Prediction of effective diffusivity and using of it in designing pulsed sieve plate extraction columns

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Chemical Engineering and Processing 47 (2008) 57–65

Prediction of effective diffusivity and using of it in designing pulsed sieve plate extraction columns Hossein Bahmanyar ∗ , Laleh Nazari, Afsaneh Sadr Engineering College, Chemical Engineering Faculty, University of Tehran, Iran Received 22 June 2007; received in revised form 20 August 2007; accepted 27 August 2007 Available online 31 August 2007

Abstract Pulsed columns are one of the most important liquid–liquid extraction equipments which have an extensive application in different industries. Determination of the mass transfer coefficients as one of the fundamental parameters is of great importance in pulsed columns design. In this paper, previous works which have been done on this subject is reviewed and a set of previous models presented in the past literature are examined. Comparison between these models and the experimental results shows that although some of these equations may have satisfactory results, they do not have enough accuracy for design. This research is based on the substitution of effective diffusivity for molecular diffusivity and attempt to access it experimentally. Using effective diffusivity in equations which have been derived from solving of continuity equation in spherical coordinates will result in more accurate prediction of mass transfer coefficients. Experimental results will lead to prediction of effective diffusivity as a function of Reynolds number. Making use of the presented model in prediction of mass transfer coefficients and calculation of column’s height is in great compatibility with experimental results and is suggested for better design. © 2007 Elsevier B.V. All rights reserved. Keywords: Pulsed columns; Mass transfer; Mass transfer coefficient; Molecular diffusivity; Effective diffusivity

1. Introduction Liquid–liquid extraction is one of the most important separation processes. It has the significant ability of separating temperature sensitive components, mixtures which have very low relative volatilities, close boiling points and totally mixtures whose separation of their components with distillation is ineffective or very difficult [1]. Different kinds of liquid–liquid contactors are being used in industries which are classified according to the mixing type of the two phases. Pulse columns are a kind of liquid–liquid contactors in which the rate of the mass transfer is enhanced by hydraulic or pneumatic pulsation of the liquids in the column. A pulsed extraction column may contain ordinary packing or special sieve plates and the two immiscible liquids contact in a countercurrent direction. The main advantages of these columns are their high efficiency, simple operation and capacity for high throughput while main-

∗

Corresponding author. Tel.: +98 2161112213. E-mail addresses: [email protected] (H. Bahmanyar), [email protected] (L. Nazari). 0255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2007.08.012

taining a high-dispersed phase flow rate up to 10 or 15 times that of the continuous phase [2,3]. In a packed tower the pulsation disperses the liquids and eliminates channeling, and the contact between the phases is greatly improved. The pulsed perforated plate column also has the economic advantages of low capital cost, reduced solvent inventory and it requires less ground space compared to mixer settlers [2]. In sieve plate pulse towers the holes are smaller than in nonpulsing sieve plate towers, ranging from 1.5 to 3 mm in diameter, with a total open area in each plate of 6–23% of the cross-sectional area of the tower. Ideally the pulsation causes light liquid to be dispersed into the heavy phase on the upward stroke and the heavy phase to jet into the light phase on the downward stroke. In the more usual case the successive dispersions are less effective, and there is backmixing of one phase in one direction which will cause to drop the column efficiency severely. Nevertheless, in both packed and sieve plate pulse columns, the height required for a given number of theoretical contacts is often less than one-third that required in an unpulsed column [1]. It might be due to the increase in mass transfer area with respect to the dispersion of one phase and also increase in mass transfer coefficients. Prediction of mass transfer coefficients in pulsed columns is one of the important

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parameters required for design. Although lots of efforts have been made up to now, there is no reliable equation to obtain mass transfer coefficients. In the next section, a study of previous correlations for mass transfer coefficients has been carried out and the method of calculating column height in presence of volumetric mass transfer coefficients is introduced and experimental plots for achieving effective diffusivity and prediction of real mass transfer coefficients are presented. 2. Mass transfer coefficients and previous work The overall mass transfer coefficient of dispersed or continuous phase is one of the fundamental and essential parameters in pulsed column design. Several equations have been presented in the past literature for calculation of the overall mass transfer coefficient of dispersed phase. One of the oldest and most important of presented equations is Newman [4] equation. Newman studied the unsteady state mass transfer inside the droplets and assumed that droplets are so small that internal circulation of them is negligible. The presented equations by Newman are not only useful for very small droplets but also it is useful when the droplets are big and have great impurities and can be considered as solid and rigid bodies. Thus the mechanism of mass transfer is molecular diffusion and the overall mass transfer coefficient is obtained on the basis of continuity equation in spherical coordinates for r direction with appropriate initial and boundary conditions. The bulk motion is assumed to be negligible and there is no reaction. So, the continuity equation in spherical coordinates for r direction reduces to:  2  ∂c ∂ c 2 ∂c + = Dd (1) ∂t ∂r 2 r ∂r c(r, 0) = c0 ;

