\;here P is in millimeters of mercury, T is in degrees centigrade, and b l , 62, and 6 3 are constants depending upon the compontm t. Therefore, from Eqiiation 27.
I’Jble I1 contains the constants [hlnent.
b l , 62,
and
63
for each com-
Nomenclature ti
I,
= niimber of components in feed = liquid flow rate from plate X, moles/unit time
XA = liquid composition vector on plate k x A i = I t h component of liquid composition vector on plate k , mole fraction If, = vapor flow rate from plate k , moles/unit time Irk = vapor coinposition vector on plate k y n i = j t h component of vapor composition vector on plate h, mole fraction ni - number of plates in tower counting from bottom D = distillate withdrawal rate, moles/unit time R = bottoms withdrawal rate, moles/unit time I , , = condensate return rate, moles/unit time Y = reflux ratio 0 = heat added F = feed rate, moles/unit time S f = feed composition vector HA = enthalpy of vapor from plate k hi, = enthalpy of liquid from plate k L! = mole fraction of jth species in free energy function n = total moles of all species in each phase c, = standard free energy per mole ofjth species 6, = change in standard free energy o f l t h species from liquid to vapor 4, I- stoichiometric restriction function for zth input species r i = ith Lagrange multiplier ?t I = partial pressure equilibrium function for I t h species a, 8, 7 , 6, 8, g, p = arbitrary proportionality (integration) constants 7 = arbitrary computer time
Literature Cited
(1) Amundson, N. R., Pontinen, .A. S., Znd. Eng. Chem. 50, 730 (1058). (2) Computer Systems, Inc., “Multi-component Distillation,” Tech. Data Sheet 80-105-001. (3) Dantzig, G. B., DeHaven, J. C., et al., “A Mathematical Model of the Human External Respiratory System,” Rand Corp., RM-2519, 1959. (4) deGroot. S. R., “Thermodynamics of Irreversible Processes,” North-Holland, Amsterdam, Interscience, New York, 1952. (5) Deland, E. C . , “Continuous Programming Methods on an Analog Computer,” Rand Corp., P-1815. 1959. (6) Deland, E. C . , “Simulation of a Biological System on an Analog Computer:” Rand Corp., P-2307, 1961. (7) Gyenstadt, J., Bard. Y.: Morse, B., Ind. En,?. Chem. 50, 1644 (19 3 8 ) . (8)‘ KO&,T.. Econometrrca24, 59 (1956). (9) Lystcr, \V. N.. Sullivan, S. I>.,Jr.. et n l . , Petrol. Rejnrr 39, 121 11 \
~
c)m\ ’- - / ’
(10) Marr. G . R., Jr.: “Distillation Column Dynamics. A Suggested Mathematical Model.” Electronics Associates, Inc.; Princeton, Computation Ccnter, Princeton, N. J., 1962. (1 1) Nelson, \+‘. I,.. “Petroleum Eng-ineering.” 3rd ed., McGrawHill, N e w York. 1949. (12) Rijnsdorp, J. E.. Maarlcveld, X., “Proceedings of Joint Symposium on Instrumentation and Computation,” Loddon, 1953. (13) Robinson. C.. Gilliland. E. R., ”Elements of Fractional Distillation,” rev. and rewritten by E. R. Gilliland, 4th ed., McGraw-Hill, New York. 1949. (14) Rossini, F. D.. Pitzer, K. S., et al., “Selected Values of groperties of Hydrocarbons.” Natl. Bur. Std.. Circ. 461 (1947). (15) \$’hit;% \V. B., Johnson. S . M.. Dantzig, G. B:, J . C h m . Phys. 28, 751 (1958). (16) \+’illianis. T. J . . Johnson, C. L., Rose, A.. Ind. Eng. Chem. 4 8 , 1172 (1956). 117) IVolf. M. B.. “Simiilation of Fractional Distillation bv ‘ kithematical Programming,” thrsis, University of California, Los .4nqeles, 1962. (18) IVorley, C. I+’., “Application of Analog Computers to Steady-State Multicomponent Distillation Calculations.” Electronics ~ssociates.Inc..’Princeton Computation Center, Princeton, N. J., 1961. RECEIVED for review October 11, 1962 ACCEPTED October 29, 1963 Research sponsored by the U. S. Air Force under Project K.AND, Contract AF 49(638)-700. Views or conclusions contained herein should not be interpreted as representing. the official opinion or policy of the U. S. .Air Forcr.
A STATISTICAL CORRELATION OF THE
EFFICIENCY OF PERFORATED T R A Y S RAY N. F I N C H 1 AND M A T T H E W VAN W I N K L E Chemical Engineering Department, The University
nf
Texas, Austin 72, Tex.
The efficiencies of perforated trays were determined for the system methanol-air-water at atmospheric pressure. The effects of gas mass velocity, liquid mass velocity, hole diameter, weir height, and tray length upon Murphree tray efficiency were determined in a series of statistically designed experiments. Tray hydraulic data were also determined for use in a model to be utilized in the scale-up of small column data for commercial design. This model relates operating and design variables to the residence times of gas and liquid upon the tray b y means of a two-constant equation. HF separation processes of distillation and gas absorption Tare so common in the chemical and petroleum industries that the reliability of predicted separation efficiencies becomes extremely important in the design of separation equipment. ‘This has become especially true in recent years as product piirity demands and new product developments have increased the number of contact stages required for many separation processes. Present address, Celanese Corp., Bay City, Tex.
