ARTICLE pubs.acs.org/IECR
Normalized Efficiency for Stagewise Operations Branislav M. Jacimovic and Srbislav B. Genic* Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia ABSTRACT: Tray efficiency in gasliquid operations is a widely discussed theme, and the shortcomings of Murphree’s, Hausen’s, Standart’s, and Holland’s efficiencies are well-known. The new parameter “normalized efficiency” is defined as the ratio of the real tray mass transfer rate and theoretically maximal mass transfer rate obtained for the counter-current plug-flow model for both phases in the case of infinite contact surface. Normalized efficiency has several advantages over the previously defined efficiencies: the range of normalized efficiency is 01; normalized efficiency values for gas and liquid phases are equal. Furthermore, normalized efficiency can be easily used for determining the required number of trays in mass-transfer operations or for estimating the overall column efficiency. The usage of normalized efficiency does not raise any dilemma about the interpretation of experimentally obtained results for trays working in distillation and sorption columns.
1. INTRODUCTION Tray (plate) efficiency estimation is still the basis of the stagewise gas (vapor)liquid column design, although nowadays there are significant efforts in computational fluid dynamics (CFD) modeling of columns and trays. Serious efforts were undertaken in the last hundred years in order to find the adequate procedure for estimation of the tray efficiency as a function of tray design variables, two-phase flow regimes, and thermophysical properties. Since Murphree’s definition of the tray efficiency (Murphree1), many researchers published their observations about tray efficiency in order to investigate the limitations and shortcomings of Murphree tray efficiency. Few of them, like Hausen,2 Standart,3 and Holland,4,5 developed other efficiency concepts, with some advantages compared to Murphree tray efficiency. 1.1. Murphree Tray Efficiency: Definition and Shortcomings. Let us consider the trayed column presented in Figure 1, in
case of transfer of one component between gas and liquid phase. Murphree tray efficiency1 for the ith tray can be defined for the gas phase (Figure 2) EMG ¼
yi yi þ 1 yout yin ¼ y ðxi Þ yi þ 1 y ðxout Þ yin
ð1Þ
xi 1 xi xin xout ¼ xi 1 x ðyi Þ xin x ðyout Þ
ð2Þ
and for liquid phase EML ¼
Gas and liquid phase Murphree tray efficiencies are tied with the following expression EMG ¼
EML EML ð1 λÞ þ λ
ð3Þ
or EML ¼
EMG λ 1 ð1 λÞEMG r 2011 American Chemical Society
ð4Þ
where λ is the stripping factor λ¼m
G L
ð5Þ
G, kmol/s, is the gas flow rate, L, kmol/s, is the liquid flow rate, and m is the slope of equilibrium line. The tray efficiency defined by Murphree has two major shortcomings: (1) EM is not normalized, i.e., it can have values between zero and infinity, so the misunderstanding of tray efficiency concept is possible,6 since the definition of the term “efficiency” refers to the ratio of the work done or energy developed by a device (machine, apparatus) to the energy supplied to it (efficiency = useful output/input) and (2) efficiencies calculated for gas and for liquid phase are equal only in case of equal slopes of operating and equilibrium lines (when stripping factor is equal to unity λ = 1) or in the case of a theoretical stage when EMG = EML = 1. Shortcomings of tray efficiency defined by Murphree can be illustrated by the examples given in Table 1. These are the published measurements for single trays gathered from the open literature. As stated above, the values of efficiency for gas and liquid phases are different and there are cases when the efficiency is greater than 1. Data presented in Table 1 clearly show that the consequences of the inadequate definition of tray efficiency given by Murphree have to be seriously analyzed. 1.2. Boundary Values of Murphree Tray Efficiency. Real flow patterns in various apparatuses are often approximated by two theoretical cases of interest: counter-current plug-flow model (CC model) and ideal mixing model (IM model). These two cases present theoretical boundaries for the mean driving force in the case of equal flow rates of both phases, inlet concentrations, phase contact surface, and mass-transfer coefficients. Counter-current apparatuses with plug-flow of both phases achieve the greatest mean driving force, while the apparatuses with ideal mixing of both phases are characterized with the least Received: May 10, 2010 Accepted: April 19, 2011 Revised: April 5, 2011 Published: April 19, 2011 7437
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Figure 2. Graphical interpretation of Murphree tray efficiency.
Table 1. Experimentally Obtained Tray Efficiencies from the Open Literature
Figure 1. Trayed column with N real trays.
