [&!! PC g0c3AP

P;q-i’4

Constant C, is a simple function of particle diameter, solid, liquid, or gaseous. Constant C, is a function of superficial area of packing, fraction of voids, and interfacial or surface tension. The different packing materials, such as Raschig rings and Berl saddles, have different relaBerl saddles are more efficient and give higher flooding rates t h a n Raschig rings. tions for C,. Packed towers were noted under certain conditions t o have higher flooding rates t h a n spray columns. The average error on 429 calculated runs of spray columns is ~ k 1 2 . 5 ~and 0 , t h e average error on 500 calculated runs of packed towers is 4~13.5%. T h e present correlations can be used directlyfor design calculations.

T

HE use of packed and spray columns in operations such as ab-

sorption, fractionation, leaching, and liquid-liquid extraction has been limited owing to a lack of reliable design data. I n the design of such columns, two factors are particulaily important; the height of packing or empty column, and the diameter required for a specified duty. The height of packing or column is determined from mass transfer considerations. The column diameter is determincd by the maximum permissible velocities of the phases involvvd. Starting with the concept that the flooding point is a particular pressure drop phenomenon, it becomes necessary t o emphasize its generality by indicating that it will be present in any apparatus in which one phase passes discontinuously through an immiscible second one. Liquid-liquid and gas-liquid packed towers should exhibit flooding in essentially the same manner as solid-liquid, liquid-liquid, gas-liquid, and solid-gas spray columns. The operative mechanism is the same except for minor differences arising from factors such as particle shape, gas expansibility, and droplet distortion. This paper presents an adequately tested generalized correlation of flooding rates for packed and spray columns for any possible combination of continuous and discontinuous phases that will be satisfactory for design purposes. SPRAY COLUMNS

Visual Flooding Point. The general character of flooding may be visualized as follows: The holdup of a discontinuous phase, solid, liquid, or gaseous, passing countercurrently through a sec1 Project conducted a t the University of Toronto while author was on leave from the Department of Chemical Engineering and the Engineering Experiment Station, Louisiana State University, Baton Rouge, La.

June 1954

ond continuous phase, liquid or gaseous, whether the latter is in motion or not, increases with its rate of flow. When this holdup becomes a substantial fraction of the free volume of the column, there refiults a tendency for the discontinuous phase to become continuous and displace the other phase. If the continuous phase is in motion, it retards the flow of the discontinuous phase through frictional drag increasing further its holdup. Since the velocities of the two phases are interdependent the effect is accelerated until the flooding point is reached. The visual flooding point may be defined as that set of flow ratcs at which either of two things may happen, depending upon the relative rates of flow of the two phases. If the holdup of the dispersed phase is high, it replaces the continuous phase, which now becomes the discontinuous phase. If the flov conditions are such that the holdups of the phases involved are comparable, thP dispersed phase map be discharged from the entrance end of the apparatus, building up to a stationary level at that end. The first mag be regarded a4 flooding with respect t o the discontinuous phase, the second xvith respect to the continuous. Any disturbance at some point in the column may produce flooding at that point a t a value other than the normal value for the column. Apparatus. The column consists of an empty vertical tube through which the two phases pass by gravity flowing in opposite directions. If the dispersed phase is lighter, it is fed at the bottom of the column through holes in a distributor nozzle or plate, rises through the downward flowing continuous phase, and is removed a t the top. The continuous phase is fed at the top and is removed at the bottom. Special end sections of the Elqin-type are necessary to minimize the disturbance of the normal flow of the discontinuous phase leaving and entering the column and to obtain reliable flooding velocity data. The interface is maintained at a fixed level.

I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY

1229

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Derivation of Theoretical Equation. The derivation will be made with reference to a liquid-liquid system. The results are directly applicable to all possible combinations of solid, liquid, and gaseous phases. Consider a lighter phase flowing upward in a countercurrent liquid-liquid spray column. Visualize the dispersed phase drops in such a way that they may be considered as forming a continuum. Consider two points arbitrarily located h distance apart in a column and apply Bernoulli's theorem separately to each phase

and 4 ( 1 - 2) 4 m c = ___ PC

Defining the wetted peripheries approximately

and

Since

Substituting Equation 16 in 14 pD =

62

DL

-

and

62 pc=-+-

The following limitations wll hr amimed:

W C

0, P C I

PC,

=

P C I P D I = PDi = P D J CCi

DL

=

17c2.L'D, = UD,, and

z1

- zz

= h.

