Journal of Scientific & Industrial Research

Vol. 62, August 2003, pp 777-789

Predicting Effective Mass Transfer Area in Randomly-packed Column - Design Considerations

Aynur Senol* Istanbul University, Faculty of Engineering, Department of Chemical Engineering, 34850 Avci lar, Istanbul, Turkey

Received: 15 November 2002; accepted: 07 April 2003

The paper describes the performance of a randomly packed distillation column depending on the effective vapor-l iquid interfacial area (ael and the flood ratio (per cent Fl) estimated by the Eckert flooding model. Capacity, pressure gradient, and efficiency data obtained on 6.25 mm size of a Raschig-type ceramic ring are used to establish the basis for the model rel iability analysis. The reliability of design models is analyzed against observed performance statistically by using a log-ratio objective function. The proposed ae approach for random packings is expected to be an improvement in data tit, clarifying the simultaneous impact of packing geometry, loading, and system properties on the performance of a moderate scale packed column. Details underlying some aspects of selection of an appropriate packed column design algorithm are discussed. An example of scaling-up a pilot-plant with respect to the resulting statistical factors is elucidated. Finally, an approach to design in terms of the derivatives of the function ae = f(per cent Fl) is interpreted analytical ly.

Keywords: Mass transfer, Randomly-packed column, Distillation column

1 Introduction

The scaling-up of a randomly packed distil lation column depends on a reasonable choice of proper variables and operating parameters to minimize production costs . Recently, a comprehensive study of designing a packed column confirmed that four major considerations need to be specified with reliabi l ity, i .e . , capacity or flood point, pressure gradient, effective interfacial area, and height equivalent to a theoretical plate (HETP) 1-4. Traditionally, determination of the packed column diameter has been attributed to two approaches based on the use of either a gas flooding percentage (per cent Fl) estimated by a Sherwood type flooding correlation or a percentage value of pressure drop line with respect to the general ized correiations2,,, and Leval• Many attempts have been made to predict the column capacity using the concept of the maximum efficient capacity (MEC) pointS. Besides considering these factors the paper discusses the efficiency of a pi lot­plant for distil lation in terms of two critical variables, effective area, and flood ratio. However, analysis is mainly focused on an analytical solution of conditions optimizing effective interfacial area (ae) and HETP as

*E-maiJ address: senol@istanbul .edu .tr Fax: 90 2 1 2 5 9 1 1 997

a function of flood ratio estimated by the Eckert6 flooding model . In the design practice the efficiency behavior of packings is usual ly interpreted against various vapor velocities or a vapor rate dependent factor such as, flood ratio, loading factor or density­corrected capacity parameterl-5.7 . The flood ratio is defined as a ratio of operating vapor superficial mass velocity to the flooding one for a given constant l iquid rate (per cent Fl = 100GIGf). This requires quantitative knowledge of the flood velocity so that the flood ratio can be quantified. Bolles and Fair4 have proposed an analytical approach to design a packed column using a flood ratio fixed between 70-80 per cent flooding. Rukovena and Koshi have extended the per cent Fl range to a 90 per cent l imit related to the MEC point .

It is conceivable that the implementation of a packed column design strategy argues an uncoupling of effective area (ae) from the volumetric mass transfer coefficient (kae) to establish a sentence structure distinguishing the divergent behaviors of k

and ae as the flood point is approached in a vapor phase control led system. Hanley et al. 7 have presented evidence of monotonical increase of the vapor phase mass transfer coefficient over the whole vapor loading range, while ae first increases with increase in vapor flux and then begins to decrease as the flood point is

778 1 SC I I N D RES VOL 62 AUGUST 2003

approached . They have concluded that the divergence of HETP near flooding depends sensi tive ly upon the dominating effect of the rate of loss of mass-transfer area in contrast to the rate of increase in the overal l mass-transfer coefficient. As HETP is inversely proport ional to both k and (lc, it turns out that the trends observed in (HETFI) at vary ing loading conditions should apply for effective area.

'Process

considerations, dealing with the competi tion between' {Ie and k about the loading point, arc stil l chal lenging since such systems show extreme behaviors of changing from a stable operating regime to the flooding one by any smal l perturbation. Examples of successful applicat ion of a rate-based model are precise evaluation of optimum flow condit ions and interfac ial area, as wel l as effective solution of non­ideal equilibrium equations regarding the val idation limits .

In this study, attempts are made to optimize a disti l lat ion plant performance by reducing a complex problem requiring simul taneous characterizat ion of the complementary effects of functional ly dependent quantit ies, HETP, k, A and {Ie, and their derivatives. A newly proposed log-basis approach for lie of moderate-type random packings al lows one to obtain criteria for improvement of the column performance. The study aims to develop a general optimization structure for dist i l lation applicat ions that would not requ ire previous validation for the packing type or size. The proposed optimization structure could then be applied to numerous packing sizes and shapes whi Ie sti II being general with regard to the system properties and flow conditions. Additional ly, a new conceptual definition for optimum de as the locus of the statistical design factors has been proposed

1.1 Eckert Flooding Model. c,Jj'ective Area Approaches

Characterization of a packed column efficiency is inti mately connected to the definitions of flooding

point and effecti ve mass-transfer area. Recently,

postu lating a continual thickening of the fal ling liquid

fi lm with increasing liquid flow and shear stress in random packings, theoretical interpretations of

flooding in terms of the interactive effects of liquid b

. d 217

holdup and pressure drop have een carne out··· .

Among the proposed versions of the empirical Sherwood flooding model for gas flood ve locity (CI) the Eckert flooding model is sti l l widely accepted as a reliable tool of designing packed columns, since it covers the performance of a large scale older random

packings and metal Pal l rings c lass ified also as a through-flow modern packingo . The model defined as

I f ' C 2 UJ\ 02 a p ot 0 y = ( I Fp ' ;(J..Id,LIw) l(gP"PL) vs x = FI.(; =

(LlCI)(PGlpd)5 has he en interpolated into a polynomial explicit form available for analysis of a distil lation plant�.

