Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study

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Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study Zoha Azizi a,∗ , Mohsen Rezaeimanesh a , Hossein Abolghasemi b,c , Hossein Bahmanyar b a

Department of Chemical Engineering, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran Oil and Gas Center of Excellence, School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran c Center for Separation Processes Modeling and Nano-Computations, School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran b

a b s t r a c t Mass transfer coefficients along a structured packed column were experimentally determined to obtain a new correlation for dispersed phase Sherwood number based on molecular diffusivity. Then in a comparative investigation, the correlation was re-established based on effective diffusivity. The applied chemical systems were toluene/acetic acid/water (T/A/W) and butyl acetate/acetic acid/water (B/A/W). The effects of droplet size and packing height on experimental Sherwood number were also discussed. It was shown that local Sherwood number could be increased up to 188% with increasing the droplet size from 6 to 9 mm in fixed dispersed phase flow rate. It was also observed that when height of packing increased from 10 to 40 cm, local Sherwood number decreased by almost 48% for constant dispersed phase flow rate. The results have shown that the proposed correlation based on effective diffusivity can estimate the experimental drop Sherwood number with high accuracy (error of less than 5%). Moreover, current research shows that replacing molecular with effective diffusivity in some theoretical models can correct their estimation. © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Drop Sherwood number; Effective diffusivity; Liquid–liquid extraction; Droplet size; Structured packed column

1.

Introduction

Liquid–liquid extraction is an important chemical engineering operation used in many industrial processes such as refining of crude petroleum and extraction of metals. Different kinds of liquid–liquid contactors are being used in industry, which are classified according to the mixing type of two phases. Packed liquid–liquid extraction columns are widely employed in industrial practice. The flow of liquids in the column is driven by the difference in densities of the two liquid phases. Liquid may be made either continuous or dispersed by the use of suitable distributors. The presence of packing in the column increases the local velocities, retards recirculation and back-mixing and improves the distribution and hold-up of the dispersed phase (Ghaffari Tooran et al., 2009). Recently,

a number of structured packings have been developed for distillation and are being considered for use in liquid–liquid extraction. A great deal of experimental effort has been expended principally for the purpose of evaluating column performance for design and scale-up. On the other hand, the design of an extraction column for a given separation method needs reliable correlations for the prediction of overall mass transfer coefficients, which is generally calculated through correlations involving the Sherwood number. Since the scale-up of extractors still depends on large quantities of pilot experiments, measuring mass transfer coefficients by single drop experiments is a promising method which is more economical and less time-consuming (Zappe et al., 2000; Wei and Fei, 2004).

∗

Corresponding author. Tel.: +98 652 235 7873; fax: +98 652 233 8586. E-mail address: [email protected] (Z. Azizi). Received 9 October 2012; Received in revised form 17 June 2013; Accepted 11 July 2013 0263-8762/$ – see front matter © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cherd.2013.07.008 Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008

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Nomenclature



=

c (c −d )gd232



Ar

Archimedes number

Bi c

constant of Eqs. (2) and (3) solute concentration in dispersed phase (kg/m3 ) initial concentration of solute in dispersed phase (kg/m3 ) equilibrium concentration of solute in dispersed phase (kg/m3 ) drop diameter (m) Sauter mean drop diameter (m) column diameter (m) effective diffusivity (m2 /s) molecular diffusivity (m2 /s) needle size of nozzle (mm) local efficiency  

c0 c* d d32 dcol Deff DM dN E

2c

(c −d )gd232 

=

E¨o

Eötvös number

Fr

Froude number

h

height of sampling from the top of the nozzle (m) number of components dispersed phase mass transfer coefficient (m/s) number of drops number of tests (observation points) molecular weight  (kg/kmol) 

k Kd n N M



=

=



U2

slip

gd32

(c −d )g4c

Mo

Morton number

Q  Re

volumetric flow rate of dispersed phase (m3 /s) enhancement factor  for masstransfer c Uslip d32 Reynolds number = c

