Review of Hydraulics Correlations for Sieve Trays without Downcomers

Review pubs.acs.org/IECR

Review of Hydraulics Correlations for Sieve Trays without Downcomers Flávio D. Mayer,*,† Liliana A. Feris,† Nilson R. Marcilio,† Vanessa Baldo,‡ and Ronaldo Hoffmann‡ †

Department of Chemical Engineering, Universidade Federal do Rio Grande do Sul, 90040-040 Porto Alegre, Rio Grande do Sul Brazil ‡ Department of Chemical Engineering, Universidade Federal de Santa Maria. Cidade Universitária, 97105-900, Santa Maria, Rio Grande do Sul Brazil ABSTRACT: The correlations developed over the past few decades for the prediction of the hydraulics characteristics of sieve trays without downcomers (dual flow) are reviewed. The parameters assessed are the froth height, the pressure drop, the capacity, and the efficiency. Discussions of each correlation are presented. economical.19,20 Liquid and vapor flow counter-currently through the holes alternately, which also provides self-cleaning action.9 The dual-flow tray has a high capacity and a low pressure drop3 because the bubbling area is larger due to the absence of downcomers. The absence of downcomers and the larger bubbling area increases the tray capacity by 20% over sieve trays with downcomers21 (generally the downcomer areas are as great as 20%; thus, this is the largest bubbling area increase). It is worth noting that new types of crossflow trays, that utilize the area under the downcomer as bubbling area, have capacities as great as those of dual-flow trays.22 Dual-flow trays are also less susceptible to fouling.18 However, dual-flow trays have less flexibility, greater sensitivity to the physical properties of the mixture,23 and lower efficiency.18 The efficiency of this tray is satisfactory at flow rates within the design values, but it is still low because the gas−liquid contact time is small.20 Dual-flow trays reach efficiency (Murphree efficiency) higher than 90%.6 In large diameter columns, there is a greater probability for instability due to unbalanced trays and liquid channeling.18 Weiland24 also states that the bigger the dual-flow tray the worse is the liquid−vapor contact, and the lowest is the efficiency. Garcia and Fair9 reported that the limited use of dual-flow trays is due to a restrictive, narrow operating range that offers high efficiency (as a result, called peak efficiency) and a lack of optimum operation parameters.

1. INTRODUCTION The trays that are used in distillation are commonly divided into two groups: those with or without downcomers.1 Trays with downcomers are also known as crossflow trays because liquid and vapor flow cross currently.2 Trays without downcomers have a countercurrent flow of liquid and vapor and are often called dual-flow.3 Trays that physically restrict the passage of liquid (i.e., the majority of the valve and bubble cap trays) can only be of the type with downcomer. The majority of correlations that predict the hydrodynamic parameters of dual-flow trays were based on analyses of sieve trays with downcomers, and two studies4,5 have considered the behavior of dual-flow trays similar to packing. According to Van Winkle,4 the dual-flow tray can be modeled as a packed column because the tray’s behavior is governed by the same factors that affect packed columns. The interaction between the liquid and vapor phases in these trays occurs counter-currently, as in packed beds. Additionally, the froth characteristics (superficial area of the froth) developed in dual-flow trays are similar to the porosity properties of packed beds. In dual-flow trays, the liquid is retained on the tray due to the dynamic balance of liquid and vapor flowing through the holes. Therefore, its turndown ratio (a measure of operational flexibility) is low, with lower efficiency when operating below 60% of the column design capacity.1 This behavior of dual-flow trays assumes peak efficiency near the nominal tray conditions. The absence of a downcomer increases the effective area and consequently increases the vapor handling capacity.6 Dual-flow trays have been used since the 19th century, when Cellier-Blumenthal built a distillation apparatus that incorporated copper saucer-shaped trays pierced with small holes.7 Different types of trays have been designed: sieve trays without downcomer (i.e., dual-flow); the turbogrid, developed by Royal Dutch Shell plc;8,9 the Kittel and Ripple Tray;1,10,11 grid trays;12 “shower” trays;13 valve trays without downcomers;14,15 and structural packing used as a dual-flow tray.16,17 The most commonly used tray without a downcomer is the dual-flow tray, which is used for its simple design and its ease of installation and maintenance and because it is not proprietary.18 This tray consists of a perforated or punched plate, making it © 2014 American Chemical Society

2. HYDRODYNAMICS OF SIEVE TRAYS WITHOUT DOWNCOMERS The main hydrodynamic characteristics of sieve trays without downcomers are the liquid height on the tray (defined by the froth height), the pressure drop, the capacity, and the efficiencyall interrelated.24 Received: Revised: Accepted: Published: 8323

March 11, 2014 April 25, 2014 April 28, 2014 April 28, 2014 dx.doi.org/10.1021/ie5010543 | Ind. Eng. Chem. Res. 2014, 53, 8323−8331

Industrial & Engineering Chemistry Research

Review

Cervenka and Kolár28 experimentally evaluated the dependency proposed in eq 4 for a homogeneous gas−liquid mixture system and obtained a value for (1 − ψ) that correlates with the experimental data:

The liquid height (i.e., froth height) on the tray results from the interaction of the liquid with the vapor flow through the tray, as measured by the pressure drop. The liquid height influences both the tray capacity and efficiency. Increasing vapor and liquid velocities (i.e., mass flow rates) provides greater liquid/vapor contact, which increases the liquid height and pressure drop and results in more efficient mass transfer between phases. Liquid entrainment can occur when the vapor velocity is high, thereby reducing the tray efficiency. If entrainment is excessive, the froth height (the height of liquid jets) can reach the tray above (i.e., froth height equals the tray spacing), causing contact between the liquid jets and the liquid from the tray above, which also adversely affects the efficiency. In an extreme situation, with an excessive vapor velocity, the tray drainage of liquid is prevented by the vapor flow, leading to the flooding of the distillation tower. According to Weiland,24 the dual-flow tray area is divided into liquid- and vapor-active areas, where liquid and vapor respectively flow exclusively, even in trays perfectly leveled. This operational aspect provides to the dual-flow tray inherent instability and usually low efficiency. Also, Weiland states that high efficiency and high capacity are opposing objectives in dual-flow trays because of the hydraulic behavior.24 The prediction of hydraulics parameters of sieve plates without downcomers has been systematically researched by many authors, including studies conducted in the Czech Republic in the 1960s and 1970s; in Japan between 1970 and the 1990s; and by the Fractionation Research, Inc., in the 1950s. The accurate prediction of hydraulics parameters is crucial for the design of the contact stage in accordance with the requirements of efficiency and capacity. This prediction involves diverse correlations that have been developed for sieve trays without downcomers and are presented and discussed as follows. 2.1. Froth Height. According to Lockett,25 many studies on froth height examined the minimum energy dispersions of the two phases on the tray, including dual-flow trays. Eduljee26 proposed correlations to predict froth height from pressure drop in sieve and grid trays, as shown in eqs 1 and 2, in agreement with the experimental data, ranging from 15 to 25%. hf ρL = 2.13Pt + 7.47 (sieve trays) hf ρL = 2.301Pt − 9.24 (grid trays)