F (G r ) = In (AB 2)+ 3.84445 - 0.9182

In(B/G f)+ 0.1282(ln (B/Gr )r = 0, ... ( I)

where A = Fr(j..lL)02p I (gPGPL), B = L(p(; I Pdo" p = pw IpL, ,LIw :: I mPa·s, and Fp == (/1' IE.1 an adjustable parameter called packing factor. hG is the l iquid-gas flow parameter. Eq. (I) coi ncides with the flooding curve for the FJ.(; ranging from 0.0 I to 8.0, showing 0.03 per cent mean deviation. Kessler and Wankat'.! have suggested a simi lar approach with s light ly d ifferent coeffic ients relating to the points and the FLG range selected.

A summary of four generalized methods, based on two-fi lm model , re liable for predicting ae of larger-scale packed distil lation columns described

I · 1410 II· . b W I ') I I' d ear ler'" Is glven y agnereta. - . n tlls stu y,

using the transfer unit concepts of fi lm theory (He;, HI., and Hoc;) the mass transfer algorithm which combines the Bravo-Fair effecti ve area model II , as wel l as the Onda ef ({I.IO wetted area (ow) approach with the Onda individual side mass transfer coefficients has been applied. The Bravo-Fair model for effective interfacial area (oJ was performed using an adjusted exponent value r = 0.88 of the liquid surface tension (a) instead of 0.5 leading to an amount of the estimated effective area larger than the wetted one regarding to the Onda aw prediction for a smal l packing (dr < 12 .7 mm).

... (2)

where Z is the height of packed bed, 111; 0" is the l iquid surface tension, NII1l; CaL = j.lLLI(PLO"ge) is the liquid capil lary number; ReG = 6C/(apj.lG) is the modified gas Reynolds number.

There is some evidence that suggests the existence of several "kinds" of area related to a packed column operationl2 . It is recognized that hydrodynamic instabil it ies in the l iquid flow due to the waves or ripples on the liquid fil m surface, gas

SENOL: EFFECTIVE MASS TRANSFER AREA IN RANDOMLY-PACKED COLUMN 779

bubbling within I iquid puddles, droplets, local ized floodingll-13,7, as well as the occurrence of the vertical density and viscosity gradients through the packed bedl4, 15 give rise to an additional amount of interfacial area beyond the wetted packing part. In fact the stationary waves with steady finite amplitudes appearing on the surface of vertically falling liquid films throughout a wall, qualified as travelling waves, are attributed to the surface tension that compensates the destabilizing effect of gravit/6,

In a preceding stud/, an explicitly formulation of dimensionless groups affecting primarily the flooding limit along with a dimensional analysis of experimental data in terms of a log-basis relation are proceeded to realize a generalized estimation method for the ratio of effective area (aJ and its upper limit (ap).

::: =C{ IO�I!��IFI r (Fi,J'

" , (3)

whL:re the liquid film number, i,e, the ratio of Froude (FrL = (lpL2/(gPL2) :1I1d Reynolds (ReL = Li(ap!ld) numbers, FiL = Frd NeL, represents the behavior attributed to the hiE group; tht� (Wi'LlRel.) term including the liquid Weber (WeL = L2i({/!'CiPd) and Reynolds numbers accounts for the kinematic wave character of the falling liquid filmll,; the Brav(l­Fair "loading term", CaLxNc(;, incorporates the simultaneous effects of vapor and liquid loads, The Onda sUlface tension term, (J/(J", accounts for the liquid/packing surface interaction; (Je is a critical I iquid surface tension for a particular materia!: 61 N/m (ceramic), 75 N/m (steel), and 33 N/m (polyethylene)lo, The (J/Oj, group designates the binary system character. Subscripts "i" and "17"

denote light and heavy components, The experimental findings rev�al that the enhanced magnitUde of interfacial area is highly sensitive to the range of changes of both variables {Ie and Z, thereby causing an interacti ve effect on (Ie 11.17, Thus the influence of these variables has been incorporated into co' Coefficients C; of Eq. (3), obtained for three systems using multi variable procedures of linpack are given in Table I, The model satisfies two limiting conditions in accordance with the theory, i,e, if per cent Fl � 0, (ae/ap) approaches 0, and if G � Gr (per cent FI � 1 00) (lL' is indefinable in nature. An important consequence of Eq. (3) is that the estimated ae is less than al' for the whole per cent Fl range, The amount of effective area is assumed to be the same for vapor-

d I· 'd 'd I' ? I' an IqUl -sie mass trans er-' .,

2 Expcrimcntal Procedure A pilot-scale column of 9 cnl inside diam

randomly filled at a depth of 1,90 m with a Raschig­type ceramic ring and equipped with a cylindrical reboiler, a pipe type liquid distributor at the top packed section containing a distribution point density of 470 points/m2, and a total condenser was used in experiments, To avoid dealing with the effect of variable packing shape and initial voidage, only a single size of packing was studied, Details about the distillation plant and associated control equipment, as well as the geometry of a packing element were reported earlier�, There are �even temperature sensors (PtIOO) and six liquid sampling ports. Experimental runs were conducted to determine rile capacity and efficiency at total reflux for several constant column pressure drops, The efficiency tests were made using three binary systems or negative [ (trichloro­ethy lene/Il-heptane), pos i t i ve 11 (n-Ill' plane/toluene), and neutral III (benzene/toluene) types clue to the ratio of liquid surface tensi0l1s of light component to heavy one (CTI/OjJ Illitial liquid mole-fraction of light component Xi = 0,::;::; for systems II <lnd III , and

Table 1- Coefficients of Eq. (3) for positive, negative and al: sy�,lell1s considered

Sy,tem Co C, ('2 (".1 C., (;-" All systems 2,552 O.():B -0.239 0.30() 0.2<)0 -0.530

