New Fraction Jetting Model for Distillation Sieve Tray Efficiency

Aug 24, 2012 - 2007, 85, 269−277) sieve tray efficiency model, that are valid for both ... In this paper, the concept of fraction jetting on a sieve...
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New Fraction Jetting Model for Distillation Sieve Tray Efficiency Prediction Anand N. Vennavelli* Fractionation Research, Inc., 424 S. Squires, Suite 200, Stillwater, Oklahoma 74074, United States

James R. Whiteley Department of Chemical Engineering, Oklahoma State University, Stillwater, Oklahoma 74078, United States

Michael R. Resetarits Fractionation Research, Inc., 424 S. Squires, Suite 200, Stillwater, Oklahoma 74074, United States ABSTRACT: The two phases on distillation trays can function in several different regimes depending on the physical properties of the system and the liquid-to-vapor ratio. Predicting the fraction of the vapor transported as jets, or fraction jetting, on a tray operating in the mixed-froth regime can bridge froth and spray regime models, explain the gradual changes in tray efficiency during the froth−spray transition, and eliminate the need to predict the froth−spray transition point when separate froth and spray regime models are used. Fraction jetting models will also facilitate multiregime models, such as the Syeda et al. (Chem. Eng. Res. Dev. 2007, 85, 269−277) sieve tray efficiency model, that are valid for both froth and spray regimes. Current fraction jetting models are empirical and developed from limited data. In this paper, the concept of fraction jetting on a sieve tray is discussed and a new semiempirical model is presented. The fraction jetting model was developed using the existing data of Raper et al. (Chem. Eng. Sci. 1982, 37, 501−506) and no new data were generated. Furthermore, the applicability of the fraction jetting model to tray efficiency models that incorporate jetting is demonstrated using the Syeda et al. sieve tray efficiency model.

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INTRODUCTION Flow regimes have long been used as a basis for fundamental tray modeling as they provide a generalization of the two-phase behavior. On a distillation tray, the flow regime describes the two phases, which could be liquid-continuous, vapor-continuous, or both. The two-phase dispersions on a tray are classified into five flow regimes: emulsion, foam, bubble, spray, and froth. This paper is focused on the two-phase mechanisms and modeling in the froth and spray regimes. In the spray regime, the vapor is the continuous phase and the liquid is the dispersed phase. Jets of vapor rise through the tray openings and disperse and entrain the liquid projecting liquid droplets into the intertray spacing. The liquid droplets are simultaneously subject to drag, gravity, and buoyancy forces and, as a result, fall back onto the tray to repeat the projection process or get entrained to the tray above. In the froth regime, liquid-continuous and vapor-continuous dispersions could coexist. The vapor transport in the froth regime, bounded by the bubble and spray regimes on a flow regime diagram, gradually changes from bubble dominated to jet dominated with increasing vapor rates. However, jetting begins well before the transition to the spray regime. For this reason, the froth regime has also been referred to as being made up of two subregimesthe bubbling froth where the vapor transport is only by bubbles and the mixed-froth where the vapor transport is by both bubbles and jets. In the froth regime at lower rates, the vapor transport is primarily in the form of bubbles. As the vapor rate is increased, intermittent vapor jets © 2012 American Chemical Society

begin to form, and eventually, the jets break through the liquid layer on the tray (Figure 1). At much higher rates, the vapor jets dominate the two-phase on the tray and the two-phase resembles a spray. The fraction jetting, which represents the amount of vapor transported in the form of jets is defined as fj = =

Vj Vb + Vj Volume of the vapor transported as jets Volume of the vapor transported as bubbles and jets (1)

The flow regime, which characterizes the dispersion, is central to fundamental hydraulic and mass transfer models on trays. To address the flow regimes in modeling, researchers have traditionally used one of the following options: (a) a semiempirical model that extrapolates into adjoining regimes1 taking advantage of the gradual change in properties or (b) a separate model for each regime2,3 coupled with correlations to detect the regime boundaries. The extrapolation approach suffers from the fact that a model developed on the assumption of a liquid continuous phase, for instance, is unreliable when the liquid becomes the dispersed Received: Revised: Accepted: Published: 11458

December 21, 2011 July 9, 2012 August 9, 2012 August 24, 2012 dx.doi.org/10.1021/ie202997t | Ind. Eng. Chem. Res. 2012, 51, 11458−11462

Industrial & Engineering Chemistry Research

Article

Figure 1. Bubbles and jets in the mixed froth flow regime on a cross-flow tray.

