a new model for estimating mass transfer coefficients for the extraction of ethanethiol with...

Applied Mathematical Modelling 31 (2007) 2515–2523

www.elsevier.com/locate/apm

A new model for estimating mass transfer coefficients forthe extraction of ethanethiol with alkaline solutions

in packed columns

Alina Barbulescu *, Claudia Irina Koncsag

Ovidius University of Constanta, Romania

Received 1 August 2005; received in revised form 1 September 2006; accepted 9 October 2006Available online 30 November 2006

Abstract

The mathematical model for estimating the mass transfer coefficients for the ethanethiol extraction from gasoline, withalkaline solutions was established by processing the data obtained in a laboratory and in a pilot plant experiment. It allowsthe estimation of overall mass transfer coefficients, taking into account the dispersed phase velocity, the concentration ofNaOH solution, the specific area and the porosity of the packing. The model’s parameters were statistical tested in order toconfirm the accuracy of the model.� 2006 Elsevier Inc. All rights reserved.

Keywords: Mass transfer; Packed columns; Modelling; Residuals; Analysis

1. Preliminaries

The thiols extraction from sour petroleum fractions is an usual process in petroleum refining industry. Theprocess is a liquid–liquid extraction accompanied by a chemical reaction of second order between the thiolsand NaOH. Since the chemical reaction is very fast, the process is controlled by the mass transfer of thiolsfrom the hydrocarbon phase into NaOH solution. So, the estimation of the mass transfer coefficients, asthe expression of the process rate, could be useful for the design and the simulation of the industrial extractionprocess.

In the present work, the authors establish a mathematical model for the estimation of the mass transfercoefficients in the case of ethanethiol extraction with NaOH solutions, analyzing and processing originalexperimental data obtained in laboratory and in a pilot plant, in extractors equipped with structured packing.We chose the ethanethiol as the key-component of the caustic refining process because it is the heaviest thiolefficiently removable from LPG by simple extraction.

0307-904X/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2006.10.003

* Corresponding author.E-mail address: [email protected] (A. Barbulescu).

mailto:[email protected]

2516 A. Barbulescu, C.I. Koncsag / Applied Mathematical Modelling 31 (2007) 2515–2523

2. Experimental results

The experiment was fulfilled in laboratory and pilot plant equipment.In laboratory, an Elgin extractor was available. It has the dimensions: 26 mm – the interior diameter and

700–800 mm – the active height. The extraction column was used in two variants: as a dispersion column andpacked with a handicraft-structured packing of metal gauze type, with the specific area ap = 60 m2/m3 and theporosity g = 0.98. The pilot scale column has a diameter of 76 mm and an active height of approximately1000 mm, from which the packing layer represents 840 mm. In this case, the structured packing was of corru-gated metal sheet type (Sulzer SMV350Y), with the specific area: ap = 340 m2/m3 and the porosity, g = 0.98.

Whatever the size of the column was, the experiment was led on the same manner: the feed was the hydro-genated gasoline enriched in ethanethiol. The ethanethiol extraction was performed with NaOH solution withthe concentrations: 5 wt%, 10 wt% and 15 wt%. The solvent-to-feed ratio was variable. The two phases (gas-oline and NaOH solution) flow in countercurrent and the concentration in raffinate (gasoline) and in extract(NaOH solution) was found by chemical analysis.

The steps followed to calculate the overall mass transfer drop-side coefficients ðKod � aÞ are:

(a) The calculation of the theoretical extraction stages NTT, where NTT is defined as the number of equi-libriums needed to reach the concentration in raffinate, for a certain solvent-to-feed ratio (S/A).The number of the theoretical extraction stages NTT was graphically determined using the distributioncoefficient, K [1].

(b) The calculation of the number of the transfer units related to the drop-side, NUTod, where the mass trans-

fer unit is defined as the height of the column along which the driving force of the mass transfer changesby a factor of e.For systems following the Nernst law (a linear correlation between the concentrations of ethanethiol-sol-ute in both the liquid phases) and for high values of the extraction factor (E = K * S/A), the number ofthe transfer units NUTod can be calculated from the number of theoretical stages NTT, by [2–4]

TableThe exin the

Conc.solutioreparti

5%, K

10%, K

15%, K

NTT

NUTod

¼1� 1

E

ln E:

(c) The calculation of the mass transfer unit height HUTod, dividing the active height of the experimentalcolumn H to the number of transfer units NUTod.