Equations 1 and 2 simplify to

+ h - ZFCPC Pz = Pi + h + WDPD Pz

=

PI

pc

(3)

PD

(4)

The terms P , and Piale identical in Equations 3 and 4. bining terms

h

(pc

- PD)

=

hAp = S F c

PC

+ BFDPD

Com-

(5)

The friction losses or pressure drop can be expressed in the usual friction equation of turbulent floir--Fanning equation

(7) This friction results from the countercurrent flow of the two phases. The friction against the wall of the column is Emall and mill be neglected here. The friction factors f~ and fc can be expressed as a function of the Reynolds number-the generalized Blasius form .fD =

C6

[4nlDPD(tDPD]

?1

(9)

Substituting Equation 8 in 7 and 9 in 6

(I1)

42 4 nao = PD

1230

(I2)

(18s)

Comparing Equations 17a and 18a, 4/Dt becomes small compared to 6xjDL for small size droplets and larger diameter columns. Furthermore, since the friction against the D, wall of the column is small and has been neglected, the contribution of the term 4 / D t to the continuous phase periphery is negligible, and the term will be dropped froin Equation 18a. Substituting Equations 10 and 11 in 5, making appropriate substitut'ions for the equivalent diameters according t n Equations 12 and 13, and canceling terms result in

-

x;

For a given pair of liquids and a fixed value of continuous phase velocity L'c, the discontinuous phase velocity I r D can be varied up to the A,, flooding point. As the discontinuous phase velocity increases approaching the floodI ing point, z,p ~ and , L'D, inI crease to maximum value. X 0 IO Similarly (1 - z)decreases to minimum value, and pc Figure 1 increases to maximum value. Assuming arbitrary values for z. Equation 19 can be solved for the dispersed phase velwity, CD. At increajing values of 2, approaching flooding, the

xhereas

B = Defining the equivalent diameters

4 Dt

UBI" pg pE+" increase,. to maximum (1 - x ) 3

CcpE-"

Hon-ever, the sum of the tmo groups remains constant. Plotting each group as a function of z results in Figure 1. The points of the curves indicated as hl represent the flooding point when the column become8 inoperable. The region to the

INDUSTRIAL AND ENGINEERING CHEMISTRY

Vol. 46, No. 6

FLOW THROUGH POROUS MEDIA left of the dotted line indicates approach to flooding point with increasing dispersed phase velocity. If the system is operated in such a way that the dispersed phase velocity is fixed and the continuous phase velocity increased, the flooding point will be reached along that portion of the curves to the right of the dotted line. In either case the flooding point should be identical. This means that for a given system the holdup and pressure drop are uniquely determined by thc operating phase velocities. Differentiating Equation 19 with respect to holdup x and noting that a t the flooding point d c J D / d x = 0 and dUc/dx = 0 will permit evaluation for any value of U C and of the corresponding value of U D that will cause flooding IIence

CcpE -* UE -*

urnr

cx

pn

Figure 2

- CDpA-n

ug-"

/*;

+ n ) p g % - 3 Pb+"]

[z(l -

(20)

24

Differentiating Equations 17a and 18a with respect to

x

term R. The error involved in calculating flooding rates is comparable to that of the exact Equation 24. This is because the term 1/22 in Equation 25 decreases as x increases, and R or better R1I4 and Ra/* approach unity. Hence

+

( C D P D U i p ~ ) l / * (Cc pc UZ. p c ) ' / * = ( 2 g & ~ ) ' / ~

(26a)

The holdup a t flooding is not introduced in this equation which is a considerable advantage. 4 dimensionless equation may be obtained by dividing arbitrarily by the second group, or Since 6 x / D ~= P D , substituting Equations 17b and 18b in 20, reducing and rearranging result in

CcpE-" pa

uZ.-

pE+"

[+ + (2 - n)] (1 - x)*

Equation 26b may be written as follows

-

C ~ p i p; - ~UB-" (2

- n)pA+"

21

(21) where

The exponent n is small, of the order 0.2. The equation will simplify considerably with little loss of accuracy if n is considered equal to 0. Equation 21 simplifies to

and

c,

=

[2 ]Vi [2 =

0 ]~ 1/4

C T X cc pc Application to Experimental Data. Equation 26c may be rearranged

x

Solving Equation 22a for

Rewriting the general Equation 19 in the simplified form

Equation 22b giving a relation for the holdup a t flooding is substituted in Equation 23.

+

x [ ( C D P D ug p D ) 1 / 4 ( C C P C u8 p C ) ' l 4 [ ( C DP D UB PD)'/* ( C c pc UZ. P C ) ~ / ~ / R ' / ' ] = 2 g s A p where

+

R =

[

23

1+-

1

(24) (25)

'Equation 24 is an exact solution of the conditions a t flooding. " T h e exact equation can be further simplified by dropping the :June 1954

The first step in the correlation of the exp-rimental data was to plot UYa us. on arithmetic coordinates (Figure 2). According t o Equation 26d, straight lines would be obtained with a slope m = -1/C1 ( p c / p D ) ' I 4 . This was the case. Hence the value of constant C1 was calculated from these plots. The mean value for all series of runs was found to be 1.80. The next step was to plot the data on log-log coordinates in the form of Equation 26c (Figure 3).

+ 1.8 [ ( ; ) ' I 4

(2)1'2] :[ E] us.

Straight lines were obtained with a slope of - I/*. In these plots, a systematic variation with viscosity of continuous phase occurred. By trial and error the spread of the data was reduced considerably by the incorporation of the term /*%' in the abscissa. From these new plots the values of the constant C. were calcu-


1231

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT

z 5

4 3 2

101 9

87

n %

4

d j 3 2

2 c

3 1 3

L 3 g o? -

-+

6 5 4

Comparison of

3 2

I I

2

3

4 5 698910

20

30 40

f301m I

2

3

4 5 67890

20

30 40 60 Bolo0

Figure 3

lated. cs.