ϕV (0) ⎧ ⎪ 0.975 − ϕ (0) hf V ⎨ = hcL 1 − ϕV (0) ⎪ ⎩ 0.975ϕV (0) ⎫ ⎡ (1 − ϕ (0))0.975 ⎤⎪ V ⎥⎬ + ln⎢ ⎢⎣ 0.025ϕV (0) ⎥⎦⎪ ⎭

Equation 4 comprises 311 foam height measurements with deviations less than 15% for 90% of the measurements. Takahashi et al.29 proposed correlations for the vertical distribution of the gas void fraction (eq 5), the froth height (eq 6), and the average gas void fraction (eq 7) for dual-flow trays by considering the geometry of bubbles formed on the plate to be partially empty spheres (i.e., “capped bubbles”): vertical distribution of gas void fraction: ⎛ 4.3Fr ϕV (z) = ⎜⎜ + ( 1.2 Fr 4.3 Fr + 1) − z /hcl ⎝

⎞0.5 ⎟ ⎟ ⎠

(5)

froth height: hf = hcl(1 +

17.2Fr )

(6)

average gas void fraction: 1−ψ = ψ

17.2Fr

(7)

These correlations were constructed from models of bubble formation on the tray presented by Azbel30 and Kim cited by Takahashi et al.29 who considered the formation of spherical and elliptical bubbles, respectively. As a result, it was found that the void fraction of gas and froth height are functions of the Froude number and are dependent on the ratio of inertial forces and gravity forces. Equations 3, 4, and 5 were further validated experimentally by Takahashi et al.31 who used trays with hole diameters between 7.1 and 30.6 mm and perforated area fractions between 0.16 and 6.23%. The average void fraction of gas was modified to incorporate the behavior of bubbles formed for two ranges of the Froude number:

(1)

Region I: (2)

As stated by Kólar,27 in quasi-steady state conditions, the properties of mixtures on a tray are functions only of the vertical distance from the tray because the phase mixture is nearly homogeneous in a horizontal plane. Using a balance of body forces and pressure, Kólar27 determined the dependency of the liquid−vapor mixture porosity (1 − ψ) on the distance from the tray level, according to eq 3:

Region II:

1−ψ = 6.5 Fr , 1 × 10−5 < Fr < Frc ψ 1−ψ = 2 3 Fr , Frc < Fr < 1 ψ

(8)

(9)

where Frc = 8.5 × 10−4

(10) 21

More recently, Furzer experimentally evaluated the foam behavior predicted by Azbel. Unlike Takahashi et al.,29 Furzer21 evaluated the distribution of the liquid volume fraction (eq 11):

ϕV (0) ⎧ ⎪ (1 − ψ ) − ϕ (0) z V ⎨ = ψ ϕ − hcL 1 − ϕV (0) ⎪ (1 ) (0) ⎩ V ⎫ ⎡ (1 − ϕ (0))(1 − ψ ) ⎤⎪ V ⎥⎬ + ln⎢ ⎢⎣ (1 − (1 − ψ ))ϕV (0) ⎥⎦⎪ ⎭

(4)

⎛ ϕL(z) = 1 − ⎜ ⎝ (1 + (3)

⎞0.5 Fr /4 ⎟ Fr /4 )2 − z /hcl ⎠

(11)

The froth height (hf) was calculated using eq 12: 8324

dx.doi.org/10.1021/ie5010543 | Ind. Eng. Chem. Res. 2014, 53, 8323−8331

Industrial & Engineering Chemistry Research hf = hcl[1 + (hcLg )0.5 Fr ]

Review

2.2. Pressure Drop. Rylek and Standart8 applied the model for pressure drop for sieve trays with downcomers proposed by McAllister et al.33 to Turbogrid trays:

(12)

Furzer21 collected data from the system water/vapor in a total reflux distillation in a column with 22 sieve trays with 20% open area and 0.32 m tray spacing. During operation, the froth height varied from 0 to 0.2 m for values of F (F-factor) ranging from 0.75 to 2.15. The relationship between the limits and values of F resulted in a flexibility (TR - turndown ratio) of 2.87. Furzer32 measured the clear liquid height in dual-flow trays, with 20% free area, for heterogeneous binary and ternary liquid systems in total reflux. The experimental data were used to correlate the froth height and the liquid height: hf = 3.730hcl − 0.019hf < 0.3m

2 ⎡ ⎛ δ ⎞⎤ ρ UVh Pt = K ⎢0.4(1.25 − τ ) + (1 − τ )2 + 4f ⎜ ⎟⎥ V ⎝ d ⎠⎦ 2g ⎣

(20) 8

According to Rylek and Standart, the values of K, obtained from experimental tests with trays with 8.0 to 22.0% free area, allowed for the accurate prediction of the pressure drop for a wide range of geometrical parameters and vapor hole velocities. In addition, Eduljee26 plotted experimental data of pressure drop from liquid and vapor velocities in a air−water system, at various isobars (ranging from 20 to 90 kg·m2), according to correlations 21 and 22. The pressure drop data agree within 10% in relation to calculated values.