Positive" 25.462 -O.OO()5 -0.439 0,027 !.O61 -2.50

Negative" 3,203 0.078 -0.394 0.3x7 0,136 2.45

" Modelled values give a supreme point

780 J SCI [ND RES VOL 62 AUGUST 2003

various X, composItiOns in the range 0.25-0.40 for system I were used. The data obtained for system I with X, = 0.33 was previously studied8. The properties of ring and testing systems are given in Table 2. Solvents (> 99 per cent, G C) were furnished from Fluka. The attainment of constant boiling temperatures for 4 h at the bottom, top, and two middle (0.6 m and 1 .20 m high) packed bed sections was the sign of steady-state reached. Liquid samples withdrawn from the column sections were analyzed

by a Hewlett-Packard G C, Model 5890A, equipped with FID and HP I -type capil lary column. All runs were performed at normal pressure (P = 1 0 1 .32 kPa). Table 3 represents the efficiency behaviors of packings varying with the column pressure gradient (Pg) attributed to system I. Furthermore, results covering the range of changes of efficiency against the variety of initial liquid composition (x,) are shown in Figure I, that actually may val idate a negligible variation of HETP with composition . However,

Table 2 - Summary of packing characteristics and system properties"

Nominal size, tip Void fraction, £ Specific area, ap System" (alah)' Slope, Ill"

(mm) (mJ/mJ) (m2/m3)

6.25 0.545 1 037.3 ( 1 .453) 0.880

" (0.7 1 0) 0.895

III ( 1 .007) 0.850

"Properties of mixture evaluated through the methods given by Reid et al.25; "I (negative) trichloroethylene/IJ-heptane, " (positive) II-heptane/toluene, JIl (neutral) benzene/toluene; C Ratio of l iquid surface tensions of light component to heavy one; tI Slope of equilibrium curve

Table 3 - Summary of the packed column efficiency properties obtained with trichloroethylene/IJ-heptane system for di fferent initial liquid compositions (Xi) and various column pressure gradients"

t'lpi' {' Column Liquid Temperature Volatility N" NOG , Xi P

section X T a kPa Mole frac. Mole frac. K (mean)

0. 1 33 0.375 Bottom 0.260 366.8

top 0.872 3 6 1 . 1 1 .404 1 8.23 9.09

0,333 0.3 1 2 Bottom 0.207 367.8

top 0.858 36 1 .2 1 .403 1 8.78 9 .5 1

0.533 0.345 Bottom 0.2 1 6 376.6

top 0.874 36 1 . 1 1 .4034 9.02 9.83

0.800 0.300 Bottom 0. 1 85 368.2

top 0.862 36 1 .2 1 .4027 9 .30 1 0.0 1

1 .333 0.290 Bottom 0. 1 38 369.6

top 0.835 36 1 .5 1 .40 1 2 9.74 1 0.26

1 .667 0.290 Bottom 0. 1 32 369.7

top 0.845 36 1 .4 1 .40 1 2 1 0. 1 1 1 0.64

2.000 0.284 Bottom 0. 1 28 369.7

top 0.860 3 6 1 .2 1 .40 1 4 1 0.57 1 1 . 1 3

2.400 0.370 Bottom 0. 1 80 368.3

top 0.9 1 5 360.8 1 .4030 1 0.99 1 1 .92

3.200 0.335 Bottom 0. 1 52 369.0

top 0,905 360.9 1 .4023 1 1 .25 1 2.06

4.000 0.350 Bottom 0. 1 58 368.9

top 0.887 36 1 .0 1 .4023 1 0.54 1 1 .29

5.333 0.3 1 5 Bottom 0. 1 24 369.7

top 0.846 3 6 1 .4 1 .40 1 2 1 0.35 1 0.86

" Total retlux condition; h Column pressure gradient; {' Initial composition of distilled binary liquid mixture, mol fraction of light

component; tI Evaluated due to the Fenske equation2(,; " Evaluated due to the Chilton-Colburn equation26

SENOL: EFFECTIVE MASS TRANSFER AREA IN RANDOMLY -PACKED COLUMN 781

pressure gradient characteristics of packing for air/water system were also evaluated in the model reliability analysis. The efficiency and pressure gradi�nt (Pg, kPalm) and the results are given in Figure 2 and 3, where HETP is obtained by the Fenske equation . The assumption inherent in this approach, fulfiled for all the systems studied, is attributed to a geometric average of the relative volatility (a) at the top and bottom in terms of the Antoine equation.

22.0 ,------------------,

Pg(kPalm)

• • • • 0.42 •

20.0 -..-, ** S * * * 0.70 (.) '-" *

ri * Ul

++ ::r: + + 1.05 18.0 - + +

A A A A 1.26 A

16.0 -+---,--.---r--r--I---,---...-�--I�-I 0.20 0.30 0.40

Figure I-Variation of efficiency with initial liquid composition (x;) for system I (Parameters are the column pressure gradient)

36.0

30.0

E 24.0

� � E- 18.0 III :I:

12.0

6.0

--,--------------� 8.0

20

b

b.

a � •• _ ••

-- . � J:L HETP * '" observed (system f)a

<> • observed (system ll) b -- modeled (eqn 3)

40 60 %FI

0*

80

6.0

4.0

2.0

100

E "OJ tl.. 0 rI3

Figure 2 - Observed efficiency and pressure gradient properties vs per cent Fl for systems I and II, per cent Fl by the Eckert flood

model. Comparison with estimates

3 Results and Discussion

3. J Design and Analytical Solution of Optimum Conditions

Section 3 . 1 outlines the steady state two-film model, as wel l as the concepts of theoretical stage (HETP) and effective interfacial area (ae) providing the basis for an analytical prediction of the optimum per cent Fl. ae was selected as a critical variable by reason of reducing the number of degrees of freedom, as compared to HETP. It is recognized that the behavior of ae along the per cent Fl range presumably tracks the behavior of HETP- I and the contribution of k to the overall impact of coupled volumetric coefficient (kae) disapprovingly is small enough to reduce a "�ocket" type divergence of efficiency near flooding?' . So, the implication for a dependently variation of ae and HETP-I assuming a mean value estimation rule for the range of changes of k and A is proceeded.