(d) The calculation of overall mass transfer drop-side coefficients ðKod � aÞ, from the HUTod values, using theformula:

Koda ¼ vd

HUTod

½s�1�;

1perimental data used for the calculation of the mass transfer unit height (HUT) at the ethanethiol extraction with NaOH solutions,dispersion column [2]

of NaOHn and thetion coefficient

Dispersedphasevelocity, vd

(cm/s)

Thiol conc.in feed, xj

(ppm)

Thiol conc.in raffinate,xe (ppm)

Thiol conc.in extract, ye

(ppm)

No. oftheoreticalstages, NTT

No. oftransferunits, NUT

Transfer unitheight,HUTod (cm)

= 48.9 0.17 1204 662 743 0.46 1.67 41.80.23 566 1118 0.36 1.31 53.50.33 780 568 0.54 1.85 37.8

= 92 0.17 711 188 258 0.74 3.04 23.00.23 205 319 0.72 2.30 30.50.33 185 458 0.82 2.46 28.4

= 102.4 0.17 745 148 178 0.80 3.50 20.20.23 159 223 0.79 2.76 25.40.33 136 320 0.75 3.51 20.0

Table 2The experimental data used for the calculation of the mass transfer unit height (HUT) at the ethanethiol extraction with NaOH solutions,in the packed laboratory column [4]

Conc. of NaOHsolution and therepartition coefficient


(cm/s)


(ppm)



(ppm)




5%, K = 48.9 0.17 1416 304 599 0.79 3.61 19.40.23 544 601 0.62 2.69 26.00.33 563 735 0.61 2.51 27.9

10%, K = 92 0.17 337 111 243 0.68 3.88 18.10.23 134 480 0.61 3.59 19.50.33 35 292 0.91 4.17 16.8

15%, K = 102.4 0.17 1638 487 906 0.71 4.69 14.90.23 675 969 0.59 4.42 15.80.33 720 1275 0.57 4.34 16.1

Table 3The experimental data used for the calculation of the mass transfer unit height (HUT) at the ethanethiol extraction with NaOH solutions,in the packed pilot column [4]

Conc. of NaOHsolution and therepartition coefficient


(cm/s)


(ppm)



(ppm)




5%, K = 48.9 0.21 992 204 998 0.81 10.12 8.30.32 211 1521 0.81 7.37 11.40.58 207 2757 0.82 4.62 18.2

10%, K = 92 0.21 651 155 631 0.77 9.03 9.30.32 127 1026 0.81 6.32 13.30.58 121 1861 0.83 3.78 22.2

15%, K = 102.4 0.21 833 152 835 0.82 8.24 10.20.43 112 1884 0.88 4.38 19.20.68 91 3094 0.91 2.94 28.6

Table 4The mass transfer coefficients calculated

NaOHconc. (%)

Dispersion column Packed laboratory column Pilot packed column

Disperse phasevelocity, vd (cm/s)

Kod � a � 103

(s�1)Disperse phasevelocity, vd (cm/s)

Kod � a � 103

(s�1)Disperse phasevelocity, vd (cm/s)

Kod � a � 103

(s�1)

5 0.17 2.78 0.17 6.02 0.21 8.30.23 3.66 0.23 6.27 0.32 11.40.33 4.31 0.33 8.34 0.58 18.2

10 0.17 5.06 0.17 6.45 0.21 9.30.23 5.36 0.23 8.26 0.32 13.30.33 8.21 0.33 13.87 0.58 22.2

15 0.17 5.77 0.17 7.84 0.21 10.20.23 6.43 0.23 10.30 0.43 19.20.33 11.67 0.33 14.46 0.68 28.6

A. Barbulescu, C.I. Koncsag / Applied Mathematical Modelling 31 (2007) 2515–2523 2517

where the superficial velocity of the disperse phase (vd) is defined as the volumetric flow of the dispersed phase(the gasoline) divided by the cross-sectional area of the column.