(13)

The average volume fraction of liquid in the foam was constant and equal to ϕL = 0.291 for hf < 0.3. Furzer32 also correlated the froth height with the physical characteristics of the liquid and vapor phases: ⎛1 1 2 ⎞ ⎟ hf = 2500.0⎜ ρV UVc ⎝2 ⎠ρ

ρV

L

⎛L ρ ⎞ V ·103⎟⎟ = x log⎜⎜ ⎝ V ρL ⎠

hf < 0.3m

ρL

⎛ V2 C ⎞ log⎜⎜ 2 · ⎟⎟ = y ⎝ ρV dh ⎠

(14)

Equation 14 can be rewritten in terms of the modified Froude number: ⎛ ρ ⎞−1/2 −6 FrM = 280.0 × 10 ⎜⎜ V ⎟⎟ ⎝ ρL ⎠

Steiner and Standart developed equations for predicting pressure drop (or holdup) based on the balance of forces and momentum in dual-flow trays. These equations consider the two-phase flow and tray design variables. Equation 23 gives the pressure drop across the tray as a function of both operating and design parameters:

(15)

Pt = ρL hf +

(LML)b2 [UVc(ρG /ρar )0.5 ]b3 b4

ρV UG2 2gτL

[ζ + (1 + (1 − τ )2 )]

(23) 35

0.42

ρL τ (δ /d)

The same authors subsequently published a work in which they experimentally evaluated the model proposed in eq 23. Equation 23 can be rewritten in terms of β1 and φ′, which are the amplitude of waves formed on the plate and the fraction of the free cross-sectional area of the plate occupied by liquid, respectively.

(16)

The terms b1, b3, and b4 were experimentally determined to be 0.006, 1.75, and 1.90, respectively. The term b2 was calculated from b2 = 0.3162τ−0.25. The froth height can be evaluated using hcL hf = [1 − (1 − ψ )]

Pt = (17)

⎡ U 2 ρ ⎤−0.2 Vc G ⎥ (1 − ψ ) = 1.0 − 0.0946⎢ ⎢⎣ ghL ρL ⎥⎦

(24)

Steiner and Standart obtained β1 and φ′ as defined by eqs 25 and 26, respectively, based on 1000 pairs of pressure drop and holdup data based on the flow rate of two phases. The trays tested comprised various conformations of holes with free areas between 3.6 and 19%.

(18) −1

9

φ′

Garcia and Fair adapted other experimental data to eq 16 to obtain new values for the parameters b1, b3, and b4, namely 0.01728, 1.0, and 1.5, respectively. Weiland24 developed a correlation to determine the froth height (eq 19), based on the principle of minimum energy applied in the assessment of dual-flow hydraulics. This correlation identifies the importance that difference in the density of liquid- and vapor-passing regions has in the froth height. ⎡ ⎛Q ⎞ ⎛Q ⎞ 1 ⎢C ρ ⎜ V ⎟ + 1 ⎜ L ⎟ ⎥ hf = V V 2 ρL (γL − γV ) ⎢⎣ 2C D ⎝ AL ⎠ ⎥⎦ ⎝ AV ⎠

2 β1 ρL UL2 1 (ζ + 1)ρG UG + 1 − β1 2g (1 − φ′)2 1 − β1 2gξφ′2 35

where the liquid−vapor layer porosity on the tray (1 − ψ) is defined by

2

(22) 34

These models were based on the theory presented by Azbel. Xu et al.23 proposed a correlation for calculating the liquid height (hcL) that considered operational characteristics and the tray design characteristics: hcL = b1

(21)

β1 =

⎡ ρ ⎤1/3⎛ U ⎞2/3 = 0.41 + 1.03⎢ G ⎥ ⎜ G ⎟ ⎢⎣ ρL ⎥⎦ ⎝ UL ⎠

(25)

1 − φ′M 1 + M − φ′M

(26)

where M = K1

2⎤

ULxUGy φ′3/2

(27)

Contrary to the predicted values from theoretical considerations suggested by Steiner and Standart,34 the value of β varied with the gas and liquid flow rates. It was also found that

(19) 8325



Review

Pt = 9.2V 0.6·τ −2.16

the value of y was zero, leaving only the determination of x and K1. The experimental value of x approached 0.83 and was used to evaluate the pressure drop data obtained from the literature, resulting in an accuracy of ±20%. Despite the use of approximately 5000 experimental data points, it was not possible to reliably determine the effects of the parameters involved in the calculation of β. Kolár27 modified eq 3 to be dependent on the pressure drop over the distance above the tray using force and momentum balances:

Matsumoto and Suzuki39 also statistically evaluated the pressure fluctuation in the froth in 245 mm diameter dual-flow trays to obtain an algorithm for the pattern recognition of the fluctuation spectrum, thereby permitting its use in online hydrodynamic estimates of this tray type. These authors also presented data of clear liquid height (hcL) as a function of gas velocity (UVc) for two types of trays with the same hole diameter (d = 5.0 mn), but with different perforated areas (13.7 and 8.2%). The sieve tray with a larger area showed a higher capacity (e.g., higher UVc before flooding). Takahashi et al.40 proposed a correlation to determine the pressure drop across the tray:

ϕV (0) ⎧ (P0 − P)(1 − ϕV (0)) z ⎨ = hcL 1 − ϕV (0) ⎩ (P0 − PH)ϕV (0) ⎪

⎪

⎡ (P − PH) ⎤⎫ + ln⎢ 0 ⎥⎬ ⎣ (P − PH) ⎦⎭

(31)

⎪

⎪

Pt = Pi − P0

(28)

Shoukry et al.36 statistically evaluated the pressure drop caused by liquid retention in different hydrodynamic regimes. In total, five regimes (bubbling, cellular foam, homogeneous froth, and two oscillating regimes) and the boundaries between the homogeneous and oscillating froth regimes were determined. Rylek and Kastánek12 developed an expression to determine the pressure drop across the froth on grid trays without downcomers: 0.5

(Pf )

= K 2FL + K3FV + K4

= β2ρL gh cL +

ρL U2Lc 2

+ (Pd + Pσ ) − ζl(ULh/τL)2 ρL /2 (32)

where the constant β2 is a function of the Reynolds number and τ and the holdup (hcL) are determined from eqs 33 and 34: 2 3 2 k1ρV UVh τL {3 − 2.4τ(1 − τL)} − 4.4ρL ULh (1 − τL)3 = 0

(33)

⎛ U ⎞1.5⎛ δ ⎞0.34 2 UVc /(ghcL ) = 2.8⎜ Vc ⎟ ⎜ ⎟ τ ·,8(ρL gd 2 /σ )0.42 ⎝ ULc ⎠ ⎝ d ⎠

(29)

where

2 {ULc /(gd)}0.63(dULcρL /μL )0.053

⎛A ⎞ K4 = ⎜ hb ⎟(Pf )0.5 ⎝ Ah ⎠

(34)

Equation 34 applies to dual-flow trays with hole diameters greater than 5.0 mm. In turn, Garcia and Fair9 stated that the balance of pressures maintained in a dual-flow tray is