IC :

BC1 :

c(rs , t) = c∗ ;

lim c(r, t) = bounded

BC2 :

n=1

Considering the mass balance for droplet we get: −d ln(1 − E) 6t

(3)

If the continuous phase resistance exists, combination of Eqs. (2) and (3) will be [4]:       6 d −4λ2n Dd t (4) Kod = − ln cn exp 6t π2 d2 n=1

where cn and λn are functions of Kc d/Dd . On the other hand the relation between contact time, the dispersed phase volumetric flow rate and hold-up is given by [5]: Qd t Aε

where cn and λn are determined by Elzinga and Banchero [7]. The above equation is applicable when the continuous phase resistance is negligible. Calderbank and Korchinski [8] recommended that with good approximation, Eq. (6) can be written as follows:   0.5 −9.0π2 Dd t E = 1 − exp (7) d2 Solving Eq. (7) with respect to t will result in: t=

−d 2 ln(1 − E2 ) 9.0π2 Dd

(8)

By substituting t in Eq. (5) the column height is calculated. Handlos and Baron [9] presented another form of correlation for calculation of dispersed phase mass transfer coefficient by assuming that the internal circulation of droplets is turbulent. This equation is useful when Red > 20 and is given by the following equation when the continuous phase resistance is negligible [9]:    −λn Dd t Pe 2 (9) E = 1 − 2 An exp 128d 2 where

Where rs is the radius of the sphere. Thus the Newman equation can be written as [4]:    c0 − c 6 1 −4n2 π2 Dd t E= (2) =1− 2 exp c0 − c ∗ π n2 d2

L=

n=1

n=1

r→0

Kd =

This equation is used for calculating the column height (when Kc → ∞). As mentioned above, the internal circulation of droplets assumed to be negligible in Newman equation. Kronig and Brink [6] presented another correlation considering the internal circulation of droplets. They assumed that the drop size is so small that the internal circulation is laminar. This equation is useful when Red < 20 and is given by [6]:   64λn Dd t 3 2 (6) cn exp E =1− 8 d2

(5)

Vt d Pe = Dd



μc μc + μ d

 (10)

Olander [10] recommended that with good approximation, Eq. (9) can be written as follows:   −2.80Dd t Pe (11) E = 1 − 0.64 exp 32d 2 and Kd is given by:  Kd = 0.00375Vt

μc μc + μ d

 +

0.075d t

(12)

By taking logarithm of Eq. (11) and solving it with respect to t we get: t=

32d 2 (ln(1 − E) + 0.446) −2.80Dd Pe

(13)

By substituting t in Eq. (5) the column height is calculated. Other investigators have presented various correlations using Newman equation and considering effective diffusivity instead of molecular diffusivity (Dd instead of Dd ). These equations

H. Bahmanyar et al. / Chemical Engineering and Processing 47 (2008) 57–65

59

Fig. 1. The schematic diagram of the pulsed column used.

are based on Newman equation but in order to modify Newman equation and considering internal circulation of droplets, effective diffusivity is used. Effective diffusivity contains the effect of all known and unknown parameters which play an important role in calculating the exact value of mass transfer coefficients. Some of the equations presented by these investigators for calculating  are described briefly below: In Johnson and Hamielec [11] equation the modifying coefficient is presented as a function of terminal drop velocity, drop diameter and molecular diffusivity: =

Vt d 2048(1 + κ)Dd

(14)

Boyadzhiev et al. [12] equation is:    2 ρc dVt 2 1  = 0.003 μc 1+κ

DE =

Davies [15] proposed a correlation in which the internal part of droplet is divided into two sections: the section with internal circulation and the stagnant section with molecular diffusion:   DOE kH Vt d = (18) = fV + (1 − fV ) Dd 2048(1 + κ)Dd Table 1 Pulsed column characteristics 175 5 Glass 2 4 5 Steel 316

(19) −2.21 ,

6k = E0c

(20)

E0c shows the start of internal circulation of droplets and is given by:

(15)