λ
example
mean driving force. The quantity of component(s) transferred between phases in real apparatuses has to be somewhere between these two theoretical boundary cases. In case when the CC model is applied to both phases, Murphree tray efficiency for gas phase can be calculated by the following equation 8 > < exp½NTUOG ðλ 1Þ 1 for λ 6¼ 1 λ1 ð6Þ EMG;CC ¼ > : NTUOG for λ ¼ 1 where NTUOG is the overall number of transfer units for the gas phase defined by NTUOG ¼
1 1 λ þ NTUG NTUL
EMG;IM
EH
literature
1
1.25
1.25
1.11
ref 22
2
0.3595
0.6
0.37
0.67
ref 20
3 4
0.08 697.6
0.65 0.012
0.13 0.89
0.67 0.89
ref 20 ref 18
5
0.5
1.0
1.0
1.0
ref 22
6
0.53
1.2
1.46
1.12
ref 21
factor, the tray efficiency data for λ < 1 are not comparable with data for λ g 1. When λ f 0, the boundary values of Murphree tray efficiencies are EMG;CC ¼ 1 expðNTUG Þ EMG;IM ¼
NTUG 1 þ NTUG
ð11Þ ð12Þ
so in both cases EMG;CC ðNTUG f 0Þ ¼ EMG;IM ðNTUG f 0Þ ¼ 0
ð13Þ
ð8Þ
The maximal (theoretical) Murphree tray efficiency is achieved for CC model when NTUOG f ¥ 8 < 1G½y y/ ðx Þ ¼ L½x/ ðy Þ x in in in in λ NA, max ¼ / / > : G½yin y ðxin Þ ¼ λL½x ðyin Þ xin
NA
ð27Þ
NA, max
so normalized efficiency is always less than 1. The normalized efficiency definition, eq 27, is analogous to the efficiency of heat exchangers,15,16 which is based on the comparison of the heat duty of the real heat exchanger to the countercurrent plug-flow heat exchanger with an infinite heat transfer surface. The normalized efficiency can be calculated separately for the gas and liquid phase. Normalized efficiency in the case of absorption for the gas phase is 8 y y in out > < λy yðx Þ for λ g 1 in in ð28Þ ηG ¼ yin yout > : y yðx Þ for λ e 1 in in and for the liquid phase is 8 x x out in > > < xðyin Þ xin ηL ¼ 1 xout xin > > : λ xðy Þ x in in
for
λg1
for
λe1
ð29Þ
In cases when the operating line lies “under” the equilibrium line (Figure 6), transfer of component A occurs from the liquid to the gas phase (distillation, stripping). Operating lines for real cases are solid and for infinite contact surface operating lines are dashed (P is the pinch point). In this case, the real mass transfer rate is NA;max ¼ Gðyout yin Þ ¼ Lðxin xout Þ
ð30Þ
and NA,max can be calculated using eq 26, so the normalized 7440
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Figure 6. Mass-transfer from liquid to gas phase (distillation, desorption).
Table 2. Boundary Values of Normalized Tray Efficiency λf0
λf¥
ηCC = 1 exp(NTUG)
ηCC = 1 exp(NTUL)
ηLM = (NTUG)/(1 þ NTUG)
ηIM = (NTUL)/(1 þ NTUL)
ηCC = (NTUG f 0) = ηIM(NTUG f 0) = 0
ηCC = (NTUL f 0) = ηIM(NTUL f 0) = 0
ηCC = (NTUG f ¥) = ηIM(NTUG f ¥) = 1
ηCC = (NTUL f ¥) = ηIM(NTUL f ¥) = 1
efficiency for the gas phase is 8 y y out in > < λyðx Þ y in in ηG ¼ yout yin > : yðx Þ y in in
for
λg1
for
λe1
and for the liquid phase is 8 x x in out > for > < xin xðyin Þ ηL ¼ 1 xin xout > > : λ x xðy Þ for in in
ð31Þ
ð32Þ
The numerical value of gas and liquid normalized efficiencies are equal and independent of the direction of mass transfer between phases, so further on, normalized efficiency will be written without subscripts η ¼ ηG ¼ ηL
for
λg1
for
λe1
ð35Þ
OG
λg1 λe1
and for the IM model 8 λ NTUOG > > < 1 þ ðλ þ 1ÞNTUOG ηIM ¼ NTUOG > > : 1 þ ðλ þ 1ÞNTU
ð33Þ
The first advantage of normalized efficiency compared to the Murphree tray efficiency is that the numerical value of normalized efficiency is equal for both phases in contact. For the CC model, normalized efficiency is 8 1 exp½NTUOG ðλ 1Þ > > > for λ > 1 λ > > 1 λ exp½NTUOG ðλ 1Þ > > < NTU OG for λ ¼ 1 ð34Þ ηCC ¼ 1 þ NTU > OG > > > > 1 exp½NTUOG ðλ 1Þ > > for λ < 1 : 1 λ exp½NTU ðλ 1Þ OG
Normalized efficiency for real cases lays between the values calculated using eqs 34 and 35. Boundary values of normalized efficiency according to eqs 34 and 35 are given in Table 2. In the case when λ f 0, normalized efficiency depends on NTUG and the flow pattern, and in the case when λ f ¥, normalized efficiency depends on NTUL and the flow pattern. The second advantage of normalized efficiency compared to the Murphree tray efficiency is that the theoretically maximal normalized efficiency equals 1 and that normalized efficiency of the real trays is always less than 1.
