Model-Based Approach To Predict and Control Hydraulic Gradient on

Article pubs.acs.org/IECR

Model-Based Approach To Predict and Control Hydraulic Gradient on Slotted Sieve Trays X. Q. Hu,† L. T. Fan,† X. G. Yuan,*,† K. T. Yu,† A. W. Zeng,† and M. A. Kalbassi‡ †

State Key Laboratory of Chemical Engineering and School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China ‡ Air Products PLC, Hersham Place Technology Park, Molesey Road, Walton-on-Thames, Surrey KT12 4RZ, United Kingdom ABSTRACT: The hydraulic gradients of the small-hole sieve trays with and without slots were experimentally determined. On the basis of the experimental results, a semitheoretical two-layer model was proposed to describe the gradient of clear liquid height on these trays. The model takes into account both the resistance and the driven forces of gases coming from the sieve holes and the slots, respectively. Two model parameters were identified by comparing the calculated results with the experimental data. By applying the model, the open area ratio of the slots on the sieve trays was optimized to give a minimum clear liquid height gradient for various sets of values of design and operating parameters, and the optimum slot open area ratio was correlated with the parameters. The approach developed in this paper can help to manipulate the hydraulic gradient by changing the slot open area ratio.

1. INTRODUCTION The sieve tray is one of the most extensively used vapor−liquid contactors in distillation and absorption operations because of its simplicity and low cost,1 and the efficiency of a sieve tray column is of great importance. Among the factors that influence the sieve tray performance, the clear liquid height (hc) and the hydraulic gradient play significant roles because they not only affect the gas−liquid interfacial area, the two-phase contact time, but also the stage efficiency.2−4 A large number of experimental measurements have been carried out to investigate the clear liquid heights,5−9 and many empirical correlations based on these measurements have been proposed.1,7,10−14 In addition, there are also much literature about the application of computational fluid dynamics (CFD) on prediction of clear liquid height of the sieve tray15−23 as well as attempts of mechanistic analysis for the froth height which is mostly based on a force balance of the droplets in the vertical direction.24 However, less attention has been paid to the hydraulic gradient, which is claimed harmful to the operation and may result in a serious decline in sieve tray efficiency, especially for trays with large diameters.25 The reason for this lies in the fact that the gas tends to flow through the lower liquid height area due to the smaller resistance, whereas at the higher liquid height area the liquid is easy to weep. As such, only a portion of the tray area is working under the desired condition, and the tray efficiency will surely be decreased. For larger diameter sieve trays, because of the larger difference in liquid height, the decreasing tray efficiency may be more severe. In the literature concerning the hydraulic gradient, Hucks and Thomson26 and Fan27,28 experimentally showed that the hydraulic gradient could be appreciable in some conditions. McDermott29 developed a numerical solution, which also demonstrated the hydraulic gradient on the sieve tray. The slotted sieve tray is one of the means to reduce the hydraulic gradient.30 On such a tray, angled slots punched at © 2014 American Chemical Society

various positions among the sieve holes deflect some of the vapor to acquire horizontal momentum to drive the liquid flow, so that a lower and more uniform froth height could be maintained.30 This kind of tray has been proved to be suitable for tray columns with large diameters and, more importantly, can be installed with rather low tray spacing.31 In such a manner, higher column efficiency can be obtained with the same column height. However, to what extent a ratio of slots can reduce the hydraulic gradient is still unknown. The small sieve-hole trays are widely used in cryogenic air separation service where the surface tension of the liquid is extremely low. However, data of the clear liquid height and the hydraulic gradient for small hole trays have been rarely reported.27 The aim of this paper is to experimentally investigate the hc and hydraulic gradient on a rectangular test tray with small sieve holes under different slots open ratio, including zero. On the basis of the experimental results, a semitheoretical model is proposed to predict the distribution profile of hc along the liquid flow direction on the tray as a function of slots ratio, outlet weir height, and flow rates of liquid and gas. Then, the best slots ratio minimizing the hydraulic gradient and realizing a uniform froth flow is calculated. As a rectangular test tray is employed in this paper, the weir constriction effect and liquid nonuniform flow on a round-shaped tray can be eliminated, so that the results are beneficial for understanding the basic hydraulic behavior and providing a building block to a sound approach to the design of efficient trays. Received: Revised: Accepted: Published: 4940