This correlation is applicable when  < 10. Lochiel and Calderbank [14] equation takes physical properties of phases into consideration more precisely:  

Vt d 2+3κ 1.45 =1+ 1− 2048(1+κ)Dd 1+(μd ρd /μc ρc )0.5 Re0.5 (17)

Vt d 2048(1 + κ)  

0.098E0 fV = 1 − exp − k

E0c =

Steiner [13] proposed the modifying coefficient as a function of Reynolds and Schmidt number as follows:  0.89 1 0.043 0.23  = 1 + 0.177Re Sc (16) 1+κ

Column height (cm) Column diameter (cm) Column material Hole diameter (mm) Hole pitch (mm) Compartment height (cm) Plate material

where fV is the fractional segmental stagnant volume and kH is the impurity coefficient and varies from 0 to 1. When kH = 0 the system is completely pure and when kH = 1 the system is completely impure.

gd 2 ρ γ

(21)

Temos et al. [16] presents the relation between eddy diffusivity and molecular diffusivity in a simple way: =1+

0.44DE Dd

(22)

where DE = 3.29 × 10

−4



ρ d Vi d μd



      ρ d Vi d μd × 1 − exp −3.29 × 10−4 (23) μd ρd Vi for Re  1 is given by: 

 1.45 2 + 3κ Vi = 1 − Vt 1 + (μd ρd /μc ρc )0.5 Re0.5

(24)

Table 2 Physical properties of systems Physical properties

Water/acetone/toluene

Water/acetone/butyl acetate

ρc (kg/m3 ) ρd (kg/m3 ) μc (×103 Pa s) μd (×103 Pa s) γ i (×103 N/m) Dd (×109 m2 /s)

993 864.192 1.11 0.573 26.92 2.7

993 879.4 1.11 0.72 12.00 3.1

T = 20 ◦ C; acetone concentration in feed stream is 3 wt.%.

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H. Bahmanyar et al. / Chemical Engineering and Processing 47 (2008) 57–65

Table 3 A sample of operating conditions (water/acetone/toluene) Run

f (1/s) Am (cm) Qc (cm3 /s) Qd (cm3 /s) Xc,Out (%) Xc,In (%) Xd,Out (%) Xd,In (%) ε (%) (hold-up) d32 (cm)

1

2

3

4

1.5 1 3.1 2.65 2.05 0 0.32 3.09 1.04 0.398

1.6 1.5 3.14 3.07 2.25 0 0.435 3.09 1.17 0.307

2.6 1.5 3.01 2.89 2.49 0 0.219 3.2 1.34 0.264

1.5 1 6.55 2.79 1.11 0 0.055 3.07 1.09 0.28

5 1.5 1 10.43 2.96 0.76 0 0.036 3.13 1.12 0.282

6

7

1.6 1.5 3.14 3.61 2.96 0 0.495 3.47 1.68 0.283

1.6 1.5 3.17 0.95 0.898 0 0.018 3.44 0.56 0.265

Table 4 A sample of operating conditions (water/acetone/normal butyl acetate) Run

f (1/s) Am (cm) Qc (cm3 /s) Qd (cm3 /s) Xc,Out (%) Xc,In (%) Xd,Out (%) Xd,In (%) ε (%) (hold-up) d32 (cm)

1

2

3

4

5

6

7

1 1.5 2.85 1.40 1.37 0 0.05 3.23 2 0.134

2 1.5 3.12 5.29 3.05 0 1.13 3.18 8.43 0.144

2 1.5 3.06 1.41 1.31 0 0.014 3.25 2.25 0.131

3 1.5 3.07 1.21 1.07 0 0.02 3.1 3.4 0.142

2 1.5 2.97 2.04 1.78 0 0.141 3.09 2.59 0.13

2 1.5 4.98 2.14 1.17 0 0.011 3.10 2.92 0.154

2 1.5 9.33 2.24 0.65 0 0.005 3.10 3.37 0.153

3. Experimental setup, chemical systems and operating conditions The apparatus used in this research is a pulsed sieve plate column which contains two separating upper and lower parts, two pump metering apparatus, two flow meters, a pulsator and four tanks. The pulsator is of reciprocating (mechanical) type which produces pulsation with the help of an electromotor and it is equipped with a frequency and amplitude controller. The schematic diagram of the apparatus is shown in Fig. 1 and its characteristics are given in Table 1. The chemical systems used are water/acetone/toluene (high interfacial tension) and water/acetone/normal butyl acetate

(medium interfacial tension). The mass transfer direction in both systems is from dispersed to continuous phase. The continuous phase and transferring component in both systems are water and acetone, respectively. These systems are proposed by the international liquid–liquid extraction federation [17]. The physical properties of chemical systems are given in Table 2. Under different operating conditions, some experiments were carried out using the pulsed column described above. In each experiment the following cases have been measured: (1) Acetone concentration of input and output streams of the column for continuous and dispersed phase using GC apparatus.