3. APPLICATION OF NORMALIZED EFFICIENCY IN STAGEWISE OPERATIONS The Murphree tray efficiency survived in the engineering practice, in spite of the obstacles noted in section 1 of this paper, simply because it can be easily used for the estimation of the number of column trays when the number of theoretical trays is known. Normalized efficiency can also be used for the same purpose as EMG and in the same elegant manner. The use of normalized efficiency for the graphical estimation of the number of trays is shown in Figure 7 for the two-component mixture. Normalized efficiency of the ith tray is 8 > xi xi 1 Ai Bi > > ¼ for λ g 1 < xðy Þ x Ai Ci iþ1 i1 ð36Þ ηi ¼ > yi yi þ 1 Ai Bi > > ¼ for λ e 1 : yðx Þ y Ai Ci i1 iþ1 7441
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Figure 7. Graphical interpretation of normalized efficiency.
and normalized efficiency of the column is 8 xN x0 > for < xðy N þ 1 Þ x0 ηcol ¼ y y 1 Nþ1 > for : yðx Þ y 0 Nþ1
λg1 λe1
ð37Þ
The number of required contact stages (N), when operating and equilibrium lines are straight and when normalized efficiency is equal for each stage, can be calculated by
8 yN þ 1 yðx0 Þ 1 1 x0 xðyN þ 1 Þ > > > þ 1 ln λ þ ð1 λÞ ln > > y1 yðx0 Þ λ λ xN xðyN þ 1 Þ > > ¼ for > > λ λη λη > > > ln ln > > λη λ λη > > > < 1 η yN þ 1 y1 1η x0 xN ¼ for N ¼ η η y y ðx Þ x xðyN þ 1 Þ > 1 0 N > > > > > y yðx0 Þ 1 1 x0 xðyN þ 1 Þ > > ln λ þ ð1 λÞ N þ 1 ln þ 1 > > y1 y ðx0 Þ λ λ xN xðyN þ 1 Þ > > > ¼ for > 1η > 1 λη > > ln ln : 1 λη 1η The form of the eq 38 is similar to the KremserBrown Souders equation for the number of theoretical stages determination. Lewis17 derived the relation between the number of trays, column efficiency (ηcol), and the Murphree tray efficiency. In a similar manner, the normalized column efficiency can be tied to the normalized tray efficiency and the number of trays in the column. In the case of straight operating and equilibrium lines and when the normalized tray efficiency is equal for each tray, the number of stages is
N ¼
8 1 1 1 > > > ln þ 1 > > λ λ 1 ηcol > > > > λ η > > > ln > > λ λη > > > < η 1η col
1 ηcol η > > > > > > 1 > > ln λ þ ð1 λÞ > > 1 ηcol > > > > > 1 λη > > ln : 1η
for
λ>1
for
λ¼1
for
λ1
λ¼1
ð38Þ
λ λη N > > 1 > > > > λ λη for > > > λη N 1 > > > > > λ λη λ > > > > N < for ¼ N þ1η > > η > > N > > > 1 λη > > 1 > > > 1η > > for > > 1 λη N > > > λ : 1η
λ>1
λ¼1
ð40Þ
λ > ¼ for < 1 þ EMG λ EML þ λ η¼ EML > EMG > ¼ for : 1 þ EMG λ EML þ λ
λg1 λe1
ARTICLE
Table 3. Murphree, Hausen, and Normalized Efficiencies ð41Þ
no.