Since C , is a function of particle diameter, 11, :L plot oi C , Table I .

D lyas made for all phase conihina,tions C,

=

0.565 D'li

(27)

where D = D,,DL,Do,solid, liquid, or gas particle diameter in inches. The equation representing all the data, of all phase combinations in spray columns, takes the final form

The viscosity factor results from the fact that the exponent u-as made equal to 0 in the preceding derivations. This is true only for turbulent f l o where ~ the friction factor is substantially independent of the Reynolds nvmber. X viscous eontinuous phase liquid .ivill increase the numerical value of the constant, C,, nhich in turn would decrease the value of constants C,. Since C, was t,aken as a constant for all fluids, the presencc of the viscosity term in the abscissa of Equation 28 in (:Beet lowers the value of C, for viscous fluids only. The magnitude of the constant 1.8 can be predicted, and it is actually a variable and a function of the holdup a t flooding. However, its variation is small-limits 1.6 to 2.0-and an average of 1.8 is satisfact'ory. of the part,iclc diameter, D , is correctly preThe exponent dicted from the exact equation. Equation 28 has been tested on all available data for solidliquid, liquid-liquid, and gas-liquid spray columns. Fartick diameter measuremcnts iT-ere taken as reportcd and checlied for consistency. Particle diameter measurements must be accurate for best results. As a consequence the results of the solid-liquid spray column have a higher degree of accuracy. The average error on 429 calculated runs was +12.5%. The calculated runs were distributed as follows (Table 1):

Phase Coiiibinations

No. Runs

References

~-

~

~-

~~~

Table I1 shom the range of variation of t,hc physical properties and other variables concerned in the calculated runs. Kquation 28 is plotted in Figure -1. Table 1 1 .

11

1232

Distribution of Calculated Runs for Spray Columns

~~~

Range of Physical Properties and Other Variables

Physical Property Continuous phase density, PC Disaersed uhase density. PD Continuous phase viscosity, ILC Dispersed pliasc \.iscosity,,uD Interiacial or auriace tension, =

Range 0.073-108.71b./cu. i t . 0.58-38 cp. 0.57-0.9op. 11.4-73 dynes/ciii.

Other S'ariablcs Solid particle diameter, D S Liquid drop diameter, D L Cas bubble diameter, D C Column diameter, Dt Column height, H

0 127-0.213 inches 0,0025-0 50 inches 0.542-1 , 9 8 4 inches 0.542-5,38 inches i d . 2.0--8.0 it.

19.5-77.81b./cu.ft.

~

PACKED TOWERS

Visual Flooding Point. The mechanism of flooding of packed towers is the aamc as that of s p r a , ~c~olurnna. A iiumbor of phenomena have been observed at, the vieual flooding point, which have been used as its criteria. In some gas-liquid towers, a t the flooding point, a layer of liquid appears on top of the packing, and the gas bubbles through this layer. In others, the liquid laycr does not appear, but the liquid is entrained in quantity by the gas t,hat flows from the packing. I n still others, this liquid phase becomes continuous a t a point just above the paclr-

INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY

Vol. 46, No. 6


D.

in.

FLOODING R A T E S

S P R A Y COLUMNS S O L I D - LIQUID LIQUID-LIQUID

2

[$e

30 40 60 B O l O O

p?]

200

8601oOO

400

I

2OCQ

x 104

Figure 4

ing support, or slugs of foaming liquid surge through the packing. For the drip point grinds, the flooding point was t’aken where the dispersed phase appeared as a layer on the top of each layer of the packing. I n liquid-liquid packed t,oaers, a t the visual flooding point, the holdup of the dispersed phase increases considerably. For a light,er dispcrsed phase, the droplets coalesce or mass at. the top and proceed down the column until they reach the entrance chamber where they remain a t a stationary level. Marked coalescence has been noted as the visual flooding point is approached. Apparatus. The tower consist,s of a packed vertical tube through which the two phases pass by gravity flowing in opposite directions. The dispersed phase, if lighter, is fed a t the bottom of tlie tower through nozzles, which are located above the packing support a short distance within t’hepacking. The tower is packed by filling the tube with the continuous phase fluid and dropping t,he nacking for free settling. Special end sect’ionsare not necessary for the packed tower. Horn-ever, care must be taken in providing a packing support with iarge free area, a t least as large as the free area of the tower proper. The interface ia maintained at a fixed level. Derivation of Theoretical Equation.