In Tables 1–3 are presented the primary experimental data and the calculation of the transfer unit heightrelated to the drop side [HUTod] in each case.

The mass transfer coefficients ðKod � aÞ calculated, using the data in Tables 1–3 are given in Table 4. Thereare nine sets of data, each set consisting of a certain concentration of NaOH solution, a certain type of columnand packing and various values of the superficial velocity.

3. Mathematical background [5,6]

Definition 1. A discrete process in time is a sequence of real random variables (Xt, t 2 Z).

Definition 2. A discrete process (Xt, t 2 Z) is said to be stationary if:

(")t 2 Z, EðX 2t Þ < þ1,

(")t 2 Z, E(Xt) is constant,(")t 2 Z, (")h 2 Z, Cov(Xt,Xt+h) = c(h),

where E(Xt) is the expectance of Xt and Cov(Xt,Xt+h) is the correlation of Xt,Xt+h.

Definition 3. The function defined on Z by

qðhÞ ¼ CovðX t;X tþhÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2ðX tÞr2ðX tþhÞ

p
is called the autocorrelation function of the process (Xt, t 2 Z).
r2(Xt) is the variance of the variable Xt.The most used estimator of q(h) is the empiric autocorrelation function, ACF:

qðhÞ ¼Pn�jhj

t¼1 ðxt � �xÞðxtþjhj � xÞPnt¼1ðxt � xÞ2

;

where xt is a realization of Xt, �x is the arithmetic mean of x1, . . . ,xn, calculated on n periods, h is the lag.

4. Mathematical model

4.1. The initial model

The data used in modelling are those given in Table 4.In [7], we gave different models for each set of data from Table 4. The best one was of Weibull type:

Kod � a � 103 ¼ A� b � e�C�vDd ;

where Kod Æ a is the overall volumetric mass transfer coefficient related to the dispersed phase, in s�1; a is theinterfacial area; it appears implicitly in the expression of the mass transfer coefficient; vd is the superficialvelocity of the disperse phase, in m/s; A, B, C, D are the parameters.

Although the Weibull model gave very good results on individual sets of data, we consider that it was notsatisfactory because the values of the parameters differ from a set of data to another one and no variation lawcould be withdrawn in order to find an unique model describing all the cases in our experiment.

The factorial analysis made for the experimental data prove that the simultaneous action of the superficialvelocity and the column packing porosity has the essential influence on the mass transfer coefficients. But, thefactorial analysis does not give the model form. The experiments proved that the NaOH concentration has animportant role in the variation of the mass transfer coefficient, by reason of the chemical reactions that evolvein system.

The experiments proved that the mass transfer coefficient is exponentially (or at least of power functiontype) dependent on the superficial velocity and is in inverse proportion with the porosity of the packing


column. Consequently, we try to develop the power function model in order to obtain a mathematical relationbetween Kod Æ a and vd, taking into account the influence of the concentration of NaOH solution and the geo-metrical characteristics of the packing:

Kod � a � 103 ¼ a 0:95þ c100

� �A1 ap

g

� �A2

ð102 � vdÞA3 ; ð1Þ

where c is the concentration of NaOH solution, wt%; ap is the specific area of the packing, in m2/m3; g is theporosity of the packing, in m3/m3.

Taking the logarithms in the formula (1), we obtain

lnðKod � a � 103Þ ¼ ln aþ A1 ln 0:95þ c100

� �þ A2 ln

ap

g

� �þ A3 lnð102 � vdÞ:

Denoting by

Y ¼ lnðKod � a � 103Þ; A0 ¼ ln a; X 1 ¼ ln 0:95þ c100

� �; X 2 ¼ ln

ap

g

� �; X 3 ¼ lnð102 � vdÞ

the previous relation can be written: Y = A0 + A1X1 + A2X2 + A3X3.So, we look for a model of the type: Y = A0 + A1X1 + A2X2 + A3X3 + e, where Y is the dependent variable,

X1, X2, X3 are independent (explicative) variables and e is the specification error of the model.Since we work with a sample of 27 successive observations, the model can be written:

yt ¼ A0 þ A1x1t þ A2x2t þ A3x3t þ et; t ¼ 1; . . . ; 27; ð2Þ
where yt, x1t, x2t, x3t, t = 1, . . . , 27 are the observed values of the variables Y, X1, X2, X3 respectively at themoment t and et, t = 1, . . . , 27 are the specification errors (the difference between the true model and the cal-culated one).
If we denote by [8]

y ¼

y1

..