Prado and Fair data are collected with the probe located at the sieve deck holes whereas the Raper et al. data were collected with the probe located in the dispersion.

phase. The two-model approach, on the other hand, assumes a single flow regime transition point with an abrupt change in two-phase behavior. The uncertainty in the predicted transition point, in addition to the likely presence of a transition zone instead of a transition point, invariably introduces model uncertainties. In either approach, simultaneous bubble and jet transport is not considered. An alternate approach, the fraction jetting approach, assumes that the change in flow regime is not sudden, but gradually changes as the dominant modes of vapor transport change. By accounting for all transport mechanisms, the regimes are naturally bridged. The fraction jetting approach has been successfully incorporated into the mass transfer efficiency models on sieve trays in the research led by Prof. James Fair4,5 and later by Prof. Karl Chuang.6 A fraction jetting model can be used to combine the relative contributions of the efficiencies of the bubbling and the jetting zones to the point efficiency using eq 2. EOG = fj Ej + (1 − fj )E b

Table 1. Comparison of Raper et al. and Prado and Fair Data system F-factor, Pa0.5 liquid load, m3/h/m-weir plate active area, m2 open hole area, % hole diameter, mm outlet weir height, mm probe location

Prado data

Raper data

air−water 0.8−3.7 5.4, 7.6, 10 0.0244 5−11 6.4−12.7 25.4−76.2 at the sieve holes

air−water 0.2−2.2 10 0.18 5−16 6.4−19.1 25−75 dispersion center

Not surprisingly, there are only two models for fraction jetting in the literature. The Prado and Fair5 model based on Prado’s data and Syeda et al.6 model based on Raper’s data. The Prado and Fair model is ub − ub,0 fj = ub,100 − ub,0

(2)

where EOG is the point efficiency, Eb and Ej are the bubbling and jetting efficiencies, and f j is the fraction jetting. The fraction jetting model becomes an important piece or submodel in the point efficiency prediction. The use of fraction jetting in efficiency modeling is not only closer to reality, but can also explain the drop in efficiency at higher rates as a result of increased jetting (which usually has a lower efficiency than bubbling) before the onset of excessive entrainment. Incorporating jetting in efficiency models can explain crowned efficiency-versus-rate patterns, provide better predictions at higher rates, and allow a single model to be used for both froth (mixed and bubble) and spray regimes. The measurement of fraction jetting is often indirect. The conductivity probe, also called the bubble probe, is used to account for all the vapor transported as bubbles. The unaccounted vapor is considered to be the vapor that bypasses the probe in the form of jets. The construction and working of the conductivity probe is described in detail elsewhere.7−9 Only two sources of fraction jetting data are available in the literature: Prado and Fair5 and Raper et al.9 Both use the conductivity probe technique with sieve trays and the air−water system. The flow rates and the tray geometries used for Prado and Fair and Raper et al. data are compared in Table 1. The

ub,0 = 0.1ρG0.5 ρL0.692 hw0.132dh−0.26ϕ0.992L0.27 ub,100 = 1.1ρG0.5 ρL0.692 hw0.132dh−0.26ϕ0.992L0.27

(3)

where ub,0 and ub,100 are the vapor velocities based on the bubbling area at 0% and 100% jetting, ρG and ρL are the vapor and liquid densities, hw is the outlet weir height, dh is the hole diameter, ϕ is the fractional tray open area, and L is the weir load. For the air−water system, the Prado et al. equation, eq 3, can be simplified as10 fj =

ubDH 0.27 103.65AF(hw 0.5L)0.27

− 0.1 (4)

The Prado and Fair model, eq 3, has eight estimated constants. The variables are the most common variables used in several froth−spray transition studies. However, the Prado data used for the Prado model included a fraction of liquid cover in addition to bubbling and jetting. The fraction liquid cover indicated the amount of inactive holes or insufficient information whether a particular hole is jetting or bubbling. 11459

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the vapor transported as jets to the ratio of the volume of the vapor transported as bubbles is proportional to the modified Froude number. The phenomenological relation can be represented in equation form as

The Syeda et al. model equation is fj = −0.1786 + 0.9857(1 − e−1.43Fb)