APc =

C ~ p b - ’h~ U8-n 2g,mE+” E2-n (1 -

,@ 2)Z-n

(Compare Equation 11)

The hydraulic radii of the flow channels occupied by the two phases are given by the relations CX

-

mD =

(Compare Equation 12)

and

mc

=

e ( l - 2) ___ (Compare Equation 11) PC

Substituting the relations for the pressure drops in the general equation derived from the Bernoulli theorem Cc

Ua-n

Lc.! p&+” - _ ~ _ _ _c_ _ p;-fi

(1 -

n

+

Cn

x)3

G;-”/A;J ph+7L Xd

= 2gCe31,,

(Compare Equation 19) (29)

Dell and Pratt ( 6 )present

a detailed derivation of the correlation of flooding rates for liquidliquid packed t,owers. Rertetti (3) has a somewhat similar simplified derivation of flooding rates for gas-liquid packed tomers. Since packed towers are a special case of spray columns, the derivation is in many respects similar to that for spray columns. Hence only t,he diffrrences from the derivation given earlier mill be pointed out in this section. The generality of the correlation escaped Bertetti (8)and Dell and Pratt ( 6 ) . The final correlation of Dell and Pral t ( 6 )has been modified. Reference mill be made to a liquid-liquid system. The results are dircctly applicable to gas-liquid systems as well. The true velocities of t,he dispersed and continuous phases are U D / a and U C / E (-x), ~ respectively, The pressure drops of the t,wo phases are expressed

June 1954

The values of the terms p~ and p, which represent the total peripheries of the two phapes per square foot of column cross-sectional area must now be evaluat,ed. These terms cannot be evaluated as accurately as those of the spray columns. These terms can be separated into two components; the periphery due to t,he packing surface and rower wall, and the periphery due to Figure 5 the interface of the two phases. As in the case of the spray col-


1233

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT umn, the friction against the wall of the tower and the packing surface is small compared to the interfacial friction of the two phases. Hence the first component can be neglected. The second component is a complex function of the shape and size of the packing. A study (18) of flows of drops through packed sections has shown that, with the exception of the case in which the entering dispersed phase drops are considerably Emaller than the smallest opening in the packing, or the case in which the density difference of the two phases is great, the entering dispersed phase drops do not retain their identity as they flow through the packed section. Instead the exit drops form at the last one or two layers of packing. Defining the wetted peripheries

Equation 33b may be remitten

(Compare Equation 260) where

and

Application to Experimental Data. ranged

Equation 33c may be ax-

(Compare Equation 17a) (Compare Equation 26d)

(31)

P C = PO

(Compare Equation 18a) where r = average periphery of dispersed phase channels, feet of fluid periphery N = number of channels The average periphery of dispersed phase channels, on a volume mean basis, is proportional t o some power fraction of the superficial area of packing and the fraction voids, or

The value of the constant C2 was calculated by plotting L'gP vs. U&" on arithmetic coordinates (Figure 6) and evaluating the slope m = -1/Cn(pc/p~)1/4 of the straight lines. The mean value for all series of rune was found to be 0.835. Next the data were plotted on log-log coordinates in the form of Equation 33c (Figure 7 ) . 1

+ 0.835 [ ( ~ ) 1 ' 4 ( ~ ) 1 / z ]

us.

"1

[$

AP

CE'

Hence

Since

x=

N K 2 ap+u p+m e

(rD?/4)

where Kpu,+%* fm = average cross-sectional area of dispersed phme channel.

where K , = K I / K ~

C P R R=

Differentiating Equations 29, 30c, and 31 Kith respect to holdup X, making appropriate substitutions for the periphery terms, simplifying by making the exponent n = 0, the holdup at flooding is expressed by

{cnpD u:,

p01114

{CDpDU*pD]l/'+

)cCPCu?PC

L

1

+'2 z l \I*'*

(32)

(Compare Equation 22b)

As B numerical approximation the term R .Nil1 be dropped from Equation 32. Substituting Equation 32 in 29 and rearranging [CD PD

C,

u:P D ] 1 / 4 + [CCP C ut pCl"'

(34)

= Caota

and evaluating the best values of C, v, and s by statistical methods. I n effect, treat the product K ~ in x Equation 33c a8 a constant. By incorporating the interfacial tension term in the constant

Substituting the expression for holdup in Equation 30b

2 -

Straight lines were obtained with a slope of I n these plots, a systematic variation with interfacial or surface tension and viscosity of continuous phase occurred. By trial and error the spread of the data was reduced considerably by the incorporation of the terms 6 1 1 4 and p&l4 in the abscissa. From these ncw plots the values of the constant C, were calculated. b s the packing size increases, the fraction void, E, increases, but KBdecreases faster (Equation 33c), so that C , should increase. Since the superficial area a of the packing varies with the fraction voids, the constant C, is evaluated by assuming

[2 g ~ € ~ " l P ] ' /(3%) ~ (Compare Equation 26a)

A dimensionless equation may be obtained by dividing arbitrarily by the second group and substituting Equation 31 for thc continuous periphery term

0.87

eO.W@

1.2

for Raschig rings

€0.78

CypBs = a o 0 3 ; l ~€or ~ ~Berl ~ saddlea Equations 35 and 36 were evaluated from the most reliablc data, using 54 and 14 points, respectively. Equation 35 does not apply t o ribbed Raschig rings. These rings have high interfacial area; a portion of which is ineffective. In addition, the following relations were evaluated from the few availabIe data and should be used with caution:

CpDPG=

6

(37)

C P L R / C g RZ%R 1.172 for Lessing rings/Raschig rings

(38)