(30)

The terms K2 and K3 are dependents of the geometric parameters of the tray and were determined from the experimental data. The term K4 was also experimentally determined; its value varied across different mixtures, ranging from 6.0 to 12.0. The applicability limits of eq 30 are

Pt = hcL + PhV

(35)

where hcL is the liquid height calculated from eq 16 with new values of b1, b3, and b4 equal to 0.01728, 1.0, and 1.5, respectively. The term PhV, which represents the pressure drop due to the passage of vapor in tray holes, is given by eq 36:

7.0 < UVh < 17.0[m·s−1] −1

PhV =

ULhmax = 0.16.τ[m·s ] 12

Rylek and Kastánek also evaluated the influence of the column diameter on the pressure drop. They found that a column diameter reduction from 1.0 to 0.4 m increased the pressure drop across the froth by a factor of 2.2. Shoukry and Kolár37 proposed a statistical model for dualflow tray operations under homogeneous froth regimes, based on the statistical determination of the pressure drop across the dispersion presented by Shoukry et al.36 In that work, Shoukry and Kolár37 also discussed the multiplicity of stationary states because the same average pressure drop across the tray may be from different operational regimes. The average pressure drop across the tray is nearly independent of holdup and operational conditions but is dependent on the gas flow and tray geometry. Mikhailenko et al.38 evaluated the hydraulic resistance in dual-flow trays as functions of gas (air) and liquid (water) flow and geometric characteristics of the tray. The pressure drop for trays with fractional free areas of 9.3 and 25%, operating in the normal liquid load region of 3.5 kg·m−2·s−1, is represented by eq 31:

Ux2ρL 2gCV2ρG

(36)

The orifice coefficient CV is calculated from eq 37 and depends on the tray free area for vapor passage (τV) and on the ratio between tray thickness (δ) and hole diameter (d): C V = 0.74τV + exp[0.29(δ /d) − 0.56]

(37)

The total fraction of holes passing vapor (x) is calculated using eq 38. The coefficients b5, b6, and b7 were 0.4668, 90, and 45, respectively. ⎛ |%flood − b6| ⎞⎤ ⎛ τ ⎞0.8⎛ Z ⎞0.2 ⎡ ⎟ exp⎢ − 0.35⎜ x = b5⎜ V ⎟ ⎜ ⎟⎥ ⎝ 0.2 ⎠ ⎝ 0.61 ⎠ ⎢⎣ b7 ⎝ ⎠⎥⎦ (38) 9

According to Garcia and Fair, the variation in the orifice coefficient accounted for 1/3 of the differences in total pressure drop for the two hole diameters evaluated (12.7 and 25.4 mm). The model for the pressure drop, fitted to the experimental data and predicted by eq 36, showed a mean absolute deviation of ±23.0% and a mean deviation of zero. Most of the points 8326



Review

Table 1. Dimension Ranges of Dual-Flow Trays and Physical Propertiesa for Application of Eq 35

a

tray type

d (m)

Ah/Ac

ρL

ρG

μL

μV

σ

dual-flow

0.0038−0.02

0.119−0.370

844−1,165

1.2

2.9−52

0.065

4.8−9.6 × 105

Source: Takahashi et al.5

that were above the mean absolute deviation of ±25.0% corresponded to the near flooding operation. Weiland24 presented a correlation to predict the pressure drop in dual-flow trays (eq 39). Equations 19 and 38 reveal that uniformity in froth density, that is, satisfactory contact between liquid and vapor, cannot occur in dual-flow trays. On the other hand, if liquid- and vapor-flow regions are segregated (high difference in froth density), pressure drop tends to decrease as well as the froth height and the efficiency. For that reason, the opposite behavior is observed between capacity and efficiency in dual-flow trays. Pt =

2 ⎛ Q ⎞2 γV 1 ⎛QL ⎞ C VρV ⎜ V ⎟ + ⎜ ⎟ γL − γV γV − γL 2C D2 ⎝ AL ⎠ ⎝ AV ⎠

Subsequently, Wallis et al.45 published a simplified form of eq 40, shown in eq 45, that is applicable to various tray types (with and without downcomers) and packings. XC 0.5 + YC 0.5 = 0.832

Equations of each tray type can be obtained using the method of Wallis et al.,45 such as that for dual-flow trays:

XC 0.5 + YC 0.5 = 0.851

Takahashi et al. evaluated the correlation shown by eq 46 using dual-flow trays with hole diameters between 2.14 and 12.10 mm and free areas between 14.65 and 63.71%. It was found that the load points are satisfactorily described for hole diameters greater than 5.0 mm. However, for trays with hole diameters less than 5.0 mm, the loading point must be correlated as

(39)

2.3. Maximum Capacity. The maximum capacity in dualflow trays can be defined by excessive entrainment or by flooding in the distillation tower. Kister18 presented the correlation proposed by Fair41 to predict flooding in different tray types. In this correlation, the flooding vapor velocity is calculated from the Souders and Brown flooding constant (CSB), defined by the flow parameter (FLV), tray spacing (from 0.15 to 0.91 m), and free area. According to Kister,18 the Fair correlation applies to dual-flow trays with free area exceeding 20%. Treybal42 represented the Fair correlation according to eq 40:

XC 0.5 + YC 0.5 = 0.516 d < 5.00 mm

⎛ ρ ⎞−3/4 Ufl = 0.0283 Z ⎜⎜ V ⎟⎟ Z < 0.3 m ⎝ ρL ⎠

2.4. Tray Efficiency. Sharma and Gupta evaluated the mass transfer on dual-flow trays with free areas of 14.5, 21, and 29.6%. The effective interfacial area was measured using an adsorption technique followed by a pseudo-first-order reaction. The volumetric mass transfer coefficient for the liquid and vapor phases are correlated according to eqs 49 and 50:

(40)

(49)

k Va = 2.61 × 10−4 ·τ −1.75·L0.6 ·U1.2

(50)

Miyahara et al. used a similar approach. These authors evaluated the interfacial area and mass transfer coefficient for the liquid phase in dual-flow trays. The model was developed for froth and transition regimes, which are equivalent to the transient jetting regime. The interfacial area (aL) for both froth and transition regimes can be calculated using eqs 51 and 52, respectively:

(41)

(42)

aLhcL = 80We 0.5/Fr 0.5 (froth regime)

(51)

aLhcL = 19.1We 0.2 (transition regime)

(52)

The mass transfer coefficient of the liquid phase (kL) for the froth and transition regimes can be calculated using eqs 53 and 54, respectively:

(43)

Correlation 39 applies to both XC and YC in the range between 0.0003 and 0.7. The term SF is the shape factor for dual-flow trays (equivalent to the packing factor [a/ε3] from the equation proposed by Sherwood et al.44 for packed columns) and is calculated using eq 44: SF = 23/(Ah /Ac)2

kL a = 4.2 × 10−2 ·τ −2.2·L0.6 ·U1.2

47

where

⎛ ρ 1 ⎞0.5 YC = UGF⎜⎜ G SF⎟⎟ ⎝ ρL g ⎠

(48) 46

Takahashi et al.5 proposed a correlation to predict the flooding velocity of several trays without downcomers (eq 41), including dual-flow trays, based on the correlation of Zens and Eckert43 for predicting the flooding velocity in packed towers. The ranges of tray dimensions and physical properties are shown in Table 1.