needle solute

The mass transfer rate is strongly affected by droplet size and hydrodynamics of the two phases. Sometimes neglecting the effects of internal circulations of drops can cause a considerable error in calculating the mass transfer inside the droplet (Ayyaswamy et al., 1990). Considering all parameters involved in prediction of mass transfer coefficient it is possible to obtain mass transfer coefficient experimentally and then an enhancement factor can be defined to correct diffusivity; replacing the molecular diffusivity with effective diffusivity obtained from that enhancement factor can modify the mass transfer coefficient. A similar job has been done on packed column in which a new correlation for effective diffusivity through enhancement factor was introduced (Rahbar et al., 2011). In the present study after a brief discussion on the behavior of mass transfer coefficient and Sherwood number along an experimental structured packed column, a new correlation for Sherwood number has been developed in which the effects of operational variables have been considered into a group of dimensionless numbers. Then it was discussed that this correlation and other theoretical models can highly be improved by applying effective diffusivity instead of molecular diffusivity whenever there are internal circulations as well as other unknown parameters that can affect the mass transfer coefficient.

c2  3



=

Kd d32 DM



Sh

Sherwood number

t T Uslip V

contact time (s) temperature (K) slip velocity of a single drop (m/s) molar volume at normal boiling temperature (cm2 /mol)  

We

Weber number

=

N S

c U2 d32 slip



Greek symbols interfacial tension (N/m)   ratio of dispersed  phase  viscosity to continuous  phase viscosity = dc i constant of Eqs. (2) and (3)  viscosity (Pa s) Pi number (=3.1415. . .)  association factor (= 2.6 for water, 1.9 for ϕ methanol, 1.5 for ethanol, and 1 for unassociated solvents) density (kg/m3 ) 

1.1.

Previous works

Several correlations have been presented in past studies for calculation of dispersed phase mass transfer coefficient. The most famous models still being considered for use to predict the mass transfer coefficient are Newman (1931) (based on molecular diffusion for unsteady state mass transfer inside a stagnant spherical drop), Kronig and Brink (1950) (based on laminar diffusion with circulation induced by relative motion of drop and continuous phase and neglecting the continuous phase resistance) and Handlos and Baron (1957) (based on eddy diffusion between internal toroidal stream lines and neglecting the continuous phase resistance). The equations from these models are, Newman (1931):

Kd = −

d 6t

 ln

6 1 exp 2 i2



i



Re < 10

(1)

Kronig-Brink (1950):

Kd = −

d 6t

 ln

3 2 Bi exp 8

 −64 D t  i M

i

Subscripts c continuous phase effective diffusivity Deff d dispersed phase eff effective i counter mixture m

−4DM 2 i2 t d2

10 < Re < 200

d2

(2) Handlos-Baron (1957):

Kd = −

d 6t





ln 2

i

 B2i

exp

−i Ut t 128(1 + )d2



Re > 200

(3)

where the constant values of i and Bi were reported by Elzinga and Banchero (1959),  = d /c and Re = c Uslip d/c . Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008

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Other investigators have replaced the molecular diffusivity with effective diffusivity, Deff , which is defined to be the product of an enhancement factor, , to molecular diffusivity, DM , in order to show the effect of internal circulation (Amanabadi et al., 2009). Inspired by this fact, Calderbank and Korchinski (1956) have modified the equation of Kronig and Brink (1950) replacing DM with DM ,

Kd = −

d 6t

 ln 1 −



 1 − exp

−42 DM t d2

1/2

(4)

in which dimensionless enhancement factor, , is 2.25. In Johnson–Hamielec equation this enhancement factor is a function of terminal velocity, molecular diffusivity, viscosity ratio and drop diameter (Johnson and Hamielec, 1960). Skelland and Wellek (1964) presented an empirical equation in which Kronig and Brink’s model have been taken into account for binary liquid–liquid extraction systems with negligible resistance in continuous phase. Steiner (1986) developed a new correlation using Johnson–Hamielec equation based on his own experimental data. Kumar and Hartland (1999) introduced a new correlation for prediction of dispersed phase mass transfer coefficient of circulating drops based on the continuous phase Reynolds number and dispersed phase Schmidt number. In a recent study (Rahbar et al., 2011), an empirical correlation for prediction of enhancement factor is developed as follows:  = 0.0272(1 + h−1 ) Re1.258

