# Mass transfer efficiency of a tall and low plate free area liquid pulsed sieve-plate extraction column

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## Abstract

Mass transfer performance is studied in a tall, thin, and low plate free area liquid pulsed sieve-plate extraction column. The 5.0 cm internal diameter column consists of eighty sieve plates with percent free area of only 13.5. The effects of pulsation intensity (product of amplitude and frequency) and dispersed phase velocity are studied on the extraction efficiency of the column for the acetic acid–kerosene–water system. A mass transfer correlation for the measurement of overall mass transfer coefficient is developed that best-fits the experimental data obtained in the present study. Mathematical analysis of the column is carried out that shows the insignificance of axial diffusivities in the column.

### Keywords

Pulsed column Pulsation intensity Mass transfer coefficient Height of transfer unit Axial diffusivity### Abbreviations

*a*Interfacial area per unit volume of the contactor, m

^{2}/m^{3}*A*Pulsation amplitude, m

*d*_{o}Diameter of sieve hole, m

*E*_{x}Axial diffusivity in aqueous phase, m

^{2}/s*E*_{y}Axial diffusivity in organic phase, m

^{2}/s*f*Pulsation frequency, s

^{−1}*H*Height of column, m

- (HTU)
_{oc} Overall height of transfer unit based on continuous phase, m

- (HTU)
_{ocp} Overall apparent height of transfer unit based on continuous phase, m

*K*_{oc}*a*Overall mass transfer coefficient based on continuous phase, s

^{−1}*K*_{ocp}*a*Overall apparent mass transfer coefficient based on continuous phase, s

^{−1}- (
*K*_{ocp}*a*)_{mod} Model or calculated overall apparent mass transfer coefficient based on continuous phase, s

^{−1}- (
*K*_{ocp}*a*)_{obs} Observed or experimental overall apparent mass transfer coefficient based on continuous phase, s

^{−1}*l*Plate spacing, m

*m*Equilibrium constant

*N*Number of data points

*N*_{p}Number of plates

- (NTU)
_{oc} Overall number of transfer units based on continuous phase

- (NTU)
_{ocp} Overall apparent number of transfer based on continuous phase

*p*Plate spacing, m

*u*_{c}Superficial continuous phase velocity, m/s

*u*_{d}Superficial dispersed phase velocity, m/s

*x*Molar concentration of solute in aqueous phase, mol/L

*x*_{e}Equilibrium molar concentration of solute in aqueous phase, mol/L

*x*_{i}Inlet molar concentration of solute in aqueous phase, mol/L

*x*_{o}Outlet molar concentration of solute in aqueous phase, mol/L

*y*Molar concentration of solute in dispersed phase, mol/L

*y*_{i}Inlet molar concentration of solute in organic phase, mol/L

*y*_{o}Outlet molar concentration of solute in organic phase, mol/L

*z*Axial position, m

*α*Fractional plate free area

## Introduction

Pulsed sieve-plate columns are essentially mass transfer devices in which two liquid phases come into contact and a solute is transferred from one liquid phase to the other liquid phase. The study of the mass transfer performance (efficiency) of a pulsed column is, therefore, essential for the design and operation of the column. Several studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] have already been performed on mass transfer characteristics of a pulsed sieve-plate tower and many correlations [4, 7, 9, 17, 18, 20, 23, 28] have been developed to predict the design and performance of a new column. However, due to the complex nature of the extraction process, the design procedures and practices (correlations) are not yet well established and usually pilot scale testing is required for such columns. Although some advanced modeling techniques [29, 30, 31, 32, 33, 34] are currently acclaimed, however, for the reliable design strategies and computer methods to be well developed, more of such experimental studies (relationships among the column efficiency and operating parameters) on a pulsed sieve-plate column have to be accomplished for the various geometries and liquid–liquid systems. The review of the literature suggests that the dispersed phase holdup and mass transfer performance are generally studied for short columns with a few number of plates and with plate free area greater than 19%. In the present study, a unique, tall and relatively thin, column with 80 sieve plates and with a very low plate free area of 13.5% is selected to study its mass transfer characteristics. Acetic acid–kerosene–water system is chosen and the effect of operating parameters on the mass transfer performance of the column is studied. The present study will be useful in understanding the working of a tall, relatively thin, and low plate free area column and in the development of more general mass transfer correlations that are required in the design, scale up, and operation of a pulsed sieve-plate extraction column.