106
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
The perforated tray is one of the simplest and most economical of the many types of gas-liquid contacting trays that have been developed. The efficiency and capacity of perforated trays are also considered to be better than those of many other types of contacting trays ( 7 7 , 2 7 . 2 4 , 3 3 ) . Several methods for predicting the separation efficiencies of gas-liquid contacting trays are available (7. 6 , 8, 7 7 , 2 6 ) . However, it is often desirable to obtain experimental efficiency d a t a for the system concerned ; in the case of perforated trays,
these data may be obtained with the conventional Oldershaw laboratory column. The use of these data for commercial column design is limited by the lack of a model for scale-up. Therefore, it was the purpose of this investigation to obtain efficiency data which could serve as a basis for developing a model for the scale-up of laboratory column efficiency data for commercial design. These data include the effects of design and operating variables upon the performance of perforated trays in a column intermediate in size between a laboratory and a commercial column. Planning the Experiments
T h e system methanol-air-water was selected for experimental study for three reasons: (1) I t was necessary to use air as the carrier gas in order to get the desired gas rates in the equipment used. (2) The lower explosion limit of methanol in air (27) allowed higher concentrations of methanol in the gas phase, which increases analytical accuracy. (3) Vapor-liquid equilibrium data were available for the desired operating conditions (5,27). Since these experiments were performed as a basis for developing a model for scale-up of a given system, system properties were not variables. Design and operating variables were the prime considerations. A literature survey showed
eight design variables and three operating variables which affect theefficiency of a given system ( 7 , 7. 9-77. 74-76, 78. 79, 28, 29, 37, 32). Because examination of previous separation studies showed that the five independent variables of gas mass velocity, liquid mass velocity. hole diameter, weir height, and tray length had the most significant effect upon separation efficiency, these variables were chosen for this study. Most of the remaining operating and design variables either could be related to the selected variables in such a way as to eliminate their effect upon efficiency or their practical range was limited by other design considerations-i.e., downcomer area is related to the gas mass velocity based on bubbling area by the total tower cross-sectional area; pitch and free area are related to hole diameter. Statistical experimental design is a powerful tool for increasing the amount of information available from a given amount of experimental runs. The Box-‘IVilson ideal central composite design (3, 4 ) in five variables as shown in Table I was selected for this work. Statistically designed experiments also enhance the economic attractiveness of efficiency data for design projects because of saving in time required for experimental work. Dechman and Van M’inkle (7) have shown the applicability of this design to experimental work on separation efficiency. Basic Equations
Table 1. Run
No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
Box Central Composite Design in Five Variables 2 1
+I +1
+I
+1
+1 i-1
+1 +1 +1
+I
+1 +1 +1 +1
+1 $1
Z3
2 4
+1 +1 +1
$1 +1 -1 -1
$1
+1
+1
$1 -1 -1
+I
+1
+I
-1 -1 -1 -1
-1
0
0
0
0 0 0 0 0 0 0 0
0
0 0 0
-1
-1 -1 +1 +l
-1 -1 -1 -1 -1 -1 -1 -1 0 0 +2 -2 0
0 0
fl -1
-1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 +2 -2 0 0 0 0
Z n
Z2 $1 $1 +1 +1 +1 +1 $1 +l
+1 +l +1
+I
$1
+I +1
+I
+I +I
-1 -1 -1
+I -1
-1 +1
-1
+1
-1
-1 +1
-1
+1
+1
+1
+I
+1 -1 -1
+1 -1
$1 +1 +1
-1
-1 -1 -1 0
0 0 0
+2 -2 0 0
0 0 0 0 0 0
+I
-1 -1
+I +1
+I
-1 $1
-1
+1
-1 +1 -1
-1 -1
+I
+1 +1
+1
-1 -1 0 0 0 0 0 0
+2 -2 0
0 0
0 0 0
Y, ~
Y;
- Y,-1 Y,t-
1
-1
-1 +1 -1
+I
E,,,,, =
-1
+I -1 -1
+1 +1 -1 -1 -1 -1 $1
Gas-liquid separation efficiencies for tray-type apparatus are usually reported as either Murphree tray (25) or over-all column efficiencies (22, 23). The defining equations for these two efficiencies are given in Equations 1 and 2.
T h e relationship between these two types of efficiencies for the case of a straight operating line and a straight equilibrium line as derived by Lewis is shown in Equation 3.
Murphree point efficiency is the more fundamentally defined efficiency with respect to mass transfer relationships; hoIvever, point efficiencies must be measured indirectly from liquid concentration profiles or liquid residence time distribution studies and Murphree tray efficiencies ( 7 ) . The relationship between the Murphree point efficiency and the Murphree tray efficiency is given in Equation 4 ( 7 ) .
-1 -1
+1
(4)
-1
0 0 0 0 0 0 0 0 +2
-2
0 0 0
0
Graphical integration of Equation 4 can readily be performed from liquid concentration profiles and liquid temperature profiles ; however, the basic assumptions of constant point efficiency across the tray, constant vapor rate through all sections of the tray, and completely mixed vapors both entering and leaving the tray become more invalid as tray length is increased. T h e basic quantity measured in this report was the Murphree tray efficiency (Equation 1). The necessary data were also obtained to enable evaluation of over-all column effiVOL. 3
NO.
2
APRIL
1964
107
Table II.
Uncoded Variables at Five Design levels
G, Level
f 2 t l 0 -1
-2
Lb./Hr. Sq. Ft.
L, Lb./Hr. Ft.
2200 1850 1500 1150 800
8900 7350 5800 4250 2700
dh I
Inch
h,, Inches
'/4
5 4
3/,c
3
1);"
2 1
'/16
'/16
Z1,
Inches
23 20 17 i4 11
Experimental Work
ciencies and Murphree point efficiencies, using Equations 3 and 4, respectively. Tray hydraulic data are usually reported as dry tray pressure drop, total tray pressure drop, clean liquid holdup on the tray, and aerated "bed" or "froth" height. With the height of the clear liquid on the tray also a pressure drop measurement, the relationship between the three types of pressure drop is as shown in Equation 5 :
(5) The total bed height, clear liquid holdup, and tray gas holdup are related by Equation 6.
BL = B ,
+ BO
(6)
Table 111.
108
Run No.
G, Gas Rate, Lb./Hr. Sq. Ft.
L , Liquid Rate, Lb./Hr. Ft.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
1850 1850 1850 1850 1850 1850 1850 1850 1850 1850 1850 1850 1850 1850 1850 1850 1150 1150 1150 1150 1150 1150 1150 1150 1150 1150 1150 1150 1150 1150 1150 1150 2200 800 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500
7350 7350 7350 7350 7350 7350 7350 7350 4250 4250 4250 4250 4250 4250 4250 4250 7350 7350 7350 7350 7350 7350 7350 7350 4250 4250 4250 4250 4250 4250 4250 4250 5800 5800 8900 2700 5800 5800 5800 5800 5800 5800 5800 5800 5800 5800
.
Thus, Equations 5 and 6 allow calculation of clear liquid holdup and tray gas holdup from the measurement of dry tray pressure drop, total tray pressure drop, and total bed height.