The opinion of achieving a sentence structure for optimization, based on the independent analysis the divergent behavior of HETP alone, by neglecting the effect of changes of k, A and ae quantities is important but not exactly a sufficient factor for a thorough design process. Unfortunately the solution of extremely complex problem pertained to a simultaneous dealing with the rate of changes of four functionally dependent variables regarding the conditions attributed to the HETP derivatives is highly difficult. However the method to formulate the optimization structure including HETP and ae as two

E � tl.." I-III :I:

8.0

100.0 � 1 * ** • J!( .-. .. ...... .. ... "J. 6.0

� ...!L HETP 4.0 '" ... observed (system III)

10.0 -lit-- Onda - Bravo-Fair (eqn 2) I!. -- modeled (eqn 3) I!. 2.0 I!.

I!. I!. I!. 1.0 +-���--'----'--'-'---r-.,-----l- 0.0

20 40 60

%Fl 80 100

:[ tl.. 0 eii �

Figure 3 - Observed efficiency and pressure gradient properties vs per cent Fl for system I I I , per cent Fl by the Eckert tlood

model. Comparison with estimates

782 J SCI IND RES VOL 62 AUGUST 2003

basic variables along with distinguishing the antagonistic changes of k and ae about the loading point was thought to be an acceptable basis for this study. Nevertheless, HETP is chosen as a basic variable establ ishing the final step of an overall design process and a main comparison criterion for estimates.

Starting from the Fenske equation for the number of theoretical plates, Z/HETP, and the transfer unit approach preferred as a fundamental basis in the film theory, it has been achieved as analytical structure of optimization related to a "hump" type divergence of both (Ie and HETP-', which justifies the derivative conditions of a supreme poin{

d(a,,) = ( ). d(O/OFl) ,

d2(a,.) 0

d(%F/)� < . . . (4)

For vapor-phase controlled mass transfer process the same remarks hold for the HETP-' quantity obtained from the height of overall gas-phase transfer unit (HoG = HG + }J-Id as,

InA ( ) HETP = A _ I H (i + AH L ' ... (5)

where A = mGIl/LIll is the ratio of slopes of equilibrium (m) to operating lines. He; = GII/(k,aJ and HL = LIl/(k.aJ are the heights of gas- and liquid-side transfer units, respectively. Incorporating Eq. (5) into Eq. (4) along with supposing a mean estimated value for the overall range of changes of mass-transfer coefficients (k, mole-fraction driving force), constant molar flow rates (Gil! and LIll), and an average }. for all Z, then, the basic optimum conditions attributed to HETP result in,

d(HETP-1 ) _ • -----0 d(%F/) ,

d 2 (HETP- ' ) I --'----1--'- \ 0 . d(O/CFI)-... (6)

Similar derivative conditions of HETP have been obtained by Mell and Spek�ljak'R using a heavy component basis along with neglected variation of {Ie·

A classical design method mainly utilizes the simple process of setting up the model and then adjusting the design variable in trying to minimize a given objective function. The usual approach to solve the problem focused on Eg. (4 and 6) is to seek an appropriate function between Ue (or HETP) and flood ratio so that the derivative of the function will be zero

at a strongly prescribed value of the independent variable (per cent Fl). The additional test with the sign of the second derivative separates the location of supreme (optimum) point. To reduce the complexity of optimization problem, an uncoupling and independent dealing only with the derivatives of a� is proceeded. From the definition of per cent FI the Reynolds number for gas is rearranged as:

Incorporating Eq. (7) into Eg_ (3) the (Ie derivative is defined.

( (5,/(5" j"C, (RC .. \C , - t O .06G, / (·a . " , . )} , .. . ( 8 ) IOO - % FI ) · " j .. '"

where Q = (/I'CO (FiL r (CaL )" (WCL / ReL)" (v/v,)" . Similarly, differentiating Eq. (2) (Bravo-Fair model) in terms of pCI' cent F! gives,

_ d (a,, ) = 0.498a (crORX /ZOA VCa )0",'12 d(%FI) fI I A I.

0.392(RCG y>YI2-1 0.06G ! I(a IJ�G ) . . .. (9)

Thus, in searching for the optimum per cent FI the derivatives of {/e to obtain the zero value, as defined by Eq. (8 and 9), need analysis. Study of a large bank of observations by Boiles and Fair� has revealed that two-film models do not accurately reflect the HETP behavior, in the 80- 1 ()O per cent FI range. By the analysis of Bravo-Fair and Onda models this tendency can also be verified (Figures 2 - 4). This leads to the modelled derivative v,ducs, d(aJid(per cent FI) and d(HETP-')!d(per cent FI), which do not change their sign about the loading point (Figures 5 and 6). The inadequacy of both models for this working range, therefore, does nol allow for a sentence structure to obtain the optimum per cent Fl. For practical reasons, in the absence of an interpolation expression for 80- 1 00 per cent FI range the evaluation of supreme point based on experiments should be utilized. The optimum per cent FI is indicative from the variation of observed {/e with per cent FI, as depicted in Figure 4.

SE OL: EFFECTIVE MASS TRANSFER AREA IN RANDOMLY-PACKED COLUMN 783

Figure 5 presents a quantitat ive asse;ssment of the Qe predictions achieved for both models, defined by Eq. (2 and 3), as well as for their derivative variations with regard to Eg. (9 and 8), respectively. The rel iabi l ity of Bravo-Fair design approaches Eq. (2 and 9) proved to be disapprovingly less accurate for the supremely ranging zone reproducing the derivative variation with no zero value, as compared to the newly proposed ones Eq. (3 and 8) yielding the optimum conditions, i .e . a zero derivative value at a strongly prescribed value of per cent Fl. The same remarks hold for HETP and its derivati ves estimated

""' "' .... t: N-5

� <:;

1200 - (sy:;tcm I, a) (system II, b) (system Ill, c)

observed • • '" (cqn J) - Bravo Fair Q • Ond" V • b

• •• 800 •

• b • • 0 0 00 o a

• a 400 t I C

�� o +l----,--,�-,-...,------,----r -'--,

20 40 60 %Fl

80 100

Figure 4 -Variation of ao with per cent Fl. Observed {/,. vs esti mates for systems I, II and III