## Experimental

Information regarding the chemicals used

Chemical | Analysis | Supplier | |
---|---|---|---|

Acetic acid | Glacial | Fischer Scientific | |

Propionic acid | ≥98% | Fischer Scientific | |

Kerosene | Sp. gravity @ 15.6 °C 0.7879 | Commercial market | |

ASTM distillation | |||

Vol% | Temperature °C | ||

IBP | 62 | ||

5% | 132 | ||

10% | 146 | ||

30% | 173 | ||

50% | 181 | ||

70% | 194 | ||

90% | 215 | ||

95% | 226 | ||

End point | 243 | ||

Water | ~100% | Laboratory |

In a typical procedure, firstly, the whole of the column was filled with the aqueous phase (continuous phase) and being heavier, the aqueous phase was allowed to flow from the top to the bottom. Then, the kerosene phase (dispersed phase) pump was started and the dispersed phase was allowed to flow from the bottom to the top of the column in countercurrent mode to the flow of the aqueous phase. A sample was collected from the aqueous phase outlet after a regular interval and analyzed for the concentration of solute to observe the steady-state condition. The steady-state outlet concentration of the solute in aqueous phase was recorded and used in the calculations. The concentration of solute in outlet kerosene phase was calculated by considering kerosene and water phases as immiscible (dilute acid concentrations). The pulsation intensities and dispersed phase velocities were varied and a set of data was collected to be analyzed.

To observe the axial distribution of acetic acid, aqueous phase samples were taken at three different heights in the column and enough time was given in between any two withdrawals so that each sample was taken at steady state.

During an experimental run, the interface was maintained at the top in order to control flooding in the column. The interface was controlled manually by throttling the discharge valve of the aqueous phase flow [35] (at the bottom).

## Results and discussion

### Effect of operating parameters on mass transfer efficiency

*m*is the equilibrium constant in the expression \(y = m \cdot x\). The value of

*m*was calculated experimentally by taking various concentrations of solute in virtually equal amounts of kerosene and water. The experimental value of

*m*was obtained as 0.0708 for the acetic acid–water–kerosene system and 0.2801 for the propionic acid–water–kerosene system.

#### Effect of pulsation intensity

_{ocp}) and mass transfer coefficient (

*K*

_{ocp}

*a*), respectively, for acetic acid–water–kerosene system. As mentioned before, the acetic acid was taken as a solute in the kerosene phase and the direction of mass transfer was from kerosene (dispersed) phase to aqueous (continuous) phase. No phase inversion was observed during the operation. It is observed that increasing pulsation intensity continuously decreases the height of transfer unit and consequently increases the mass transfer coefficient for constant values of superficial velocities of the continuous and dispersed phases. It can be seen from Fig. 2 that the rate of decrease of HTU

_{ocp}is initially higher and steadily decreases with an increase in pulsation intensity and may reach a steady minimum value. It is observed that in the first half of the range of

*Af*studied, the height of transfer unit decreases from 4.14 to 3.57 m (13.7% decrease) and in the lower half it decreases from 3.57 to 3.30 m (7.61% decrease). This suggests that with an increase in pulsation intensity, initially height of transfer unit decreases more rapidly and then decreases at a relatively lower rate. Cohen and Beyer [1] for boric acid–water–isoamyl alcohol system, Smoot and Babb [9] for both acetone–water-1,1,2-trichloroethane system and acetic acid–water–methyl isobutyl ketone (MIBK) system, Gourdon and Casamatta [15] for acetone–water–toluene system, Venkatanarasaiah and Varma [17] for both n-butyric acid–kerosene–water system and benzoic acid–water–kerosene system, Li et al. [22] for nitric acid–30% TBP (in kerosene)–water system, He et al. [23] for caprolactam–water–benzene system, Jahya et al. [24] for acetone–water–toluene system, Usman et al. [25] for acetic acid–water–ethyl acetate system, and Torab-Mostaedi et al. [27] for both acetone–water–toluene system and acetone–water–butylacetate system have observed a continuous and generally a similar increase in mass transfer performance with an increase in pulsation intensity.