(5)

in which the Reynolds number is calculated in each experiment considering the physical properties of thecontinuous  phase and slip velocity of drops

=

c Uslip d32 c

. Its main

advantage over similar equations in literature is involvement of packing height which is really an important factor in design. In a work done by Ubal et al. (2010), numerical simulations of mass transfer were performed for a circulating single drop with applications in liquid–liquid extraction. However, it is fair to say that even the mass transfer process on the scale of a single drop is not fully understood. However, many correlations are published for prediction of Sherwood number. Although many of them have a good theoretical basis, they do not practically have accurate results. For instance, since stagnant drop models do not consider any circulation in drop and practically there are some movements in droplets hence the predicted Sherwood numbers of these models are considerably lower than experimental ones. Even though circulating drop models give better results than stagnant drop models as they consider internal circulation of drops (Rahbar and Bahmanyar, 2012), we can obtain the mass transfer coefficient with much better precision by considering the effect of packing height together with internal circulation in Sherwood number. There are also some other works on this topic but most of them could not be applied for comparison with present study since they have been presented for a specific column or a specific condition. For example Jie et al. (2005) studied on high viscosity solvents or in a study on this topic Koncsag and Barbulescu (2008) presented a correlation in which there were parameters which were specific for their own experiment.

2.

Experimental study

2.1.

Set up description

Experimental set-up contains a structured packed column in bench scale. The schematic figure of this set up is shown in Fig. 1(a). This column is made of a Pyrex glass tube. At the lower end of the column, there is a discharge valve and a glass entrance nozzle which can be used to connect to different nozzles for dispersed phase inlet. The continuous phase (batch) is fed from top and dispersed phase whose flow rate is controlled by a rothameter enters through the bottom of column. To draw the samples of the drops in different heights separate valves have been provided. Structured packing as shown in Fig. 1(b) and (c) has been used for which characteristic parameters are described in Table 1. Before each run, the column was filled with continuous phase up to the sampling valve, i.e. at the heights of 10, 25 and 40 cm. Opening its valve, dispersed phase was let to leave the nozzle in a single drop flow condition. Here the single drops moved along the packing and were drawn to the open valve at each height. They scarcely moved beside the wall and the wall effect can be negligible. The column was operated isothermally at 20 ◦ C.

2.2.

Chemical systems

The chemical system consisted of distilled water saturated with toluene or butyl-acetate as continuous phase, while dispersed phase involved toluene or butyl acetate saturated with water and the solute was specific amount of acetic acid. The mass transfer direction is from dispersed to continuous phase. The physical properties of the systems are tabulated in Table 2. According to this table, these two systems have almost similar physical properties but are different in interfacial tension, which is lower in system 2. The phases’ densities are determined by use of a scale in order of 0.0001 g based on 1000 cc, and the viscosities are evaluated by a laboratory LAUDA viscometer which has resolution of 0.001 mPa s. Interfacial tensions are based on the literature (Murphy et al., 1957; Jufu et al., 1986). The modified Wilke-Chang correlation (Eq. (6)) was used to calculate the molecular diffusivity (Poling et al., 2001):

DSm = 7.4 × 10−5

(ϕM)

1/2

T

m VS0.6

ϕM =

k

xi ϕi Mi

(6)

i=1 i= / S

2.3. Calculation of experimental dispersed phase mass transfer coefficient From mass balance for a single drop, we get (Slater and Godfrey, 1994): Kd = −

d ln(1 − E) 6t

(7)

where E=

c0 − c c0 − c∗

(8)

Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008

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Fig. 1 – (a) Schematic diagram of the apparatus, (b) a piece of Penta-Pack TM PS-500M1 structured packing, and (c) characteristics of packing.