Table 5 Calculated height using equations presented in the previous works (water/acetone/toluene) Run

1 2 3 4 5 6 7

Newman

Kronig and Brink

Handlos and Baron

Height (m)

Error (%)

Height (m)

Error (%)

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

33.64 17.02 15.48 33.52 39.55 11.71 26.88

−1822.40 −872.62 −784.73 −1815.25 −2159.79 −569.25 −1436.13

13.73 6.91 6.37 14.10 16.73 4.76 11.46

−684.78 −295.12 −264.03 −705.87 −856.01 −171.82 −554.85

0.1222 7.82 × 10−2 9.95 × 10−2 0.17 0.19 7.11 × 10−2 0.22

93.01 95.53 94.31 90.42 89.29 95.94 87.71

0.27 0.21 0.24 0.47 0.55 0.17 0.42

84.83 88.01 86.04 72.89 68.46 90.53 76.22

2.09 0.59 0.31 1.39 1.63 0.42 0.97

−19.17 66.21 82.35 20.63 6.63 76.21 44.82

H. Bahmanyar et al. / Chemical Engineering and Processing 47 (2008) 57–65

61

Table 6 Calculated height using equations presented in the previous works (continued) (water/acetone/toluene) Run

Johnson and Hamielec

Steiner

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

1 2 3 4 5 6 7

0.54 0.34 0.44 0.75 0.84 0.31 0.97

69.18 80.37 78.78 57.13 52.01 82.15 44.79

1.17 0.92 1.08 2.12 2.47 0.73 1.87

33.07 47.53 38.11 −21.23 −41.32 58.39 −6.77

9.20 2.59 1.37 6.21 7.32 1.83 4.34

−425.75 −48.39 21.74 −255.04 −318.32 −4.44 −147.74

4.74 2.61 2.69 5.34 6.18 1.99 5.07

−170.95 −49.17 −53.62 −204.97 −253.03 −13.49 −189.41

6.27 3.69 3.66 7.66 8.99 2.66 6.34

−258.19 −110.89 −108.92 −337.57 −414.22 −51.99 −261.98

12.02 5.14 3.95 10.72 12.64 3.56 8.25

−586.61 −193.52 −125.44 −512.39 −622.27 −103.36 −371.39

Run

Boyadzhiev Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

Temos Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

1 2 3 4 5 6 7

0.12 0.09 0.17 0.23 0.24 0.11 0.47

93.27 94.60 90.21 86.93 86.12 93.52 72.98

0.56 0.68 1.03 1.83 2.11 0.62 1.77

68.26 61.18 41.02 −4.53 −20.39 64.77 −1.09

34.28 5.39 1.65 15.69 18.46 3.89 9.25

−1858.65 −208.42 5.71 −796.55 −954.85 −122.02 −444.26

5.80 4.28 6.39 9.88 10.76 4.43 14.48

−231.59 −144.71 −264.87 −464.64 −515.09 −153.00 −727.29

17.53 12.72 13.07 27.15 31.86 9.41 22.64

−901.71 −626.80 −646.81 −1451.59 −1720.68 −437.71 −1193.53

33.50 16.61 14.02 33.04 38.98 11.44 26.32

−1814.31 −848.85 −701.24 −1788.08 −2127.61 −553.86 −1403.86

Run

Lochiel and Calderbank Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

0.62 0.39 0.53 0.88 0.98 0.37 1.17

64.72 77.18 69.93 49.80 44.03 78.83 32.90

1.42 1.16 1.40 2.72 3.17 0.93 2.42

18.87 33.56 19.87 −55.37 −80.89 46.78 −38.14

13.43 3.64 1.82 8.86 10.44 2.57 6.11

−667.27 −108.07 3.80 −406.44 −496.63 −46.75 −249.16

9.84 5.84 6.61 12.32 14.04 4.80 13.07

−462.21 −233.85 −277.51 −603.73 −702.24 −174.43 −646.58

16.08 10.08 10.16 21.17 24.86 7.34 17.59

−818.56 −475.91 −480.51 −1109.63 −1320.44 −319.67 −905.53

30.45 13.96 11.01 28.57 33.69 9.65 22.31

−1640.18 −697.92 −529.15 −1532.47 −1825.65 −451.65 −1174.79

1 2 3 4 5 6 7

Davies

(2) Mean dispersed phase hold-up (ε) along the column using shut-down procedure. (3) Sauter mean diameter of droplets (d32 ), in the way that first pictures of four different points of the column has been taken in each experiment using a very high-resolution Rico type camera and has been compared with a standard and specific particle inside the column. The size of droplets is exactly determined in this way.

