Prediction of Effective Diffusion Coefficient in Rotating Disc Columns and Application in Design

SEPARATION SCIENCE AND ENGINEERING Chinese Journal of Chemical Engineering, 17(3) 366ü372 (2009)

Prediction of Effective Diffusion Coefficient in Rotating Disc Columns and Application in Design Marzieh Amanabadi, Hossein Bahmanyar*, Zohreh Zarkeshan and Mohamad Ali Mousavian Engineering College, Chemical Engineering Faculty, University of Tehran, Iran Abstract A rotating disc column (RDC) with inner diameter 68 mm and 28 compartments is used in this study. Parameters including Sauter mean diameter, hold-up and mass transfer coefficient are measured experimentally under different operating conditions. The correlations in literature for molecular diffusion and enhancement factor equation including eddy diffusion, circulation and oscillation of drops are evaluated. A new equation for the effective diffusion coefficient as a function of Reynolds number is proposed. The calculated values of mass transfer coefficient and column height from the previous equations and present equation are compared with the experimental data. The results from the present equation are in very good agreement with the experimental results, which may be used in designing RDC columns. Keywords liquid-liquid extraction, rotating disc column, mass transfer coefficient, effective diffusion coefficient

1

INTRODUCTION

Rotating disc column (RDC) is widely used for liquid-liquid extraction. The performance of these columns indicates that they are more efficient and possess better operational flexibility than the conventional sieve plate, packed and spray columns. An important application of these contactors is in the petroleum industry for furfural and sulfur dioxide extraction, propane deasphalting, solfolane extraction and for caprolactum purification [1]. In order to obtain a suitable design for RDC columns, a number of hydrodynamic parameters, axial mixing and mass transfer should be considered. In these columns, new drops are generated from breakage of bigger drops or coalescence of smaller drops [2, 3]. The variation of droplet sizes and dispersed phase hold-up along the column height due to droplet interactions have been studied by using the droplet population balance model [46]. Another most important parameter in design is the mass transfer coefficient. The fundamental process for the rate of mass transfer in extraction columns is still not sufficiently well understood nor adequately modeled [7]. Passage of drops from the continuous phase is under the influence of hydrodynamics and has a distinct effect on the mechanism and amount of mass transfer. With regard to the dispersion in the column, the inside of the drop may be stagnant, circulating, or oscillating [8, 9]. Therefore, the mass transfer mechanism inside the drop will be based on the existence or non-existence of circular flows. In the following sections, by introducing a number of equations, the mass transfer coefficient will be calculated and the column height will be specified. A new equation for effective diffusivity is proposed for calculating the mass transfer coefficient and column height, and the results are compared with the calculated values from other equations in literature

and experimental data. 2

PREVIOUS WORK

One of the oldest equations for the mass transfer coefficient, used for stagnant drops with molecular diffusion mechanism, is the Newman equation [10]. With the continuity equation and following initial and boundary conditions, Eq. (2) is obtained [11]. § w 2C 2 wC · Dd ¨  ¸ © wr 2 r wr ¹

wC wt

(1)

I.C: C(r,0) C0 B.C.(1): C(rs,t) C* B.C.(2): lim  r , t is bounded. r o0

d ª 6 ln « 6t ¬ S2

f

1

º

¦ n2 exp  4n2 S2 Dd t / d 2 »

(2) ¼ where rs is the drop radius. In Eq. (2) the resistance of continuous phase is neglected. By applying the resistance in the continuous phase, Eq. (2) becomes [7]: Kd

K od

n 1

d ª 6 ln « 6t ¬ S2

f

º

n 1

¼

¦ Cn exp  4On2 Dd t / d 2 »

(3)

where Cn and On are functions of Kcd/Dd. The equation, obtained by Kronig and Brink [12], for drops in which the mass transfer mechanism involves both molecular diffusion and composed internal circulation, is presented as follows: K od



d ª3 f 2 § 64On Dd t · º ln « ¦ Cn exp ¨ ¸» 6t ¬ 8 n 1 d2 © ¹¼

(4)

For the drops with toroidal internal circulation,

Received 2008-08-25, accepted 2009-03-02. * To whom correspondence should be addressed. E-mail: [email protected]

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Chin. J. Chem. Eng., Vol. 17, No. 3, June 2009 Table 1

Equations for enhancement factor in the literature

Investigator

Correlations R

Johnson & Hamielec [14]

R 1

Lochiel & Calderbank [15]

dU t 2048(1  N ) Dd R

Davis [16]

dU t [2048(1  N ) Dd ]