(5)

where Fb is the vapor F-factor based on the bubbling area, and f j is the fraction jetting. The Syeda fraction jetting model was developed from Raper et al.’s data and has three fitted constants. The Syeda model only considers the vapor F-factor. However, the Syeda model captures the nonlinear rate of fraction jetting dependence on the F-factor by using an exponential relationship. The Syeda fraction jetting model is artificially limited to a fraction jetting value of 0.8071 for high F-factors due to its empirical model structure. In summary, both fraction jetting models are empirical, based on air−water data, and only the Syeda et al. model is developed from fraction jetting data in the dispersion. The fraction jetting model is a critical component used in the recent sieve tray efficiency models to relate the contributions of jetting and bubbling to the observed overall effect. However, only limited modeling effort has been expended to date. In addition, the two available fraction jetting models must be extrapolated to systems other than air−water to be fully useful for efficiency modeling. Therefore, there is clearly a need for a new fraction jetting model with a more phenomenological basis.

Vj Vb

where Vj is the volumetric vapor flow as jets (m /s) and Vb is the volumetric vapor flow as bubbles (m3/s). From eqs 1 and 7, V 1 1 −1= b ∝ fj Vj Fr′ β 1 −1= fj Fr′

(8)

where f j is the fraction jetting, and β is the proportionality constant. The proportionality constant β is the value of the modified Froude number where Vb = Vj. If Fr′ > β, the volume of the vapor transported as jets is greater than that transported as bubbles and vice versa. Rearranging eq 7, the model structure for the new fraction jetting model may be obtained as fj =

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Fr′ β + Fr′

(9)

RESULTS AND DISCUSSION Although two sources of fraction jetting data are available in the literature, the Prado and Fair data are inherently different from the Raper et al. data because of the different measurement locations employed. The Prado data were obtained with the conductivity probe located at the holes. Moreover, the authors report, in addition to bubbling and jetting, a fraction of liquid cover, which indicates that there is not enough information about whether a particular hole is bubbling or jetting. The measurements of the Raper et al. data were based on the conductivity probe placed at the midpoint of the froth. For the fraction jetting model to be used in capacity and efficiency correlations, the data of Raper et al. measured in the midpoint of the froth are deemed herein to be more appropriate. Therefore, for estimating β, the experimental fraction jetting data of Raper et al. were chosen. The fraction jetting data were plotted against the modified Froude number, which was calculated from Raper’s data and a clear liquid height calculated from the Bennett et al. correlation.14 A least-squares estimation of β resulted in

NEW MODEL STRUCTURE The premise of the fraction jetting concept is that the formation of vapor jets at increasing vapor rates is responsible for the changes in the hydraulic and mass transfer properties of the two-phase. Moreover, a key feature of the new fraction jetting model is that the transition point is not a specific operating point; instead, there is an operating range over which fraction jetting increases from zero to one. The jet penetration model, proposed by Lockett,11 provides the theoretical basis for describing the phenomenon of jetting. According to the jet penetration model, the force of the weight of the liquid above a jet tends to form a liquid bridge across the jet and acts to collapse it. The force of the gas momentum acts to resist the liquid bridge formation. A stable jet is formed when the gas momentum exceeds the liquid weight force. The forces favoring and resisting jet formation are well represented by the modified Froude number. The modified Froude number is the ratio of the inertia of the vapor (square root of the kinetic energy of the vapor) to the liquid resistance to the vapor (square root of the potential energy of the liquid). The modified Froude number, defined in terms of the vapor velocity based on the bubbling area, ub, and the clear liquid height, hcl is

fj =

ubρG0.5 (ρL ghcl )0.5

(7) 3

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Fr′ =

∝ Fr′

Fr′ 0.0449 + Fr′

(10)

The goodness-of-fit of the new correlation is shown in Figure 2. Equation 11 defines the mean absolute relative error used to quantify the model fit. The model describes the Raper et al. data to a mean absolute relative error of 24%.

(6)

Therefore, it follows that the modified Froude number is the appropriate dimensionless number for correlating fraction jetting. The modified Froude number, defined in eq 6, was previously used by Hofhuis and Zuiderweg12 and Colwell13 as an independent variable in their mixed-froth regime hydraulic correlations. The modified Froude number can be interpreted as the propensity of the vapor to be transported in the form of jets rather than bubbles. In other words, the ratio of the volume of

Mean Absolute Relative Error =

∑ n

fj,exp − fj,calc fj,exp

(11)

The new model properly follows the Raper et al. data over the entire experimental range. Furthermore, the model allows all fraction jetting values between 0 and 1 depending on the Ffactor and the clear liquid height. The physical significance of 11460

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Figure 4. Fraction jetting predictions for the IC4/NC4 system 1138 kPa, 8.3% open hole area.