⎛ 1 ⎞0.5 XC = ULF⎜ SF⎟ ⎝g ⎠

(47)

The flooding velocity (Ufl) is then calculated using eq 48, with the froth height (hf) equivalent to the tray spacing (Z):

UVF = [(0.0744Z + 0.0117)(log10 FLV −0,1) + 0.0304Z

YC = exp(2.9/ln XC)

(46)

40

γL

0.5 ⎛ σ ⎞0.2 ⎛ ρ − ρV ⎞ ⎟⎟ + 0.0153]⎜ ⎟ ·⎜⎜ L ⎝ 20 ⎠ ⎝ ρ ⎠ V

(45)

(44) 8327

Sh =

kLhcL = 3.37Sc 0.5Ga 0.25 (froth regime) D

(53)

Sh =

kLhcL = 0.07Sc 0.5Ga 0.5 (transition regime) D

(54)



Review

On the basis of the theory of double films, it follows that

The volumetric mass transfer coefficient from the liquid phase (kLaL) for the froth and transition regimes can be calculated using eqs 55 and 56, respectively: 2 kL′ aLhcL = 270Sc 0.5Ga 0.25We 0.5Fr −0.5 (froth regime) DA

(55)

2 kL′ aLhcL = 1.34Sc 0.5Ga0.25We 0.2 (transition regime) DA

(56)

1 1 λ = + NOG NG NL

with NG and NL calculated by eqs 63 and 64, respectively. According to Xu et al.23 the efficiency predicted by this model had a mean absolute relative error of 7.1% and a 95% agreement with the calculated values within ±15%. These authors explain in detail the relationship between the F factor and froth height with efficiency. The increase of the F factor (which describes the vapor loading point) results in a greater efficiency due to increased turbulence in the tray. This increase also favors an increase in froth height, which causes a corresponding increase in the contact time between phases and, consequently, in the mass transfer. This characteristic differentiates dual-flow trays from those with downcomers. Furthermore, an increase in the free area reduces the froth height, thus resulting in lower efficiency. Thus, to maximize the efficiency with the tray capacity, the authors suggest using plates with a free area between 15 and 25%. Xu et al.23 also evaluated the behavior of foam height and efficiency as a function of the mixture concentration. Mixtures in which density and surface tension vary significantly with the concentration are more affected with respect to efficiency. Reducing the density and surface tension causes a decrease in froth height, thus reducing the efficiency of the plate. This should be considered especially for trays located at the end points of the distillation column, where the concentration significantly varies. Garcia and Fair9 evaluated the efficiency of dual-flow trays using previously developed models of point efficiency applied to sieve trays with downcomers.50,51 According to these authors, the mass transfer characteristics of dual-flow trays may be considered similar to those of cross-flow trays, although they clearly emphasize that the spray zone contributes significantly to the mass transfer, which occurs only in dualflow trays. This significant contribution of spray zone to mass transfer was also cited by Jiang et al.52 The application of this model9 to experimental data for dual-flow trays showed a mean deviation of ±25.0% between the predicted and experimental values. The predicted efficiency was lower than that of the experimental values, likely because the model did not consider the mass transfer in the spray zone located above the froth height. In this work, Garcia and Fair9 proposed a lower efficiency factor for cases of liquid entrainment of flows greater than that at peak efficiency (γ) or low-load (γ′), according to eqs 65 and 66, respectively.

The model fit to the experimental data for the volumetric mass transfer coefficient from the liquid phase (kLaL) and significantly changed by ±50%. Kastánek and Rylek48 developed an expression to determine the number of transfer units from pressure drop on froth, as presented in eq 57. The term K5 is a constant and was experimentally determined over several tests. The authors reported that K5 = 0.0705 fit well with experimental data, with maximum relative deviations of −13.5% and +5.9%. NV = K5·Pf′

(57)

where the apparent pressure drop in the froth is calculated using eq 57:

Pf′ = Pt − Pd

(58) 49

Koltunova et al. evaluated several correlations to predict dual-flow tray efficiency and compared the results with those obtained experimentally in columns operating under pressure or vacuum. The deviation of the predicted data ranged from 20 to 40%. Most of this variation was due to errors in determining operational parameters, e.g. reflux ratio or product concentration. Xu et al.23 suggested a correlation for determining the number of transfer units for liquid and vapor phases based on the similarity of the hydraulic behavior of dual-flow trays and agitated vessels: NG = k G′ tGaL = C1

NL = kL′ t LaL = C2

0.4 (1 − ψ )(DGtG)0.5 (ρL2 ρG )0.2 U Vc

ψ 0.5τ Cσ 0.6μ0.1

(59)

0.4 (1 − ψ )(DL t L′ )0.5 (ρL2 ρG )0.2 U Vc

ψτ Cσ 0.6μ0.1

(60)

where tG =

hcL UVcψ

h A t L′ = cL c QL

(61)

1− γ=

(62)

The coefficients c, C1, and C2 from eqs 59 and 60 were determined experimentally to be 1.32, 0.27, and 7.00, respectively. The experiments were conducted in a 300 mm diameter column with five trays spaced at 457 mm and operating at total reflux. Three sets of trays with 12.7 mm hole diameters and free areas of 20.0, 28.0, and 37.0% were tested. Data were collected from intermediate trays to avoid interference with column end conditions. The separation efficiency of the dual-flow tray can be measured thus:

NOG = −ln(1 − EOG)

(64)