1000

750

� 500 � -=--<l 250

,-----------------, -45.0 (l�(systc� • • ob�ervcd •• •• ---- (eqn 3) • --~ � 50 9 0FI 75 1 25

o observed --b-- (eqn 8) -- «qn9)

- -30.0

-15.0

100 0.0

15.0

f e � .s;, '0

Figure 5 - Comparison of observed {(c with predictions through Eq. (2 and 3) for system II. Variation of d(a.,)/d(per cent Fl)

with per cent Fl

by these models on the basis of the Onda k coefficients, as depicted in Figure 6 . It is expected to apply the obtained opt imum per cent FI to any packed system, so long as the disti l lated systems are identical with respect to the (O/Oi,) ratio, i . e . systems are either positive or negative , and a geometrical ly proportional plant in terms of the (dJdp) ratio is tested . The observed etc in Figures 4 and 5 was obtained from experimental data of HETP and A, and the Onda l iquid- and vapor-side mass transfer coeffic ients, kL and kG, according to Eq. (1 0) using the m�asured system properties.

. . . ( 1 0)

where HL = LI(kLQePL); HG = G/(kc;C1cPMd. The ratio of slopes, A = mGn/Lnh was correlated using the Yoshida and Koyanagi'9 method for an average slope of equ i l ibrium curve, 111 = NOLL,J(NOGGm), The number of transfer units, NOL = fdx/(x - x*) and NOG = Jdy/(y' - y) , were estimated by the S impson method; x and y are the operating l iquid and vapor mole fractions; y' and x· are the equi l ibrium values of x and y, respect ive ly_ Qo was evaluated using a mean m value given in Table 2 and Ln/Grn = I.

Consequent ly the inadequacy of the estimated derivative variation through Eq. (2 and 9) for 80- 1 00 per cent FI range wou ld call for assumptions of a

. � E � .,; f-u.l ;C

100 -,----- .-----------,- -0.0024

10

- "

;------------ " �� . . �..:"

HETP (systems I and rl)� h Q • obselved (H) a (eqn 3) (lI)a ---- Bravo-Fai, (II) a

25

* observed (Il b Bravo-Fair (J)b

50 %Fl 75

Q Q Q Q �Q d(HETP-

')/d(%FlJ (system TI) Q

o observed --9-- (egn 3) --+-- Bravo-Fair

-0.0016

100

1 0 ... , L-________________________ �

Figure 6 - Comparison of observed efticiency with estimates through Eq. (2 and 3) for systems I and II. Variation of

d(HETP-I)/d(per cent Fl) with per cent FI

784 J SCI IND RES VOL 62 AUGUST 2003

more complex criterion should be adapted to the main structure (CaLxReG) of model to extend the prediction to the flooding zone. A preliminary analysis of, formulating a perceptible criterion structure was focused on categorizing and coupling functional ly the dimensionless-group variables l isted, DG = J(dcldp; OJ/Oh; hiE; 0" 100c; per cent Fl), where h is the l iquid holdup, m'/m3 ; E is the packing voidage, m3/m3 ; O"and o"c are the surface tension of l iquid and its critical value for a packing material, respectively, N/m. A positive system (Oj/O"h < I) provides for a more interfacial area during distil lation20. Also, variation in the column diam (dC> probably has a l ittle further effect for a (dJdr) ratio larger than 8-10, quoted as a minimum value6. 7. 3.2 Model Reliability Analysis

The reliability of packed column design models has been analyzed analytical ly by application of a log­ratio objective function (OF) proposed by Bolles and Fair4.

. . . ( 1 1 )

where X is OF of model rel iabil ity; Yobs is observed value of performance, i .e. flood velocity (Gr) , (lc, HETP, or Pg; Ymod is model predicted (modelled) value. In the design practice of disti llation applying a new class rate-based algorithm, success is

assumed to be critical at only one end of the confidence band. Using the overall design factor (F 00)' the model normalization factor (Fill = exp(X), and the safety factor (F, = exp(tS ) gi ven by Bolles and Fair4 the fol lowing expression for upper (Fod =

FmxFs; Y= HETP, Pg) and lower (FOtI = Fn/Fs; Y= Gf, ae) bounds critical i s obtained.

y X ±15 Y (F F±lh, desn = e e mod = m s ]I mod = FOd Y,,"><1' ... (12)

where Ydesn = Design value ; Ymod = Model led value; x = LX)" is mean (bias) of the objective function; Xi = OF for the i-th observation ; n = Number of observations; s = [I, (x - Xi)' /(n _I)]" is standard deviation of the OF; t = Student 's t for a certain degree of confidence. The overall design factor, Fod, is the overal l correction factor that must be applied to the model to achieve a specified probabi lity of success. For a selected 95 per cent probabil ity of success (I chance in 20 failures) and an infinite number of degrees of freedom related to a 90 per cent confidence band Student's t = 1 .645 ( ref. 21).

The resulting statistical factors of a pilot-scale column, as well as a comparison with the results of Bolles and Fair for the commercial plant are presented in Table 4. These factors indicate the need to overdesign as a result of insufficient model rel iabil ity,

Table 4 - Summary of reliability of flood velocity (Gr), pressure gradient (Pg) and effective area (aJ models

Model Eckert Bemer-Kalis" Onda Eq. (2) Eq. (3)

Present work Bolles-Fair"

Il (runs) 14' 48 103' 21" 21" 21"

X -0.3505 -0.204 0.0127 0.0409 0.4268 0.1225

5 0.1273 0.167 0.1212 0.1213 0.0419 0.0216

Fill 0.7043 0.82 1.0128 1.0417 1.5324 1.1303

F, 1.2337 1.32 1.2214 1.2216 1.0715 1.0363

FOlt 0.5709 0.62 1.2370' 0.8527 1.4301 1.0907

Student's I 1.6500 1.65 1.6500 1.6500 1.6500 1.6500

" P = 0.29A.-z (P - 1 / 2 G )2 F!( I _ __ h_ ]5 , where h=0.34ap(,uJpL)2n Re�n (Bemer-Kalis modeI22);" Estimates due to � 'f' G p 2X5/J¢£