The column’s height has been calculated with the presented equations in Section 2. In order to have a better comparison and conclusion, all three droplet velocities (slip velocity (VS ), terminal velocity (Vt ) and characteristic velocity (VK )) have been taken into account in calculation of height and Reynolds number. The relation between phase velocities and dispersed phase holdup in pulsed columns is usually given by slip drop velocity: VS =

Tables 3 and 4 show a sample of operating conditions.

Qd Qc + Aε A(1 − ε)

(25)

Table 7 Calculated height using equations presented in the previous works (water/acetone/butyl acetate) Run

1 2 3 4 5 6 7

Newman

Kronig and Brink

Handlos and Baron

Height (m)

Error (%)

Height (m)

Error (%)

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

1.90 0.31 2.20 1.35 1.42 3.69 3.82

−8.80 82.57 −25.79 22.93 18.93 −111.39 −118.42

0.80 0.13 0.94 0.57 0.59 1.58 1.64

51.13 92.83 46.29 67.20 66.39 9.61 6.13

8.95 × 10−2 1.46 × 10−2 0.12 0.11 6.16 × 10−2 0.14 0.15

94.89 99.16 93.32 93.59 96.48 91.99 91.44

0.13 1.79 × 10−2 0.15 8.10 × 10−2 9.99 × 10−2 0.19 0.20

92.77 98.98 91.34 95.37 94.29 88.95 88.47

0.21 0.02 0.19 6.51 × 10−2 0.13 0.26 0.27

88.07 98.65 88.89 96.28 92.69 85.02 84.41

62

H. Bahmanyar et al. / Chemical Engineering and Processing 47 (2008) 57–65

Fig. 2. Deff vs. Re using slip velocity (water/acetone/toluene). Fig. 4. Deff vs. Re using slip velocity (the two systems together).

The characteristic drop velocity is required in correlating phase velocities and dispersed phase hold-up. The ratio of the characteristic and terminal velocities is related to column geometry, pulsation frequency and amplitude, perforation size and plate free area [18]:    1 d VK = 1− 1 + c2 H D0 Vt   fAm (fAm )2 − − (26) 1 + 0.95H 1 + 28.3H c2 = 0.0268 + 0.0365e

(27)

fAm is in mm/s and H is in mm. Vt is the terminal velocity and can be calculated from Klee–Treybal equation [19]. The relationship between the free area (e), the hole diameter (D0 ) and the pitch (dp ) is given by the following equation: πD2 e= 0 12dp2

(28)

4. Calculation of height using the presented equations in the past literature Considering the operation conditions for the two systems and using the equations presented in Section 2, the height of the pulsed column has been calculated. The results are given in

Fig. 3. Deff vs. Re using slip velocity (water/acetone/butyl acetate).

Tables 5–8. The percentage of the errors shows the accuracy of the equations. Results for different operating conditions (pulsation frequency and amplitude, dispersed and continuous phase volumetric flow rate) show that Newman and Kronig and Brink equations and Temos, Davies and Steiner equations for calculating , are not suitable to use in pulsed column design, because although the continuous phase resistance assumed to be negligible, the calculated height is much higher than the actual height of the column. Lochiel and Calderbank, Johnson and Hamielec equations are more suitable for calculating . Furthermore results show that calculated height using Handlos and Baron and Boyadzhiev equations are much smaller than the actual height of the column. 5. Achieving effective diffusivity for the chemical systems in different operating conditions The experiment was repeated for different operating conditions and for the pulsed column with specified height. The solute concentration, mean diameter of droplets and the dispersed phase hold-up were measured and the contact time (t = l/V) was calculated. Experimental values of the mass transfer coefficients were used in Newman’s equation to find effective diffusivity which is replaced with molecular diffusivity. After calculating Deff using different droplet velocities, the best curve of Deff versus Re for both systems is drawn in a logarithmic scale. The results for both

Fig. 5. Deff vs. Re using characteristic velocity (water/acetone/butyl acetate).