Figure 2. Fit of the new fraction jetting model using the sieve tray data of Raper. Raper’s valve and bubble cap data are also shown for comparison.

the numerical constant in eq 10 is indicative of the change in dominant mode of vapor transport from bubbling to jetting. Correlations based on the modified Froude number have been used to predict the gas holdup fraction or the liquid holdup fraction on trays.12,13 The correlation by Hofhuis and Zuiderweg is shown in Figure 3. If the β value of eq 10 is

Figure 5. Fraction jetting predictions for the C6/C7 system 165 kPa, 8.3% open hole area.

difference between the model predictions occurs at high Ffactors in the lower pressure C6/C7 system. The difference in predictions for the C6/C7 system is significant because jetting is most dominant at high F-factors. The lower predictions of the Syeda et al. model are, in part, due to the artificial limitation of its empirical model structure. The new fraction jetting model overcomes that limitation and can predict fraction jetting values greater than 0.8071 as shown in Figure 5.

Figure 3. Correlation of hold-up fraction with the modified Froude number. Source: Hofhuis and Zuiderweg (1979).2

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applied to the Hofhuis and Zuiderweg correlation plot, it coincides with an inflection zone where a rate of change occurs. This provides further evidence that the modified Froude number at the numerical value of 0.0449 is indicative of a change in the dominant mode of vapor transport from bubbles to jets. In addition, the fraction jetting models are used in efficiency correlations for hydrocarbon systems. The predictions of the new fraction jetting model are compared to the predictions of the Syeda et al. fraction model for two hydrocarbon systems: isobutane/n-butane (IC4/NC4) system at 1138 kPa in Figure 4 and cyclohexane/n-heptane (C6/C7) system at 165 kPa in Figure 5. As shown in Figure 4, the model predictions do not differ much at the relatively moderate F-factors encountered in the IC4/NC4 system except at the lower rates where jetting may not be dominant. However, as shown in Figure 5, the largest

APPLICATION TO THE EXISTING SIEVE TRAY EFFICIENCY MODEL The ability of the new fraction jetting model to replace the existing Syeda et al. fraction jetting model in the Syeda et al. efficiency model is demonstrated as follows. The Syeda et al. sieve tray efficiency predictions with the new fraction jetting model are compared to the Syeda et al. sieve tray efficiency predictions with their original fraction jetting model in Figure 6 for the isobutane/n-butane (IC4/NC4) system at 1138 kPa. Figure 6 includes the following efficiencies: 1. measured at FRI 2. calculated by the authors of this paper using the Syeda fraction jetting model and the Syeda efficiency model 3. calculated by the authors of this paper using the new jetting model of this paper and the Syeda efficiency model 11461

dx.doi.org/10.1021/ie202997t | Ind. Eng. Chem. Res. 2012, 51, 11458−11462

Industrial & Engineering Chemistry Research

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Eb Ej EOG Fb fj Fr′ g hcl hw L ub ub,0 ub,100 Vb Vj

Greek Letters

Figure 6. Efficiency predictions for the IC4/NC4 system 1138 kPa, 8.3% open hole area.

β ε1 ϕ ρG ρL

4. calculated and reported by Syeda based on the Syeda jetting and efficiency models. As shown in Figure 6, the new fraction jetting model can effectively replace the Syeda et al. fraction jetting model for the IC4/NC4 system at 1138 kPa. A key feature of the Syeda sieve tray efficiency model is its ability to predict the crowned efficiency patterns observed in the experimental data. Using the new fraction jetting model in place of the Syeda fraction jetting model retained the crowned pattern prediction capability as seen in Figure 6. The result reconfirms that the new fraction jetting model can effectively replace Syeda’s fraction jetting model in Syeda’s sieve tray efficiency model.