.

yt

..

.

y27

0BBBBBBBB@

1CCCCCCCCA

; x ¼

1 x11 x21 x31

..

. ... ..

. ...

1 x1i x2i x3i

..

. ... ..

. ...

1 x127 x227 x327

0BBBBBBB@

1CCCCCCCA

; A ¼

A0

A1

A2

A3

0BBB@

1CCCA; e ¼

e1

..

.

ei

..

.

e27

0BBBBBBB@

1CCCCCCCA

the model can be written in the form

y ¼ xAþ e: ð3Þ
Using the least squares method, the solution of Eq. (3) is
A ¼

A0

A1

A2

A3

0BBB@

1CCCA ¼ x0xð Þ�1x0y () A ¼

2:717

5:201

0:086

0:752

0BBB@

1CCCA; ð4Þ

where x 0 is the transposed of x.The estimated value of yt is yt ¼ A0 þ A1x1t þ A2x2t þ A3x3t and the residual is et ¼ yt � yt, t = 1, . . . , 27.

Remark. It must distinguish between the specification error of the model, et which is and remains unknownand the residual, which is known.

The variance of the error, r2(e) can be estimated by

r2ðeÞ ¼ e0en� k � 1

; ð5Þ

where n = 27 is the number of observations, k = 3 is the number of explicative variables and e is the vectorthat contains the residuals, et, t = 1, . . . , 27.


Making the calculus it can be seen that the residuals sum is zero.Using (5), the variance–covariance matrix can be determined. Its principal diagonal contains the variances

of the regression coefficients.The variance of y explained by the regression of y on x is measured using the determination coefficient

R2 ¼ 1�P27

t¼1e2tP27

t¼1ðyt � �yÞ2¼ 0:863 ð6Þ

and the modified determination coefficient

R2 ¼ 1� n� 1

n� k � 1� ð1� R2Þ ¼ 0:845; ð7Þ

where �y ¼P27

t¼1yt

27¼2.17322.

Since these values are closed to 1, the fitting quality is good.

4.1.1. Tests on the model coefficients

In what follows we fix the significance level a = 5%.

(i) We verify if the explicative variables have significant contributions to the explanation of the dependentvariable, testing the hypothesis: H0: Ai = 0, i = 0,1,2,3. Student test is used. Let us define

t�Ai¼ Ai

rAi

��; i ¼ 0; 1; 2; 3: ð8Þ

This value is compared with that given in the tables of Student test, for n � k � 1 = 23 liberty degree andat the significance level a = 5%, t23 = 2.069.If t�

Ai> t23, then the hypothesis H0 is rejected; thus Ai is significantly nonzero.

This is our case, because t�A0¼ 14:501; t�

A1¼ 4:8046; t�

A2¼ 6:93; t�

A3¼ 6:373 are greater than 2.069.

(ii) The second step is to analyze the global significance of the model, e.g. the set of the explicative variableshas an influence on the variable y. It means to test the hypothesis: H0: A0 = A1 = A2 = A3 = 0.
Let us denote by Fk,n�k�1 the value of the Fisher repartition, for k = 3 and n � k � 1 = 23 liberty degree,at the significance level a = 5% and
F � ¼ R2=k

ð1� R2Þ=ðn� k � 1Þ¼ 48:294: ð9Þ

Since F * > F3,23 = 3.028, then the hypothesis H0 is rejected.The results from (i) and (ii) indicate that the coefficients were well chosen.