November 15, 2013 January 30, 2014 February 27, 2014 February 27, 2014 dx.doi.org/10.1021/ie403883z | Ind. Eng. Chem. Res. 2014, 53, 4940−4952

Industrial & Engineering Chemistry Research

Article

2. EXPERIMENTAL SECTION 2.1. Experimental Installation. Figure 1 shows the apparatus photograph, sketch of the slot, and schematic view

slotted open area to the total open area of the perforations. The size of the slot opening is 1 mm in height and 6 mm in width, with a bulging length of 4 mm. The slots on the tray are in an equilateral triangle distribution. Figure 2 shows the geometric

Figure 2. Geometric structure of the experimental tray.

structure of the test tray. The trays used in this experiment were in a rectangular shape, which was aimed at decreasing the effect of liquid nonuniform flow which occurred on the round-shaped trays on the precision of the measurement. The experimental sieve tray material was aluminum, which was typical of that commonly used in cryogenic distillation for air separation. The geometrical parameters for the test tray shown in Figure 2 are given in Table 1. The diameter of the sieve holes is 1 mm, Table 1. Parameters for Experimental Installation

Figure 1. Schematic view and photograph of experimental apparatus.

of the experimental installation. Air and water were used as the gas and liquid fluids, respectively, in our experiment. Air was introduced in the bottom of the column by a blower. A gas distributor tray was installed below the test tray to acquire a uniform entering gas flow. Water was pumped into the top of the column by a centrifugal pump from a water tank and then introduced on the tray by the inlet downcomer. The lower part of the inlet downcomer was packed with 25 mm Pall Rings in order to distribute uniformly the liquid in the inlet of the tray. The flow rates of the gas and liquid phases were measured by the Pitot tube and vortex flow meter, respectively. The measuring range of the gas kinetic factor (F) based on the bubbling area is from 1.78 to 3.69 kg0.5·s−1·m−0.5. The measuring range of the liquid flow rate per unit weir length (Lw) is from 15 to 45 m2·h−1. Outlet weir heights of 0, 20, 35, and 50 mm were used to investigate the outlet weir height effects. An ordinary sieve tray and slot trays with three different slot ratios were used as the test trays: α = 3.30%, 6.65%, and 10.00%, where the slot ratio α is defined as the ratio of the

parameter

unit

value

tray spacing tray path length weir length bubbling area tray thickness length of inlet calming section length of outlet calming section diameter of sieve holes hole spacing holes pitch ratio of perforation area to total bubbling area inlet downcomer clearance

mm mm mm mm2 mm mm mm mm mm

450 900 600 780 × 600 1 60 60 1 3.2 equilateral triangular 8.856 33

% mm

which is a typical value for a cryogenic distillation tray. As shown in Figures 1a and 2, nine U tubes were used to measure the clear liquid height on the test tray. As probes, the mouths of all the U tubes were arranged evenly on the central line of the test tray along the liquid flow path. A screen mesh at each of the probe mouths and capillary tubes in the U tubes was used to stabilize the measurement readings. The readings of the clear liquid heights in the experiments for various structural and operating conditions were recorded by visual inspection through the U tubes. The recordings took place at 5 min after the levels of the liquid in the U tubes became stable. 2.2. Experimental Results and Discussion. For each of the U tubes, 2 readings were recorded separately for the same experimental condition, and the average was taken as the experimental clear liquid height for the corresponding position on the tray. The standard deviations of the measurements were within 2.0 mm. 4941

dx.doi.org/10.1021/ie403883z | Ind. Eng. Chem. Res. 2014, 53, 4940−4952


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Figure 3. Comparison of fall in hc on sieve trays with and without slots, hw = 0.