DoE Dd fv

­° ª (2  3N ) º 1.45 ½° 0.5 » ®1  « 0.5 ¾ 1  P U / P U  d d c c ¼ Re ¿° ¯° ¬

Ud DE Pd

§ U dU 3.29 u 104 ¨ d i © Pd Ui

the dominant mechanism of mass transfer is eddy diffusion and the equation by Handlos and Baron [13] is used as follows: K od



d ­ f 2 ª OnU t t º ½ ln ® 2¦ Cn exp « »¾ 6t ¯ n 1 ¬128 1  k d ¼ ¿

(5)

Many arbitrarily chosen combinations of prediction models for mass transfer coefficient in dispersed phase have been recommended for use in RDC design. Nearly all the recommended combinations have resulted from the combination of mass transfer coefficient for dispersed phase from one of the above mentioned theoretical models, which can give the best agreement between predicted and experimental separation efficiencies. The equation for effective diffusivities or enhancement factor applicable in the rigid drop model is obtained by empirical combination of molecular diffusivity and an equivalent diffusivity. Many equations have been presented for determining the enhancement factor, several of which are listed in Table 1. All these equations will be used in the experimental results section, and the results obtained from these equations will be compared with the experimental data. 3

APPARATUS AND MEASUREMENT

The experimental apparatus is shown in Fig. 1. The RDC column was made of glass and its rotors and stators were made of stainless steel. The continuous

(10)

6k

(11)

2 1 · 0.003  Uc dU t Pc §¨ ¸ ©1 N ¹

2

(12)

0.89

(13)

Dd  0.44 DE

DOE

Temos et al. [18]

(9)

gd 2 'U / J

1 · R 1  0.177 Re0.43 Scd0.23 §¨ ¸ ©1 N ¹

Steiner [8]

(8)

2.21 ª º 1  exp «  ¨§ 0.098 Eo ¸· » k ¹ ¬ © ¼

Eoc

R

(7)

kHU t d ª º f v  1  f v « » ¬ 2048(1  N ) Dd ¼

Eo

Boyadzhiev et al. [17]

(6)

(14)

§ 3.29 u 10 Ud dU i ·ª ¸ «1  exp ¨ Pd ¹ «¬ © 4

2  3N º 1.45 ½° °­ ª U 0.5 » ®1  « 0.5 ¾ t °¯ ¬1   Pd Ud / Pc U c ¼ Re ¿°

·º ¸» ¹ »¼

(15)

(16)

phase was fed at the top of the column to flow countercurrently to the dispersed phase fed at the bottom. The flow rates of two phases were measured using calibrated rotameters. The rotor speed was measured using Digital Tachometer photo/contact type and adjusted to the desired value (about 1.2 to 2 times of the critical value at which drops start to break). The holdup of dispersed phase was measured by shutdown method, in which the inlet and outlet flows were stopped simultaneously after reaching the steady state. The dispersion was then coalesced at the interface. The holdup was then measured by determining the change of interfacial height. The Sauter mean diameter, d32, was obtained by photographic technique at a few stages of the column height and calculated by Eq. (17), N

d32

¦ ni di3 i 1 N

¦ ni di2

(17)

i 1

In each experiment for a specific column height L, flow rates of dispersed and continuous phases, after measuring the holdup of dispersed phase and Sauter mean diameter, the slip velocity and contact time can be calculated using the following equations: Qd Qc  (18) Vs AH A 1  H

t

L / Vs

(19)

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Chin. J. Chem. Eng., Vol. 17, No. 3, June 2009

Figure 1

The apparatus used in this research

Table 2 Column height/ cm

Column active height/cm

110

76

Characteristics of the column

Inner diameter of Inner diameter of Outer diameter of column/cm stator/cm rotor/cm 6.6

4.5

Mass transfer coefficients and enhancement factor are calculated using previous mentioned equations. With sampling from the valves in different sections of the column and measuring the solute concentration, C, the experimental values of mass transfer coefficient can be found from Eqs. (20) and (21). Using these equations with the Newman equation, the definition of effective diffusivity can be shown in Eq. (22). 6tK od ln 1  E  (20) d C  C0 E (21) C *  C0 § C*  C · 6 f § 4On2 Deff t · (22) C exp ¨¨ * ¸¸ ¦ ¨ ¸ n 2 d2 © ¹ © C  C0 ¹ S n 1 The other geometric values of the column are listed in Table 2.

4

CHEMICAL SYSTEM

The chemical system used in this study is water/acetic acid/carbon tetrachloride, in which water is the continuous phase, acetic acid as the solute and carbon tetrachloride as the dispersed phase. The mass transfer is from the dispersed to continuous phase. Physical properties of the system are shown in Table 3.