Bolles and Fair�; 'Data for air/water and organic systems I, II and Ill; d Systems 1 and II; "Lower bound critical, F"d = F"/F,, and 95 per cent contidence; f Upper bound critical, F",I = Fill X F,

SENOL: EFFECTIVE MASS TRANSFER AREA IN RANDOMLY -PACKED COLUMN 785

e.g. a safety factor of I .234 for the Eckert flood model means that a pilot scale column designed with this model must have, on an average, a cross section 23.4 per cent larger than the modeled value to achieve a reasonable (95 per cent) confidence of success. Table 4 reveals that the standard deviation of the Eckert flood model is remarkably close to 0. 1 5 , indicating the consistency of predictions for a wide range of packing type and size and various scal ing plants. The rel iability tests covering the Berner-Kalis pressure gradient model 22 (Table 4) were performed concerning air/water system and iteratively-developed shape factors, i .e. the effecti ve voidage factor (<f!) of 0.52 and the constriction factor ( x ) of 0.475 for the packing studied. The convenience of Berner-Kal is model was also studied by plotting estimates against observed performance (Figure 7) .

Among the tested effective area models the proposed approach Eq. (3) yields a slightly more reliable overall design factor (Fod) of 1 .09 1 , as compared to 1 .430 and 0.853 for Bravo-Fair and Onda approaches, respectively (Table 4). A comparatively low accuracy of prediction for Eq. (2) may be attributed to the influence of additional control ling factors in excess of the effect of (J and (CaLxReG) quantities, presumably being responsible for an increased area beyond the wetting part. The HETP estimates in Figures 2, 3, and 6, based on the Onda k coefficients manifest quite conflict predictions, yielding Fucl factors of 0.9 1 7, 0.699 and

E '(0 c... C -0" Q) 0:; -0 0 E

0...\,0

2.5 -,----------------�

2.0

1 .5

1 .0

0.5

0.0

*

* i<k

* *

* * * *

0.5 1 .0 1 .5 2.0 pg-observed, (lcPa/m)

"*

2.5

Figure 7- Reliability of Berner-Kalis pressure gradient model evaluated against observed performance for air/water system

1 . 1 57 for Eq. (2 and 3), and the Onda model, respectively. Consequently, Eq. (3) was found to be accurate for all the systems considered, reproducing the experimental data with a mean relative error e(ae), per cent = 27.3 and root-mean-square-deviation x(ae/m- I ) = 1 48.9, as compared with e, per cent = 6 1 .7 and X = 364.4 for Bravo-Fair model . The Onda model gives a moderate confidence of e(ae), per cent = 2 1 . 79 and x(ae/m- I ) = 1 35 .8, but d isapprovingly not convenient for designing a distil lation system, since the predicted HETP increases with increase in per cent Fl over the entire working range, i .e. a trend of changes indicative for absorption rather than d istillation, as depicted in Figure 3. In fact, besides the accuracy of the model prediction, an important concern is whether the proposed model actually tracks the trend of a pocket type divergence of HETP and its derivative variation from the positive region to the negative one as the flood point is approached. However, it is essential that this phenomenon should have a significant impact on the implementation of a simu lation algorithm applying a rate-based method, e.g. a packed column design method of Mori et al. 2' . As evident from Figures 2, 3 and 6 the new model matches the observed HETP reasonably through the whole per cent Fl range except for system III (benzene/toluene) .

3.3 Design Application Using Statistical Factors

It is observed from Figure 4 that the supremely variation of ae with per cent Fl indicatively represents a maximum point. In the region above the loading point the effective mass transfer area decreases abruptly along with an unavoidable accumulation of l iquid and a large change in vapor pressure in the packed bed.

In this section, an attempt has been made to interpret analytically both optimum per cent Fl and de. Experimental data of a pilot-plant obtained with trichloroethylene/n-heptane system pertain ing to the top column conditions at the beginning of flooding is considered. The significant properties of vapor and l iquid estimated by the methods given by Reid et al. 25 are : Liquid composition, x = 0.862 mole-fraction; operating pressure, P = 1 0 1 .32 kPa; liquid load (superficial) , L = 1 .22 kg/m2/s ; vapor load (mass), U = 7.753 X IO-J kg/s ; PL = 1 1 96 .4 kg/m'; PG = 4.29 1 kg/m'; ilL = 0.4 1 4 mPa· s.

Execution of the Eckert flood model with a packing factor (Fp) of 6407 .9 m2/m' gives a model led

186 J SCI IND RES VOL 62 AUGUST 2003

gas flood velocity, Gr.mod = 1.383 kg/m2/s. Application of Eq. (12) for Gf with statistical parameters from Table 4 results in a design mass flood velocity, which for a selected 95 per cent probabi lity level should favorably fal l to a scope of 90 per cent confidence band (t = 1 .65) restricted between the fol lowing bounds of Gf•des,,: Upper bound = 1 .383FmxFs = 1 .202 kglm2/s ; Mean = 1 .383Fm = 0.974 kg/m2/s ; Lower bound = 1.383FdFs = 0.790 kg/m2/s, with the most probable value of 0.974 kg/m2/s. Because the maximum hydraulic lim� of a packed column is restrained by the flood point, or a condition defined as the 1 00 per cent MEC pointS. i .e. a point related to an infinite slope of the pressure drop curve (Figure 2 and 3), the column cross-section with respect to the design flooding load of 0.974 kg/m2/s is :

d = [ U 4 )0.; = (7.753 X IO-3 4 )0 ; 1 00 = l o Cm . . . . (13) e Gr.des" n 0.974 n

Using the concept of Rukovena and Kosh/, the design with a proposed load of a 90 per cent MEC can be evaluated, assuming that one the modeled flood velocity is the 1 00 per cent MEC: de = [4/n(U 1 0.9 X Gf.mod)]05 = [4 1 n: (7.753 X 1 0'3/0.9 X 1.383»)0 5 X 1 00 = 8.9 1 cm.

