rodynamics of Flow in Packed T. &I. REED, 111, AND M. R. FENSKE School of Chemistry and Physics, The Pennsyluania State College, State College, Pa.
A relationship correlates variables, such as packing dimensions, phase properties, and operating conditions, involved in the hydrodynamics of countercurrent twophase flow through packed distillation columns. Data obtained in columns packed with single-turn helices are well represented by the correlation. Various other types of laboratory distillation column paclcings follow approximately the same correlation. Uses for simplified forms
of the general equation are suggested. The effects of vapor density on pressure drop, operating pressure on throughput, and phase densities on the flood point are discussed. The value of exponent n in the expression = kG"is examined. It is shown that values of n often found experimentally to be greater than the usual turbulent maximum of 2 are consistent with and required by the conditions under which a distillation column operates.
T
HE hydrodymmics which describe the operation of packed columns through which a liquid phase and a vapor phase flow concurrently or countercurrently differ from the usual fluid flow treatments in that there are two separate fluid streams flowing. The liquid phase flow is more or less in the form of a thin film on the surface of the packing material and should be treated as open channel flow. The vapor stream fills the entire volume of the conduit through which i t flows. This stream can be treated in the familiar manner with modifications. The conditions of flow and the properties of one stream affect the conditions of flow of the other stream-for instance, a change in the liquid volume, called the operating holdup, present on the packing will change the volume of the interstices available for the flow of the vapor phase. This holdup in turn is a function of the liquid rate. Furthermore, the holdup depends upon the physical properties of the liquid and probably upon those of the vapor. I n general, then, the volume of conduit available for the flow of the vapor stream is a function of the liquid properties, the vapor properties, and the flow rate of the liquid phase, as well as of the physical dimensions of the packing in any particular case. CORRELATION OF EXPERlMENTAL DATA
A relationship, Equation 1, is developed below which correlates some of the variables involved in the hydrodynamics of the vapor stream in packed columns. A relationship of this sort is of value in throughput-pressure drop calculations that need t o be made in designing fractionating columns for vacuum or pressure work when data are available only a t atmospheric pressure. The relationship developed permits the extrapolation of values obtained in a n y given packing, and using a given distilling material, to cases where data on any particular packing and other distilling materials may not be available. The expression derived is :
PaMaP(fo
- HI3 =
R Tahp?Sa
[SI"
(1)
in vihich fo
G H k h
M
= fractional free space of the dry packing = mass rate of throughput per unit of total cross section
= volume of liquid holdup a t boiling point per unit
packed volume = dimensionless proportionality constant = length or height of packing = molecular weight of vapor
P,
arithmetic average of absolute pressures at top and bottom of packing R = perfect gas law constant S = surface of dry packing per unit packed volume T, = absolute temperature a t pressure P, A T = pressure drop in the vapor stream over the total packed length, h = absolute viscosity of vapor a t T , pa =
This equation is arranged in two dimensionless products. Consistent units of mass, length, time, and force must be used throughout in evaluating the terms. A considerable amount of pressure drop and holdup data obtained by this laboratory in testing small-diameter fractionating columns over the past 15 years has been correlated by Equation 1. A complete compilation of the experimental values used is given in reference ( 9 ) . Some of the dimensions of the various columns and packings with which the data were obtained are given in Table I. These distillation columns were packed with various sizes of single-turn wire helices. Although most of the information was obtained a t atmospheric pressure using benzene, (6) and '/*-inch inside diameter some experiments with 3/32-in~h (8) helices gave points over a wide range in absolute pressure a t the top of the column using benzene, n-heptane, and tetraisobutane. Figure 1 is a logarithmic plot of these data according to Equation l. The dimensionless term comprising the entire left-hand side of the equation is the ordinate, Y , and the dimensionless group G/p,S of the right-hand side of the equation is the abscissa, X. Tests made with and without preflooding the packings are included. Points for either one of these conditions fall on the same line. A total of 290 points were calculated from the experimental data. The points obtained with benzene and nheptane as distilling materials lie approximately on a straight line of slope 1.59. The error in the holdup term, H , and consequently in (fo - H ) caused by holdup in parts of the column outside the packing may become relatively large in columns of small packed volume. Because of this, the values for short columns of small diameter should probably be accepted with reservation. An inspection of the plot reveals that a majority of the points obtained using tetraisobutane approximate a line of slope 1.59 slightly above the one drawn for the benzene%-heptane materials. The equations for each of these lines may be written as: Y = 1.67 X1.6@for benzene and similar compounds Y = 3.5 Xl.59 for tetraisobutane
654
INDUSTRIAL AND ENGINEERING CHEMISTRY
April 1950
TABLEI. Helixa I.D., Inch 3/64 1/16 1/16 1/16 1/16 3/32 ?/S o/32 3/32 3/32 3/32 3/32
1’4
1/4 3/32
PACKED
WHICH TEST DATAWERE
COLU&.INS IN
OBTAINED Wireb Gage 40 36 34 36 34 30 26 26 30 30 30 30 24 24 24 24 24 24 30
Inside Column Diam., Cm. 1.1 1.1 11.1 .5 2.54 2.54 2.8 2 .. 5 8 3 3.5 3.6 3.5 4.76 4.76 4.76 4.76 4.76 45 .. 70 68
Packed Height, Cm. 26.7 26.7 26.7 113 108 264 1250 1160 405 406 405 4Q5
Absolute Pressure MaterialC at Top. Distilled Mm. Hg BE. 730 Bz. 730 Bz. 730 Bz . 730 Bz 730 Bz 730 Bz . 730 Bz Be. 730 96 Bz 196 RZ. 730 Be . 3900
.. . .