⎛ E w + ⎜1 − ⎝

1− γ′ =

Ew Ep Ew ⎞ ⎟ Ep ⎠

(65)

E Ep

⎛ E + ⎜1 − ⎝

E⎞ ⎟ Ep ⎠

(66) 9

The results obtained by Garcia and Fair showed that a greater free area and tray spacing resulted in greater capacity and lower efficiency. Garcia and Fair9 also found that a larger hole size resulted in greater efficiency and a greater pressure drop on the plate. The tray spacing was found to have no influence on the pressure

(63) 8328



Review

drop if the operation condition was below the flood point. In addition, the authors observed an increased efficiency in columns of smaller diameter, contrary to observations of the capacity. On the basis of these observations, Garcia and Fair9 suggested that dual-flow trays with free areas between 15 and 20%, 12.7 mm hole diameters, and tray spacings from 0.5 to 0.7 m were more economically advantageous combinations of designed variables for this tray. The last work published about the efficiency of dual-flow trays was presented by Domingues et al.3 The authors developed an efficiency prediction model from the model proposed by Garcia53 cited by Domingues et al.3 The modified model involved replacing the fraction of active holes that are jetting (FJ) with the fraction FJ (FJmodified), as proposed by Prado et al.,54 which uses a 60% value for the fraction of holes that are jetting. In this work, the model from Domingues et al.3 uses the point efficiency (EOG). Considering that the liquid is completely mixed in dual-flow trays,3 resulting in Lewis’s “case 1” for Peclet number equals zero, therefore EOG is the same as Murphree efficiency (EMV).25 From Lewis “case 1”,55 EOG and EMV can be correlated according eq 67. EMV =

exp(λEOG) − 1 λ

and vapor by passing the same hole,21 which does not occur in trays with downcomers. The alternating passage of liquid and vapor through the holes causes an oscillatory movement, with a greater oscillation frequency for smaller diameter holes.21 The approach proposed by Shoukry et al.36 presents an important contribution to the study of dual-flow trays because it allows a differentiated assessment for each hydrodynamic regime in this tray. In addition, precautions should be taken in the evaluation of efficiency through Lewis’s cases because of the instability in the liquid and vapor flow in dual-flow trays.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +55 3220-8691. E-mail: fl[email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support is acknowledged by CAPES (Coordination of Improvement of Higher Education Personnel) and the Ministry of Science, Technology and Innovation (MCTI).

■

(67)

In addition, the model from Domingues et al.3 was compared with experimental data from two distillation columns operating with hydrocarbons (a butene-1 fractionator and a propylene fractionator) using dual-flow trays. The data predicted by the modified model of Domingues et al.3 showed an improved correlation than did the model presented by Xu et al.23 The approach used by Domingues et al.3 to evaluate the Murphree efficiency is valid only to Lewis’s “case 1”, that is, vapor is completely mixed between trays and the direction of liquid flow is not relevant. According to Weiland,24 the liquid and vapor flow in dual-flow trays is inherently segregated, and this behavior is more pronounced as the higher the column diameter is. Therefore, the Lewis’s “case 1” can be stated with more accuracy only to columns with small diameter. For larger diameters, the relationship between both Murphree and overall efficiency must be evaluated through Lewis’s cases 2 or 3.

3. CONCLUSIONS The majority of models that predict froth layer behavior were originally based on the Azbel theory, excluding the work of Xu et al.,23 which found that the radial gradient of concentration in the liquid and vapor phases is negligible when no channeling occurs. The tests performed to validate these models showed good fit, which was not obtained for mixtures with different behaviors. Similar predictions were found for the pressure drop, capacity, and efficiency. It should be noted that many models ignored or did not disclose the hydrodynamic regime during operation, which affected the application of these models for differentiated operating conditions. In this regard, many correlations were developed from data obtained in tests with inert mixtures (i.e., without mass transfer) at room temperature. The most important consideration found in the works on dual-flow trays refers to the conditions presented by Shoukry et al.36 and Shoukry and Kolár,37 in which the multiplicity of steady states was developed during the operation of this tray type. This condition arises from the competition between liquid 8329

NOMENCLATURE a = effective interfacial area per volume of the column [m2· m−3] aL = effective interfacial area per volume of liquid [m2·m−3] A = area passing liquid (AL) or vapor (AV) [m2] Ah = total hole area [m2] Ac = column cross-section area [m2] b1−4 = coefficients from eq 16 [ - ] b5−7 = coefficients from eq 38 [ - ] c = coefficient in eqs 59 and 60 [ - ] C = plate factor: 1.0 for grid tray; 1.33 for sieve tray [ - ] CD = coefficient of discharge [ - ] CV = orifice coefficient [ - ] C1 = coefficient in eq 59 [ - ] C2 = coefficient in eq 60 [ - ] d = hole diameter [m] dh = hydraulic diameter of the opening [mm] D = coefficient of molecular diffusion [m2·s−1] E = low-load efficiency [ - ] EOG = point efficiency [ - ] EOG = Murphree efficiency [ - ] Ep = peak efficiency [ - ] Ew = efficiency corrected for liquid-in-vapor entrainment [ - ] f = Fanning friction factor F = capacity factor, which is a measure of liquid or vapor kinetic energy (= Uc(ρ)0.5) [Pa−0.5] Fh = capacity factor based on hole velocity, used to correlate the loading point (= UVh(ρV/SG)0.5) [Pa−0.5] FLV = flow parameter Fr = Froude number (Fr = UVc/ghcL) [ - ] Frc = critical Froude number [ - ] FrM = modified Froude number [ - ] g = gravitational constant [m·s−2] Ga = Galilei number (= gh3cL/v2L) [ - ] hcL = holdup [cm] hf = froth height [cm] k = mass transfer coefficient [kmol·m−2·s−1] k′ = mass transfer coefficient [m·s−1] k1 = dry tray coefficient proposed by McAllister et al.,33 [ - ] K1 = constant in eq 27 [ - ] dx.doi.org/10.1021/ie5010543 | Ind. Eng. Chem. Res. 2014, 53, 8323−8331