The four figures from Figure 3−6 show the variations of hc along the liquid flow path for different values of F, Lw, and slot ratio on the test trays with an outlet weir height of 0, 20, 35, and 50 mm. From Figure 3a we can see that there exists an obvious hc fall in the liquid flow direction on the conventional sieve tray without outlet weir. Figure 3a−d demonstrates that with increasing slot ratio both hc and hc fall decrease. When α = 10.00%, the hc values are almost the same all over the tray. Figures 4, 5, and 6 give the hc profiles for outlet weir height of 20, 35, and 50 mm, respectively. It can be seen from these figures that, first, under the same slot ratio, with increasing outlet weir height from 20 to 50 mm, the hc fall became less serious but still kept similar trends of variations with respect to other parameters, compared to the results without outlet weir shown in Figure 3. Second, a negative hc gradient, which meant that hc at the tray inlet was smaller than that at the outlet, appeared at α = 6.65% for all three weir heights. This indicates that the slot generated horizontal momentum to the liquid, which was determined by the slot ratio, can effectively reduce the hydraulic gradient, and at the same time on optimal value for the slot ratio, which gives zero hydraulic gradient, should exist. At this optimal slot ratio, the driven force coming from liquid height gradient is completely replaced by the horizontal momentum, and beyond the optimum, the excessive horizontal momentum may cause a negative gradient. Because for the case of α = 6.65% the hc differences appear minus the value for all weir heights, by consulting Figure 5d, it could be expected that for α = 10.00% the hc differences will also have a negative value,

so we did not measure the entire outlet weir height range; instead, we only measured the 35 mm case as a representative.

3. HYDRAULIC GRADIENT MODELING 3.1. Momentum Balance Analysis. On the basis of previous experimental results, a semitheoretical two-liquid-layer model is established to describe the hc as well as the resistance of the liquid on the sieve tray. First, the model proposed by Syeda et al.32 for the gas−liquid two-phase flow on sieve trays is adopted. Syeda et al.32 identified four regimes for the two-phase flow on a sieve tray: jets, bubbles, liquid splashes, and drops, which are schematically shown in Figure 7. Second, the space above the sieve tray was divided into the four corresponding layers as shown in Figure 7. Different from the drop and splashes layers where the gas is the continuous phase, in the jets and bubbles layers the liquid is continuous phase and, thus, we assume that the clear liquid height and its gradient are determined by the gas−liquid interactions in these two layers. In jets and bubbles layers, the horizontal momentum balance of the liquid per unit volume can be given by ρL g

dhc du + αFs = ρL u + (1 − α) ∑ hf ds ds

(1)

where α is equal to zero for a conventional sieve tray. The first term on the left-hand side of eq 1 represents the momentum generated by the hydraulic gradient and the second term the momentum contributed by the horizontal component of the gas flow through the slots. On the right-hand side of eq 1, the 4942



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Figure 4. Comparison of fall in hc on sieve trays with and without slots, hw = 20 mm.

influence on the interaction between the liquid flow and the gas jets. For this purpose, we assumed C0 = 1.1Cp, where Cp = 1 for the case of liquid flow around one cylinder and should be experimentally determined, as will be discussed in section 3.3, for our case of multijets with slots. The jet height h in eq 1 can be calculated by35

first term represents the horizontal momentum increase of liquid and the last term the momentum lost of the liquid due to the resistance of the gas flow through the sieve holes. By assuming that the momentum of the horizontal gas flow from the slots is totally transformed into that of the liquid, the force generated by the gas flow represented by Fs in the source term can be given by eq 2 Fs =

h = 1.1 × 10−3D0.20Reh 0.46

ρG UG(us − u) hc

To apply eq 4 we must assume that the increase in α had no influence on h. In the bubbles layer, the interaction between liquid and gas should include the lift force, virtual mass force, and drag force.15,36−38 For gas−liquid two-phase flow on distillation trays, Zhang and Yu38 and Wang et al.19 summarized the resistance forces of gas bubbles as the resistance due to the momentum exchange between the rising gas and the horizontally flowing liquid, which is given in eq 5

(2)

Use of eq 2 implies that the value of the horizontal gas velocity us through the slots is assumed to be the same as that of the gas uh through the sieve holes. The force ∑hf in the last source term in eq 1 represents the resistance of the gas to the liquid, which should be estimated differently for the jets layer and the bubbles layer. In the jets layer, the resistance can be regarded as the result of liquid flow around the cylinders of the gas jets and can be calculated by eq 333 Fp =

Fm =

1 C ρ u2 2 0 L

h

(4)