H. TIB. TIB. TIB. TIB. Bz
540 540 540 540 540 229
.
250 730 250 100
50 730 10
All helices, except ‘/+inch and l/s-inoh i.d.. were single-turn stainless steel. ‘/&-inoh helices were single-turn aluminum, and i/*-inoh single-turn nickel. b Brown and Sharpe wire gage used throughout. C BB.,benzene; H., n-heptane; TIB., tetraisobutane. Q
That two lines are obtained for two compounds that differ appreciably in molecular weight seems to i n d i c a t e t h a t t h e r e are effects of liquid properties for which no account is made in the equation developed. E x p e r i m e n t a l values for the operating h o l d u p , H , i n 1/4-inch aluminum helices (8) show that, under similar conditions with respect to absolute pressure and throughput in grams per hour per square centimeter, H using tetraisobutane has only about one half to three quarters the value of H using n-heptane. It is known that liquid properties influence the holdup (IS). H is a critical number, because it occurs in a term t o a third power in Equation 1. C o n s e q u e n t l y , small effects of liquid properties on H may give divergent values in the correlation. The effect of the ratio of column diameter to packing diameter has not been included in the final relationship. This d i m e n s i o n l e s s ratio is probably of some significance when its value becomes small. According to Rose (IO)in the flow of a single fluid through porous beds with values of this ratio below approximately 10, the resistance of the bed is about 0.8 t o 0.9 that in a bed of infinite
65.5
diameter composed of the same particles. This effect seems t o be of minor importance with countercurrent liquid-vapor flow in helix packings in ordinary columns, inasmuch a s the data do not reveal any trends suggesting the influence of this so-called “wall factor.” Recent data collected by Struck ( 1 1 )indicate t h a t relationship 1 may be applied not only t o single-turn wire helices, but also t o all types of packings used in distillation columns. Struck examined the various characteristics such as distillation efficiency, pressure drop, and holdup of 0.25-inch Raschig rings, 0.25-inch McMahon packing (7), 0.16 X 0.16 inch Cannon packing (6), and 6/az-inch inside diameter wire helix packing in a 0.75 X 32 inch column using n-heptane at atmospheric pressure and ndecane a t lower pressures for holdup, pressure drop, and throughput measurements at 10, 20, 50, 100, and 730 mm. of mercury absolute pressure at the top of the packing. The points given by these data when treated according to Equation 1 are shown on Figure 2. The line for Raschig rings is aPProximately Parallel t o and slightly below the line for the wire helix data. Struck’s values for the helix packing and McMahon packing, both of which are made of wire material, are very close to the line found for the helix packings of Figure 1.
30000
20000
10000
8000
6000 4000
3000
2000
IO00
800
$1
600
II
2. IO0
80 60 40
30 20
10 8
6 4
0
X
Figure 1.