C,,/C,,,

a.

p

9

~

drip~ point ~ gridu G ~

~for ~

1.088 for balls/'Raschig rings

(39)

The data of Raschig rings and Berl saddles include packings of all sizes and materials, stainless steel, carbon, and porcelain. The equation representing all the data for the liquid-liquid and gas-liquid systems in packed towers takes the find form

(33b) (Compare Equation 26b) 1234


Vof. 46,No. 6


30 SOUARE

28

TYPICAL 26 24 22 20 18

= , (ft/R,

16 14

I2 10

8 6 4

2

0 0

1

2

3

4

5

6

7

20

8

9

40

30

50

UL2(ft/hr)

Figure 6 The values of the constant C, for the various packing materiah are given by Equations 35, 36, 37, 38, and 39. As in the case of the spray columns, the viscosity factor results from the fact that the exponent n mas made equal to 0 in the preceding derivations. The interfacial or surface tension of

the phases affects the flooding rates by affecting drop coalescence ROOT P L O T S and formation and the proportionFLOODING R A T E S ality constant Ka entering the periphery term. The greater the interfacial tension, the smaller is the size of formed drops, and the lower the tendency to coalesce. Both factors increase the numerical value of the proportionality factor Ks. Since the product K8z in Equation 33c is treated as constant, the presence of the interfacial term in effect lowers the value of C, for systems of high interfacial tension. The magnitude of the constant 0.835 can be predicted] and it is actually a variable and a functionof the holdupat flooding. Its variation is smalllimits 0.74 to 0.9-and an average of 0.835 is satisfactory. Equation 40 has been tested on all available data for liquid-liquid 60 70 80 90 and gas-liquid packed towers. -4 number of investigations of gasliquid towers were examined but could not be included in the final calculations because of insufficient data on physical properties and other variables. The fraction voids appearing in Equation 40 are the effective voids a t the flooding point. This is more closely represented by the “wet” or “drained” fraction voids. Some investigators used towers of small diameter compared to packing size. Clearly this arrangement

Figure 7

June 1954


ENGINEERING, DESIGN. AND PROCESS DEVELOPMENT ~~

Table I V .

Distribution of Calculated Runs for Packed Towers

Table V.

Packing Phase Combinations Type Liquid-liquid Raschig rings (carbon, porcelain, stainless steel) Raschig rings (porcelain, stainless steel) Raschig rings (carbon, porcelain) Raschig rings (stoneware)

Gas-liquid

Raschig rings (clay, porcelain) Raschig rings (clay) Berl saddles (porcelain) Berl saddles (clay)

Nopinal size, No. inch runs

References

'/a

133

(2, 4-6, 1 7 )

dig

120

'12

37

314

?O

112

56

1

10 3 6

111

11%

( I , I O , 90. 86')

%Ti

Total

has abnormally high fraction voids which results in higher flooding ratcs, since the operation of the tower approaches that of the spray column. This is t'he case with the data of Elgiii and Weiss (8) and SheriTood et al. (23'). Spray columns, generally speaking, have higher flooding rates than packed ton-ers. The average error of 500 calculated runs was =k13.5%. The distribution of the calculated runs appears in Table Is'. tion of the physical properties Table V shows the range of v and other variables concerned in the calculated and examined runs. Equation 40 is plot,ted in Figure 8.

Superficial area of packing, a Fractional voidage of packing, Column diameter, Dt Coilinin height, packed, €I

43.1-99.6 0.009-5.7 cp. 0.5-35.0 cp. 8.8-73.0 dynes/cni.

18-285 sq. ft./cu. i t . 0.6-0.823 cu. ft./cu. it. 1.8-8.75 inch i.d. 2.0-5.0 it

Assuming that C D = C,, Equations 24 and 26a are solved for Cn or C, by trial and error. Hence CC = CD

=

1.92 Equation 24 (exact)

CC = CD = 2.00 Equation 26a (approximate)

The difference of 4% betmen the txvo values is relatiucly small and within the range of experimental error. Since the exponent n rvas made equal t o 0, it EolloTw that

f c = c c = 2.0

A comparison is niade of t,he exact (Equation 24) and approximate (Equation 26a) equations in ,predict,ing flooding rates. For this purpose the following data arc taken from Price (16).

a,

e

Range 0.0054-99.0 lb./cu. it.

System: Benzene dispersed into water Nozzle 1 U o = 178 ft./hr. C C = 218 ft./hr. p c = 62.0 lb./cu. ft. po = 54.2 lb./cu. ft'.