CmρG UGu hc

(5)

where Cm is the momentum exchange coefficient and was found to be 0.8.39 In eq 1, if hc < h, only the jet resistance Fp is considered, that is, ∑hf = Fp; the final expression is given in eq 6a. For hc > h, both the jet resistance Fp and the bubble resistance Fm should be considered, that is, ∑hf = Fp + Fm; eq 1 becomes eq 6b. In eq 1, the liquid velocity u is equal to Lw divided by hc; the

(3)

where C0 is a constant that has been estimated to be 1.134 for the resistance of liquid flow around one cylinder but should be experimentally modified for our cases of multicylinder configuration as shown in Figure 7. Additionally, the horizontal momentum changed by gas through the slots may have a strong 4943



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outlet then we assume that the hydraulic drop condition holds, that is

derivation from eq 1 to eqs 6a and 6b is a simple mathematical transformation. Derivation of eq 6a from eq 1 is given in the Appendix, and derivation of eq 6b can follow a similar procedure

ϕeLw 2 u f0 2 Fr = C hFrf0 = C h = Ch =1 gHf0 ghc0 3

⎡1 dhc αGuh 2 ⎤ αG hc − hc ⎥ = ⎢ C0S(1 − α) + dx L LLw ⎦ ⎣2 ⎞ ⎛ g /⎜ 2 hc 3 + 1⎟ ⎠ ⎝ Lw

where Ch is a general correction factor that we introduce to correct the hydraulic drop condition for the froth flow. For liquid flow, Ch = 1; for froth flow, Ch should be experimentally correlated, as discussed in section 3.3. With eq 7 hc at the outlet can be estimated as

(6a)

⎡ C S(1 − α) dhc C G(1 − α) αG hc + hc + m hc =⎢ 0 dx 2h L L ⎣ −

⎞ αGuh 2 ⎤ ⎛ g 3 hc ⎥ /⎜ 2 hc + 1⎟ LLw ⎦ ⎝ Lw ⎠

(7)

⎧ 2 ⎞1/3 ⎛ ⎪ 1/3⎜ ϕeLw ⎟ without outlet weir ⎪C h ⎜ g ⎟ ⎝ ⎠ ⎪ hc0 = ⎨ ⎪ ⎛ ϕ L 2 ⎞1/3 ⎪ ϕ h + C 1/3⎜ e w ⎟ with outlet weir h ⎜ g ⎟ ⎪ e w ⎝ ⎠ ⎩

(6b)

By eqs 6a and 6b, the distribution profile of hc along the liquid flowing path for a given column dimension is represented as a function of the operating conditions like liquid and gas flow rates and also of the slot’s open area ratio. The solutions of eqs 6a and 6b require a boundary condition, which will be given in the next section. 3.2. Hydraulic Drop Condition for Froth Flow. In an open channel flow, the liquid flow at the outlet without a weir can be regarded as hydraulic drop, the Froude number (Fr) of which is equal to 1.40 For our case, we assume that this is also true for the froth flow at the outlet of a tray without outlet weir. However, this time the Froude number should be corrected. If Frf0 is defined as the Froude number for the froth flow at the

(8)

Then, eq 8 can be used as a boundary condition for solving eqs 6a and 6b. 3.3. Experimental Correlation. Two model parameters, Cp and Ch, are introduced in the model given by eqs 6a and 6b and thus need to be experimentally correlated. Note that these two parameters can hardly be constant and should be influenced by both operating conditions (F and Lw) and design parameters (hw and α). Such influential factors complicate the correlation. In the present work, we assume that both 4944