Nguyen24 has proposed �m implausible in nature approach to design de with a 50 per cent modelled flood velocity value. The implementation of a design method, . improved by Bol les and Fair4, has been fulfiled by using a flood ratio fixed between 70 and 80 per cent Fl. Assuming a flood ratio of 73 per cent, the estimated de in terms of the model led load (1.383 kg/m2/s) i s : de = [4/n(UlO.73 X Gf•mod)]0 5 = [41n: (7.753 X 1 0'3/0.73 X 1.383)]° 5 X 1 00 = 9.89 cm. Correspondingly, a valuable design algorithm of Koch-Gl itsch, Inc. (Koch-Glitsch Tower Design Software - KTD 7.0 version) performs perfectly an appropriate result for de. On the other hand, from Eq. ( 1 4) the designed de with respect to the upper and lower critical confidence levels of Fad must be approximately 7 to 32 per cent greater than the model predicted value, with the most probable increase of 19 per cent, according to the mean of objective function, assuming one hopes for a 95 per cent probabi lity of success.

de (designed) = (FOd )-0 5 de (model led) . . . . ( 1 4)

Assuming that the designed optimum load does not exceed 90 per cent of the statistical ly defined lower bound of flood velocity, application of Eq. ( 1 2) resu lts i n a mass velocity of: Gopt = 0.9xFouxGr.mod = 0.9xO.57 1 x 1 .383 = 0.7 1 1 kg/m2/s . With a designed optimum load of 0.7 1 I kg/m2/s the 90 per cent confidence band in terms of flood ratio (per cent Fl) is defined as follows:

Lower bound: (0.7 1 1 1 1 .202) 100 = 59.6 per cent

Mean: (0.7 1 1 /0.974) 1 00 = 73.0 per cent

Upper bound: (0.7 1 1 /0.790) 1 00 = 90.0 per cent

The results indicate that the optimum flood ratio is probably between 59 and 90 per cent, with expected value of 73 per cent Fl. This is considered a good design loading because it is desirable to avoid the region above 90 per cent Fl, where the efficiency decreases beyond control . The observed values are: optimum per cent Fl = 80.5 per cent ; de = 9 cm; minimum HETP = 1 7 cm due to the Fenske equation .

4 Conclusions

Characterization of the mass transfer efficiency with ae, k and per cent Fl variables provides a basis for the description of some aspects of the packed column performance. The evaluated efficiency results for three binary systems give evidence of three critical factors, i .e . the flood ratio, system properties, and packing geometry which affect significantly the magnitude of effective interfacial area. The implication for an elevated amount of effective area in distil lation, as compared with the wetted one, is attributed to a tendency toward rippling, wave, and droplet formation in the fal ling l iquid film can be val idated by observing various packings. Figure 4 i l lustrates that the observed and predicted behaviors of ae are very sensitive to the vapor and liquid loads, as well as to the system properties. Inherent to the uncoupling ae from the mass transfer coefficient the effect of changes of k on a "pocket"-like divergence of efficiency near loading are assumed to be negligible, as compared to ae closely approximating the behavior of HETP" . Consequently, a complex challenging problem of treating simultaneously with the derivatives of HETP, k, ae, and .Ie quantities has been reduced to the analysis of two dependent changing variables, HETP and ae. The way to formulate the optimization problem, including these design variables, is critically discussed . Some aspects

SENOL: EFFECfIVE MASS TRANSFER AREA IN RANDOMLY-PACKED COLUMN 787

of designing statistically the de variable in terms of the per cent Fl is expounded.

The proposed ae approach, Eq. (3), reflecting the combined effect of loading, packing geometry, and system properties appears to be improvement in data fit for moderate random packings. A modified version of the Kuramoto-Sivashinsky equation 1 6 for travel l ing l iquid waves including an additional kinetic energy term of gas flow is expected to reflect the influence of the factors mentioned.

Acknowledgments

The author is grateful to Prof. Dr Emin Ulusoy for the technical support during the study. The author also thanks Koch-Glitsch, Inc. , for providing KTD software.

References

I Kister H Z, DistiLiation design (McGraw-Hil l , New York) 1 992.

2 Wagner I, Stichlmair J & Fair J R, Mass-transfer in beds of modern, high-efficiency random packings, Illd Eng Chem Res, 36 ( 1 997) 227-237.

3 Stichlmair J, Bravo J L & Fair J R, General model for prediction of pressure drop and capacity of countercurrent gas/liquid packed columns, Gas Sep Purif, 3 ( 1 989) 1 9-28.

4 Bolles W L & Fair J R, Improved mass-transfer model enhances packed-column design, Chem Eng, 89 ( 1 982) 1 09-1 1 6.

5 Rukovena F & Koshy D T, Packed disti l lation tower hydraulic design method and mechanical considerations, Ind Eng Chem Res, 32 ( 1 993) 2400-2407.

6 Eckert J S, Trays and packings: Selecting the proper disti l lation column packing, Chem Ellg Prog, 66 ( 1 970) 39-44.

7 Hanley B, Dunbobbin B & Bennett D, A unified model for countercurrent vapor/ liquid packed columns: 2. Equations for the mass-transfer coefficients, mass-transfer area, the HETP, and the dynamic liquid holdup, Ind Eng Chem Res, 33 ( 1 994) 1 222- 1 230.

8 Senol A, Mass transfer eftlciency of randomly-packed column: Modeling considerations, Chem Eng Process, 40 (200 1 ) 4 1 -48.

9 Kessler D & Wankat P, Correlations for column parameters, Chem Ellg, 26 ( 1 988) 72-74.

1 0 Onda K , Takeuchi H & Okumoto Y , Gas absorption with chemical reaction in packed columns, i Chem Eng ipn, 1

( 1 968) 56-62. I I Bravo J L & Fair J R, Generalized correlation for mass

transfer in packed disti l lation columns, lnd Ellg Chell! Proc Des Dev, 21 ( 1 982) 1 62- 1 70.