DI SCUSSl ON

20

Range of Physical Properties and Other Variables

Physical properties Continuous phase density, PC Dispersed phase density, PD Continuous phase viscosity, Nc Dispersed phase viscosity, N D Interfacial or surface tension, u

Hence the T slue of the tvo-phase, interfacial, friction constant is about 400 times gieater than a corrrsponding average value for friction against the ~ s a lin l smooth pipes iyhich justifies the elim-

2 3 f t . m

30 40 60 8 0 0 0

200

400

\ \

PACKED

TOWERS

LIQUID -LIQUID

GAS

- LIQUID

Figure 8

1236


Vol. 46, No. 6


PACKED TOWERS

Figure 9

ination of the latter pressure drop from the equations. The predominance of the two-phase, interfacial, friction is the basis for the close correspondence of all these types of contacting devices. The final correlations for the spray columns and packed towers, Equations 28 and 40, respectively, can be expressed on a common basis for purposes of comparison, after making simplifications

y

=

1

+ 0 835 [(

(%)''I

us.

c, u

1116

Equations 28 and 40 should be used as follows when calculating flooding rates:

1. Assume or set the ratio UD/C'C 2.

Calculate the group Y = 1

+ 1.8 [(E)'''

(2)"*I

or the corresponding packed tower group

Spray columns

3. Calculate C', or C, observing the limits of the respective equations 4. Solve for L'c

Packed t onws

Z 0.887 + R R

+H

Wire helices Raschig rings $SS

#JB

= + R R (1.432 e".773)

Berl saddles #JLR

Z 1.088 # J R R Balls

S 0.785 + R R Drip point grids

+DPG

S 1.172 + R R

Lessing rings Equation 41 is plotted in Figuw 9. There is some evidence indlcating that a t very high continuous phase rates, when L ' D / l l C -+ 0, or at very high discontinuous phase rates, when UDICC m, the mechanism of flooding changes gradually. Hence Equations 28 and 40 1%-ouldbe applicable within limits. T o facilitate application of these equations, limits were established from the range of the examined experimental data. These limits were incorporated in the curves of Figure 4 on spray columns. For the packed towers the limits are presented separately as a plot of ---f

June 1954

5. Calculate CO Equation 41 brings out an important fact. Generally, spray column8 have higher flooding rates than packed towers. However, a tower packed Ivith large packing, 3/4-inch Raschig rings or bigger will have a higher flooding point than a corresponding spray column equipped with small distributor nozzles. The following interpolated values are givcn as an illustra,tion: Spray Column, Cc, Ft./Hr. (10) D L = 0.18 D L = 0.12 inch(' inchQ 40 64

81

100 121 144 b C

d

88.4 79.2 69.6 61.0 32.5 46.0

100.0 88.4 78.5 68.9

60.9 51.9

Packed Tower, CD,Ft./Hr. ( 6 ) Raschig rings 1-inclic a/c-inchd

1-inchb

133 5 123.0 110.0 98 0 86.5 76.6

173.0 151.0 130.0 111 1 93.5 77.5

177.0 154.2 132.8 113.3 95.2 78.2

Carbon tetrachloride into glycerol solution. Benzene into water. Dibutyl Carbitol into water. blethyl-isobutyl ketone into water.

I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY

1237

ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT Although a direct comparison is not possible, nevertheless, disregarding differences in physical properties, the packed tower has higher flooding rates in these instances. Another point of interest arises from a comparison of Equations 35, 36, 37, 38, and 39, for calculating the constant C, of the packed tower correlation. Such a comparison serves as a measure of performance of the various packing materials. The greater the numcrical value of the constant C,, the higher the dispersed flooding velocity, for a fixed continuous phase velocity, all other factors being the same. For large values of a and small value of E corresponding to small size pacliings, Raschig rings give higher dispersed phase flooding velocities, that is, they are more efficient than Berl saddles. For the large packing sizes the relation is reversed. Actually all nominal sizes of Bcrl saddles are more efficient than Raschig rings. Bubble, droplet, and particle diameter measurements must hc accurate. The diameter must be determined at each flooding point for each system. An error of 10% in diameter deicrmination will result in an estimated error of 10% in predicting flooding rates. The diameter used in these correlations is an equivalent diameter, or diameter of a sphere having the same volume as the bubble, droplet, or particle in question. For Equations 28 and 40 it is not necessary to know the holdup in order to calculate the flooding rates. By rearrangement of the exact equation, an equation involving the holdup at flooding and only one constant, the two-phase, interfacial, friction factor can be derived. This friction constant plotted as a function of a modified Reynolds number could include such factors as bubble and particle shape, bubble and droplet distortion that affect the two-phase, interfacial, pressure drop, and flooding rates. Such a friction factor has not so far been determined by any other means. Finally it is possible to arrive at an equation predicting the holdup and pressure drop at flooding. Computations backed by experimental work are continued along these lines. CONCLUSIONS

The new correlations presented in this paper for the dekrmination of flooding rates of spray columns and packed towere, of all possible phase combinations, give rreults satisfactory for design purposes. Most published data and some unpublished one8 were studied and used in the computations. When using Equations 28 and 40 for design purposes the value, UD/C’C,the ratio of the flow rates of the phases is set by the conditions of the problem. Hence the maximum value of Cc, the continuous phase flow rate, is calculated, and the column diameter is selected. Using this procedure, the average error on 429 calculated rum of sprag columns is &12.5%, and the average error on 500 calculated iuns of packed towers is zk13.570. The mavimum error seldom exceeded 25.0y0, Undcr certain conditions packed toners have higher flooding rates than spray columns. As a packing material, Berl saddles have a higher efficiency than Raschig rings. Although data on gas-solid spray towers were not available, there is no reason to expect that Equation 28 will not predict flooding rates for this system. Following are two examples illustrating the use of Equations 28 and 40 in design calculations: Example 1. Calculate the diameter of a liquid-liquid spray column given the following data: System: Ethylene dichloride dispersed into mater. F l o rates: ~ Ethylene dichloride 30.0 cu. ft./hr. Water 40.0 cu. ft./hr. Average physical propesiiea: P D = 77.5 Ib./cu. ft. pc = 62.3 lb./cu. ft. /.6c = 0.9 cp. Distributor nozzle has holes 0.228 inch in diameter: D L = 0.22 inch.