3
GIpaS
Correlation of Some Variables of Countercurrent Liquid-Vapor Flow in SmallDiameter Distillation Columns Packed with Single-Turn Wire Helices
656
INDUSTRIAL AND ENGINEERING CHEMISTRY
The Cannon protruded packing gives points which are in the same region as those for the wire-type packings, but when considered by themselves these points lie on a line of much greater slope than the value of 1.6 found for the n-ire-type packings and for Raschig rings. The Cannon packing differs considerably from the other types of packings. Raschig rings and &ire packings probably have similar relationships between the free space, f o , and surfare per unit packed volume, S . Cannon packing probably differs from the others in this respect and requires some other form or function for these factors in the relationship to obtain a correlation exactly equivalent to those for the other packings. It is surprising and indicative of the fundamental validity of Equation 1 t o find that the correlation developed for wire helix packings applies t o within a maximum factor of 2 or 3 jn the ordinate for many types of packing 15 hich differ so profoundly in construction. For the present i t can be stated t h a t Equation 1 vi11 predict the approximate hydrodynamic behavior of all the common types of packings used in small scale distillation at pressures above 10 mm. of mercury. At pressures much below this value arid particularly below 1 mm. of mercury, caution diould he exercised in the application of this or any other simple relationship between quantity of flow and the conditions of flow. At such very low pressures the phenomena of "slip flov-" anti "molerular flow"' enter ( I )
Vol. 42, No. 4
of a compressible fluid such as a vapor two variables in particular-vapor velocity u and vapor density p-are implicit functions of the absolute pressure. These terms must be rpduced to functions of P before further use can be made of tht> general equation in t,his case, The relat,ionships between each of these variables and the absolut,e pressure may be derived froni the perfect gas law. They are: p
= PM/R?'
(3)
GRTjjPX
= G/fp =
1.1
(4)
The factors p and u in Equabion 2 may be replaced by their equivalent expressions from Equations 3 and 4 to yield an equation in terms of some of the more fundamental variables:
rlP
=
k ( P M / RT)"
-'( p )
2 --)I
(GRT / f P M )?L ( D ) -3d h
(.?I
To conform with the requirement that the final correlation kit: useful over a large pressure range, the correct average value of' P must be used. By integration of all the differential pressurch drops over t,he packed length, h, it) is possible to show that, thtl value of P required by Equation 5 is the arithmetical average of the absolute pressures at the top and bottom of t,he pac,king. A rearrangement of the terms in Equation 5 yields the following differential expression: k&T(p)2-" ( G ) PdP = M (f) ( D )3 -p- dh Assuming certain reasonable conditions outlined below COW cerriing the several variables appraring in this equation, it m w he integratcd:
DERIVATION O F EQU,4TIOh- 1
-4, general equation may t i e obtained by the methods of dimensional analysis for the differential pressure drop, dP, in ii flowing fluid arising from energy losses over a length of path dh: dP =
k ( ~ ) ~ - I ( p )-'I(u)'l(D)"-3dh 2
(2)
in which
D = "hydraulic radius" of the conduit d h = differential elenient of conduit height or length k = proportionality const,ant = velocity term exponent u = actual linear velocitv of the flowing s h a m p = absolute viscosity of the flowing fluid p = density of the flowing fluid The right-hand side of this equation is a general expression for the friction or energy-loss term in the Bernoulli equation. I n vapor flow t,hrough parked distillation columns the kinetic a.nd potential energy terms are iiegligihle. As a n illustration of the griirral nature of this equation the Fanning equation, Tvhich is the usual hasis for the treatment of fluid flow friction data, may be ohtained immediately from Equation 2 by a simple grouping of the appropriate factors.
dP = X-(I~upj,~)"-2(pu2dh/D) The usual form of the Fanning equation is
dP = 2F X (pu2dhjU) MMHg
where Ii' is a function of the Reynolds number, Dup/p. By comparison of these last two equations it is seen that F = ( D u p / ~ ) " - ~ and IC = 2 in the Fanning equation. Various forms of Equation 2 have been useful in many problems of hydrodyna,mics. I t will now be shown how this equation yields Equat,ion 1, the relationship useful in countercurrent liquid-vapor flow t,hroughpackings. As a step toward generality it is required that the final equation be useful over the entire range of absolute pressures normally encountered in distillation. Because packed distillation columns may be operated at' absolute pressures ranging from 10 to 20 mm. of mercury up to several atmospheres, and in general the pressure drop t'hrough a packed column increases with decreasing pressure for a given mass f l o rate, ~ ~ the final equation must apply in cases where t'he ratio of the pressure drop t o the absolute operating pressure (at the top of the packing) is both high (greater t,han 1) and low (practically zero). I n t.he flow
730
4
Figure 2.