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by solving eqs 6a and 6b. Thus, parameters Cp and Ch can be optimized by a criterion given by ∑i(hc,the,i − hc,exp,i)2, where i = 1−9 refers to the different measurement positions for the clear liquid height and hc,exp,i is the experimentally measured clear liquid height at the measuring position i. By some trials we find that suitable searching spaces for these two parameters are from 0 to 0.5 with an interval of 0.005 for Cp and from 0 to 3 with an interval of 0.01 for Ch. Thus, we have 30 401 combinations of values of Cp and Ch within the searching domain. At the same time, the total numbers of experimental data available given in Figures 3−6 is 225, and for each of them 30 401 estimations of hc are envisaged to find one pair of optimal values for Cp and Ch. It means that we have to solve eqs 6a and 6b 30 401 × 225 times for all combinations of Cp and Ch and for all operating and design conditions used in the experiments. For such a large number of combinations, a searching strategy (for example, simulated annealing or others) could be implemented to reduce the number of calls of eqs 6a and 6b. In the present work, however, the enumeration method, calculation of all combinations, is chosen to search for the best Cp and Ch. The reason we chose such a method lies in that, first, eqs 6a and 6b are easy to solve and the total time for all enumerating computing was in an order of several hours, which is regarded as acceptable in our work; second, enumeration calculation can guarantee a global optimum for parameters Cp and Ch. The MATLAB software was used in the enumerating computing. 3.3.2. Data Fitting for Correlations for Cp and Ch. Data fitting is to create quadratic polynomial functions that correlate the data of Cp and Ch generated in section 3.3.1 with

Figure 7. Schematic representation of two-phase distribution on the sieve tray.

parameters are related to all operating conditions (F and Lw) and design parameters (hw and α), that is Cp = Cp(F , Lw , hw , α)

(9a)

C h = C h(F , Lw , hw , α)

(9b)

Then the correlation was performed in two steps: first, data of Cp and Ch were generated by solving eqs 6a and 6b and correlation with the available experimental data to establish relations 9a and 9b digitally; then the numerical relations of eqs 9a and 9b were fitted with quadratic polynomials. 3.3.1. Data Generation for Correlation. We know that under a given set of operating conditions (F and Lw) and design parameters (hw and α) if Cp and Ch are given then predicted clear liquid heights hc,theo,i for every location corresponding to the measurement points as shown in Figure 1 can be calculated 4945



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parameters F, Lw, hw, and α. The Stat-Studio software was used in this correlation work. Our trials indicated that, in order to obtain an adequate function for Cp, the tray without outlet weir and that with the weir should be correlated separately. Then the resulted correlations are given by eq 10a for the tray without outlet weir and eq 10b for that with outlet weir. Cp = 0.0861 + 0.0032Lw − 0.047F + 2.838α − 0.0006Lw F − 0.0402Lw α − 0.1891Fα + 0.0071F 2 − 21.42α 2

(10a)

Cp = 0.1723 − 0.0801F + 1.143α + 0.0076hw + 0.001Lw F − 0.021Lw α − 9.82e − 5Lw hw − 0.0045Fhw + 0.0417αhw − 5.46e − 5Lw 2 + 0.0227F 2 − 27.55α 2 + 1.493e − 4hw 2

Figure 9. Correlation equation calculated Ch against experimental fitting value.

(10b)

A comparison between the experimentally correlated values generated in section 3.3.1 and the calculated values by correlations 10a and 10b for parameter Cp is shown in Figure 8. As can be seen from this figure, the correlations can fit adequately to the data.

C h = −1.384 + 0.0343Lw + 0.1226F + 8.352α + 0.1144hw − 0.4381Lw α + 0.0008Lw hw − 0.0273Fhw + 0.1112αhw − 0.0007Lw 2 + 93.41α 2 − 0.0008hw 2

(11b)

It should be noted that eqs 10a, 10b, 11a, and 11b were all fitted with limited experimental data, so in the application of the correlations care should be made about the extents with which the correlations remain valid. The value ranges of the parameters used to produce the correlations are summarized in Table 2 and should be regarded as the suggested applicable Table 2. Parameters’ Value Ranges for Correlating eqs 10a, 10b, 11a, and 11b α

hw, mm

Lw, m2·h−1

F, kg0.5·s−1· m−0.5

10a and 11a 11b and 11b

0−0.1 0−0.1

0 20−50

15−45 15−45

1.78−3.38 1.78−3.74

extents for the correlations. Also, in our experiments air and water were used as the working fluids. For other experimental systems, the physical properties such as density and viscosity may have an influence on both parameters.

Figure 8. Correlation equation calculated Cp against experimental fitting value.