Normalized efficiency of theoretical stage (ηTS) is obtained for EMG = EMG = 1 in the form
ηTS
8 λ > > < 1 þ λ ¼ 1 > > : 1þλ
for
λg1
for
λe1
λ
EMG
EMG,max
EML
EH
η
ηTS
literature ref 22
1
1
1.25
¥
1.25
1.11
0.56
0.50
2
0.3595
0.6
1.56
0.37
0.67
0.49
0.74
ref 20
3
0.08
0.65
1.09
0.13
0.67
0.62
0.93
ref 20
4
697.6
0.012
¥
0.89
0.89
0.89
1.00
ref 18
5
0.5
1.0
1.0
1.0
1.0
0.67
0.67
ref 22
6
0.53
1.2
2.13
1.46
1.12
0.73
0.65
ref 21
ð42Þ
5. APPLICATION OF NORMALIZED EFFICIENCY IN THE ANALYSIS OF TRAY EFFICIENCIES Table 3 contains the efficiency data based on the experimental values from Table 1 and calculated using eqs 9, 41, and 42. Some comments and conclusions are required concerning the data from Table 3. Examples 1 and 6: Normalized efficiencies of these trays are greater than the efficiencies of the theoretical stage. Examples 1 and 4: Concerning the high values of EMG, EML, and EH in example 1, one may conclude that this tray is a very efficient one. When compared to example 4, tray efficiencies EMG, EML, and EH are greater than in example 1, so one may conclude that the tray from example 4 is less efficient than the tray from example 1. The reality is quite the opposite: normalized efficiency of the example 4 tray has significantly (about 59%) greater efficiency than the tray from example 1. Example 4: Although EMG is rather low, this tray has the greatest normalized efficiency of all trays presented in Table 2. The value of λ in this case is very high but not unusual (steam stripping of toluene from water 18 ), and in these cases E MG approaches zero (if λ f ¥ then EML f η and E MG f 0). Example 3: Although the EML value does not look satisfactory, normalized efficiency is quite good. Murphree gas efficiency in this case approaches normalized efficiency (EMG f η) because the stripping factor is close to zero. Examples 2 and 3: Hausen’s efficiency EH = 0.67 is the same in both examples, so it may be concluded that both trays show equal separation effects. Normalized efficiency in example 3 is 27% greater than in example 2, which means that EH cannot be used for tray efficiency comparison as successfully as η. Example 5: This tray has achieved the efficiency of the theoretical stage η = ηTS = 0.67. Examples 16: Comparison of tray efficiencies is very simple when normalized efficiency is used. Trays from examples 16 are ranked in the following order: η(example 4) > η(6) > η(5) > η(3) > η(1) > η(2). 5.1. Example 7: Analysis of Mac Farland et al. Correlation19 Using Measured Data Taken From Oi20. Let us reconsider
example 2 using two widely cited tray efficiency correlations of Mac Farland et al.19 For the column and tray data taken from Oi,20 using λ = 0.3595, calculated Murphree tray efficiencies are EMG = 2.14 that correspond to EML = 2.98 and EMG = 3.20 that correspond to EML = 1.36. The question is, are these values possible concerning that EMG is much greater than 1 and EML is less than zero?
Corresponding values of normalized efficiencies calculated by eq 41 are η = 1.21 and η = 1.49, respectively. Although it was proved that η must be less than 1, Mac Farland et al.’s19 correlation provides unreal greater values. The maximal Murphree tray efficiency value, according to eq 9, for λ = 0.3595 is EMG,max = 1.56. This means that calculated efficiencies EMG = 2.14 and EMG = 3.20 are not theoretically possible. Although it is claimed by Mac Farland et al.19 that both correlations predict tray efficiency to a good extent (average deviations for their correlations are 13.2% and 10.6%), it is obvious that their correlations are seriously limited.