6
810
20
30 40
A 0
B i\
G ci
6 0 60100
D
v
200 300
Data from Struck ( 1 1 ) on Four Tlpes of Paclcings Treated by Equation 1 Dashed line A. B. C.
for helix packings from Figure 1 Cannon protruded Raschig rings McMahon saddles D . Single-turn wire helices
INDUSTRIAL AND ENGINEERING CHEMISTRY
April 1950
TABLE 11. PHYSICAL DIMENSIONS OF DISTILLATION COLUMN PACKINGS Helix I.D., Inch 3/64 3/64 1/16 1/16 1/16 1/16 3/32 3/32 1/8 5/32 6/32 1/4 1/4
Material of Fabrication
Wire l)iameter
___
R. 8. s. gage Cm. A. Single.-Turn Wk 0.0080 0.0080 0.0127 0,0127 0.0127 0.0160 0.0160 0.0255 0.0405 0.0405 0.0405 0.0511 0.0511
40 40 36 36 36 34 34 30 26 26
S. steel
.....
8. steel
S. steel Aluminum Aluminum
26
24 24
. B. Nominal Size, Inch Cennon packIng
0 . 1 6 X 0.16
McMahon Saddles
0.25
Rasohig rings Stainless steel.
0.25
Inside Column Diameter, Cm. e Helices 1.1 1.3 1.1 1.3 1.5 2.54
Fractional Free Space
Surface
per Unit
Packed Volume Sq.Cm./Co. 60
6.9
1:88 4.76 1.38 4.76
Other Packings Material of Fabrication Protruded nickel
1.88
0.918
22.3
100 X 100 1.88 mesh stainless steel screen Carbon 1.88
0.937
22.4
0.690
5.4
or
p
, =~k RTa(Pa,)'-"(G)'f
iM(f)" (D) a --n
(6)
It has been shown (3) t h a t the hydraulic radius, D, and the surface of packing per unit packed volume are related by the following equation : a
D
=
f/S
(7)
The actual fractional free space during the countercurrent flow is less than the fractional free space of the dry packing, fo, by the amount, H , of liquid present in a unit packed volumethat is,
f=fo-H
(8)
By substitution in Equation 6 of this last expression for f together with the substitution for D given by Equation 7 the fmal relationship among the variables is obtained. The final equation may be put in any of several forms, all of which consist of power relationships between two dimensionless ratios. The form used in the correlation of Figures 1 and 2 has been given as Equation 1.
657
surface in contact with the vapor. For want of a more accurate value the surface per unit packed volume, S, used in the numerical computations was that obtained from the dimensions and density of the dry packing particle and from the dry free space. ~5' is equal to the surface per dry packing particle times the number of particles per unit volume of packed column. T h e surface of a packing fabricated from wjre is equal t o the surface of the wire before i t is made into the packing particle. Free space and surface area values are listed in Table 11. Without doubt the surface of the dry packing is not the true surface with which the vapor stream is in contact when liquid is flowing over the packing. The particles of packing are covered to a considerable extent by the liquid phase, so t h a t the liquid on the packing and in the form of droplets, perhaps, contributes to the total friction surface and alters t h e pore dimensions in the packing. It is an experimental fact t h a t in single-turn helix packings operating near maximum throughput the liquid holdup is independent of the dry packing surface per unit volume, I n l/la-inch inside diameter helices with a dry surface of about 43 sq. cm. per cubic centimeter of packed volume the holdup is 0.25 cc. of liquid per unit packed volume a t near-maximum throughput of 300 grams per hour per square centimeter of column cross section, while in 1/4-inch inside diameter helices with a dry surface of only 11.3 sq. cm. per cubic centimeter the holdup is 0.24 cc. of liquid per cubic centimeter of packed volume a t the near-maximum throughput of 750 grams per hour per square centimeter of column cross section. An approach t o the problem of the determination of actual interfacial surface may perhaps be made with the aid of diffusion concepts and rectification efficiency measurements carried out simultaneously with the measurement of the hydrodynamic variables-of packings. SIMPLIFIED FORMS O F EQUATION 1
Because of its generalized nature, Equation 1 is unwieldly. It is worth while to examine some of the results of simplifying this expression by neglecting some of the less important or effective variables. Various degrees of simplification may be obtained, as shown in Table 111.