Apparatus. The basic piece of experimental equipment was a single-tray gas absorption tower (Figure l ) , constructed of Lucite acrylic plastic for clear observation of tray action. This tower was 6 inches wide, 283/4 inches long, and 50 inches high (inside dimensions), 18 inches below and 32 inches above the tray. The tower was rectangular to represent a section of a tray and was equipped for different parts of the experiment with five 20-gage stainless steel perforated trays with hole diameters of I/p,, l / 8 , 3/16, l/q, and "16 inch which correspond to the five levels of variables in the experimental design as shown in Table 11. Each tray had a free area of 7% based on total tower cross-sectional area and was constructed to fit between the Lucite flanges in the middle of the tower. The trays were equipped with five sets of weirs and five tray length spacers, which correspond to the five levels of weir height and tray length shown in Table 11. The air was introduced into the system with a Turboblower. A butterfly blast gate was used for flow control. The air then passed through a humidification chamber where live steam was introduced to preheat and saturate the air. A Venturi was used for measuring the air flow rate. After passing
Experimental Results
h, Weir d, Hole Height, Z , Liquid Diameter, Inch Inches Path, Inches INDEPENDENT VARIABLE SETTINGS
0.250 0.250 0.250 0.250 0.125 0.125
4 4 2 2 4 4
0,250 0.125
2 4
0.250 0.250 0.125 0.125 0.125 0.125 0.250 0.250 0.250 0.250 0.125 0.125 0.125 0.125 0.1875 0.1875 0.1875 0.1875 0.3125 0.0625 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875
2 2 4 4 2 2 4 4 2 2 4 4 2 2 3 3 3 3
l&EC PROCESS DESIGN A N D DEVELOPMENT
3
3 5 1
3 3 3 3 3 3
20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 20 14 17 17 17 17 17 17 17 17 23 11 17
17 17 17
(2, Volume
F Factor G/3600
1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 2.30 0.835 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57
=
45
Liquid Rate = L,/490
15.00 15.00 15.00 15.00 15.00 15.00 15.00 15.00 8.67 8.67 8.67 8.67 8.67 8.67 8.67 8.67 15.00 15.00 15.00 15.00 15.00 15.00 15.00 15.00 8.67 8.67 8.67 8.67 8.67 8.67 8.67 8.67 11.84 11.84 18.16 5.51 11.84 11.84 11.84 11.84 11.84 11.84 11.84 11.84 11.84 11.84
through the contacting tray, the gas left the tower via a 6-inch Demistor pad and a 6-inch galvanized exhaust flue. The liquid stream was fed to the tower with a 10-gallonper-minute, 50-foot-head centrifugal pump. The flow rate of the solution was controlled with a gate valve for coarse control and a globe valve for fine control. T h e liquid was measured with a rotameter before entering the inlet downcomer of the absorption tower. After flowing across the contacting tray, the liquid left the tower via the outlet downcomer into a 20-gallon stainless steel mixing tank. The solute methanol was fed into this mixing tank from a 55-gallon methanol feed barrel via a centrifugal pump and a rotameter. Another 10-gallon-per-minute, 50-foot-head centrifugal pump was used to recirculate the liquid in this tank to ensure uniform composition. From the mixing tank the solution was transferred into a second 20-gallon stainless steel tank which served as a feed tank for the tower. The feed tank was equipped with hot water coils to preheat the feed to the desired temperature. Temperatures were measured with three copper-constantan thermocouples connected to a precision potentiometer. The thermocouples were mounted in glass protecting tubes filled with mercury for good heat transfer characteristics. The thermocouple and protecting tube for measuring liquid temperatures were constructed so that the temperature on each part of the tray could be measured. Differential pressure measurements were made with two inclined-tube manometers. One manometer measured the gas Venturi pressure difkrential and the other measured the differential pressure across the perforated tray.
I
Figure 1 . Schematic of experimental equipment
Materials and Analysis. The methanol was supplied by Monsanto Chemical Co. The assay of the original lot was 99.9570 methanol. During the course of emptying a 55-gallon drum of methanol, some water was picked u p ; however, in no case did the final analysis of a drum fall below 99.75y0. T h e only detectable impurity a t any time was water.
Table 111. (Continued) Rdn N o .
APt Inches
APD, Inches
Inches
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
4.44 4.70 4.32 4.36 4.85 4.79 4.55 4.59 4.14 4.52 4.00 4.13 4.61 4 58 4.43 4.54 3.15 3.14 2.45 2.48 3.26 3.28 2.64 2.64 2.85 2.90 2.29 2.32 2.98 2.96 2.52 2.49 5,48 2.25 3.40 3.05 3.15 5.41 3.84 2.89 3.29 3.17 3.19 3.38 2.73 2.78
3.05 3.34 3.12 3 31 3 77 3 74 3 64 3.78 3.05 3.37 3.12 3.30 3.57 3.87 3.77 4.12 1.16 1.28 1.11 1.13 1.42 1.66 1.51 1.59 1.12 1.35 1.15 1.43 1.36 1.48 1.41 1.57 4.41 0.60 2.05 2.10 1.85 4.32 2.09 2.09 2.06 1.97 2.06 2.13 1.61 1.61
1.39 1.36 1.20 1 05 1 08 1 05 0 91 0.81 1.09 1.15 0.88 0.83 1.04 0.71 0.66 0.42 1.99 1.86 1.34 1.35 1.84 1.62 1.13 1.05 1.73 1.55 1.14 0.89 1.62 1.48 1.10 0.92 1.07 1.65 1.35 0.95 1.30 1.09 1.75 0.80 1.23 1.20 1.15 1.25 1.12 1.17
APl,
E O A ,70 EMV, % MEASURED A N D CALCULATED DEPENDENT VARIABLES 73.5 65.9 63.9 52 5 66 5 63 3 59 5 52.0 68.3 62.1 65.6 59.7 71.4 65.2 66.4 53.1 82.2 75.5 57.6 54.1 86.1 71.4 67.0 48.1 85.3 69.6 60.8 49.7 88.0 73.9 62.9 48.9 50.0 78.0 57.0 57.5 62.6 66.6 68.7 53.2 68.9 53.8 60.8 61.5 59.3 58.3
90.2 88.4 85.1 80 3 86 5 87 0 82 2 80.0 83.6 82.5 81.9 80.8 85.6 84.4 82.5 75.9 95.6 94.5 84.6 84.9 96.7 93.1 89.6 80.9 95.1 89.9 82.5 77.5 96.1 91.8 83.8 77.0 81.8 95.1 85.1 75.2 85.2 87.4 88.6 78.8 86.8 82.9 84.1 84.4 83.1 82.5
Ehiv*,
%
84.8 85.4 80.2 77.7 82.7 84.3 77.8 77.2 77.4 78.9 74.1 76.0 79.9 81.9 75.9 72.4 92.8 92.6 81.5 82.5 93.2 91.2 86.4 78.5 91.4 87.4 78.5 74.5 92.4 88.7 78.8 73.3 69.6 92.3 81 9 69.1 81.4 82.1 84.1 73.4 80.1 80.5 79.4 79.9 79.1 79.2
VOL. 3
NO. 2
Material Balnnce, %
100.0 94 8 93.9 104.0 99.3 92.8 97.8 100.0 92.6 100.0 103.3 98.4 93.4 93.3 90.9