H. Bahmanyar et al. / Chemical Engineering and Processing 47 (2008) 57–65

63

Table 8 Calculated height using equations presented in the previous works (continued) (water/acetone/butyl acetate) Run

1 2 3 4 5 6 7 Run

Johnson and Hamielec

Steiner

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

0.40 0.07 0.52 0.50 0.27 0.63 0.67

77.12 96.24 70.01 71.23 84.34 64.03 61.47

0.57 0.08 0.68 0.36 0.44 0.87 0.91

67.63 95.41 61.11 79.22 74.61 50.36 48.13

0.93 0.11 0.87 0.29 0.57 1.18 1.23

46.62 93.93 50.09 83.31 67.49 32.69 29.85

0.62 0.09 0.74 0.51 0.45 1.12 1.17

64.82 94.32 57.82 70.75 74.43 35.79 32.93

0.68 0.11 0.79 0.47 0.51 1.24 1.28

61.18 93.99 54.63 73.24 70.66 29.43 26.85

0.77 0.11 0.85 0.44 0.55 1.35 1.39

55.66 93.51 51.48 74.86 68.64 23.15 20.40

Boyadzhiev

Temos

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

1 2 3 4 5 6 7

0.87 0.15 1.29 1.94 0.55 1.11 1.23

50.27 91.62 26.14 −10.95 68.74 36.77 29.76

1.74 0.22 2.18 1.01 1.44 2.11 2.23

0.49 87.52 −24.26 42.12 17.81 −20.47 −27.27

4.74 0.38 3.58 0.65 2.36 3.88 4.07

−170.59 78.14 −104.61 62.63 −34.72 −121.47 −132.81

1.88 0.30 2.18 1.35 1.39 3.62 3.75

−7.49 82.76 −24.76 23.07 20.17 −106.79 −114.12

1.89 0.30 2.19 1.34 1.41 3.67 3.79

−8.36 82.68 −25.35 23.36 19.20 −109.56 −116.59

1.90 0.30 2.19 1.33 1.42 3.69 3.81

−8.74 82.61 −25.62 23.78 19.04 −110.71 −117.74

Run

Lochiel and Calderbank

1 2 3 4 5 6 7

Davies

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

Height using VS (m)

Error (%)

Height using Vt (m)

Error (%)

Height using VK (m)

Error (%)

0.55 0.09 0.73 0.69 0.38 0.86 0.92

68.33 94.79 58.29 60.24 78.42 50.83 47.19

0.79 0.11 0.95 0.51 0.62 1.21 1.26

54.88 93.61 45.81 71.04 64.63 30.99 27.88

1.25 0.15 1.20 0.40 0.78 1.64 1.71

28.41 91.57 32.33 76.86 55.31 6.28 2.35

1.66 0.27 1.96 1.27 1.22 3.10 3.23

5.23 84.75 −11.73 27.71 30.39 −77.29 −84.47

1.74 0.27 2.02 1.22 1.31 3.28 3.39

0.46 84.30 −15.62 30.37 25.38 −87.33 −93.88

1.83 0.28 2.08 1.18 1.34 3.41 3.53

−4.52 83.77 −18.68 32.54 23.44 −94.94 −101.62

systems are represented in Figs. 2–6. It is noticeable that in Figs. 2 and 3 the variation of Deff versus Re has better results. Considering figures mentioned, it is obvious that the data dispersion is fewer if the slip velocity is used and the curves display the variation of Deff as a function of Re in a more clear way (Figs. 2–4). While if terminal or characteristic velocity is used, the data dispersion is much more and it is difficult to draw the best curve through them. In Fig. 5 the curve for

Fig. 6. Deff vs. Re curve using terminal velocity (water/acetone/toluene).

water/acetone/normal butyl acetate system with VK , and in Fig. 6 the curve for water/acetone/toluene with Vt is displayed as an example. Considering Fig. 4, the following equation is proposed for predicting the effective diffusivity: Deff = 4.5151 × 10−9 exp(0.0067Re) in which Re is defined as Re = ρc VS d32 /μc .

Fig. 7. Comparison of calculated and actual height using Vt , VK and VS (water/acetone/toluene).

64

H. Bahmanyar et al. / Chemical Engineering and Processing 47 (2008) 57–65 Table 9 Comparison of calculated and actual height using VS (water/acetone/toluene)

Fig. 8. Comparison of calculated and actual height using Vt , VK and VS (water/acetone/butyl acetate).