Compartment height/cm

Maximum number of compartments in active region

2.3

28

3.3

Table 3 ˉ3

5

Physical properties of system ˉ3

ȡc/kg·m

ȡd/kg·m

1000

1590

ȝc/kg·m 1·s ˉ

ˉ1

0.001

ȝd/kg·m 1·s ˉ

ˉ1

0.00095

ˉ1

Ȗ/N·m

0.045

EXPERIMENTAL RESULTS AND DISCUSSION

The operating conditions and hydrodynamic parameters measured are listed in Table 4. Table 4 Run 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Operating conditions

N/ Qc/ Qd/ d32/ minˉ1 L·minˉ1 L·minˉ1 mm 875 970 1070 1164 1361 862 810 943 860 1075 1113 970 970 970

0.3 0.3 0.3 0.3 0.35 0.5 0.75 0.85 1.0 1.0 1.0 1.0 1.2 1.3

0.3 0.3 0.3 0.3 0.35 0.5 0.75 0.85 1.0 1.0 1.0 0.6 0.5 0.35

0.96 0.91 0.87 0.83 0.76 0.96 0.99 0.92 0.96 0.86 0.84 0.90 0.90 0.90

Holdup×102

Kod×106/ ˉ m·s 1

4.48 5.20 6.19 7.42 13.17 7.21 9.42 12.54 12.88 18.34 19.7 10.9 9.43 7.72

9.2570 8.5265 7.8210 6.6470 6.1873 9.9118 10.310 10.410 9.9585 9.0685 8.9243 8.9256 8.4315 7.8500

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Chin. J. Chem. Eng., Vol. 17, No. 3, June 2009

5.1 Calculation of mass transfer coefficient and column height using the equations in literature

By using the equations mentioned in the previous sections, and substituting t (t L/Vs) for different droplet slip velocities, the mass transfer coefficient was calculated for each run and compared to the experimental data. The error percentage shows the precision of each equation. The calculation results are shown in Figs. 24.

fraction of solute measured at height H, and compared to the actual values. The results are shown in Figs. 57. X dh Qd dX d H (23) ³ K od aA X d ,in  X *  X 1  X d 2 d

d

Figure 5 Calculated height using Newman, Boyadzhiev, Steiner and Temos equations ƹ actual height;Ƶheight using Vs by Newman equation; Ʒ height using Vs by Boyadzhiev equation; × height using Vs by Steiner equation; height using Vs by Temos equation

Figure 2 Calculated Kod using Newman, Boyadzhiev, Steiner and Temos equations ƹ Newman;Ƶ Boyadzhiev;ƷSteiner; × Temos

Figure 6 Calculated height using Davis, Lochiel & Calderbank, Johnson & Hamielec, and Kronig & Bring equations ƹ actual height;Ƶheight using Vs by Davis equation;Ʒheight using Vs by Lochiel & Calderbank equation; × height using Vs by Johnson & Hamielec equation; height using Vs by Bring & Kronig equation

Figure 3 Calculated Kod using Davis, Lochiel & Calderbank, Johnson & Hamielec, and Kronig & Bring equations ƹ Davis;ƵLochiel & Calderbank;ƷJohnson & Hamielec; ƽ Kronig & Bring

Figure 7 Calculated height using Handlos & Baron equations ƹ actual height;Ʒheight using Vs by Handlos & Baron equation

Figure 4

Calculated Kod using Handlos & Baron equations

With calculated mass transfer coefficients, specific area and superficial velocity of dispersed phase in each run, the column height was calculated by Eq. (23), in which a is the specific area and Xdh is the mole

By considering Figs. 27, the previous equations can be divided into three categories: (1) The equations which predict Kod much lower than the actual values. The Newman equation [10] is for stagnant drops, the mechanism of mass transfer is molecular diffusion and no resistance in the continuous phase. Steiner [8] reduced Eq. (3) to its first term of the summation series and evaluated an enhancement factor. Temos et al. [18] proposed an overall diffusivity, DOE, which was a combination of molecular diffusion in dispersed phase