92.5 103.1 102.7 106.1 96.8 100.7 95.3 105.8 96.7 91.9 104.8 94.1 97.3 90.4 100.7 95.2 106.1 89 3 97.3 94.7 102.7 95.3 95.7 104.0 96.0 98.3 90.3 95.5 100.8 94.5 97.6
APRIL 1964
109
A modified Ostwald viscometer was used to analyze the methanol-water solutions from the gas absorption tray. A constant temperature bath was used to thermostat the viscometer and a stopwatch for timing. Calibration was made with prepared solutions. The precision of liquid analyses was *0.0001 mole fraction. A Beckman GC-2 chromatograph was used to analyze the vapor samples containing methanol, air, and water. The column was a 12-foot Carbowax 400 on Teflon a t 111' C. An absolute calibration technique gave partial pressures of methanol and water cs. recorder signal. The total pressure of each sample was measured and the partial pressure of air was calculated by difference, since the sampling pressures were low enough to assume ideality. Procedure. Experimental runs were begun by assembling the proper tray, weir, and weir spacer to give the desired level of hole diameter, weir height, and length of liquid travel required by that particular experimental run. The hot water was started through the feed tank coils. The feed and recycle pumps started recycling the liquid around the tower. Methanol was added to the solution with the solute pump and rotameter until the desired inlet concentration was reached. Vapor sample bombs were installed and their evacuation was started. Zero readings were taken on the Venturi draft gage and on the pressure drop manometer. Air was started through the tower with the blower. Steam was admitted to the humidifier to begin saturation of the air. All rates were now adjusted until both gas and liquid streams were maintained a t the desired conditions of contacting. The dry plate pressure drop was ready from the pressure drop manometer. The liquid feed line was opened and the recycle line was closed to begin contacting. ,411 rates were readjusted to the desired contacting conditions. Vapor temperatures above and below the tray, liquid temperatures on and off the tray, and outlet liquid composition were measured each 15 minutes until steady-state conditions, approximately x , = 0.0425 and to = 91.0' F., had been reached. This line-out time took from 50 minutes to 2 hours, depending on the conditions used and the operator's control. When steady-state conditions were reached, liquid and vapor samples were taken. Vapor sample bombs were purged a t least 15 minutes after steady-state conditions were reached. Liquid samples were taken from serum cap plugs at five positions along the tray: (1) before inlet weir, (2) three points along the active length of tray, and (3) from the outlet downcomer. Four vapor samples were taken: two above and two below the active tray. After sampling, the unit was shut down, atmospheric pressure was recorded, and the samples were analyzed. PROCEDURE FOR BED HEIGHT EXPERIMESTS. Total bed height \vas determined in the equipment used for efficiency and pressure drop studies and a t the same temperatures and methanol concentrations. Bed height experiments were performed as a central composite design in three dimensions. Preliminary experiments revealed that total bed height was affected only by gas mass velocity, liquid mass velocity, and weir height for a given system and concentration. The equipment was started as for efficiency and pressure drop determinations and brought to the desired conditions of operation. Less time was required to reach hydraulic than thermal and mass transfer steady-state conditions. Bed height was measured to the nearest half inch a t the tray inlet, center of the tray, and the tray outlet Jvith a meter stick placed next to the Lucite walls. The reported bed height is the arithmetic average of the three points. After measurements \vex made twice and averaged, the column flow conditions were changed to the next desired point. The liquid was recirculated through the column bypass line while the column was opened to make weir height changes. ResuIts
The results of the efficiency and pressure drop experiments are presented in Table 111. The calculated quantities, Murphree vapor point efficiency, over-all column efficiency, clear liquid holdup on the tray, and material balance were also reported as F factor based on bubbling area and volume liquid velocity for comparison with other data and correlations. Clear liquid holdup on the tray. A P l . was calculated by 110
I & E C PROCESS D E S I G N A N D DEVELOPMENT
Equation 5. Over-all column efficiency was calculated using Equation 3. The Murphree vapor point efficiency was calculated from the measured concentration and temperature profiles using Equation 4. The material balance for each run was calculated as the ratio of the measured change in methanol concentration of the gas passing through the tray to that calculated from the change in methanol composition of the liquid flowing across the tray. This ratio was expressed as a percentage and was reported as an estimate of the accuracy of the analytical measurements of this study. Total bed height measurements are presented in Table IV. The last column shows bed heights which are adjusted within the experimental error of 1 0 . 5 inch to agree with the model equation presented in the new scale-up method proposed below. This adjustment brings the average deviation of the model equation (2.9 efficiency %) into close agreement with the estimated accuracy of the measured Murphree tray efficiencies (2.5 efficiency %). This points out the need for better methods of measuring total bed heights.
Table IV.
Total Bed Height Measurements Hole diameter. T r a y length.
Run h'o. 1 2 3 4 5 6 7 8 9 10 11
16 17 18
Gas Mass Velocity, Lb./Hr. Sq. Ft. 1850 1850 1850 1850 1150 1150 1150 1150 2200 800 1500
1500 1500 1500
3/16 inch 17 inches
Liquid Mass U'eir Height, Velocity, Lb./Hr. Ft. Itiches 7350 4 7350 2 4250 4 4250 2 7350 4 7350 2 4250 4 4250 2 3 5800 5800 3 8900 3
5800 5800 5800
3 3 3
Bed Ao.