Fig. 9. Comparison of (water/acetone/toluene).

calculated

and

actual

height

using

VS

Run

Actual height (m)

Calculated height (m)

Error (%)

1 2 3 4 5 6 7

1.75 1.75 1.75 1.75 1.2 1.39 1.5

1.59 2.12 2.22 1.06 0.72 1.80 1.31

9.26 −21.32 −26.77 39.19 40.36 42.40 12.44

Table 10 Comparison of calculated and actual height using VS (water/acetone/butyl acetate) Run

Actual height (m)

Calculated height (m)

Error (%)

1 2 3 4 5 6 7

1.75 1.75 0.45 1.01 1.2 1.39 1.5

1.76 1.84 0.49 1.06 1.21 1.36 1.56

−0.38 −5.08 −9.02 −5.32 −0.72 2.26 −4.07

The experiments were reported for different operating conditions. In calculating height with slip velocity the data along the column are also used. The parameters described in Section 3 were measured in different heights of the column. Then the height of that part is calculated and compared with its actual height. The following figures (Figs. 9 and 10) and tables (Tables 9 and 10) present the results.

6. Calculation of column height using Deff 7. Conclusion In this section the column height is calculated using Deff curves and equation obtained in the previous section in Newman equation and the results are compared with actual values. The results of height calculation using Vt , VK and VS for both systems are shown in Figs. 7 and 8. In Runs 2, 4 and 5 of Fig. 8 the points were far from the curve and have been eliminated. The effect of slip velocity in comparison with other velocities (Vt and VK ) for two systems can be seen in Figs. 7 and 8. In these figures column height calculated by using VS , Vt , VK and Deff are compared with actual column height. The best results were obtained for VS .

Fig. 10. Comparison of calculated and actual height using VS (water /acetone/butyl acetate).

According to the results, the following conclusions are obtained: • Newman, Kronig and Brink equations are not suitable to use in pulsed column design, because although the continuous phase resistance is assumed to be negligible, the calculated height is much higher than the actual height of the column (a decrease in phase resistance will result in a decrease in calculated height for the column). • Calculated heights using Handlos and Baron equation is much less than the actual height and have a great deviation from real values. • Using Dd (Deff ) instead of Dd in calculating pulsed columns height will reduce the error percent and will give us more desirable results. Among the equations which have been proposed for calculating , Lochiel and Calderbank, Johnson and Hamielec equations give more acceptable results. • Using the presented equation to predict Deff as a function of Reynolds number, as a reliable substitution for molecular diffusivity, and also making use of slip drop velocity (VS ) instead of other velocities and comparison between the obtained results from this model and the models presented by other investigators is recommended for more precise design of these unit operations.

H. Bahmanyar et al. / Chemical Engineering and Processing 47 (2008) 57–65

Appendix A. Nomenclature

A Am An c c0 c2 cn c* d d32 di dp D0 Dd DE Deff DOE e E E0 E0c f fV g H k kH Kc Kd Kod L ni Pe Qc Qd r  Re Sc t V VK VS

column cross-sectional area (m2 ) pulsation amplitude (cm) Taylor constants in Eq. (9) solute concentration in dispersed phase (g/cm3 ) initial concentration of solute in dispersed phase (g/cm3 ) constant of Eq. (26) Taylor constants in Eqs. (4) and (6) equilibrium concentration of solute in dispersed phase (g/cm3 ) droplet diameter (m) mean diameter of droplets (m) the ith group droplet diameter (m) pitch of the plate holes (m) plate holes diameter (mm) molecular diffusivity of transferred component in dispersed phase (m2 /s) effective diffusivity in Handlos and Baron equation, Eddy diffusivity in Temos equation (m2 /s) effective diffusivity of transferred component in dispersed phase (m2 /s) overall effective diffusivity (m2 /s) percentage of free area local efficiency (=(c0 − c)/(c0 − c* )) E¨otvus number E¨otvus number when the droplet has critical diameter pulsation frequency (1/s) fractional segmental volume of drop which is stagnant gravity acceleration (m2 /s) compartment height (mm) the dimensionless number in Eq. (20) (=E0c /6) empirical constant of Eq. (18) which varies between 0 and 1 mass transfer coefficient based on continuous phase (m/s) mass transfer coefficient based on dispersed phase (m/s) the overall mass transfer coefficient based on dispersed phase (m/s) column height (m) the number of the ith group droplets Peclet number (=(Vd/Dd )(1/(1 + k))) volumetric flow rate of continuous phase (cm3 /s) volumetric flow rate of dispersed phase (cm3 /s) r direction in spherical coordinates (m) the modifying coefficient of transferred component molecular diffusivity due to the internal circulations Reynolds number (=ρc Vd32 /μc ) Schmidt number (=μ/ρDd ) the resident time of dispersed phase in column, contact time (s) droplet velocity characteristic velocity of droplet (m/s) slip velocity of droplet (m/s)