4.1.2. Tests on the errors

(i) Independence test

In order to determine if there exists a correlation of first order between the errors, the test Durbin Wat-son is used. Let us consider:

DW ¼P27

t¼2ðet � et�1Þ2P27t¼1e2

t

¼ 2:179: ð10Þ

From the tables of Durbin–Watson statistics, we determine at the significance level 5%, the critical val-ues, d1 = 1.16 and d2 = 1.65 (which depend on the observation and on the explicative variables number).Since d2 = 1.65 < DW < 4 � d2, it results that there is not a first order correlation between the errors.


(ii) Normality test

In order verify the normality of the errors, the test Lilliefors is used.

�e ¼P27

t¼1et

27; s2 ¼

P27t¼1ðet � �eÞ2

27; s ¼

ffiffiffiffis2p

; zt ¼yt � �y

s; t ¼ 1; . . . ; 27: ð11Þ

Let z1 6 z2 6 � � � 6 z27 be the values of zi, increasing ordered and

D27 ¼ maxt21;...;27

jF 0ðztÞ � F 27ðztÞj; ð12Þ

where F0(zt), t 2 1, . . . , 27 is the value of the normal repartition function at zt,

F 27ðziÞ ¼0; i 6 0

i27; 1 6 i 6 26 is the selection repartition function:

1; i ¼ 27

8><>:

The hypothesis that must be verified is: H0: the errors have a normal distribution.The hypothesis is admitted because 0.152 = D27 6 D27,0.95 = 0.168, where D27,0.95 is the value given inthe tables of Lilliefors test, at the fixed significance level.So, the errors have a normal repartition.

(iii) Homoscedasticity testThe test Bartlett is used to verify the homoscedasticity of the errors.The hypothesis which must be tested is: H0: the errors have the same variance. To do it, the selectionvalues are divided in i = 3 groups, each of them containing ni = 9 data.Let us define s2

1; s22; s2

3 the selection variance of the groups, s2 is the selection variance of the sample(calculated with a formula analogous with (11)) and

X 2 ¼�Pi

j¼1nj lns2

j

s2

1þ 1i�1

Pij¼1

1nj� 1

n

� � : ð13Þ

If X 2 < @2ði� 1Þ, at the significance level a, than the hypothesis H0 is accepted.@2ði� 1Þ ¼ @2ð2Þ is the value given in the tables of the repartition @2, i � 1 = 2 liberty degrees. In thiscase, s2 = 0.040675, s2

1 ¼ 0:022581; s22 ¼ 0:0094117; s2

3 ¼ 0:078208.Since X 2 ¼ 8:469 > 5:991 ¼ @2ð2Þ, the homoscedasticity hypothesis is rejected.Since s2

3 is bigger than s21 and s2

2, the analysis of the values of the third column in Table 1 was done. Thesixth value in the last column is an aberrant value (its deviation from trend is the biggest). Following astandard procedure in the time series analysis this value was removed [6,8] and the model werereconsidered.

4.2. The new model

Following the same way, with 26 data, we obtain

– from formula (4): A0 ¼ 2:823; A1 ¼ 5:144; A2 ¼ 0:0747; A3 ¼ 0:832,

– from formula (5): r2ðeÞ ¼ 0:0178,

– from formulas (6) and (7): R2 = 0.945, R2 ¼ 0:938.

Till now it is clear that the model is better than the previous one, since the values R2and R2 are greater thanthose obtained in the first case and r2ðeÞ is smaller.


4.2.1. Tests on the model coefficients

Passing to the significance test for the coefficients, at the significance level 5%, we obtain

(i) t�A0¼ 22:446; t�

A1¼ 7:793; t�

A2¼ 9:619; t�

A3¼ 11:246.

t�Ai> 2:074 ¼ t22; i ¼ 0; . . . ; 3, so the coefficients are not zero.

(ii) Using (9), at the significance level 5%, we determine: F * = 126.527 > F3,22 = 2.351, e.g. the model isglobally significant.

From (i) and (ii) it results that the coefficients were well chosen.

4.2.2. Tests on the residuals

(i) The test Lilliefors applied at the significance level a = 5%, gives:

TableThe va

0:1331 ¼ D26 6 D26;0:95 ¼ 0:171;

e.g. the errors have a normal repartition.

5lues of ACF

Fig. 1. The graph of ACF.