6. CONCLUSIONS Murphree tray efficiency is commonly used for expressing the rate of mass transfer in trayed columns. This paper shows that there are serious problems in the interpretation of the intensity of mass transfer on the basis of Murphree tray efficiency, as a result of its inadequate definition: (1) gas and liquid phase Murphree tray efficiency of one tray have different values, unless λ = 1 or the case of the theoretical stage; (2) range of theoretically maximal Murphree tray efficiency depends on the stripping factor (for gas phase EMG,max = 1/(1 λ) for λ < 1, EMG,max f ¥ for λ > 1); (3) values of Murphree tray efficiency for λ > 1 and λ < 1 for various trays are not comparable, except when both phases are ideally mixed; (4) for the analysis of the tray mass transfer rate beside Murphree tray efficiency, the value of the stripping factor must be known. Boundary values of the Murphree tray efficiency in the case of counter-current plug-flow defined by eq 6 must be used as the control value during the tray efficiency estimation. Obviously, some researchers have not taken into account these boundary values (analysis is shown in example 7). Other efficiency concepts like Hausen’s, Standart’s, and Holland’s also show shortcomings. In order to improve the interpretation of the effects of mass transfer on trays, a new concept of normalized efficiency was introduced. Normalized efficiency has several advantages: (1) range of normalized efficiency is 01; (2) normalized efficiency is equal for the gas and liquid phases (there exists a single value of normalized tray efficiency); (3) there are no dilemmas about the interpretation of the tray efficiency values, experimentally obtained or calculated, on different trays working at various regimes. Also, it must be noted that normalized efficiency can be used as easily as the Murphree tray efficiency for the determination of the required number of trays for mass transfer operations, in the case of one component transfer but also for multicomponent systems. ’ AUTHOR INFORMATION Corresponding Author
*Fax: þ381 11 3370364. Phone: þ381 11 3302360. E-mail:
[email protected]. 7443
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’ ACKNOWLEDGMENT We thank the Ministry of Science and Technological Development of Serbia for a partial support to this study through the Project of Energy Efficiency. ’ REFERENCES (1) Murphree, E. V. Rectifying Column Calculations. Ind. Eng. Chem. 1925, 17, 747. (2) Hausen, H. On the Definition of the Degree of Exchange of Rectification Plates in the Case of Mixtures of Two and Three Components Substances. Chem. Ing. Tech. 1953, 25, 595–597. (3) Standart, G. L. Generalized Definition of Theoretical Plate or Stage of Conducting Equipment. Chem. Eng. Sci. 1965, 20, 611–622. (4) Holland, C. D.; McMahon, K. S. Comparison of Vaporization Efficiencies with Murphree-type Efficiencies in Distillation I. Chem. Eng. Sci. 1970, 25, 431–436. (5) Holland, C. D. Fundamentals of Multicomponent Distillation; McGraw-Hill: New York, 1981. (6) Boles, W.; Fair, J. R. Distillation. Ind. Eng. Chem. 1970, 62, 81–90. (7) Ho, G. E.; Prince, R. G. H. Hausen Plate Efficiency for Binary Systems. Trans. Inst. Chem. Eng. 1970, 48, 101–106. (8) Pavlechko, V. N.; Levdanskii, E. I. Complex Model of the Efficiency of Rectification Plates. Comparison with Other Models with Respect to Experimental Data. J. Eng. Phys. Thermophys. 2002, 75, 17–21. (9) Pavlechko, V. N.; Minakhmetov, A. V.; Nikolaev, N. A. Analysis of Mass Exchange Efficiency in Rectifying Columns. Russ. J. Appl. Chem. 2009, 82, 435–438. (10) Standart, G. L. Comparison of Murphree-type Efficiencies with Vaporization Efficiencies. Chem. Eng. Sci. 1971, 26, 985–988. (11) Medina, A. G.; Ashton, N.; McDermott, C. Murphree and Vaporization Efficiencies in Multicomponent Distillation. Chem. Eng. Sci. 1978, 33, 331–339. (12) Savkovic-Stevanovic, J. Murphree, Hausen, Vaporization, and Overall Efficiencies in Binary Distillation of Associated Systems. Sep. Sci. Technol. 1984, 19, 283–295. (13) King, C. J. Separation Processes; McGraw-Hill: New York, 1980. (14) Green, D. W.; Perry, R. H. Perry’s Chemical Engineers’ Handbook; McGraw-Hill: New York, 2008. (15) Kuppan, T. Heat Exchanger Design Handbook; Marcel Dekker: New York, 2000. (16) Hausen, H. W€arme€ubertragung im Gegenstrom, Gleichstrom und Kreuzstrom; Springer-Verlag GmbH: Berlin, Germany, 1976. (17) Lewis, W. K. Rectification of Binary Mixtures. Ind. Eng. Chem. 1936, 28, 399. (18) Kunesh, J. G.; Ognisty, T. P.; Sakata, M.; Chen, G. X. Sieve Tray Performances for Steam Stripping Toluene from Water in a 4-Ft Diameter Column. Ind. Eng. Chem. Res. 1996, 35, 2660–2671. (19) Mac Farland, S. A.; Sigmund, P. M.; van Winkle, M. Predict distillation efficiency. Hydrocarbon Process. 1972, 51, 111–114. (20) Oi, L. E. Estimation of Tray Efficiency in Dehydration Absorbers. Chem. Eng. Process. 2003, 42, 867–878. (21) Hubner, W. Der Einfluss der Konzentration auf das Verst€arkungsverh€altnis von Rektifizierb€oden. Chem. Ing. Tech. 1972, 44, 546–552. (22) Shilling, G. D.; Beyer, G. H.; Watson, C. C. Efficiencies in Ethanol-Water Fractionation. Chem. Eng. Prog. 1953, 49, 128–134.
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