Review

Subscript

K2−4 = constants in eq 29 [ - ] K5 = constant in eq 57 [ - ] L = mass liquid velocity through holes [kg·s−1·m−3] m = slope of equilibrium line [ - ] M = molar weight [kg·kmol−1] N = number of transfer units [ - ] P0 = pressure at x = 0 in relation to tray [Pa] Pf = pressure drop of froth [Pa] P′f = apparent pressure drop of froth [mmH2O] Pi = pressure below the tray [Pa] PH = pressure at x = H in relation to tray [Pa] Pd = dry pressure drop (= ζd{UVh/(1 − τ)}2ρV/2) [Pa] Pi = vapor pressure below the tray [Pa] Pt = pressure drop across the tray [Pa] Q = volumetric flow rate of liquid (QL) or vapor (QV) [m3· h−1] Sc = Schmidt number (= νL/DA) [ - ] SF = shape factor at flooding [ - ] SG = liquid specific gravity [ - ] Sh = Sherwood number (= kLhcL/DA) [ - ] t = mean residence time [s] t′ = time defined in eq 62 [s] ULc = superficial liquid velocity [m·s−1] ULF = flooding liquid velocity [m·s−1] ULh = liquid velocity through the holes [m·s−1] UVc = superficial vapor velocity [m·s−1] UVF = flooding vapor velocity [m·s−1] UVh = vapor velocity through the holes [m·s−1] Ux = velocity through x fraction at total number of holes [m· s−1] V = mass vapor velocity through holes [kg·s−1·m−3] x = fraction of total holes passing vapor at any instant [ - ] X = x-axis in flooding velocity diagram 0.5 We = Weber number (= hcLρ0.5 V UVρL UL/σ) [ - ] Y = y-axis in flooding velocity diagram; z = distance from the plate [cm] Z = tray spacing [m]

■

air = air b = blocked F = flooding h = hole L = liquid V = vapor

REFERENCES

(1) Caldas, J. N.; Lacerda, A. I.; Veloso, E.; Paschoal, L. C. M. Internos de Torres: Pratos E Recheios; Interciência: Rio de Janeiro, 2007; p 510. (2) Rousseau, R. W. Handbook of Separation Process Technology; John Wiley & Sons: New York, 1987; p 1048. (3) Domingues, T. L.; Secchi, A. R.; Mendes, T. F. Overall Efficiency Evaluation of Commercial Distillation Columns with Valve and Dualflow Trays. AIChE J. 2010, 56, 2323. (4) Van Winkle, M. Distillation; McGraw-Hill: New York, 1967; p 684. (5) Takahashi, T.; Akagi, Y.; Fujita, K. Flooding Velocities of Plate Columns without Downcomer. J. Chem. Eng. Jpn. 1973, 6, 97. (6) Walas, S. M. Chemical Process Equipment - Selection and Design; Butterworth-Heinemann: Newton, MA, 1990; p 782. (7) Wright, F. B. A Practical Handbook on the Distillation of Alcohol from Farm Products, 2nd ed.; Camelot Press: New York, 1907; p 293. (8) Rylek, M.; Standart, G. Studies on Distillation. IV. Hydraulics of Plates without Downcomers and Dry Plate Pressure Drop. Collect. Czech. Chem. Commun. 1965, 30, 1041. (9) Garcia, J. A.; Fair, J. R. Distillation Sieve Trays without Downcomers: Prediction of Performance Characteristics. Ind. Eng. Chem. Res. 2002, 41, 1632. (10) Hutchinson, M. H.; Baddour, R. F. Ripple Trays: A New Tool for Vapor-Liquid Contacting. Chem. Eng. Prog. 1956, 52, 503. (11) Badínský, M.; Rylek, M. Dry Pressure Drops of Ripple and Kittel Plates. Collect. Czech. Chem. Commun. 1969, 34, 3156. (12) Rylek, M.; Kastánek, F. Pressure Drop of Foam Bed on Grid Trays. Collect. Czech. Chem. Commun. 1972, 37, 254. (13) Kraft, W. W. Vacuum Distillation of Petroleum Residues. Ind. Eng. Chem. 1948, 40, 807. (14) Aérov, M. É.; Bystrova, T. A.; Berezhnaya, K. P.; Koltunova, L. N.; Poain, L. S.; Darovskikh, E. P. Hidraulics and Mass Transfer on Valve Trays without Downcomers. Chem. Technol. Fuels Oils 1969, 5, 49. (15) Billet, R. Separation Tray without Downcomers. Chem. Eng. Technol. 2001, 24, 1103. (16) Xu, Z. P.; Afacan, A.; Chuang, K. T. Prediction of Packed Sieve Tray Efficiency in Distillation. Chem. Eng. Res. Des. 1996, 74, 893. (17) Karchur, J. P.; Afacan, A.; Chuang, K. T. Use of Structured Packings as a Dualflow Tray in Distillation. Chem. Eng. Res. Des. 2004, 82, 813. (18) Kister, H. Z. Distillation Design; McGraw-Hill: New York, 1992; p 710. (19) Wankat, P. C. Equilibrium Staged Separations; Prentice Hall: NJ, 1988; p 707. (20) Trambouze, P. Materials and Equipment; Petroleum Refining, Vol. 4Edition TECHNIP: Paris, 1999; p 754. (21) Furzer, I. A. Froth Heights on Dualflow Trays with a Ternary Azeotropic System of Ethyl Acetate-Etahnol-Water. Ind. Eng. Chem. Res. 2000, 39, 1430. (22) SUPERFRAC High Performance Trays; Koch-Glitsh: Wichita, KS, 2013; p 16; http://www.koch-glitsch.com/Document%20Library/ SUPERFRAC.pdf (23) Xu, P. Z.; Afacan, A.; Chuang, K. T. Efficiency of Dualflow Trays in Distillation. Can. J. Chem. Eng. 1994, 72, 607. (24) Weiland, R. H. Hydraulic Stability of Dualflow Trays. In AIChE Spring National Meeting. New Frontiers in High Capacity Tray Technology, Houston, TX, 2001; p 12.