1 2 Shi M G & Mersmann A, Effective interfacial area in packed columns, Gel' Chem Eng, 8 ( 1 985) 87-96.

1 3 Henriques de Bruto M, von Stockar U, Bangerter M A, Bomio P & Laso M, Effective mass-transfer area in a pilot plant column equipped with structured packings and with ceramic rings, lnd Eng Chem Res, 33 ( 1 994) 2400-2407.

1 4 Benneker A H, Kronberg A E & Westerterp K R , Intluence of buoyancy forces on the tlow of gases through packed beds at elevated pressures, AlChE i, 44 ( 1 998) 263-270.

1 5 Puranik S S & Vogelpohl A, Effective interfacial area in packed columns, Chem Eng Sci, 29 ( 1 974) 501 -507.

1 6 Kutateladze S S, Selected works (in Russian) (Nauka, Novosibirsk) 1 980.

1 7 Vidwans A D & Sharma M M, Gas-side mass transfer coefficient in packed columns, Chem Eng Sci, 22 ( 1 967) 673-684.

1 8 Mell R T & Spekuljak Z, Optimum design of packed distillation towers, lnd Ellg Chem Proc Des Dev, 22 ( 1 983) 230-236.

1 9 Yoshida F & Koyanagi T, Mass transfer and effective interfacial areas in packed columns, AlChE i, 8 ( 1 962) 309-3 1 6.

20 Zuiderweg F J & Harmens A, Intluence of surface phenomena on the performance of disti l lation columns, Chem Eng Sci, 9 ( 1 958) 89- 1 03 .

2 1 Johnson N L ' & Leone F C, Statistics alld experimental design ill engilleerillg and the physical sciences (John Wiley & Sons, New York) 1 964.

22 Bemer G G & Kalis G A J, A new method to predict hold-up and pressure drop in packed columns, Trans Inst Chem Eng, 56 ( 1 978) 200-204.

23 Mori H, Ito C, Oda A & Aragaki 'T, Total retlux simulation of packed column disti l lation, i Chem Eng iPII, 32 ( 1 999) 69-75.

24 Nguyen H X, Computer program expedites packed tower design, Chem Eng, (November 1 978) 1 8 1 - 1 84.

25 Reid R C, Prausnitz J M & Poling B E, The properties of gases alld liquids, 4th ed (McGraw-Hil l , New York) 1 987.

26 Seader J D & Henley E J, Separatioll process principles (John Wiley & Sons, New York) 1 998.

788 J SCI IND RES VOL 62 AUGUST 2003

5 Nomenclature

A

B

C

DG

e

F,f FLG Fp

FiL Per cent Fl

FrL G

g gc HG; HL

HOG HETP h

kG kL

L

M

m

n

P

Pg

Q

= Quantity as defined in Eq. ( I ) = Effective interfacial area (m2/m3) = Specific surface area of dry packing (m2/m3) = Wetted area of packing (m2/m3) = Quantity as defined if} Eq. ( 1 ) = Constant, or coefficient = Capil lary number for the l iquid, IlLLI(PLcrgc) = Dimensionless group = Column diameter (m) = Nominal size (or diameter) of packing (m) = Relative mean error, e = (IOO/n)Ln/(Yobs - Y mod)/Yobs/, (per cent) = Function symbol = Liquid-gas flow parameter = Packing factor, aIEl (m" )

= Film number for the l iquid, FrJReL = a/Il�/(gPL2) = Flood ratio, (G/Gf)1 00 (per cent) = Froude number for the liquid, apL2/(gpL2) = Gas superficial mass velocity (kg/m2/s) = Acceleration of gravity (rn/s2) = Force/mass conversion factor (dimensionless) = Heights of gas- and l iquid-side phase transfer units (m) = Height of an overall gas phase transfer unit (m) = Height equivalent to a theoretical plate (m) = Liquid holdup (m3/m3) = Individual mass transfer coefficient for gas-phase (kmoUm2/s/Pa) = Individual mass transfer coefficient for l iquid-phase (rn/s) = Mass transfer coefficients for the l iquid- and gas-side phases, respectively, related to the mole

fraction concentration (kmollm2/s) = Liquid superficial mass rate (kg/m2/s) = Molar mass (kg/kmol) = Slope of the equi librium curve = Number of overall transfer units based on gas- and l iquid-side phases = Number of observations = Total pressure (Pa) = Pressure gradient (Palm) = Quantity as defined in Eq. (8) = Modified Reynolds number for the gas, 6G/(ap/1G)

SENOL: EFFECTIVE MASS TRANSFER AREA IN RANDOMLY-PACKED COLUMN

ReL = Reynolds number for the liquid, L/(apJ1L)

S = Standard deviation of objective function U = Vapor mass velocity (kg/s) WeL = Weber number for the l iquid, L2/(apapd

X = Objective function of model reliability

X = Arithmetic mean (bias) of objective function .

x; x = Liquid phase mole fractions, operating and equil ibrium, respectively x = Constriction factor

Y = Independent variable used in objective function, Eq. ( 1 1 ) y; y = Vapor phase mole fractions, operating and equi librium, respectively Z = Height of packed bed (m)

Greek letters

a E

A

J1

P a

¢ X lJf

Subscripts

desn f G (OG) h L (OL)

= Relative volatility = Fraction void in packing (m3/m3) = Ratio of slopes of equilibrium to operating lines, mGrr/Lm

= Viscosity (Pa·s) = Density (kg/m3) = Surface tension of l iquid (N/m) = Critical surface tension for a packing material (N/m) = Correction factor for packing voidage = Root-mean-square-deviation

= Correction factor for density (PwlpL)

= Design = Flooding = Gas (overall gas) = Heavy ( less volatile) component = Liquid (overall liquid) = Light (more volatile) component

m = Referred to a molar quantity mod = Modelled value of performance obs = Observed value of performance opt = Optimum W . = Water

789