Table 1 – Characteristic parameters for the structured packed columns used. Characteristic parameters [Unit] Column diameter, dcol [m] Column height [m] Type of packing [–] Number of packing sections used in the column [–] Length of one packed element [m] Diameter of one packed element [m] Inclination of corrugated sheets [–] Void fraction within “packed channels” [–] Specific surface for the flow [m2 /m3 ] Height of the structured packed section [m] Diameter of nozzles [mm] Number of nozzles in each run [–]

Value/Property 0.10 1.25 Penta-Pack TM PS-500M1 2 0.20 0.091 45o 0.94 500 0.40 0.4, 0.8, 1.75, and 2.0 1

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Table 2 – Physical properties of systems at 20 ◦ C. Physical property 3

c (kg/m ) d (kg/m3 ) c (mPa s) d (mPa s)  (mN/m) DM (m2 /s)

System 1 (T/A/W)

System 2 (B/A/W)

1009.7 882.7 1.016 0.611 27.5–30.1 2.92 × 10−9

1010.2 895.9 1.013 0.684 12.4–13.2 2.66 × 10−9

in which c0 , c, and c* are solute concentration in primary drop (before contact), concentration in specific position and the equilibrium concentration, respectively. In each experiment for different heights of the column by measuring acetic acid concentration, mean drop diameter (d32 ), and the contact time between two phases (t), the mass transfer coefficients (Kd ) were calculated by Eq. (7). The Sauter mean drop diameters, d32 , were determined by photographing the drops using a high-resolution Powershot G9 type camera with a background reference size located inside the column, followed by analysis with Photoshop software. At least 200 drops were analyzed for each determination. Then from Eq. (9) the Sauter mean drop diameter can be calculated (Slater, 1994).

Two different volumetric flow rate of dispersed phase (Q1 = 21.8 ml/min and Q2 = 15.6 ml/min) have been used, which were kept fixed by a rotameter during two series of the tests in order to have the desired range of spherical droplet size. Moreover, four different needle sizes of nozzles (dN = 0.4, 0.8, 1.75 and 2.0 mm) are used to study the effect of droplet size on mass transfer. Table 3 illustrates how the tests have been implemented for full factorial experimental design; with two different flow rates, Q, three different heights of sampling, h, and four different needle sizes of nozzles, dN , the number of observation points would be 2 × 3 × 4 = 24 for each chemical system. To verify the repeatability, 135 tests have totally been implemented on both chemical systems.

 3 ni di d32 =  2

3.

ni di

(9)

The acetic acid content of the dispersed phase was determined by a titration technique using normal NaOH and phenolphthalein as the indicator. The rising time of a drop, t, was determined by use of a stop watch and measuring time of a single drop from the moment it left the nozzle until being collected from the valve. Thus, with regard to mass transfer coefficient one can obtain the experimental Sherwood number using Eq. (10): Sh =

d2 Kd d32 = − 32 ln(1 − E) DM 6tDM

(10)

Results and discussion

The experimental values of dispersed phase mass transfer coefficients versus the height of packing in different nozzles are presented in Tables 4 and 5 for systems T/A/W and B/A/W, respectively. According to these tables, the dispersed phase mass transfer coefficient decreases with increase of the packing height, i.e., when height of packing increases from 10 to 40 cm, mass transfer coefficient decreases by almost 46% for the dispersed phase flow rate of 21.8 ml/min, and by almost 45% for the dispersed phase flow rate of 15.6 ml/min. We know that by drop movement along the column, the mass transfer occurs and the acetic acid concentration inside the drop decreases,

Table 3 – The characteristics of tests (observation points) for full factorial experimental design for each chemical system. Observation point number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Q (ml/min) 15.6 15.6 15.6 15.6 21.8 21.8 21.8 21.8 15.6 15.6 15.6 15.6 21.8 21.8 21.8 21.8 15.6 15.6 15.6 15.6 21.8 21.8 21.8 21.8

h (m)

dN (mm)

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.40 0.80 1.75 2.00 0.40 0.80 1.75 2.00 0.40 0.80 1.75 2.00 0.40 0.80 1.75 2.00 0.40 0.80 1.75 2.00 0.40 0.80 1.75 2.00

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Table 4 – Mass transfer coefficient (Kd ) versus height of packing for four different nozzle sizes and fixed volumetric flow rate of dispersed phase (Q), system (T/A/W). h (mm) (packing height)

Nozzle diameter (mm)

(a) Kd (cm/s) in Q = 15.6 ml/min. 100 250 400 (b) Kd (cm/s) in Q = 21.8 ml/min. 100 250 400