Greek letters

β1 = amplitude of waves formed on the tray [ - ] β2 = aeration term [ - ] γ = specific gravity [ - ] δ = tray thickness [m] ζ = dry plate resistance coefficient [ - ] ζd = resistance coefficient due to vapor (= k[{1 − τ(1 − τL)}2 + 0.4{1.25 − τ(1 − τL)}] [ - ] ζL = resistance coefficient due to liquid [ - ] ζσ = resistance coefficient due to superficial tension [ - ] λ = striping factor (ratio of slope of equilibrium line to slope of operating line) [= mV/L] μ = viscosity [kg·m−1·h−1] ν = kinematic viscosity [m2·s−1] ρ = density [kg·m−3] σ = superficial tension [kg·h−2] τ = tray free area [-] τL = ratio of hole area of dropping liquid flow per total hole area of a sieve tray [-] τV = ratio of hole area of dropping vapor flow per total hole area of a sieve tray [-] φ′ = tray free area occupied by liquid [ - ] ϕV = vertical distribution of the vapor void fraction ψ = mean liquid fraction on the tray (= hcl/hf) [ - ] 8330



Review

(25) Lockett, M. J. Distillation Trays Fundamentals; Cambridge University Press: New York, 2009; p 226. (26) Eduljee, H. E. Hydraulics of Grid and Sieve Type Shower Plates. Br. Chem. Eng. 1966, 11, 1519. (27) Kolár, V. The Structure of Gas-Liquid Mixtures on Sieve Trays of Separation Columns. Chem. Eng. Sci. 1969, 24, 1285. (28) Cervenka, J.; Kolár, V. The Structure and Heigh of the GaseousLiquid Mixture on Sieve Plates without Downcomers. Chem. Eng. J. 1973, 6, 45. (29) Takahashi, T.; Matsuno, R.; Miyahara, T. Theoretical Consideration of Gas Void Fraction and Froth Height on a Perforated Plate. J. Chem. Eng. Jpn. 1973, 6, 38. (30) Azbel, D. S. The Hydrodynamics of Bubbler Processes. Int. Chem. Eng. 1963, 3, 319. (31) Takahashi, T.; Miyahara, T.; Shimizu, K. Experimental Studies of Gas Void Fraction and Froth Height on a Perforated Plate. J. Chem. Eng. Jpn. 1974, 7, 75. (32) Furzer, I. A. Froth Heights on Dual-Flow Trays with a Heterogeneous Binary Azeotropic System and a Heterogeneous Ternary System with a Homogeneous Azeotrope. Ind. Eng. Chem. Res. 2001, 40, 4951. (33) McAllister, R. A.; McGinnis, P. H., Jr.; Plank, C. A. PerforatedPlate Performance. Chem. Eng. Sci. 1958, 9, 25. (34) Steiner, L.; Standart, G. Studies on Distillation. XV. Liquid Holdup on Plates without Downcomers. Collect. Czech. Chem. Commun. 1967, 32, 89. (35) Steiner, L.; Standart, G. Studies on Distillation. XVI. Pressure Drop across Plates without Downcomers. Collect. Czech. Chem. Commun. 1967, 32, 101. (36) Shoukry, E.; Cermák, J.; Kolár, V. On the Hydrodynamics of Sieve Plates without Downcomers. I. Dynamic Nature of the GasLiquid Dispersions. Chem. Eng. J. 1974, 8, 27. (37) Shoukry, E.; Kolár, V. On the Hydrodynamics of Sieve Plates without Downcomers. II. A Model of Plate Operation and the Phenomenon of Multiplicity of Steady State. Chem. Eng. J. 1974, 8, 41. (38) Mikhailenko, G. G.; Bol’shakov, A. G.; Ennan, A. A.; Golivets, G. I. Investigation of the Resistance of Perforated Dual-Flow Trays at High Loadings. Int. Chem. Eng. 1974, 14, 641. (39) Matsumoto, S.; Suzuki, M. Statistical Analysis of Fluctuations of Froth Pressure on Perforated Plates without Downcomers. Int. J. Multiphase Flow 1984, 10, 217. (40) Takahashi, T.; Miyahara, T.; Tano, T.; Akagi, Y. Hydrodynamic Characteristics of Sieve Trays Having Large Free Areas without Downcomers. J. Chem. Eng. Jpn. 1986, 19, 339. (41) Fair, J. R. How to Predict Sieve Tray Entrainment and Flooding. Petro/Chem. Eng. 1961, 33, 45. (42) Treybal, R. Mass Transfer Operations; McGraw-Hill: New York, 1968; p 784. (43) Zenz, F. A.; Eckert, R. A. New Chart for Packed Tower Flooding. Pet. Refin. 1961, 40, 130. (44) Sherwood, T. K.; Shipley, G. H.; Holloway, F. A. L. Flooding Velocities in Packed Columns. Ind. Eng. Chem. 1938, 30, 765. (45) Wallis, G. B.; Takahashi, T.; Akagi, Y. Comments on the Paper “Flooding Velocities of Gas-Liquid and Liquid-Liquid Contactors Os Column Type”. J. Chem. Eng. Jpn. 1975, 8, 341. (46) Sharma, M. M.; Gupta, R. K. Mass Transfer Characteristics of Plate Columns without Downcomer. Trans. Inst. Chem. Eng. 1967, 45, 169. (47) Miyahara, T.; Kurihara, M.; Asoda, M.; Takahashi, T. GasLiquid Interfacial Area and Liquid-Phase Mass Transfer Coefficient in Sieve Plate Columns without Downcomer Operating at High Gas Velocities. J. Chem. Eng. Jpn. 1990, 23, 280. (48) Kastánek, F.; Rylek, M. Turbogrid Trays Efficiencies. Collect. Czech. Chem. Commun. 1970, 35, 3367. (49) Koltunova, L. N.; Bystrova, T. A.; Aérov, M. É.; Sobolev, O. B.; Krushnevich, T. N.; Rashevskii, A. A.; Dereveneva, I. P. Calculation of Efficiency for Trays without Downcomers. Chem. Technol. Fuels Oils 1973, 9, 853.

(50) Garcia, J. A.; Fair, J. R. A Fundamental Model for the Prediction of Distillation Sieve Tray Efficiency. 1. Database Development. Ind. Eng. Chem. Res. 2000, 39, 1809. (51) 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, 1818. (52) Jiang, B.; Liu, P.; Zhang, L.; Sun, Y.; Wang, H.; Liu, Y.; Fang, Z. Hydrodynamics and Mass-Transfer Analysis of a Distillation Ripple Tray by Computational Fluid Dynamics Simulation. Ind. Eng. Chem. Res. 2013, 52, 17618. (53) Garcia-Martinez, J. A. Fundamental Model for the Prediction of Distillation Sieve Tray Efficiency: Hydrocarbon and Aqueous Systems; The University of Texas: Austin, TX, 1999. (54) Prado, M.; Johnson, K.; Fair, J. Bubble-to-Spray Transition on Sieve Trays. Chem. Eng. Prog. 1987, 83, 32. (55) Lewis, W. K., Jr. Rectification of Binary Mixtures. Ind. Eng. Chem. 1936, 28, 399.

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