Vt Xc Xd

65

terminal velocity of droplet (m/s) solute concentration in dispersed phase (g acetone/ g water) solute concentration in dispersed phase (g acetone/ g toluene (or g butyl acetate))

Greek symbols γ interfacial tension (N/m) ε dispersed phase hold-up κ ratio of dispersed phase viscosity to continuous phase viscosity λn Eigen value of Taylor series in Eqs. (4), (6) and (9) μc continuous phase viscosity (Pa s) dispersed phase viscosity (Pa s) μd ρ difference of dispersed and continuous phase densities (kg/m3 ) ρc continuous phase density (kg/m3 ) ρd dispersed phase density (kg/m3 ) References [1] W.L. McCabe, J.C. Smith, P. Harriot, Unit Operations of Chemical Engineering, 6th ed., McGraw-Hill, Boston, 2001, pp. 747–756. [2] K. Gottliebsen, B. Grinbaum, D. Chen, G.W. Stevens, The use of pulsed perforated plate extraction column for recovery of sulphuric acid from copper tank house electrolyte bleeds, Hydrometallurgy 58 (2000) 203– 213. [3] Y. Wang, S. Jing, G. Wu, W. Wu, Axial mixing and mass transfer characteristics of pulsed extraction column with discs and doughnuts, Trans. Nonferrous Met. Soc. Chin. 16 (2006) 178–184. [4] A.B. Newman, The drying of porous solids: diffusions and surface emission equations, Trans. Am. Inst. Chem. Eng. 27 (1931) 203–220. [5] J.C. Godfrey, M.J. Slater, Liquid–Liquid Extraction Equipment, 2nd ed., John Wiley and Sons, New York, 1994, pp. 227–735. [6] R. Kronig, J.C. Brink, On the theory of extraction from falling droplets, Appl. Sci. Res. A-2 (1950) 142–154. [7] E.R. Elzinga, J.T. Banchero, Film coefficients for heat transfer to liquid drops, Chem. Eng. Prog. Symp. Ser. 29 (55) (1959) 149–161. [8] P.H. Calderbank, I.J.O. Korchinski, Circulation in liquid drops: a heattransfer study, Chem. Eng. Sci. 6 (2) (1956) 65–78. [9] A.E. Handlos, T. Baron, Mass and heat transfer from drops in liquid–liquid extraction, AIChE J. 3 (1957) 127–136. [10] D.R. Olander, The Handlos–Baron drop extraction model, AIChE J. 12 (1966) 1018–1019. [11] A.I. Johnson, A.E. Hamielec, Mass transfer inside drops, AIChE J. 6 (1960) 145–149. [12] L. Boyadzhiev, D. Elenkov, G. Kyuchukov, On liquid–liquid mass transfer inside drops in a turbulent flow field, Can. J. Chem. Eng. 47 (1969) 42–44. [13] L. Steiner, Mass-transfer rates from single drops and drop swarms, Chem. Eng. Sci. 41 (8) (1986) 1979–1986. [14] A.C. Lochiel, P.H. Calderbank, Mass transfer in the continuous phase around axisymmetric bodies of revolution, Chem. Eng. Sci. 19 (7) (1964) 471–484. [15] J.T. Davies, Interfacial renewal, Chem. Eng. Prog. 62 (7) (1966) 89–94. [16] J. Temos, H.R.C. Pratt, G.W. Stevens, Comparison of tracer and bulk mass transfer coefficients for droplets, in: Proceedings of the ISEC, vol. 93, Elsevier, Amsterdam, 1993, pp. 1770–1777. [17] T. Misek, Recommended Systems for Liquid Extraction Studies, EPCE Publication Series, Inst. Chem. Engrs., Rugby, UK, 1978. [18] A.A. Hussain, T.B. Liang, M.J. Slater, Characteristic velocity of drops in a liquid–liquid extraction pulsed sieve plate column, Chem. Eng. Res. Des. 66 (1988) 541–554. [19] A.J. Klee, R.E. Treybal, Rate of rise or fall of liquid drops, AIChE J. 2 (1956) 444–447.