4. MODEL VALIDATION In this section the models given by eqs 6a and 6b developed in the previous section are validated by comparing the calculated hc values with the experimentally measured ones. The same experimental data reported in section 2 were used here for the validations. 4.1. Comparisons for Conventional Sieve Tray (α = 0). 4.1.1. Without Outlet Weir and α = 0. A comparison of hc for conventional sieve trays (α = 0) without outlet weir is shown in Figure 10. As can be seen from these two figures, the modelpredicted hc values agreed generally well with the experimentally measured ones for various operational conditions. The two sets of data suited typically well approaching the outlet. At the liquid inlet side, unstable experimental measurement was with some measured hc much higher than predictions for a higher liquid−gas flow ratio (Figure 10a). An explanation

The parameter Ch for the hydraulic drop condition for the froth flow was shown to be most sensitive to the height of outlet weir in our trials. For the cases without a weir, Ch turned out to be a constant of 0.001 for all of our experimental conditions, and so, correlation 11a was obtained. The correlation of Ch for the tray with an outlet weir is given by eq 11b. A comparison between the experimentally correlated values generated in section 3.3.1 and the calculated values by correlations 11a and 11b for parameter Ch is shown in Figure 9. As can be seen from this figure, the errors for most predicted values by the correlations were within 30%.

C h = 0.001

parameter equation

(11a) 4946



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Figure 10. Comparisons of the predicted hc with experimental data for different operating conditions for tray without slot (α = 0) and outlet weir.

the hc measurement near the tray inlet may not be accurate because of the effect of an additional portion of the dynamic head of the vertical liquid flow. Such an unstable reading was smoothed out by increasing the gas flow by comparing Figure 10a and 10b. Even with the uncertainty, the model could give satisfactory predictions for most of the measurements, with an average relative error of 16.73%. 4.1.2. With Outlet Weir and α = 0. A comparison between the predicted hc and the experimental measurements for a conventional sieve tray with outlet weir is shown in Figure 12. An outlet height of 35 mm is used as an example to show this comparison. As can be seen from these two figures, the weir simply generated higher hc but with lower hydraulic gradient comparing with the cases without an outlet weir. In addition, the average relative error of the predicted hc was 10.39%. As expected, the gas flow rate could increase the froth height but decrease hc as demonstrated by the experimental data in Figure 12. Such a typical fluid dynamic behavior of the froth flow on a sieve tray was captured by the proposed model. It is noted that an increase of 36% in the gas flow rate could lower hc even with a doubled liquid flow rate, as can be seen by comparing Figure 12a and 12b.

is that the liquid pouring out from the bottom of the inlet downcomer complicated the local liquid flow by generating a circulating zone,21,22 as shown schematically in Figure 11. In

Figure 11. Schematic diagram of liquid flow directions at different positions on the tray.

the region near the inlet downcomer there could be vertical velocity components due to the circulation of the liquid. Thus,

Figure 12. Comparisons of predicted hc with experimental data for different operating conditions for a tray without slot (α = 0) but with outlet weir. 4947



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Figure 13. Comparisons of predicted hc with experimental data for different operating conditions for a tray with slots (α > 0) but without outlet weir.

4.2.2. With Outlet Weir and α > 0. A comparison of the predicted hc with the experimental results for a slotted sieve tray with an outlet weir is shown in Figure 14. A 35 mm high outlet weir is used as an example to show this comparison. From Figure 14a, 14b, and 14c, we can see that under the same gas and liquid flow rate and outlet weir height with increasing slot ratio the gradient of hc in the liquid flow direction was decreased. Moreover, even a negative hc gradient appeared for a large slot ratio, as shown in Figure 14b and 14c. The results shown in these figures indicate that horizontal momentum produced by the gas through the slots did affect the hydraulic gradient and so the uniformity of the froth flow on the tray. In addition, such an influence of α on hc and its gradient was captured well by the model. Again, the unstable hc readings near the liquid inlet also happened for this case, an explanation of which has been discussed in section 4.1. For the prediction of hc by the model, the average relative error is 8.68%.