The height of transfer unit and, therefore, mass transfer performance may depend on drop size and dispersed phase holdup. Increase in pulsation intensity increases the energy supply to the column and, therefore, decreases the diameter of the dispersed phase drops which in effect increases the mass transfer surface area and, therefore, mass transfer efficiency [37, 38]. The increase in dispersed phase holdup may increase the residence time [37] and, therefore, may aid in increasing the mass transfer performance. However, a higher value of the dispersed phase holdup such as in the mixer-settler regime not necessarily means a decrease in height of transfer unit and, therefore, increase in dispersed phase holdup not always an indication of an increase in mass transfer efficiency. Figure 2 suggests that height of transfer unit continuously decreases and corresponding mass transfer coefficient continuously increases with an increase in pulsation intensity. In our previous work [39] on the dispersed phase holdup in the same column as studied in the present contribution and for the system kerosene–water (no solute addition) under virtually the same operating conditions, the mixer-settler regime was observed in the range of *Af *equal to 3.01 × 10^{−3} to ~17.0 × 10^{−3} m/s. Comparing these findings to Fig. 2, for the mixer-settler regime, therefore, one may say that though dispersed phase holdup decreases with an increase in *Af* but mass transfer efficiency, i.e., *K* _{ocp}a increases (HTU_{ocp} decreases) with an increase in *Af*. It is, therefore, concluded that as the diameter of the dispersed phase drop decreases more rapidly than the holdup and contributes more towards mass transfer, the overall effect is the increase in mass transfer efficiency [17]. In support of that, it is important to mention here that unlike dispersed phase holdup, the average drop size of the dispersed phase always decreases with an increase in pulsation intensity [26, 40, 41, 42, 43, 44]. Further, it may be deduced from Fig. 2 that the effect of pulsation intensity on mass transfer efficiency is more pronounced in the mixer-settler region compared to the effect for the region beyond the mixer-settler region (dispersion region).

#### Effect of dispersed phase velocity

*u*

_{c}+

*u*

_{d}), Venkatanarasaiah and Varma [17] for both n-butyric acid–kerosene–water system and benzoic acid–water–kerosene system, He et al. [23] for caprolactam–water–benzene system, Jahya et al. [24] for acetone–water–toluene system, Usman et al. [25] for acetic acid–water–ethyl acetate system, and Torab-Mostaedi et al. [27] for both acetone–water–toluene system and acetone–water–butylacetate system have observed a continuous increase in mass transfer performance with increase in dispersed phase velocity. Similar to a change in pulsation intensity, a change in dispersed phase velocity also affects both the dispersed phase drop diameter and the dispersed phase holdup. However, unlike increase in pulsation intensity an increase in dispersed phase velocity continuously decreases the dispersed phase holdup.

### Variation of acetic acid concentration in the axial direction

### Effect of solute type

Effect of solute on the HTU_{ocp} and *K* _{ocp} *a*

Solute | | | | | | HTU | |
---|---|---|---|---|---|---|---|

Acetic acid | 0.6661 | 0.1106 | 0 | 0.24 | 0.0708 | 7.342 | 5.18 × 10 |

Propionic acid | 0.54 | 0.0322 | 0 | 0.22 | 0.2801 | 1.097 | 3.47 × 10 |

Comparison of mass transfer efficiencies for different pulsed column systems

System | | | | | | Af × 10 | | | HTU | References |
---|---|---|---|---|---|---|---|---|---|---|

Kerosene-acetic acid–water | 50.0 | 13.5 | 50 | 4.1 | 80 | 19.33 | 3.803 | 1.647 | 7.342 | This work |

Kerosene-propionic acid–water | 50.0 | 13.5 | 50 | 4.1 | 80 | 19.33 | 3.803 | 1.647 | 1.097 | This work |

Kerosene-n-butyric acid–water | 43 | 28.0 | 100 | 2.0 | − | 22.0 | 3.15 | 3.15 | 2.478 | |

Kerosene-benzoic acid–water | 43 | 46.0 | 100 | 2.0 | − | 22.0 | 3.15 | 3.15 | 0.701 | 17] |

### Correlation development for mass transfer coefficient

Yadav and Patwardhan [45] have reviewed the published correlations for mass transfer coefficient. Using a large number of experimental data (from literature), they tested the validity of the available correlations for predicting mass transfer coefficients. They have concluded that none of the available mass transfer correlations can satisfactorily represent the experimental data and, therefore, the design of a pulsed sieve-plate column should be based on pilot scale testing of the new system. However, for the experimental data generated in the present study, the correlation of Venkatanarasaiah and Varma [17] was tested. The equation was chosen because of its simplicity having a few variables and with a few parameters to be fitted. Moreover, as both for the dispersed phase holdup and slip velocity, the equations of Venkatanarasaiah and Varma [17] were found in relatively better agreement to the experimental data obtained on the same column in our previous study [39]. It is important to mention here that the equation of Venkatanarasaiah and Varma [17] has been developed for overall apparent mass transfer coefficient, i.e., it is based only on the measurable values of the inlet and exit concentrations and the equilibrium relationship for the system and without considering any axial mixing in the system. The mass transfer coefficients obtained in Sect. 3.1 are also apparent values and not corrected for the axial mixing. Based on the results obtained, a new mass transfer correlation is also developed.