0.4

0.8

1.75

2.0

0.0155 0.0088 0.0086

0.0197 0.0138 0.0104

0.0237 0.0162 0.0129

0.0275 0.0209 0.0147

0.0192 0.0099 0.0089

0.0203 0.0142 0.0115

0.0251 0.0199 0.0145

0.0304 0.0224 0.0163

Table 5 – Mass transfer coefficient (Kd ) versus height of packing for four different nozzle sizes and fixed volumetric flow rate of dispersed phase (Q), system (B/A/W). h (mm) (packing height)

Nozzle diameter (mm)

(a) Kd (cm/s) in Q = 15.6 ml/min 100 250 400 (b) Kd (cm/s) in Q = 21.8 ml/min 100 250 400

0.4

0.8

0.0170 0.0123 0.0083

0.0180 0.0127 0.0104

0.0190 0.0148 0.0107

0.0233 0.0149 0.0109

0.0246 0.0161 0.0113

0.0250 0.0165 0.0132

0.0290 0.0180 0.0139

0.0298 0.0213 0.0151

thus the driving force of mass transfer decreases as height of packing increases. This reason is not generally true unless we can assume that the continuous phase resistance is negligible; but since we have a single drop flow of dispersed phase, the continuous phase remains almost pure and the assumption is valid. Consequently, the local efficiency and mass transfer coefficient decrease. Additionally, when using larger needle sizes of nozzles while the dispersed phase flow rate is constant, larger drops are produced inside of which internal circulation becomes significant (Skelland, 1985). This phenomenon resulted in increase of mass transfer such that mass transfer coefficient increases by 89% for flow rate of 21.8 ml/min, and by 105% for the flow rate of 15.6 ml/min. It should be noted that either breakage or coalescences along the packed bed are negligible as proved by photograph analysis. Furthermore, for the flow rates we used we had desired range

Nozzle diameter=0.4 mm Nozzle diameter= 1.75 mm

Nozzle diameter= 0.8 mm Nozzle diameter= 2 mm

Nozzle diameter=0.4 mm Nozzle diameter= 1.75 mm

800

800

700

700

Nozzle diameter= 0.8 mm Nozzle diameter= 2 mm

600 Sh

Sh

600 500 400

500 400 300

300 100

2.0

of droplet sizes with spherical shapes throughout the range of dispersed phase flow rate. The droplet size range during the tests was 6–9 mm. In Figs. 2 and 3 the experimental results of Sherwood number along the packed column in different nozzles with fixed volumetric flow rates are shown. While height of packing increases from 10 to 40 cm, local Sherwood number decreases by almost 47% for dispersed phase flow rate of 21.8 ml/min, and by almost 49% for flow rate of 15.6 ml/min. It is apparent from Eq. (10) that there is a direct relation between the experimental Sherwood number and mass transfer coefficient. Thus, the decreasing trend of mass transfer coefficient leads to reduction in Sherwood number. It is also shown that when using larger needle sizes of nozzles, local Sherwood number could be increased up to 188% for the flow rate of 21.8 ml/min, and up to 182% for the flow rate of 15.6 ml/min.

900

200

1.75

200

(a) 0.1

0.2

0.3 h [m]

0.4

100

(b) 0.1

0.2

0.3

0.4

h [m]

Fig. 2 – The local Sherwood number versus height of packing with four different nozzles and in fixed volumetric flow rate ((a) 15.6 ml/min and (b) 21.8 ml/min), for system (T/A/W). Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008

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Nozzle diameter=0.4 mm Nozzle diameter=1.75 mm

Nozzle diameter=0.8mm

Nozzle diameter=0.4mm

Nozzle diameter=0.8mm

Nozzle diameter=2mm

Nozzle diameter=1.75mm

Nozzle diameter=2mm

500

1000 900 800 700 600 500 400 300 200 100

400

Sh

Sh

300 200 100

(a)

0 0.1

0.2

0.3

0.4

(b) 0.1

0.2

h [m]

0.3

0.4

h [m]

Fig. 3 – The local Sherwood number versus height of packing with four different nozzles and in fixed volumetric flow rate ((a) 15.6 ml/min and (b) 21.8 ml/min), for system (B/A/W).