4.2. Comparisons for Slotted Sieve Tray (α > 0). 4.2.1. Without Outlet Weir and α > 0. The validations of the model for a slotted sieve tray (α > 0) without outlet weir are shown in Figure 13. As can be seen from these figures, the model predicted h c values were consistent with the experimentally measured data approaching the outlet, whereas near the inlet some measured hc values were much higher than the predicted ones. A possible explanation for this phenomenon could be the same with the cases of a conventional sieve tray without outlet weir which has been discussed in section 4.1. By comparing Figure 13a, 13b, and 13c it can be seen that under the same gas and liquid flow rate hc decreased with increasing slot ratio. This is mainly because the gas flow direction when flowing through the slot is deflected from vertical to horizontal due to the shape of the slot (please see Figure 1b); thus, more slots would provide larger horizontal momentum, which increased the liquid velocity and thus decreased hc. This dynamic behavior was successfully predicted by the proposed model. For this case, the average relative error of the predicted hc is 13.74%. 4948



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Figure 14. Comparisons of predicted hc with experimental data for different operating conditions for tray with slots (α > 0) and outlet weir.

5. IDENTIFICATION OF THE BEST SLOT RATIO Our experiments and model predictions demonstrated that the slot ratio was relevant to the hydraulic gradient on the test tray. Therefore, we define here αbest as the best value of α that gives hc at the inlet equal to that at the outlet. Note that the hydraulic gradient is also a function of parameters F, Lw, and hw. Thus, αbest should also depend on these parameters. Figures 15 and 16 give simple dependences of αbest on F, Lw, and hw separately based on a interpolation of the experimental data. As expected, the increasing gas flow rate could smooth out the hydraulic gradient, and as a result, the slot ratio needed to keep the clear liquid flat can be decreased. In addition, the increasing liquid flow rate increased not surprisingly αbest. However, Figure 15 implied that if the liquid flow rate was higher than a critical value (about 35 m2·h−1 in the case of Figure 15), the hydraulic gradient, or the slot needed to maintain small gradient, would be reduced. A comprehensive investigation should be needed to explain this, but a qualitative explanation could be that the hydraulic drop induced by a higher crest over weir dominated the driven force

Figure 15. Variation of αbest with F and Lw. 4949



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Figure 16. Variation of αbest with hw.

for froth flow. Figure 16 gives the variation of αbest with hw. It can be seen that αbest decreased fast at lower hw before it became flat for hw higher than 20 mm. An explanation is that, for such a large tray, with increasing hw the froth velocity gradient in the vertical direction is small; as a result, with increasing liquid residence time, froth flow was slowed down and led to a lower hydraulic gradient; then less slots were needed to drive the clear liquid flat. However, when hw is increasd to a critical value, the froth velocity profile in the vertical direction would be independent to hw. On the basis of the foregoing analysis, αbest can be correlated with F, Lw, and hw, and the resulting correlation is given in eq 12.

Figure 17. Correlation equation calculated αbest against experimental fitting value.

αbest = exp(2.254 × 10−2Lw − 3.153 × 10−4Lw 2 − 0.3764F + 2.602 × 10−2F 2 − 4.104 × 10−2hw + 5.807 × 10−4hw 2 − 2.110)

(12)

Note that the application range of the parameters in correlation 12 should be the same as that shown in Table 2, with the slot sizes shown in Figure 1b. The result shows that within the operating condition for eqs 11a and 11b the best α ranges from 3.30% to 9.40%. Figure 17 gives a comparison between the correlation calculated αbest and the experimental fitting results. As can be seen from this figure, the error for most predicted values by the correlation was within 14%. Figure 18 gives the effect of α on the hc difference between the tray inlet and outlet (Δhc) for particular operating conditions and weir height. It can be seen that with increasing α the difference between hc at the inlet and that at the outlet turned from positive to negative. The approximate αbest corresponding to zero Δhc is 0.041, which is shown in Figure 18. The corresponding αbest estimated by eqs 11a and 11b is 0.037, which should be a good approximation.