Proportionality constant of eq 8 Liquid volume fraction in the dispersion, fraction, 0−1 Fractional tray deck open area, fraction, 0−1 Vapor density, kg/m3 Liquid density kg/m3

Abbreviations

C6/C7 cyclohexane/n-heptane IC4/NC4 iso-butane/n-butane

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REFERENCES

(1) Chen, G. X.; Chuang, K. T. Prediction of point efficiency for sieve trays in distillation. Ind. Eng. Chem. Res. 1993, 32 (4), 701−708. (2) Zuiderweg, F. J. Sieve Trays - A View on the State of the Art. Chem. Eng. Sci. 1982, 37 (10), 1441−1464. (3) Bekassymolnar, E.; Mustafa, H. Clear liquid height on sieve plates in the froth, mixed and spray regimes. Chem. Eng. Res. Des. 1991, 69 (1), 14−20. (4) Garcia, J. A.; Fair, J. R. A fundamental model for the prediction of distillation sieve tray efficiency. 2. Model development and validation. Ind. Eng. Chem. Res. 2000, 39 (6), 1818−1825. (5) Prado, M.; Fair, J. R. Fundamental model for the prediction of sieve tray efficiency. Ind. Eng. Chem. Res. 1990, 29 (6), 1031−1042. (6) Syeda, S. R.; Afacan, A.; Chuang, K. T. A fundamental model for prediction of sieve tray efficiency. Chem. Eng. Res. Des. 2007, 85 (A2), 269−277. (7) Burgess, J. M.; Calderbank, P. H. The measurement of bubble parameters in two-phase dispersion-2. Chem. Eng. Sci. 1975, 30, 1107− 1121. (8) Raper, J. A.; Hai, N. T.; Pinczewski, W. V.; Fell, C. J. D. Mass transfer efficiency on simulated industrial sieve trays operating in the spray regime. Inst. Chem. Eng. Symposium Series No. 56 1979, 57−74. (9) Raper, J. A.; Kearney, M. S.; Burgess, J. M.; Fell, C. J. D. The structure of industrial sieve tray froths. Chem. Eng. Sci. 1982, 37 (4), 501−506. (10) Prado, M.; Johnson, K. L.; Fair, J. R. Bubble-to-Spray Transition on Sieve Trays. Chem. Eng. Prog. 1987, 83 (3), 32−40. (11) Lockett, M. J. Distillation Tray Fundamentals; Cambridge University Press: Cambridge, 1986. (12) Hofhuis, P. A. M.; Zuiderweg, F. J. Sieve plates: dispersion density and flow regimes. Inst. Chem. Eng. Symp. Series 1979, 56, 2.2/1. (13) Colwell, C. J. Clear Liquid Height and Froth Density on Sieve Trays. Ind. Eng. Chem. Process Des. Dev. 1981, 20 (2), 298−307. (14) Bennett, D. L.; Agrawal, R.; Cook, P. J. New Pressure-Drop Correlation for Sieve Tray Distillation-Columns. AlChE J. 1983, 29 (3), 434−442.

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CONCLUSIONS A fraction jetting model based on a modified Froude number has been developed from Raper et al.’s data. The model describes Raper’s data over the entire experimental range. The new fraction jetting model has a phenomenological basis, is not artificially limited to the fraction jetting value of 0.8071, and the physical significance of the fitted constant can be explained in terms of a change in the dominant mode of vapor transport from bubbles to jets. Furthermore, the new model can be used in place of the existing fraction jetting models of the multiregime tray efficiency models. The new fraction jetting model is expected to lead to improved models for predicting multiregime sieve tray efficiencies. The methodology and the model structure developed in this work provide a basis for further improvement of the model with the availability of additional data.

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Efficiency of the bubbling zone, fraction, 0−1 Efficiency of the jetting zone, fraction, 0−1 Point efficiency, fraction, 0−1 F-factor based on the bubbling area (= ubρG0.5) Pa0.5 Fraction jetting, fraction, 0−1 Modified Froude number, eq 6 dimensionless Acceleration due to gravity, m/s2 Clear liquid height, m Outlet weir height, m Weir load, m3/s-mweir Vapor velocity based on the bubbling area, m/s ub at 0% and 100% jetting, m/s Volume of the vapor transported as bubbles, m3/s Volume of the vapor transported as jets, m3/s

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge Fractionation Research, Inc. (FRI) for the financial support to this study. NOMENCLATURE Ab Tray deck bubbling area, m2 dh Hole diameter, m 11462

dx.doi.org/10.1021/ie202997t | Ind. Eng. Chem. Res. 2012, 51, 11458−11462