4.

Data correlation

4.1.

Correlating the Sherwood number based on DM

The aim is to find a correlation for local Sherwood number. The Sherwood number is directly proportional to mass transfer coefficient and generally restate the amount of mass transfer in a dimensionless form. We have tried to define the effective parameters on mass transfer coefficient as dimensionless numbers. If we consider mass transfer coefficient as a function of drop diameter (d32 ), viscosity (), density (), interfacial tension (), packing geometry (h and dcol ) and slip velocity (Uslip ), then we have 7 variables with 3 reference dimensions of Mass, Length, and Time, thus according to Buckingham Pi theorem we can group them as (7 − 3 = 4) dimensionless numbers. There are various dimensionless numbers which holds these variables and can be grouped to have a predictive correlation for Sherwood number. Among dimensionless numbers of Reynolds 2 d /), Eötvös (E¨ o = (c − (Re = c Uslip d32 /c ), Weber (We = c Uslip 32 d )gd232 /), Morton (Mo = (c − d )g4c /c2  3 ), Archimedes (Ar = 2 /gd ), etc., we tried to find c (c − d )gd232 /2c ), Froude (Fr = Uslip 32 the most effective and independent ones. Considering their equations, it is apparent that the Archimedes number (Ar) is related to Reynolds number. The interaction between gravitational and viscous forces can also be defined by Eötvös (Eö) number. The Morton number (Mo) is a dimensionless number used together with the Eötvös number to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase. The Morton number can also be expressed by using a combination of the Weber number, Froude number and Reynolds number (Mo = We3 Fr−1 Re−4 ). With this condition, a general expression of grouped dimensionless numbers to predict the drop phase Sherwood number is as follows: Sh = A ReB WeC E¨oD

 h E dcol

predicted by model and experimental values. According to the strength of EViews computing program in such area, it is being used for establishment of different combinations of the dimensionless numbers above. The parameter R2 measures the success of the regression in predicting the values of the dependent variable within the sample. In standard settings, may be interpreted as the fraction of the variance of the dependent variable explained by the independent variables. The statistic will equal one if the regression fits perfectly, and zero if it fits no better than the simple mean of the dependent variable. Another criterion, Prob, is defined in the program, which tells about the presence probability of each variable in the defined correlation. The value of Prob is also between 0 and 1, but conversely, the closer to zero indicates that the variable is more effective, i.e. if the variable’s Prob were zero, it definitely should be present in the correlation. Thus this criterion measures the probability of drawing a t-statistic as extreme as the one actually observed, under the assumption that the errors are normally distributed, or that the estimated coefficients are a symptotically normally distributed (QMS, 2002). In the computing program, first a general correlation as Eq. (11) was given with the values of dimensionless numbers calculated from our 24 observation points for each chemical system defined in Table 3. The resulted values for the parameters of A, B, C, D and E is shown in Table 6. It can be concluded from this table that the use of this combination does not seem to be much appropriate since the Probs are close to 1. It was also observed that the data reproduced by Eq. (11) is way off compared to experimental Sherwood number with average absolute diversion (AARD) of 91.43%. It is noted that the AARD for the N data is calculated as follows:





1 experiment − model × 100 N experiment N

%AARD =

(12)

i=1

(11)

Having divided by the inside diameter of the column, the height of packing has also become dimensionless in the above expression. In this research in the process of correlating, the method of least square has been applied which is one of the most appropriate methods and has an extensive application in modeling area. It is based on minimizing the sum of the squares of residuals -the difference between values

Additionally, residual plot based on Eq. (11) for system 2 as a sample is shown in Fig. 4, which also displays the calculated versus experimental Sherwood number in comparison. According to this figure we can conclude that Eq. (11) may not be useful due to increase of residuals with Sherwood number. In fact Buckingham’s theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most ‘physically meaningful’. Hence, other combinations of the mentioned dimensionless numbers have been tried in

Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008

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Table 6 – The average results of coefficients in the general model for both systems. Sh = A ReB WeC Eo˝ D (h/dcol )E

A B C D E

Coefficient

Prob.