8.3 7.3 7.5 7.2 7.2 5.2 6 7 4.7 9.7 5.2 6.7
Inches Adjusted 8.4 7.5 7.6 7.2 6 8 5 5 6 2 4 9 9.4 4.9 6.9
6.6 6.2 6.2
6.6 6.6 6.6
Independent Variables. The independent variables measured were gas mass velocity, liquid mass velocity, hole diameter, weir height, and length of liquid path. The operating variables of gas rate and liquid rate were measured with a Venturi and a rotameter, respectively. The precisions of these measurements were 0.5% for gas rate and 0.3% for the liquid flow rate. The estimated accuracies were 1.57, for gas rate and 1.0% for the liquid rate. The design variable measurements were all linear, with an estimated accuracy of 1.0% for the hole diameter, 0.2570 for weir height, and 1.0% for tray length. Dependent Variables. The dependent variables measured were total tray pressure drop, dry tray pressure drop, hlurphree tray efficiency, liquid concentration profile, and liquid temperature profile across the tray. Both pressure drop measurements were made with a 30 ' inclined water-filled manometer. The precision was 0.05 inch of water, while the estimated accuracy was 0.20 inch of water. The Murphree tray efficiency measurements were a function of the analytical measurements. The precision of these measurements may be calculated from the four repeated "zero" points of the design. This precision was found to be 0.7 efficiency %. The accuracy
estimated from the analytical accuracy was 2.5 efficiency %. T h e estimated accuracy for the liquid concentration profiles was 0.0003 mole fraction methanol; for the liquid temperature profiles, 0.25' I:. T h e precision and accuracy of bed height measurements were estimated to be k 0 . 5 inch. Discussion
'The linear form of the regression Equation 7 :
Y
= A,
+ A1X1 + A n X i . . . +
A11X12
A22X2'
+
+ A12X1X2 + . . . .
(7)
was adequate to correlate the dependent variables of Murphree tray efficiency? Murphree point efficiency, over-all column efficiency, and tray liquid pressure drop \vithin the estimated experimental error. Neither the linear nor the quadratic form of the regression Equation 7 was adequate to correlate d r y and total tray pressure drop within the estimated experimental error. The total bed height data were correlated satisfactorily by including the quadratic term in gas mass velocity in the regression equation. Table J\ sho\cs the regression equation coefficients and the standard deviations for the seven dependent variables reported. All figures in this section are plots of the regression equation. The +2 and -2 experimental points of the independent variables considered are shown on the plots to illustrate the deviation a t the extremes of the experimental design. Effect of Gas and Liquid Mass Velocities on Murphree Tray Efficiency. The effects of gas and liquid mass velocities on Murphree tray efficiency are shown in Figure 2. For the system methanol-air-water, over the ranges of gas and liquid rates studied, the hlurphree tray efficiency decreased with increased gas rate and increased with increased liquid rate. This result was in agreement with Umholtz and Van M'inkle (\?7, 3.?), who studied the over-all column efficiency of smalldiameter perforated tray columns. T h e Murphree tray efficiencies of bubble caps and sieve trays axe reported to decrease gradually with gas rate over the range of F factors from 0.3 to 2.6 ( 7 ) . T h e liquid-phase-controlled system n-pentane-fixylene was the only exception for the three systems studied. Hellums et al. ( 7 6 ) sho\sed an increasing over-all column eficiency for the system, n-octane-toluene in a 6-inch-diameter perforated tray column; however, the gas rates employed were lower than those used in this work. Kocatas (79) reported a parabolic eficiency curve with respect to gas rate over a very wide range of gas rates. Gerster, Bonnett, and Hess (72) reported relatively little effect of gas rate upon efficiency for the
e00
Figure 2.
1 t60 1500 G A S Y A L S VELOCITY. LIS./
I 850
2200
humidification of air in a 13-inch-diameter bubble cap column a t low gas rates. Kirschbaum (78) reported a rapidly rising efficiency for perforated plates a t loiv gas rate, followed by a long gradually decreasing efficiency for further increased gas rates. A bubble cap column a t the same conditions did not exhibit a rapidly rising efficiency curve. Mayfield et al. (24) reported no effect of gas rate on the efficiency of a 6-inch perforated tray column over a narrow range of gas rate. The effect of liquid rate shown in this work is intermediate to that of several other investigators in this field. Kocatas ( 7 9 ) reported a stronger positive effect of increasing liquid rate upon the over-all column efficiency of perforated trays; however! he was operating a t external reflux ratios (L/'D) of 4 or less for all runs except one total reflux run. Hellunis e t al. (76) reported no effect of reflux ratio on the over-all column efficiency of perforated trays for an external reflux ratio of 2 or greater, but for a reflux ratio of 1, the efficiency dropped about 10%. The A.1.Ch.E. report ( 7 ) stated a smaller effect of liquid rate upon the Murphree tray efficiency of bubble caps than reported in this work; however, the liquid rates employed in that work included one rate considerably higher than those presented here. Gerster et al. ( 7 3 ) in earlier work reported that the length of liquid film transfer unit rose rapidly from zero liquid rate and showed a slight maximum xvith respect to liquid rate. The effect of gas and liquid rates over the entire range of perforated tray operation is:
As the gas rate is increased from zero, a point is reached a t which the liquid is begun to be supported on the tray. If the gas rate is increased further, the efficiency rises rapidly from almost zero to a maximum. After passing the maximum, the efficiency falls off slowly a t a n almost constant rate with increased gas rate. Further increase of the gas rate will result in reaching the limit of tray operation of either flooding or entrainment, after which the efficiency falls off rapidly with increased gas rate. As the liquid rate is increased from zero, the efficiency begins to rise rapidly. As the liquid rate is further increased, the rate of increase of efficiency with liquid rate becomes smaller until the effect of liquid rate on efficiency becomes negligible. Further increase of liquid rate results in reaching the limit of tray operation of either flooding or dumping, after which the efficiency drops off rapidly. 'Thus, there is a region of tray operation, bounded by certain gas and liquid rates. within which the efficiency varies sloivly and is near the maximum attainable for a fixed set of design variables. This region, designated as the operating range of tray efficiency, was qualitatively described by Eolles ( 2 ) for bubble cap plates. All experimental points in this work were
000
HR.- sa. Fr.