TABLE111. RECOMMENDED VALUES OF EXPONENTn SINGLE-TURNWIRE HELICESI N RELATIONSHIPSY = Variables Neglected in Equation 1
Form of Y
Form of
x
FOR
kx"
Value of n
DISCUSSION O F VARIABLES I N EQUATION 5
T o simplify the final results, a n isothermal condition was assumed in the above integration. Values of T and p corresponding to Pahave been used in the calculations. At total reflux in a fractional distillation column the numerical value of the mass throughput rate, G, is the same for both the vapor and the liquid superficial flow rates. I n the usual case the heats of vaporization of the compounds distilled are of approximately the same value throughout the length of a column and i t is reasonable to take G a s constant over the length of packing in a n y one test. I n instances where G should depend upon the position in the column, would no longer be linear in h, and the operating holdup would not be distributed uniformly with respect to the length of the column. T h e hydraulic radius, D,is measured by f and S i n Equation 7. No attempt has been made to determine or measure the actual
It may be shown experimentally that the effect of these successive simplifications is not only to change the value of the proportionality factor, k , but also to change the value of the exponent, n. The experimental values of n for single-turn wire helix packings corresponding to the various simplified forms of the expression are listed i n Table 111. As the equation becomes more simplified, the value of n increases. For the simplest expression, the familiar = kG", the experimental value of n for helix packings varies from about 1.7 to 2.2. P R E S S U R E DROP AND VAPOR DENSITY
It has been suggested (6) t h a t the effect of vapor density on pressure drop in packed distillation columns be correlated by plotting log against log ( G / d i ) . A better correlation involving vapor density is t h a t listed in Table 111-namely,
658
INDUSTRIAL AND ENGINEERING CHEMISTRY
I
I
I
t
HROUGHPUT AT TOP, GM./(HR)(cM~)
I
40 6 0 80100
200 300
500
1000
Figure 3. Pressure Drop X Average Vapor Density as a Function of Throughput for 3/32-In~hStainless Steel Helices Helices m a d e of No. 30 B. & S. gage wire
-
p a A p = kGn, where n has a value of about 1.8 for \Tire helix
packings. Hagy (6) obtained pressure drop-throughput data for Ca and C? hydrocarbons distilling in a 3.5-em. inside diameter stainless steel distillation column packed with 3 / d n c h stainless steel helices to a height of 4 meters. His data extend over a n absolute operating pressure range of 100 to 3900 nim. of mercury a t the top of the column with vapor densities ranging from 0.0004 to 0.0136 gram per cc. A different line is obtained for each is plotted against log G. However, each pressure when log of these lines may be brought into approximate coincidence by multiplying the pressure drop by the average vapor density in each case. The result of this correction is shown in Figure 3. It is clear that most of the effect of absolute pressure on the pressure drop arises from the effect of pressure on vapor density.
z p
THROUGHPUT AND OPERATING PRESSURE
Equation 1 may be used to determine the effect of operating pressure on the throughput in a distillation a t any given pressure drop. Simplification of this expression to a relationship between throughput G and average pressure Pa gives G = k P a l f n . For helix pacltings a value of 2 is recommended for n in this relationship. Figure 4 is a logarithmic plot of the ratio of G a t any pressure P t o G at 1 atmosphere against the average operating pressure, Pa,in atmospheres for wire helix packings. FLOOD POlNT AND PHASE DENSITIES
As a first approximation i t may be assumed that the pressure drop in any given packed column attains the same maximum value when the flood point is reached a t any operating pressure if the liquid phase has substantially the same density. Data supporting this view are listed in Table IV. From these values i t is shown by Figure 5 that the maximum throughput in volume of liquid per unit time per unit column cross section a t total reflux is directly proportional to about the 0.32 power of the vapor density-liquid density ratio in a distillation column packed
Vol. 42, No. 4
with R iie-type helices. The assumption about the pressure drop made here is, no doubt, not strictly valid, especially when the liquid phase varies markedly in density. The properties of the materials flowing and the flow conditions certainly have an effect on the value of the energy loss, a t least on that part of the loss involved in frictional drag on the surface of the liquid phase which is just sufficient to cause flooding ( 1 3 ) . I t is generally supposed that the minimum frictional drag on the liquid surface per unit length of path required to produce flooding conditions is proportional to the liquid density. This effect is to be distinguished from those others of phase densities discussed below, Because of the usual interdependence of liquid rate and vapor rate in distillation, the frictional drag on the liquid surface probably depends upon a power of the liquid density which is less than unity-. Although the liquid densities for the data of Table IV show a maximum variation of 20%, the pressure drop was found t o be fairly constant in each packing at the flood point, indicating that this effect of liquid density is relatively minor. B similar conclusion can be drawn from the correlation of Figuie 3, JThere the vapor density was