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Chin. J. Chem. Eng., Vol. 17, No. 3, June 2009

and eddy diffusivity. In their model an interfacial velocity at the drop equator was used. DOE was then used for laminar circulation model. In these models, the mass transfer is due to molecular diffusion or laminar circulation, so it is possible that the calculated values of Kod are smaller than the experimental data (Fig. 2) and consequently the calculated column height are larger than the actual values (Fig. 5). (2) The equations which predict Kod higher than the actual values. Davis [16] used a combination model for drops, divided into two regions: a stagnant cap with molecular diffusion and a circulating region with effective diffusivity. Lochiel and Calderbank [15] considered a flow around fluid spheres at intermediate Reynolds number. Johnson and Hamielec [14] modified Eq. (3) to define an enhancement factor. Their experiments involved the transfer of ethyl acetate into vigorously circulating water drops. Kronig and Brink [12] developed a model for mass transfer in circulating drops for zero external resistance. In these models with the mechanism of internal circulation, it is possible that the calculated values of Kod are larger than the experimental values and then the column height calculated will be smaller (Figs. 3 and 6). (3) The equations which predict Kod much higher than the actual values. Handlos and Baron [13] proposed a model with the mass transfer mechanism in vigorously circulating drops. Due to the turbulence inside drops, the mass transfer coefficient calculated is much larger than the experimental values and the column height calculated is much smaller than the actual value (Figs. 4 and 7). Boyadzhiev et al. [17] presented an equation for enhancement factor based on their experimental data and those by Johnson and Hamielec [14]. As seen in Figs. 2 and 5, The equation shows the least error and may be considered an appropriate equation. Considering all above results, we intend to find an effective diffusivity as a function of hydrodynamic parameters such as Sauter mean diameter and slip velocity, in which the holdup of dispersed phase are considered. 5.2

fective diffusion coefficient was calculated and the best curve of Deff vs. Re for different droplet velocity was drawn in a logarithmic scale. In order to obtain more accurate results, calculations were carried out using following different droplet velocities and the results are shown in Figs. 810. (1) Slip velocity given in Eq. (18). (2) Characteristic velocity

Vk

Vs 1  H

(24)

(3) Terminal velocity [7] Vt

d 2 g 'U 18Pc

Vt

§ g 2 'U 2 · 0.249 ¨ ¸ © U c Pc ¹

for

Re

dVt Uc

Pc

İ 10

(25)

Re ! 10

(26)

1/ 3

for

Figure 8 Deff vs. Re using slip velocity

Figure 9 Deff vs. Re using characteristic velocity

A new equation for effective diffusion coefficient

The error from the calculated mass transfer coefficient and column height shows the necessity for a more appropriate equation in column design. As shown in the previous sections, one of the equations with the best theoretical background is the Newman equation, which is based on the continuity equation with appropriate initial and boundary conditions. Thus this equation is used in this section for finding the effective diffusivity, Deff, and it is preferred to other equations like Boyadzhiev equation based on experimental data. Due to the main effect of effective diffusion coefficient on mass transfer coefficient and column height, by replacing t and experimental mass transfer coefficient in the Newman equation, the ef-

Figure 10 Deff vs. Re using terminal velocity

With Figs. 810, the equation for Deff can be found based on Re. The mass transfer coefficient and column height calculated using the effective diffusion coefficients (Figs. 810) are given in Tables 5 and 6. As seen in Tables 5 and 6, if the Reynolds number is defined based on slip velocity, the error percentage is less compared to those with characteristic