^{®}program. The following objective function was minimized:

*i*represents the

*i*th value, and

*N*is the total number of data points.

#### Testing of Venkatanarasaiah and Varma’s equation [17]

*K*is dependent on the solute-liquid–liquid system used. Therefore, the equation was fitted against the experimental data and

*K*was taken as the only parameter to be fitted. The value of

*K*was found to be 0.0170. Figure 5 shows the validity of Venkatanarasaiah and Varma’s equation [17] for the experimental data of the present study, whereas Table 4 shows the values of the statistical parameters to show the goodness of the fit. Clearly, the equation is found not appropriate to fit the experimental data and quite a low value of R

^{2}is obtained.

#### Development of the correlation for the mass transfer coefficient

*R*

^{2}was improved from 0.504 to 0.795. However, a low value of the exponent of

*u*

_{ c }suggested to remove the variable

*u*

_{ c }and to modify the equation. In the next attempt, the variable

*u*

_{ c }was eliminated from the equation and the modified equation was subjected against the experimental data. Virtually the same values of SSE and

*R*

^{2}were obtained and, therefore, suggested the equation to be retained as the final best-fit equation. The final equation with its parameters that described the experimental data was obtained as:

*R*

^{2}values of Eq. 7 are given in Table 4 and Fig. 6 shows the scatter diagram between the measured

*K*

_{ocp}

*a*and the

*K*

_{ocp}

*a*values calculated from Eq. 7. The average percentage error as obtained from Eq. 8 was only 10.76%.

### Mathematical modeling of the pulsed column

*z*was selected in the body of the pulsed sieve-plate column, as shown in Fig. 7, and a mass balance was applied across the differential element.

The following equations (for the dispersed phase and the continuous phase) were obtained in terms of overall mass transfer coefficient based on the continuous phase:

As the system studied did not involve any heat effects, therefore, energy balance was not required in modeling the column.

#### Mass transfer in the absence of axial diffusivities

As for a tall column with a small diameter such as that studied here, mass transfer process due to axial diffusivities was expected to be negligible in comparison to the mass transfer by convection (bulk liquid flows); therefore, the first term of both Eqs. 9 and 10 was neglected and the following equations were obtained as desired.

Rearranging Eqs. 11 and 12 and writing *K* _{oc} *a* as *K* _{ocp} *a*, i.e., in terms of overall apparent mass transfer coefficient, it may be shown that

The solution requires the values of *u* _{ c }, *u* _{ d }, *K* _{ocp} *a*, and an equilibrium relationship between molar concentration of solute in each of the continuous and dispersed phases.

In Sect. 3.4.2, the experimental data obtained in the present study were used to obtain a relationship (Eq. 7) of mass transfer coefficient in terms of pulsation intensity and dispersed phase velocity which was used to obtain the value of *K* _{ocp} *a* in solving Eqs. 13 and 14. The values of *u* _{ c } and *u* _{ d } were obtained from the experimental data. The expression \(y = 0.0708x\) was used for the equilibrium value, where 0.0708 is the equilibrium constant for acetic acid–kerosene–water system.

The differential equations along with the auxiliary equations and boundary conditions were solved in POLYMATH (an established mathematical software) using RKF-45 (Runge–Kutta-Fehlberg-45) routine and concentration values of acetic acid in both the continuous (water) and dispersed (kerosene) phases were obtained at five different points in the column. It is important to mention here that the given problem was a countercurrent problem, so the initial values of outlet concentration of acetic acid in aqueous phase were found by trial and error. Initially, the experimental value of the outlet concentration of acetic acid in aqueous solution was used as a guess value and then the inlet concentration of acetic acid in water was compared with the experimental value (which was zero). If the predicted inlet aqueous phase concentration was different than zero, then a new initial value was used. The procedure was carried on till the predicted and actual values were numerically equal.