1238

Calculations:

‘C‘se Equation 28. 1

[(E):

+ 1,8 C,

1‘4

=

(2)”‘1

= 2.655

0.565 D L ” ~= 0.386

X lo8 X 15.51i2 rc = 4.17 ___ x 0.0212 62.3 X 0.974 I;c = 219 ft. lhr. continuous phase velocity at flooding

Csing a safety factor of 1.25, the cross-sectional area of the column is: csa = 1.25 X 40 cu. ft.jhr.1’219 ft./hr.

=

0 228 sq. ft.

This corresgonds to a diameter of 0.530 ft. or about 7 inches (6.46 actual).Example 2. Calculate the diameter of a liquid-liquid packed torer for the system of Examale 1. Packing, /*:inch Raschig rings a = 96 sq. ft./cu. ft. E = 0.749 cu. ft./cu. ft u = 27 dynes’cm. Calculations: Cse Equations 35 and 40. 1

(z)1i2]

+ 0.835 [(E)’”

=

1.767

1.216 X 1.230

-1

4.17 X lo8 X 0.12 X 15.5 96 X 62.3 X 0.974

[

li?

X 0.104

lic = 71.0 ft./hr. continuous phase velocity a t flooding Using a safety factor of 1.25> the cross-sectional area of the tower is csa

=

1.25 X 40 cu. ft./hr./71.0 ft./hr. = 0.705 sq. Et.

This corresponds to a diameter of 0.947 ft. or about 1 ft. (11.37 inches actual). (Compare with spray column diameter.) NOMENCLATURE

A , B, Y,R

= groups

a

= superficial area of packing, sq. ft./cu.

d

= derivative

C’, C, C,, C2 = constants D = diamekr, ft., inches F j

=

ft.

friction loss, ft,.

= friction coefficient, resistance

= acceleration due to gravity, 4.17 X 18 ft./sq. hr. H , z = height of column, ft. h = difference in height of column, ft. K1, K2, K1 = proportionality constants kI = flooding point m = hydraulic radius of dispersed or continuous phase channels, ft. nz, n, p , q, s, u, v = exponents iY = number P = pressure, lb./sq. ft. = periphery of dispersed or continuous phase channels, p f t . of fluid periphery/sq. ft. of total cross-sectional area (csa) or sq. ft. of fluid contact surface area/cu. ft. of total volume or/ft. = periphery of dispersed or continuous phase channels, r ft. of fluid periphery = superficial velocity, based on ompty column, cu. ft. of U fluid/sq. ft. csa/hr. or ft,./hr. 1Vo = aork, Et. = fract,ional holdup of dispersed phase, cu. ft. of dispersed z phasejcu. ft. of total volume = fractional voidage of packing, cu. ft, free volume/cu. ft. e total volume p = viscosity, cp. p = density, lb./cu. ft. u = interfacial or surface tension, dynes/cm.

gc


Yol. 46, No. 6

FLOW THROUGH POROUS MEDIA Laddha, G. S., Smith, J. M., Chena. Eng. Progr., 46, 195 (1950). Molstad, M. C., Abbey, R. G., Thompson, 8.R., and McKinney, J. F., Trans. Am. Inst. Chem. Engrs., 38, 387 (1942). Perry, J. H., “Chemical Engineers’ Handbook,” 3rd ed., Section 10 and 11, New York, McGraw-Hill Book Co., 1950. Pratt, H. R. C., and Glover, S. T., Trans. Inst. Chem. Engrs., 24,

= packed tower group

= spray column group

$

Subscripts C

D G L P

S 8

1

= = = =

= = = =

1,2 =

continuous phase dispersedphase gasbubble liquiddrop packedtower solid particle spray column tube section 1, section 2

54 (1946).

Price, M. M., Jr., thesis, Chemical Engineering Dept., Louisiana State University, 1947. Rosenthal, H., thesis, New York University, 1960; cf. R. E. Treybal, “Liquid Extraction,” 1st ed., p. 304, New York, McGraw-Hill Book Co., 1951. Row, S. B., Koffolt, J. H., and Withrow, J. R.. Trans. A m . Inst. Chem. Engrs., 37,559 (1941).

REFERENCES

Bain, W. A., and Hougen, 0. A., Trans. Am. Inst. Chem. Engrs., 40,29 (1944).

Ballard, J. H., and Piret, E. L., IND.ENG. CHEM.,42, 1088 (1950).