found to be the most significant variable. On the other hand, the higher pressuie drops a t the flood points correspond, in the majority of cases, to the higher liquid densities. More comprehensive data using a wide range of liquid densities are necessary before this effect of liquid density can be definitely established. The effects of vapor 40 density and liquid density on the flood point or maximum throughput of a dis20 t i l l a t i o n column are o b s c u r e because the v a p o r d e n s i t y is IO affected to a consid0 erable degree by the 5z pressure, temperature? a 05 and molecular weight 2 of the material. The liquid density, on the 03 other hand, is depende n t mainly on t h e 02 weights and kinds of the individual atoms in the molecule and not on the number of 01 P. IN ATMOSPHERES atoms alone. F r o m these considerations, Figure 4. Chart for Calculaas a first approximation of Effect of Absolute Prestion, i t appears that stire on Throughput in Columns Packed with-_Single-Turn Wire the flood point in disHelices t i l l a t i o n columns at Based on relationship G j G a t m o s . = total reflux should be (Po/Patrnos.)o.j. GlGstrnos. = ratio of throughput a t any average pressure correlated as the liquid t o throughput a t 1 atmosphere presvolume flow rate, sure,Patmos. = p a t t o p +o.j(pressure drop) Ti,,,., a t the boiling point versus the quotient of the vapor density, po, divided by the liquid density, PI, a t the boiling point. This relationship is similar to the more generally established flood point correlations (4)for countercurrent flow in that i t involves the density ratio factor, pa/pi. However, this correlation does not take into account any variation in the pressure drop at flooding caused by the higher drag on the liquid surface at flooding required by greater density liquids. This frictional drag that is supposedly required multiplied by a function of the may enter as the ordinate, V,,., liquid density. As a first approximation the exponent on the liquid density in this case may be taken as unity. If this is done the ordinate becomes the mass flow rate. This modification has not been included on Figure 5. v)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
April 1950
POIKTS TABLEIv. FLOOD
IN
A simple mathematical analysis reveals more clearly this interesting dual character of the exponent in Equation 9.
WIRE HELIX PACKINQS
(Distilling Material n-Heptane-Methylcyelohexane Mixture) Column Packed DiamHeight, Pi, eter,Cm. Cm. Mm. Hg
3.5 3.5 3.5 2.5 4.8
Grnsx.
Pa, Mm. Hg
Vm'max.
-
G./ Liters/ Apmsx. Hour X Hour X Mm. Hg/ Sq. Cm. Sq. Cm. Meter
pl,
p./pl
X 108 G./Co.
A. a/wInch I.D. Stainless Steel Helices of No. 30 B. & S. Gage Wire 110 200 0.280 17.5 0.76 405 103
390 750 390 370 B. l/r-Inoh I.D. Aluminum Helices of No. 24 Diameter 4.76 Cm. I.D. Packed 405 465 264 539
Distilling Material %-Heptaneane mixture Tetraisobutane
735 3700 730 730
100 250 730
100 250
740 3700 735 741
107 256 736 15 58 107 257
440 550 730 300 350 480 470
0.590 1.260 0.590 0.560
12 13 20 17
0.72 0.66 0.59 0.66 0.66
4.8 22.8 4.8 4.8
B. & S. Gage Wire. Column Height 540 Cm.
0.61 0.81 1.10
0.41 0.505 0.70 0.77
13 12 11 10 15 13 13
659
0.76 1.77 4.8 0.20 0.73 1.33 3.33
0.72 0.68 0.66 0.725 0.695 0.685 0.61
Takin Equatioa 1 and restricting attention to a sQfe packing and flowing material, it is seen that Ap is determined by the throughput, G, and by the liquid holdup, H , c
AP = kih(G,H) (10) It has been established (4, IS) that the operating holdup of a packed column is independent of the gas rate in countercurrent liquid-gas flow b e low the flood point. The holdup is determined by the liquid rate, the liquid properties, and the packing dimensions. Equation 10 then becomes
-
Ap = Rz+z(G,L) (11) It is thegeneral practice to express the logarithm of Ap in terms of the logarithm of G and of L, so that Equation 11 is transformed into
P VALUE OF n IN RELATIONSHIP A T = kCn
p = log A p
kG"
relating pressure drop per unit height in a packed distillation column a t total reflux to vapor throughput G in mass per unit time per unit total column cross section, experimental values of n are encountered which seem unreasonably high. Data collected in this laboratory show that for helix packings n ranges from 1.7 to 2.2 with the most frequent values in the neighborhood of 1.9. For Raschig rings, on the other hand, n is usually greater than 2.2 and has a range of values from 2 to 2.4. If these values of n greater than 2 were assumed to apply to the complete Equations 1 or 2, the exponent on the viscosity p would be negative, indicating that as the viscosity of the flowing fluid increased the pressure drop required to maintain a given flow rate would decrease. Obviously, exponent n in Equation 9 has a somewhat different significancethan exponent n in Equations 1and 2. The difference lies in the fact that the more complete Equation 2 through the presence of the factor (fo - H ) takes into account the effect of throughput on the conduit dimensions through which the vapor flows. No such correcting factor enters Equation 9. A part of the task of n then in Equation 9 is to show the effect on the pressure drop of this change in conduit dimensions with throughput. In addition, this exponent must account for the usual increase in pressure drop accompanying an increase in vapor flow rate through any conduit. The sum of these two effects then gives the value of n. That many of the experimental values are greater than 2 is no longer surprising.