(ii) To apply Bartlett test, we divide the residuals in three groups, with n1 = n2 = 9 data and n3 = 8 elements.In this case: s2 = 0.01458, s2

1 ¼ 0:0259; s22 ¼ 0:00848; s2

3 ¼ 0:008686.Since X 2 ¼ 3:4389 < 5:991 ¼ @2ð2Þ, the errors have the same variance.

(iii) The independence testUsing Durbin–Watson test for n = 26 residuals, we determine: DW = 1.613.At the significance level a = 5%, the critical values, d1 = 1.14 and d2 = 1.65 were determined from thetables of the Durbin–Watson statistics. Since d1 < DW < d2, we cannot decide on the independence ofthe errors.

In order to verify if the errors are independent, the autocorrelation function was used. The values of ACFand its graph are given in Table 5 and Fig. 1.

In Fig. 1 it can be seen that the values of the ACF are inside the limits of the confidence interval, at a con-fidence level 95%.

The values of the Box-Ljung statistics (Table 5, the column 5) are less that @2ð5Þ and the probabilities toaccept the independence hypothesis (Table 5, the last column) are not too small. So, we can decide that theerrors are independent.

5. Conclusions

In this paper, we found two models for the mass transfer coefficients at the extraction of ethanethiol withalkaline solutions in packed columns, connecting the mass transfer coefficient, the concentration of NaOHsolution, the geometrical characteristics of the packing and the dispersed phase velocity. The second modelwas obtained from the first, removing an aberrant value. It is better because the errors satisfy the normality,the independence and the homoscedaticity conditions. In both cases, the medium error was 0% and the errordispersion was less than 5%. Between the different models determined for this problem [7] it is also the mostrobust and has the advantage that it has the same form as the model describing the mass transfer coefficientfor other thiols [9].

The experiment has been replicated many times. The data used in our paper was a set of the data collected.It was used for modelling purposes because between the sets of data there are insignificant differences. Thefinal model obtained is also satisfied by the other sets of data.

The model is valid in all the range of dispersed phase velocity (up to the flooding capacity) and for the usualconcentration of NaOH in industrial applications (5–15 wt%). We recommend it only for the structured pack-ing. It could also be used for the design and the simulation of the industrial extraction process.

References

[1] C. Koncsag, Extraction of Mercaptans from gasoline. Phase equilibrium in the system gasoline–Mercaptans–caustic solutions, in:Proceedings of Romanian Conference on Chemistry and Chemical Engineering (RICCCE XI), Section: Physical Chemistry, Bucures�ti,1999, pp. 81–86.

[2] C. Koncsag, C. Stratula, The liquid–liquid extraction in structured packing columns. I. The mass transfer study, Revista de Chimie 54(1) (2003) 7–12.

[3] C. Koncsag, E. Graure, Preliminary study of mass transfer for liquid–liquid extraction on a new type of structured packing, AnaleleUniversitatii ‘‘Ovidius’’, Seria Chimie XII (2001) 63–67.

[4] C. Koncsag, G. Maioru, et al., Study of the extraction in a contactor equipped with a new type of packing, Studii s�i CercetariS�tiintifice, Universitatea Bacau 4 (2003) 87–92.

[5] J.J. Droesbeke, B. Fichet, Ph. Tassi, Series chronologiques. Theorie et pratique des modeles, ARIMA, Economica, Paris, 1989, pp. 14–60.

[6] C. Gourieroux, A. Monfort, Series temporelles et modeles dynamiques, Economica, Paris, 1990, 326–368.[7] C. Koncsag, A. Barbulescu, Mathematical model for the estimation of mass transfer coefficients at the extraction of thiols with alkaline

solutions in packed columns, Bulletin Scientific of ‘‘Politehnica’’ University Timisoara, Chemistry and Environment Series 49 (63, 1–2)(2004) 30–35.

[8] R. Bourbonais, Econometrie, Dunod, Paris, 2000, pp. 47–175.[9] A. Barbulescu, C. Koncsag, Mathematical modelling of the propanethiol–buthanethiol extraction for sour alkaline fraction,

International Journal of Pure and Applied mathematical Modelling, in press.

a new model for estimating mass transfer coefficients for the extraction of ethanethiol with...

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