In Fig. 9, for *u* _{d} = 7.173 × 10^{−3} m/s, the two profiles actually crosses between 0.5 and 0.6 fractional height (axial position in the column to the total height of the column). This concentration cross in profiles does not mean the mass was not transferred at or after this point or the direction of mass transfer was reversed after this point. Concentration difference in two different liquid phases is not like temperature difference for heat transfer where no temperature difference means no transfer of heat. It is the effective concentration difference which is important and which may be calculated by the equilibrium relationship. The effective concentration difference is, therefore, not *y* – *x*, but *y* – *y* _{e}, where *y* _{e} is *mx* _{ e }. Now as *m* is quite a small fractional value (*m* = 0.0708) the effective concentration difference is positive throughout the column length.

^{−3}m/s and for varying dispersed phase velocities and compares the model concentration profiles to the experimental concentration profiles obtained in the present study. Though not perfect, but the model profiles give virtually the same trend and show rather good representation of the axial concentrations.

#### Mass transfer in the presence of axial diffusivities

The model equations, Eqs. 9 and 10, involve axial diffusivities. The equations were rearranged to obtain the following expressions:

*y*is taken as

*y*

_{ i }and

*x*is taken as

*x*

_{ i }, respectively.

Equations 17 and 18 together with boundary conditions were solved to fit the experimental axial concentrations. The fitting was obtained so that the SSE between the experimental and predicted axial concentrations was minimized. In each case, the %error between the experimental and model outlet concentrations in the dispersed phase was kept less than 5% and the inlet concentration of acetic acid in water was required to have a value of virtually equal to zero.

*K*

_{ocp}

*a*has already ~11% error as discussed earlier.

Diffusivities and mass transfer coefficients obtained with and without the use of axial diffusivities

(m/s) | (m/s) | (m | (m | (s | (s |
---|---|---|---|---|---|

19.33 | 1.647 | 0.110 | 0.046 | 0.0519 | 0.0518 |

19.33 | 4.524 | 0.105 | 0.012 | 0.111 | 0.1109 |

19.33 | 7.173 | 0.099 | 0.022 | 0.256 | 0.2593 |

26.00 | 1.647 | 0.140 | 0.05 | 0.0519 | 0.0516 |

26.00 | 4.524 | 0.180 | 0.12 | 0.135 | 0.1346 |

26.00 | 7.173 | 0.800 | 0.992 | 0.222 | 0.2224 |

## Conclusions

The mass transfer performance in a liquid phase pulsed sieve tray column of 50 mm diameter and 4 m height, with 80 trays with a plate free area of 13.5% was investigated. The mass transfer performance appears to be a strong function of both the pulsation intensity and dispersed phase velocity. The number of transfer units (NTU_{ocp}), the height of transfer unit (HTU_{ocp}), and the apparent mass transfer coefficient based on continuous phase (*K* _{ocp} *a*) is calculated. Mass transfer coefficient increases with an increase in pulsation intensity and dispersed phase velocity. Variation of the concentration of acetic acid in aqueous phase along the length of the column is discussed as a function of dispersed phase velocity and pulsation intensity. Mass transfer coefficient of propionic acid as the solute in the kerosene phase is observed to be greater than when acetic acid is the solute in the kerosene phase.

For the experimental data of mass transfer coefficient, the equation of Venkatnarasaiah and Varma [17] for mass transfer coefficient is tried which, however, is found not successful. Venkatnarasaiah and Varma [17] worked on a column with plate free area ranging between 23 and 46%. This may be the reason that the data in the present study are not fitted well by their equation.

A new correlation that is, at least, applicable for the liquid–liquid system and the column studied in the present work is developed. The new equation gives a %error of only 10.8 as calculated from Eq. 8.

Mathematical modeling of the pulsed column is carried out with and without using the axial diffusivities. Based on the model without incorporating axial diffusivities, the concentration profiles for acetic acid in continuous (aqueous) phase and dispersed phase for various operating conditions are shown and in some cases compared with the experimental data. Based on the model considering axial diffusivities, model axial diffusivities and mass transfer coefficients are calculated. Virtually, similar mass transfer coefficients with and without the use of axial diffusivities in a model are obtained and virtually the similar concentration profiles are obtained that may show the insignificance of axial diffusivities in the column. The results may support the assumption of negligible axial diffusivities in the model as applied in the determination of mass transfer coefficients and in developing the corresponding mass transfer correlation (Eq. 7).

## Notes

### Acknowledgements

The authors would like to acknowledge Chakwal group of industries for funding the project. Ms. Madiha, Ms. Zona, Mr. Sohaib, Mr. Abdullah, Mr. Mudassar, and Mr. Salahuddin also deserve our acknowledgements for their assistance in different ways.

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