(121
-
In the use of the simple expression =
= wg,o
where
g = log
G
1 = log L
When G is related to L by a known relationshi as in distillacurve or line tion, Equation 12 is the algebraic expression of on the ordinary logarithmic plot of Ap versus G. Differentiation of this equation with respect to g gives the slope of the logarithmic plot and, thus, exponent n in Equation 9:
tk
Since in distillation L = rG where r/(l 1 = g + l o g r a n d d l / d g = 1. Then
or
- T)is the reflux ratio,
n = nz 3- n,
(13)
This last equatioIighows that the slope, n, of the ordinary logarithmic plot of Ap versus G used in distillation is equal to the sum of two slopes. nz = (dp/ag)i is the slope of the plot of log A p versus log G for constant values of the liquid rate, L, as parameter-Le., the slope n if there were no change in holdup. n, = (dp/bE), is the slope of log Ap versus log L for constant values of vapor flow rate G as parameter. This latter value accounts for the effect of change - in conduit diameters on the pressure drop. Using data (12) obtained with the system air-water floaing- countercurrently in a &inch diameter tower packed with f/s X l/z inch Raschig rings a value of n in Equation 13 may be calculated from values of n1 and nLI. Water was passed down over the packing a t various constant rates. At each constant water rate the pressure drop in the air stream was measured over a range from low air flow rates to and above the flood points in each case. The slopes of log pressure drop versus log air rate are about 2.0 in the range below the incipient flooding values. Thus, ni = 2.0. Replotting these data as log pressure drop versus log liquid rate a t constant air rate, a value for the slope of this plot is obtained as 0.2 for air velocities around 1 foot per second. Thus, ng = 0.2. According to these data and Equation 13 the pressure drop in this column if used for a distillation would vary as the 2 0 0.2 = 2.2 power of (fa& ) x I O 3 the vapor flow rate. Data obtained in this laboratory for '/z X '/z inch Raschig rings in columns distilling Flood Points in Helix-Packed Columns as Function of Phase Densities hydrocarbons show that this exponent is about 2.3. This is satisfactory agreement, considering the difference A, 0 n-Heptane-methyloydohexane mixture in the properties of the liquid phase in the two cases. 0 Tetraisobutane
+
Figure 5.
.
660
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY NOMENCLATURE
TL
Fundamental dimensions are ( M ) = mass, (8) = time, and ( t ) = temperature. F = friction factor in Fanning equation = fractional free space or voids of packing, .f subscript o for value in dry packing G = mass rate of vapor throughput, subscript “max.” for value at flood point g = l0gG H = total volume of liquid holdup _ _per unit packed volumeh = height or length of packed section k,kl,k, = proportionality constants L = superficial mass rate of liauid flow I = log L M = molecular weight n = velocity or vapor mass throughput exponent P = absolute pressure; subscript a for arithmetical average, subscripts 1 and 2 for top and bottom of packing, respectively = total pressure drop in vapor stream, Pz p1 & = pressure drop per unit length, A P / h ; subscspt “max.” for value a t flood point p = log A p R = perfect gas law constant r = ratio L/G S = surface of dry packing per unit packed volume T = absolute temperature, subscript a designating T a t Pa
-
( L ) = length,
packing
M P
(ML-20-’)
Dimensionless
= actual linear velocity of vapor through -
v = liquid volume flow rate per unit total
Diniensionless Dimensionless
Vol. 42, No. 4
$,$1,+P
column cross section, subscript “max.” for value at flood point = .4bsolute viscosity of vapor, subscript a designating p a t To = absolute density of fluid; subscript a for arithmetic average vapor density and subscript I for liquid density = function notation
(L)
LITERATURE CITED
( M L-2 0 -1) (11.1)
(1) Brown, G. P.,
DiSardo, A , , Cheng, G. K., and Sherwood, T. K . ,
J . A p p l i ~ dPhgs., 17, 802 (1946). (2) Cannon, XI. R., IND. ENG.CHEM.,41, 1953 (1949). (3) DallaValle, J. M., “Micromeritics,” p. 206, New York, Pitman (4)
Publishing Corp., 1943. Elgin, J. C., and Weiss, F. B., IND.ENG.CHEM.,31, 435 (1939).