Effect of flow rates on Murphree tray efficiency
GAS
Figure 3.
Ill0 MASS
I850
I500 VELOCITY.
2200
L B S . / H R . - S Q . FT.
Effect of weir height on Murphree tray efficiency VOL. 3
NO. 2
APRIL
1964
111
800
1150 1100 I a50 GAS MASS V E L O C I T Y . L B S . / H R . - S Q . FT.
2200
800 GAS
1150 I 500 1850 MASS V E L O C I T Y . L b . / HR.-SQ. rl.
2200
Figure 4. Effect of hole diameter on Murphree tray efficiency
Figure 5. Effect of tray length on Murphree tray efficiency
in this operating range of tray efficiency. The results presented here agree satisfactorily with those of other investigators when the relative ranges of tray operation are considered. Effect of Weir Height on Murphree Tray Efficiency. Increasing weir height increases the Murphree tray efficiency, as shown in Figure 3. This result is in agreement with the following investigations ( 7 , 7 7 . 72, 76, 78, 37. 32. 34). Kocatas (79) found no effect of weir height upon over-all column efficiency ; however, his experiments were performed a t relatively low liquid rates and high gas rates. Under these conditions. the small amount of liquid upon the tray is suspended more by gas flow than by weir height. This area of operation lies on the edge of the operating region of tray efficiency. Effect of Hole Diameter on Murphree Tray Efficiency. T h e effect of hole diameter on Murphree tray efficiency was negligible for hole diameters from l/16 to 5/16 inch (Figure 4). Hellums et al. (76) reported an increase of efficiency with decreased hole diameter; however, the gas rates employed in that study were lower than those presented here. In effect. Hellums e t al. (76) were operating in the rapidly rising portion of the efficiency curve, where trays with smaller hole diameters are able to support the liquid upon the tray a t lower gas rates than the trays of larger hole diameter because of the capillary surface tension effect. This means that the trays of smaller hole diameter are able to rise to the operating region of tray efficiency a t lower gas rates than the trays of larger hole diameter. In the operating range of tray efficiency, the hole diameter effect disappears. Hellums et al. show this convergence a t the maximum gas rates studied in that work. Effect of Tray Length on Murphree Tray Efficiency. The effect of increasing tray length on Murphree tray efficiency is shown in Figure 5. The magnitude of increase in Murphree tray efficiency with increased tray length for the system methanol-air-water is 6 efficiency 70 per foot of tray length according to the linear coefficient of Z1 shown in Table V. This effect is not truly linear; therefore, the linear regression equation should not be extrapolated beyond the range of these experiments. The effect of increased tray length was only to increase the time of liquid contact upon the tray. This result agreed in principle with the A.1.Ch.E. report ( 7 ) . Hellums et al. (76) reported an increase of 6 to 7 efficiency % in the over-all column efficiency of perforated trays for a 6-inch over a 3-inch diameter column with the system n-octane-toluene. Kirschbaum (78) reports increases of from 8 to 18 efficiency % in Murphree tray efficiency for a 400-mm. over a 110-mm. diameter perforated tray column. T h e system distilled was ethanol-water.
Murphree Point Efficiencies from Measured Concentration Profiles. The measured concentration profiles of the tray liquid were used to calculate the Murphree point efficiency for each experimental run. T h e effects of the five independent variables upon Murphree point efficiency were in the same direction as their effects upon Murphree tray efficiency, but slightly larger for the point efficiencies. The hole diameter effect was absent from the hfurphree point efficiency, just as in the case of the tray efficiencies. The effect of tray length should have been entirely absent from Murphree point efficiencies; however. a slight positive effect was found. This effect was within the experimental error of the method and was probably due to some extent to the temperature variation across the tray and to the assumption of a completely mixed vapor above and below the tray. The concentration gradients across the tray were rather small and therefore the point efficiencies were only slightly less than the Murphree tray efficiencies. The ratio of Murphree tray efficiency to Murphree point efficiency in this lvork varied from 1.02 to 1.10: Lvhich is in good agreement with the ratios predicted for this system by the A.1.Ch.E. method ( 7 ) ; however, the absolute values of Murphree tray efficiency and, consequently, the point efficiencies: were from 15 to 2 5 efficiency o/c higher than those predicted by the A.1.Ch.E. method ( 7 ) . This discrepancy can be attributed to two major reasons:
112
I&EC PROCESS DESIGN A N D DEVELOPMENT
The efficiencies of perforated trays are actually higher than those of bubble cap trays. The A.1.Ch.E. report states that gas phase point efficiencies for perforated trays are 6 to 8 efficiency % higher for perforated trays than for bubble cup trays a t high \veir heights and low gas rates. Kirschbaum (78) reports Murphree tray efficiencies of 15 to 30 efficiency % higher for perforated trays over bubble cap trays. Both tray types were compared for the system ethanol-lvater in a 430mm.-diameter column. Mayfield et al. (24) reported Murphree tray efficiencies of 11 to 15 efficiency yo higher for perforated trays over bubble cap trays in distilling mixtures of alcohols; hohvever, there \vas enough difference in design and system property variables to prevent exact comparison. Jones and Pyle (77) reported efficiencies from 0 to 10 efficiency 76 higher for perforated trays over bubble cap plates. The free area in that work was twice as great for the bubble cap plates as for the perforated plates. Lee (27) reported tray efficiencies of 5 to 15 efficiency higher for perforated trays than for bubble cap trays. The effect of gas phase Schmidt number between the system methanol-air-water and ammonia-air-water is not so great as predicted by the A.1.Ch.E. method. The Schmidt number of the ammonia-air-water system a t 20° C. is reported as 0.61 by the A.1.Ch.E. The Schmidt number of the methanol-air-water system a t 33' C. is 0.93. The reason for this wide difference is the larger gas phase viscosity and smaller
x,
gas phase density and diffusivity for the system methanol-airwater. It may be that in both cases the diffusivity used was that of the methanol-air binary, whereas both systems contained 4 to 5%, Ivater vapor.