Figure 18. Effect of α on the hc difference between the tray inlet and outlet (Δhc).

and shown to be able to describe hc on the conventional sieve tray and slotted sieve trays. Because the interactions between liquid and gas on the trays complex and depend on operating and structural parameters, two correcting factors were needed to introduce into the model. Models for correlating the two parameters with the experimental data were developed with adequate accuracy. The models developed in this paper were used to estimate the best slot ratio to minimize the hydraulic gradient, and a correlation was also produced to calculate such best slots. The work in this paper demonstrated that modeling the behaviors of sieve trays with slots and their effects on the hydraulic gradient could be achieved by incorporating the experimentally correlated factors embedding the complexities into theoretical model, and such a theoretical−empirical approach remains useful to characterize the complex transport phenomena on distillation trays. It should be noted again that, first, the experimental data used to produce the model were obtained from experiments

6. CONCLUSION AND REMARK Experiments were conducted to investigate the hydraulic gradient for small hole trays under different slots open ratio on a rectangular tray. The experimental results showed that the slot ratio determined effectively the hydraulic gradient on the tray. On the basis of the experimental results, a onedimensional semitheoretical model was developed to describe 4950



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C0 = constant, 1.1, dimensionless D = hole diameter, mm F = gas kinetic factor based on the bubble area (F = UG(ρG)0.5), kg0.5·s−1·m−0.5 Fm = liquid flow resistance per unit volume in the bubbles layer, kg·m−2·s−2 Fp = liquid flow resistance per unit volume in the jets layer, kg·m−2·s−2 Fr = Froude number, dimensionless Fs = slots gas driving force per unit volume, kg·m−2·s−2 g = acceleration of gravity, 9.8 m2·s−1 G = mass flow rate of the gas phase, kg·s−1 h = jet height, m hc = clear liquid height, m hc,exp = experimental clear liquid height, m hc,the = theoretical clear liquid height, m hf = liquid flow resistance per unit liquid volume, kg·m−2·s−2 Hf = froth height, m hw = outlet weir height, mm L = mass flow rate of liquid phase, kg·s−1 Lw = liquid volume flow rate per unit weir length, m2·h−1 Reh = gas hole Reynolds number, dimensionless s = distance from outlet weir, m S = length of liquid flow path, m UG = gas velocity based on bubbling area, m·s−1 uh = gas hole velocity, m·s−1 us = gas slot velocity, m·s−1 x = distance from outlet weir, dimensionless

with a rectangular tray. To apply to an industrial round section column, the proposed model needs to be tested and corresponding corrections should be made. Second, the model parameters in this work were all correlated with the air−water system. Although physical properties like liquid and gas densities and liquid viscosity are involved in the model, it needs to be checked before applying to other systems.

■

APPENDIX Derivation of eq 6a from eq 1 ρL g

dhc du + αFs = ρL u + (1 − α) ∑ hf ds ds

(1)

When hc < h we have eq 13

∑ hf

= Fp =

C0ρL u 2 2hc

(13)

We know u = (Lw/hc), thus

L dh du = − w2 c ds hc ds

(14)

Substituting eqs 13 and 14 into eq 1 yields ⎛ gh 3 ⎞ dh αρ uG αρ uGuh 1 ⎜ c2 + 1⎟ c = (1 − α)C0 + G hc − G 2 2 ρL Lw ρL Lw ⎝ Lw ⎠ ds hc 2

(15)

Greek letters

We know SuGρG G = L ρL Lw

α = slots area ratio based on total open area, dimensionless ρ = density, kg·m−3 Φe = froth density, dimensionless

(16)

Subscripts

Substituting eq 16 into eq 15 ⎛ gh 3 ⎞ dh αGuh 2 1 αG ⎜ c2 + 1⎟ c = (1 − α)C0 + hc − hc d 2 LSLw s LS ⎝ Lw ⎠

■

(17)

Defining x = s/S, eq 17 becomes ⎡1 dhc αGuh 2 ⎤ αG hc − hc ⎥ = ⎢ C0S(1 − α) + dx L LLw ⎦ ⎣2

■

⎞ ⎛ g /⎜ 2 hc 3 + 1⎟ ⎠ ⎝ Lw

best = optimum ratio 0 = outlet f = froth G = gas L = liquid

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(6a)

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-22-27404732. Fax: 86-22-27404496. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS The authors acknowledge Professor K. E. Porter for his advice and help in the theoretical aspect of model development. NOMENCLATURE Ch = model parameter, dimensionless Cm = momentum exchange coefficient, dimensionless Cp = model parameter, dimensionless 4951



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Model-Based Approach To Predict and Control Hydraulic Gradient on

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