(5) Forsythe, W. L., Jr., Stack, T,G., Wolf, J. E., and Conn, A. L., Ibid., 39, 714 (1947). (6) Hagy, J. D., R1.B. thesis, Peiinsylvania State College, 1941. (7) McMahon, H. O., IND.ENG.CHEM.,39, 712 (1947). (8) Petroleum Refining Laboratory, Pennsylvania State College, (9) (10)
private communication. Reed, T. M., 111, M.B.thesis, Pennsylvania State College, 1948. Roae, H. E., Proc. Inst. Mech. Engrs. (London),Applied Mech.
Gvoup, 153, 141 (1945). (11) Struck, R. T., Ph.D. thesis, Pennsylvania State College, 1948. (12) W-hite, A. M.,Trans. Am. Inst. Chem. Engrs., 3 1 , 3 9 0 (1935). (13) Zenz, F. A., Ibid., 43, 415 (1947). RECEITEDJuly 25, 1949.
Isopiestic Liquid-Va
uilibria
CALCULATED FROM OTHER EQUILIBRIA OF BINARY SYSTEMS SCOTT E. WOOD’ Yale University, New Haven, Conn., and Illinois Institute of Technology, Chicago, I l l .
T
the variation of the activity A n exact thermodynamic method developed by ScatHE calculation of one coefficients with temperachard for the calculation of one set of equilibria data for set of equilibria data liquid mixtures from another set is applied to the calculature. Scatchard (5)has defor liquid mixtures from anveloped a general thermction of boiling point diagrams. In particular, the boiling other set hm always predynamic method which perpoint diagram of the benzene-methanol system at 1sented a problem. One such mits the exact solution of atmosphere pressure has been calculated from vapor calculation of particular insuch a problem and it is the pressure data. The calculations necessary to obtain the terest in distillation is the purpose of this paper to ilboiling point diagram at any pressure and to allow for the calculation of the liquidlustrate this method. I t is general dependence of the activity coefficients on the temvapor equilibria a t constant believed that this is the first perature are discussed. The method of Calculation to be pressure from other equitime that this method has used for ternary systems is outlined. libria, such as the liquidbeen applied to the calculavapor equilibria at constant tion of a boiling point diatemperature, liquid-liquid gram from a knowledge of other types of equilibria. In particular equilibria, or liquid-solid equilibria. Carlson and Colburn ( I ) , the boiling point diagram of the benzene-methanol system at among others, have made use of the isothermal activity coefficients I-atmosphere pressure has been calculated from the vapor pressure for the calculation of distillation diagrams from the isothermal data of Scatchard, Wood, and Mochel ( 7 ) in which the virial form vapor pressures. The activity coefficients are, in general, funcof the equation of state of a gas is used, and the activity cotions of the temperature and pressure but for approximate calcuefficients are considered to be functions of the temperature. lations many times it is sufficient to assume that they are indeThe calculations necessary to obtain the boiling point diagram pendent of these variables. Moreover, it i s necessary for a simple a t any pressure and to allow for the general dependence of the calculation to assume that the vapor phase follows the ideal gas activity coefficients on the temperature are also outlined. The laws. In any case the solution is obtained by a method of apmethod of calculation to be used for ternary systems i s discussed. proximations. If, however, the solution must be of sufficient accuracy or the CALCUL4TIONS FOR BINARY SYSTEMS pressure of distillation is sufficientIy high that the deviations from The condition for equilibrium between two phases is that the the ideal gas law must be considered, transcendental equations chemical potential of a component must be identical in the two are obtained making the calculations much more difficult,. Also phases and that the temperature and pressure of the two phasea it may be necessary for the sake of accuracy to take into account must also be identical. Since the absolute values of the chemical potentials of the components in their standard states are not 1 Present address, Illinois Institute of Technology, Chioagb, Ill.