Bertetti, J. W., Trans. Am. Inst. Chem. Engrs., 38, 1023 (1942). Blanding, F. H. and Elgin, J. C., Ibid., 38, 327 (1942). Breckenfeld, R. R., and Wilke. C. R., Chem. Eng. Progr., 46, 187 (1950).

Dell, F. R., and Pratt, H. R. C., Trans. Inst. Chem. Engrs., 29, 89 (1951).

Elgin, J. C., and Foust, H. C., IND.ENG. CHEM.,42, 1127 (1950).

Elgin, J. C., and Weiss, F. B., Ibid., 31, 435 (1939). Holmes, R. C., in “Chemical Engineers’ Handbook” (J. H. Perry, editor), 3rd ed., p. 686, New York, McGraw-Hill Book Co., 1950. Johnson, A. I., thesis, Chemical Engineering Dept., University of Toronto, 1950.

Sakiadis, B. C., “An Introductory Study of Flow of Drops Through Packed Sections,” Report, Chemical Engineering Dept., University of Toronto, September 1953. Sarchet, €5. R., Trans. Am. Inst. Chem. Engrs., 38,283 (1942). Schoenborn, E. M., and Dougherty, W. J., Ibid., 40, 51 (1944). Sherwood, T. K., Evans, J. E. and Longcor, J. V. A,, Ibid., 35, 507 (1939); IND. ENG.CHEM., 31, 1144 (1939).

Sherwood, T. K., and Pigford, R. L., “Absorption and Extraction,” 2nd ed., Chap. 1-11 and X, New York, McGraw-Hill Book Co., 1952. Sherwood, T. K., Shipley, G. H., and Holloway, F. A , IXD.ENQ. CHEM.,30,765 (1938). Treybal, R. E., “Liquid Extraction,” 1st ed., Chap. 10, New York, McGraw-Hill Book Co., 1951. White, A. M., Trans. Am. Inst. Chem. Engrs., 31, 390 (1935); undergraduate researches of -4.E. New; E. L. Laxton and R. L. Huber; W. B. Rose and F. D. Higby; University of Worth Carolina, 1934. Winning, hI. D., thesis, University of Toronto, 1952. RECEIYED for review November 23, 1953,

ACCEFTQDMarch 2 2 , 1954.

Permeability of Kaolinite ALAN S . MICHAELS

AND

C. S. LIN

MASSACHUSETTS INSTITUTE OF TECHNOLOGY. C A M B R I D G E 39. M A S S .

P e r m e a b i l i t y of kaolinite was f o u n d t o decrease markedly as t h e polarity of t h e permeating f l u i d increased. T h e permeability t o dry nitrogen gas, f o r example, was almost 20 t i m e s greater t h a n t o d i l u t e aqueous sodium versenate solution, when measured a t equal void ratios. Variation of permeability w i t h void r a t i o was f o u n d t o disagree w t h t h e KozenyCarman equation. However, t h e specific surface area of t h e clay, as calculated f r o m t h i s equation, was f o u n d t o be a linear f u n c t i o n of void r a t i o for a l l permeant fluids f o r void ratios greater t h a n u n i t y . Replacement of a polar permeant f l u i d by a nonpolar one b y gradual desolvation of a confined clay bed resulted in only a s m a l l increase in permeability; signific a n t (10 t o 30%) increases were noted o n l y when water o r sodium versenate solutions were t h e i n i t i a l permeant fluids. T h e m o s t i m p o r t a n t factor controlling kaolinite permeability appears t o be t h e degree of dispersion or disaggregation t a k i n g place in t h e original suspendi n g fluid. Colloidal effects, such as adsorbed l i q u i d surface films, appear t o have l i t t l e effect on permeability. Electroosmotic counterflow m a y be of some importance when aqueous solutions are used as permeant fluids, b u t t h i s phenomenon can account f o r less t h a n 30% of t h e t o t a l resistance t o fluid flow. Thus, w h i l e t h e permeability of compacted clay can be reduced by permeating t h e bed w i t h liquids of h i g h dispersive power, t h e possibility of signific a n t l y increasing t h e permeability of such a clay by altering t h e permeant f l u i d seems remote.

F

UNDAMENTAL variables governing the flow of fluids through porous media have been the subject of considerable atudy for many years, and for many systems success has been realized in correlating measured physical properties of permeant and porous solid with the experimentally determined permeabilities. The most important properties of porous beds which appear to correlate with bed permeability are porosity, size, and &ape of the particles composing the bed, specific surface area bf the solid, and particle orientation (9, 12, 23, $6, SS). These correlations appear to apply with considerable accuracy to

Jane 1954

porous masses composed of rigid particles of relatively large size, but fail rather badly when used to describe the permeability characteristics of beds composed of extremely small particles (of the order of 1 micron or less). It is perhaps significant that the greatest difficulties in correlation of permeability data are encountered with masses of anisometric particles-eg., clayand gelatinous substances--e.g., the heavy metal hydroxidea. One of the most widely used and most thoroughly tested correlations between the permeability and structure of porous be& is that proposed by Rozeny (18) and later modified by Carman


1239

Generalized Correlation of Flooding Rates - Industrial & Engineering

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