397.4004 −0.055907 0.060767 1.125563 −0.478452

1.0000 1.0000 1.0000 0.9997 0.0000 0.989945 0.988117 25.67054

R-squared Adjusted R-squared S.E. of regression

Exprimental Sh

2500

Model Sh

Residuals

Sherwood Number

2000

1500

1000

500

0 0

5

10

15

20

25

Observaon Point Number Fig. 4 – Comparison of experimental results with calculated values from general model (Eq. (11)) and residuals based on DM for system 2 (B/A/W).

Sh = 250.847 Re0.06 E¨o0.786

 h −0.539 dcol

2

R = 0.99 AARD = 6.14%

(13)

The Probs given by the program were all zero. Considering the values of R2 it is found that the proposed correlation has good agreement with experimental data of each chemical system. As understood from the physical property table (Table 2), the proposed correlations can be used for systems of similar physical properties with possible difference in their interfacial tension. Weber and Eötvös numbers both depend on physical properties of phases including interfacial tension, hence have the same trend, may be interpreted as a conflict, with the experimental dataset. In fact, passing from Eq. (11) to Eq. (13) to have better R2 and also AARD might be hidden in the removal of this conflict. As shown in Fig. 5, the error produced by the proposed correlation is not considerable and can guaranty the accuracy of results for both systems. Corresponding residual plot of Fig. 6 indicates that most of the residuals are randomly scattered around zero.

4.2.

Correlating of Sherwood number based on Deff

Drops dispersed in liquid-liquid extraction systems are generally not rigid entities. Circulation patterns can affect them internally and also externally (Kronig and Brink, 1950; Kumar

and Hartland, 1999). These circulation patterns convert the mass transfer process from a purely diffusive one, to an advective-diffusive process (Juncu, 2001), leading to a considerable complication in the spatio-temporal distribution of solute. As discussed by Ubal et al. (2010), the circulating drop equilibrates faster than the rigid drop which is by definition wholly diffusive. For the circulating drop, material is transported rapidly along streamlines, and only needs to diffuse between the surface and an internal stagnation point of the streamline pattern. This gives a lesser diffusion distance than the rigid drop, where material diffuses between the surface and the center, and hence a more rapid equilibration time.

Sys#1 (T/A/W)

1200.00 Calculated Sherwood Number

the program and with the given Probs, %AARD, R2 value, and also considering the resulted residuals finally the correlation below (Eq. (13)) has been selected.

Sys#2 (B/A/W)

1000.00

+10%

800.00

-10%

600.00 400.00 200.00 0.00

0

200

400

600

800

1000

1200

Experimental Sherwood Number Fig. 5 – Comparison of experimental results with calculated values (Eq. (13)) based on DM .

Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008

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Sys#1 (T/A/W)

Sys#2 (B/A/W)

100.00 80.00 60.00

Residuals

40.00 20.00 0.00 -20.00

0

5

10

15

20

25

-40.00 -60.00 -80.00

Observaon Point Number

Fig. 6 – Residual plot for the calculated Sherwood number from Eq. (13).

ShDeff =

d232 Kd d32 =− ln (1 − E) Deff 6tDM

(14)

Sh = 7.264 Re0.046 E¨o0.017

 h −0.091 dcol

2

R = 0.92 AARD = 1.79%

(15)

With above equation the Probs were again almost zero. According to the comparison made between predicted values Sys#1 (T/A/W)

1

Sys#2 (B/A/W)

9 8 7

+5%

6 5

-5%

4 3 2 1 0

in which ShDeff is the Sherwood number based on Deff and the values of  are those calculated in our own recent work for both systems (Rahbar et al., 2011). According to our literature survey this replacement is a new proposition in defining Sherwood number. By the modified results of Sherwood number, a new correlation is resulted from computing program with almost equal coefficients for both systems as follows,

Sys#1 (T/A/W)

10 Calculated Sherwood Number

In the present section, we desire to replace molecular diffusivity (DM ) with effective diffusivity (Deff ) in the available correlations to study the effect of other unknown parameters, such as internal circulations interfering in determination of mass transfer coefficient and also of Sherwood number. We are trying to obtain an equivalent correlation for Sherwood number considering effective diffusivity. Thus the values of experimental Sherwood number should be re-obtained from Eq. (14):