Over-all Column Efficiencies. Over-all column efficiencies were calculated from Equation 1. The four were about 20 efficiency yGlower than the Murphree tray efficiencies because of the low K V / L values for the system and operating conditions used. The five independent variables affected the over-all column efficiencies i n the same direction as the Murphree tray efficiencies; however. the weir height and tray length effects were considerably less for over-all column efficiencies than for Murphree tray efficiencies. Pressure Drop iMeasurements. Dry and wet tray pressure drops were measured and the equivalent height of clear liquid on the tray \vas calculated by taking the difference betLveen the two measured quantities. The largest percentage of total or wet tray pressure drop was the dry tray pressure drop. The dry tray pressure drop is a function of gas mass velocity and hole diameter. Seither the !inear nor the quadratic form of the regression equation ( 3 ) gave a satisfactory fit to the experimental data: because the dry tray pressure drop was inversely rather than directly proportional to the hole diameter. T h e experimental data for dry tray pressure drop Lvere predicted !vel1 rvithin the accuracy of the experimental data by the semiempirical orifice equation of Kolodzie, Smith, and Van Winkle (20, 30). The effect of inverse hole diameter was found in the total tray pressure drop also, since the dry tray pressure drop was a large percentage of the total. An increase in gas mass velocity decreased the tray liquid pressure drop, Jvhile an increase in lveir height had the reverse effect. ,4n increase in liquid rate increased the tray liquid pressure drop. Increased hole diameter and increased tray length had slightly positive effects. .4lthough these effects are marginal Lvith respect to estimated experimental accuracy, it is believed that they are real effects and should be used when greater accuracy is required. The increase of tray liquid pressure drop LLith tray length undoubtedly corresponds to the slight hydraulic gradient found on perforated trays. The above results agree \vith the A.1.Ch.E. study on bubble caps ( 7 ) in direction but differ in magnitude. The A.1.Ch.E. report states that the clear liquid holdup on perforated trays is less than that of bubble cap trays; hoivever, no data or equation are presented for perforated trays. Therefore, the A.1.Ch.E. equation for clear liquid holdup for bubble cap
trays and the linear regression for perforated trays, excluding hole diameter and tray length effects, are given below in similar units for comparison of the clear liquid holdup on these two tray types.
B,
=
1.65
B , = 0.98
- O.65F
+ 0.020Q +
- 0.56F + 0 . 0 3 7 4
0.19h, (bubble caps)
(8)
0.22hW (perforated plates)
(9)
+
Equation 9 will remove one of the restrictions of the A.1.Ch.E. prediction method when applied to perforated plates. Both Equations 8 and 9 are in inches of water and should be converted to inches of clear liquid for other systems by the density ratio of the t\vo systems concerned. Total Bed Height Measurements. The liquid rate had a small positive effect. I t was necessary to include a squared term in gas rate to give a satisfactory fit of the regression equation to the bed height data. The trends given above are in agreement with the A.1.Ch.E. report on bubble caps ( 7 ) ; however, the magnitude of total bed heights is less for perforated trays than for bubble caps, mainly because of percentage volume of the total bed taken up by the bubble caps. The equation for total bed height for the ammonia-air-water system on a bubble cap tray ( I ) is compared below with the regression equation for the methanolair-water system on perforated trays found in this work.
+ 0.084Q + 3.24F (bubble caps) 4.51 + 0.74hW+ 0.0934 - 4.55F +
Bi = 0.73 h, B,
=
2.30F2 (perforated plates)
+64.3 $85.4 +81.5 6.34
+ -
0.31 0.70 APi 0.39 EOA +31.8 Esiv f72.1 EMV" +72.4 BT 4.51 Coeflcient of 22 in BT equation. Apt
Apd
+
+
$0.842 +I ,045 -0.204 -3.2 -3.03 -3.56 +O. 969 f0.00241 +O ,00299
-0,000583 -0.0094 -0.00866 -0.0102 -2.63
$0.102 -0.013 f0.116 -0.32 +I . 7 1 +2.32 + O , 294
CODED -0.201 -0.295 i-0.095
-0,14 -0.0 -0.01
...
U N c oD E D SO.000658 -3.2 -0,0000084 -4.72 +O. 000075 $1.52 -o.ooo~o~ -2.24 $0.001 1 -0.00 f0.0015 -0.16 +0.00019
(1 1)
The slope of Equation 11 is positive with respect to gas rate over the range of gas rates studied in these experiments. The A.1.Ch.E. report ( 7 ) did not propose a general equation for the total bed height of bubble caps, and wisely so, since Zuiderweg and Harmens (36) have shown that the total bed height, and consequently the Murphree tray efficiency, is a strong function of the direction and magnitude of the surface tension gradients of a particular system. Therefore, a t present, no generalized equation is available for bed heights of either bubble cap or perforated trays. With present methods, the total bed height measurements are accurate to only f 0.5 inch.
Table V. Regression Coefficients and Standard Deviations for linear Fitn Zero Point Hole Weir Constant, '40 Gas Rat?, A1 Liquid Rate, A2 Diameter, Ax Height, A4 Tray Length,
++ 2.40 3.60 + 1.20
(10)
...
Ab
$0.208 -0.012 +o .22 4-0.694 f3.78 f4.07 & O . 74
-0.018 -0.07 +o. 053 +4.76 + I .51 + O . 61 + O . 307"
+o 208
-0.006 -0.023 +O ,0177 $1.59 +0.5 +0.2 +0.000877a
-0.012 $0.22 +6.94 f3.78 4-4.07 +o. 74
VOL. 3
NO. 2
Standard Dev., IAYI
0.326 0.316 0.117 5.66 2.89 3.07 0.44
APRIL
1964
113
Proposed Scale-up Method for Tray Efficiency Data
The results presented here were the basis for the following model for scale-up of small column efficiency data for industrial design. Postulates
40
ao
20
1. The efficiency of perforated trays is a function of the contact time of liquid and gas upon the tray and of the diffusional properties of the system. 2. ‘Ihe efficiency of perforated trays may be expressed as a weighted average of the gas and liquid contact times upon the tray as defined below.
where
10
0
0.6
I
I
I
I .4
2.2
3.0
1 0
Figure 6. Evaluation of A , and A L for system methanolair-water in a perforated tray column
3. The weight coefficients, A , and A , of Equation 12, are functions of diffusive properties of the gas and liquid phases, respectively, for the systems. T h e proposed Equation 12 may be evaluated simply by dividing the equation by 0, to reduce it to the linear form y = mx 6.