371

Chin. J. Chem. Eng., Vol. 17, No. 3, June 2009 Table 5 6

Calculated mass transfer coefficient with present model

Run

Kod×10 ˉ (using Vs)/m·s 1

Error/%

Kod×106 ˉ (using Vk)/m·s 1

Error/%

Kod×106 ˉ (using Vt)/m·s 1

Error/%

1

9.2075

0.532

8.6590

6.44

8.9515

3.29

2

8.4277

1.15

9.9570

6.67

7.615

10.68

3

7.9580

2.85

7.2170

7.71

6.529

16.51

4

6.8363

2.84

6.5460

1.51

5.730

13.79

5

5.5630

10.09

5.5550

10.20

4.526

26.84

6

10.140

2.30

9.7390

1.738

9.141

7.7750

7

11.313

9.71

1.1068

7.34

1.008

2.18

8

10.335

0.710

1.0338

0.686

7.935

23.77

9

11.472

15.20

1.1530

15.78

9.165

7.961

10

9.5613

5.44

1.0012

10.40

6.471

28.63

11

9.2400

3.53

9.7815

9.60

6.133

31.27

12

9.5410

6.89

9.4165

5.49

7.571

15.17

13

9.1061

8.00

8.8791

5.30

7.571

10.19

14

8.5493

8.90

8.2256

4.78

7.571

average of abs. error

5.58

Table 6

6.68

3.54 14.40

Calculated height with present model

Run

Height (using Vs)/m

Error/%

Height (using Vk)/m

Error/%

Height (using Vt)/m

Error/%

1

0.766

0.53

0.814

6.89

0.788

3.41

2

0.771

1.17

0.816

7.15

0.853

11.96

3

0.784

2.93

0.825

7.82

0.912

19.78

4

0.740

2.77

0.773

1.53

0.884

16.00

5

0.84

11.22

0.848

11.37

1.041

36.69

6

0.74

2.25

0.775

1.77

0.826

8.43

7

0.694

8.85

0.709

6.84

0.779

2.22

8

0.967

0.71

0.767

0.69

0.999

31.19

9

0.661

13.19

0.658

13.63

0.828

8.65

10

0.722

5.16

0.690

9.42

1.067

40.12

11

0.735

3.41

0.695

8.763

1.108

45.50

12

0.712

6.44

0.722

5.21

0.898

17.88

13

0.705

7.40

0.723

5.04

0.848

11.35

14

0.699

8.18

0.727

7.85

0.790

3.67

average of abs. error

5.30

6.71

velocity and terminal velocity. Therefore, utilization of slip velocity for the Reynolds number and in the present equation may be most suitable for the column design. An equation for effective diffusion coefficient with the least standard deviation is suggested as follows. Deff

0.4755 u 109 Re0.6501 Re

UcVs d32 Pc

(27) (28)

Figures 11 and 12 show the final results using above equations. Comparison of these figures with Figs.

18.34

27 shows that the proposed equation is much better. 6

CONCLUSIONS

Among the equations used for calculating mass transfer coefficient and column height including enhancement factor, Boyadzhiev equation [17] is in good agreement with the experimental results. An new experimental equation is proposed in this work for calculating effective diffusion coefficient based on Reynolds number Re ȡcVsd32/ȝc, which can be used for designing RDC.

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Chin. J. Chem. Eng., Vol. 17, No. 3, June 2009 Vt Xd Xdh

X d*

Figure 11 equation

Comparison of calculated Kod using present

Ȗ İ ț Ȝn ȝc ȝd ȡc ȡd ǻȡ

terminal velocity of drop, m·s 1 mole fraction of solute in dispersed phase mole fraction of solute in dispersed phase at the column height H equilibrium mole fraction of solute in dispersed phase ˉ interfacial tension, N·m 1 dispersed phase holdup viscosity ratio (dispersed/continuous phase) eigen value ˉ ˉ continuous phase viscosity, kg·m 1·s 1 ˉ ˉ dispersed phase viscosity, kg·m 1·s 1 ˉ continuous phase density, kg·m 3 ˉ dispersed phase density, kg·m 3 ˉ difference of dispersed and continuous densities, kg·m 3 ˉ

REFERENCES 1 2

Figure 12 Comparison of calculated height using present equation Ʒ height using Vs by suggested equation;Ƶactual height

3

4

NOMENCLATURE A a C C* Cn C0 Dd DE Deff DOE d d32 Eo Eoc fv g Kc Kd Kod kH L Qc Qd R Re r, rs Sc T U Ut Vk Vs -

column cross sectional area, m2 ˉ specific area, m2·m 3 ˉ solute concentration in dispersed phase, kmol·m 3 ˉ equilibrium concentration of solute in dispersed phase, kmol·m 3 Taylor constant ˉ initial concentration of solute in dispersed phase, kmol·m 3 2 ˉ1 diffusion coefficient, m ·s ˉ effective diffusion coefficient, m2·s 1 2 ˉ1 effective diffusion coefficient, m ·s ˉ overall effective diffusion coefficient, m2·s 1 drop diameter, m mean diameter of drop, m Eotvos number ( gd 2 'U / J ) Eotvos number for drop with critical diameter fractional segmental volume of stagnant drop ˉ gravitational constant, m2·s 1 ˉ mass transfer coefficient based on continuous phase, m·s 1 ˉ mass transfer coefficient based on dispersed phase, m·s 1 ˉ overall mass transfer coefficient based on dispersed phase, m·s 1 contamination coefficient varied between 0 to 1 column length, m ˉ volume flow rate of continuous phase, m3·s 1 3 ˉ1 volume flow rate of dispersed phase, m ·s internal enhancement factor for mass transfer Reynolds number ( Uc du / Pc ) drop radius, m Schmidt number [ Pc /( U c d ) ] contact time, s ˉ velocity, m·s 1 ˉ terminal velocity, m·s 1 ˉ characteristic velocity of drop, m·s 1 ˉ slip velocity of drop, m·s 1

5

6

7 8 9

10 11 12 13 14 15

16 17

18

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