0

2

4 6 8 Experimental Sherwood Number

10

Fig. 7 – Comparison of experimental results with calculated values (Eq. (15)) based on Deff . (Eq. (15)) and the experimental Sherwood number (Eq. (14)) shown in Fig. 7, the modified correlation can well predict the Sherwood number with average absolute relative diversion of 1.79%, which reveals better agreement with experimental results than the previous correlation (Eq. (13)). In other words, effective diffusivity can improve the proposed correlation. Here, the mass transfer is not controlled by molecular diffusivity, but by effective diffusivity. Corresponding residuals are plotted in Fig. 8. With a comparative look it is found Sys#2 (B/A/W)

0.8 0.6

Residuals

0.4 0.2 0 -0.2

0

5

10

15

20

25

-0.4 -0.6 -0.8

Observaon Point Number

Fig. 8 – Residual plot for the calculated Sherwood number from Eq. (15). Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008

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Table 7 – The %AARD results of two theoretical Sherwood number based on DM and Deff comparatively. Type of system

ShNewman Based on DM

%AARD

T/A/W B/A/W

89.0 88.8

that the random residuals are more concentrated around zero in Fig. 8 than in Fig. 6. It is important to note that from Fig. 7, the range of Sherwood numbers has been compacted because of using effective diffusivity. The new range is what we expect from single drop experiment. Moreover, two models of Newman and CalderbankKorchinski have been selected, in both of which the effective diffusivity has been used. Actually, among the many theoretical models available in literature these two have been chosen to be compared with our proposed models since they are found to be more general – e.g. Newman model has a strong mathematical basis – and more similar with respect to our experimental condition, while others could not be applied for comparison with present study since they have been developed for a specific column or a specific condition. Table 7 shows the %AARD of predicted values for both models using DM and Deff comparatively. It is obvious from Table 7 that the prediction made by these models is far way off. Newman studied the unsteady state mass transfer inside the droplets and assumed that droplets are so small that internal circulation of them is negligible. Thus the presented equations by Newman are useful for very small and rigid droplets. Calderbank-Korchinski model is an approximation form of the Kronig-Brink’s model which itself considers the small droplet sizes with laminar internal circulation. Packing cause internal circulation and increases the mass transfer coefficient (Azizi et al., 2010; Rahbar and Bahmanyar, 2012), additionally the droplet size range in this experiment was 6–9 mm, and according to what we know from literature (Skelland, 1985), when droplet size is more than 2 mm the internal circulations play a significant role in changing the mechanism of mass transfer. That’s why these correlations are not fitted with our experimental data, while the effective diffusivity can highly improve their predicted values.

5.

Conclusions

Several tests have been carried out in the packed column to obtain mass transfer coefficient and also Sherwood number, accompanied by a Sherwood number correlation and attempting to modify it through effective diffusivity replacement. The ultimate results indicate that when height of packing increases, mass transfer coefficient and Sherwood number decrease. With the increase of needle size of nozzles, larger drops with internal circulations are produced which can raise both of the mass transfer and Sherwood number. Results of the proposed models have proved a good compatibility with experimental data, and when using effective diffusivity instead of molecular diffusivity their prediction improves significantly. Defining effective diffusivity for theoretical models can also highly improve the predicted values. It is possible to extend the applicability of the well-known Newman, CalderbankKorchinski, and also our proposed equation to real drops, independent of their internal behavior. This approach can

ShCalderbank-Korchinsky Based on Deff 0.040 0.045

Based on DM 82.7 89.0

Based on Deff 6.35 6.67

be used to obtain more precise design of structured packed columns.

Acknowledgements This work as a research project entitled “Investigation of Mass Transfer Coefficient in Liquid-Liquid extraction Packed Columns” was supported by a grant from the Islamic Azad University of Mahshahr Branch.

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Please cite this article in press as: Azizi, Z., et al., Effective diffusivity in a structured packed column: Experimental and Sherwood number correlating study. Chem. Eng. Res. Des. (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.008