+
c
Equation 15 may be used for graphical or least squares evaluation of A , and A,. The values of A , and A , for the system methanol-air-water have been evaluated by the method of least squares, as shown in Figure 6. The numerical values were 1000 and 1.12% per second, respectively. BG and B, are the height of gas holdup and clear liquid holdup on the tray, respectively. They are related to the total holdup on the tray or foam height by Equation 6. For the system methanol-air-water, the foam height and clear liquid holdup on the tray were measured and correlated by Equations 9 and 11. Then B, may be calculated by Equation 6 . BG represents the length of gas contact in the two-phase bed on the operating tray. B , represents the height of clear liquid on the tray, which defines the liquid mass velocity on the tray in terms of clear liquid-Le.,
=
L -
00
Figure 7. Evaluation of AG and n-pentane in bubble cap column
AI, for system
.
BL
This model for efficiency divides the total efficiency into a part due to gas phase contact time and a part due to liquid phase contact time. The relative size of A , and A , in Equation 12 is an indication of which phase resistance is controlling. For this particular system A , was very large compared to A,. The order of magnitude of the ratio A,/A, was very nearly 1000 to 1. This indicates that the system was very largely gas-phase-controlled. With this type of system the contribution to efficiency of liquid contact time is small, so that an increase of tray length has a small effect on efficiency, whereas an increase in weir height and a decrease in gas rate substantially increase efficiency. However, an increase of column diameter at the same total gas flow would decrease the gas mass velocity and thus increase efficiency while decreasing preqsure drop. O n the other hand, a strongly liquid-phasecontrolling system would have a larger value of A , relative to A,. Thus increased tray length would have a significant effect upon tray efficiency. 114
l&EC PROCESS DESIGN A N D DEVELOPMENT
40
SO
20
IO
d UO
I
I
0.0
16.0
bg
I
240
0
x IO2
OL
Figure 8. Evaluation of AG and A, for system acetonebenzene in bubble cap column
I
1.0
a0
I
I
5.0
1.0
-x 00
9.0
loe
*L
Figure 9. Contour plot of efficiency as function of g a s and liquid contact times Limits of troy operation of a perforated tray
B,, B,, and B, are measures of the total, gas, and liquid holdups on the tray, respectively. For gas rates commonly used in distillation and gas absorption columns, the gas holdup is a much greater portion of the total holdup than the liquid holdup. Zuiderweg and Harmens (36) have shown that the Murphree tray efficiency of perforated trays is a strong function of the direction and magnitude of the surface tension gradients of the system. Thus, until a n adequate equation for calculating total bed heights is available, the model presented here should be used only for extrapolation of small column data and not for apriori predictions of perforated tray efficiency. T h e effect of tray length as shown by Equation 14, is to increase directly the liquid contact time upon the tray. T h e effect is not a linear increase of Murphree tray efficiency with tray length, but direct increase of that portion of efficiency due to liquid contact time. Thus, if all other factors remain constant, an increase of iray length from 1 foot to 2 feet doubles the contribution of liquid contact time to Murphree tray efficiency. T o double this contribution again, the tray length must be increased from 2 feet to 4 feet. From this, it can readily be seen that the increase of Murphree tray efficiency due to increased tray length falls off rapidly with tray length. As the tray length is reduced to zero, the contribution due to liquid contact time becomes zero and the first term of the equation becomes the original Murphree vapor point efficiency. Although it is not the purpose of this work to study the effect of system properties upon Murphree tray efficiency, it was thought advisable to test the model for a few systems where Murphree tray efficiency and the necessary bed height and clear liquid holdup data were available (7). T h e results of these tests are shown in the linear plots of EMV/8,(Figures 7 and 8). These tests are in reasonable agreement with the model, considering the accuracy of total holdup measurement. They show that the model is satisfactory for bubble cap trays as well, although a bubble cap model tray would have to be considerably larger than a perforated model tray to prevent wall effects. This increased size of the model tray reduces the economic attractiveness of scale-up for bubble cap trays. This method should prove very useful in the scale-up of small column data for industrial design. However, the following limitations should be noted. Adequate tray spacing should be used in the small column model, as premature flooding and entrainment would limit the range of the model, but not the prototype.
The effect of free area was not considered in this study and was fixed a t 7%. If the same percentage free area is used in model and prototype, the method is valid. If not, the effect of free area on BG and BL should be considered, as there is considerable disagreement between previous investigators as to the magnitude or even the existence of this effect ( 7 , 9, 76, 37, 32). Although variation of column diameter is not needed to evaluate the coefficients in this method, it is recommended that a t least a 3-inch-diameter model be used, because smaller diameters make the determination of length of liquid path extremely difficult to measure accurately. This method does not apply in the sharply rising region of the efficiency curve a t low gas rates, since large amounts of liquid bypassing through the holes will reduce both gas and liquid contact times as calculated by this model. Extreme care must be used in measuring total bed heights, since foam height is difficult to measure quantitatively (probably i O . 5 inch a t best). The gas height, BG,becomes sensitive to measured foam height, especially a t low foam heights. Fortunately, the foam height is most stable and easiest to measure a t this point. The same concentration must be used in both model and prototype, since the system properties which affect coefficients A0 and A L may be functions of the concentration of a given system. The pressure of both model and prototype should be the same, since both system temperature and gas density are affected by system pressure. The gas residence time, e,, includes the gas density. However, the system properties which affect coefficients A 0 and AI, may be altered by a change in system pressure. The model presented here also agrees with the limitations of tray operation as described in Figure 7. As B, becomes equal to the dry tray pressure drop, liquid begins to bypass through holes, thus decreasing the efficiency. The effect of hole size becomes significant in this range, as shown by Hellums et al. (76). Eventually tray dumping will entirely limit operation. If the gas rate is increased until BI becomes equal to the tray spacing, “priming” will occur. Before this, however: entrainment will probably limit operation a t approximately Bl = T / 2 . If the liquid holdup on the tray is decreased until no liquid is present, the efficiency must go to zero. The efficiency contour surface as shown in Figure 9 will begin to drop off sharply before complete inoperability is encountered. As the liquid holdup is increased, the range of gas rate over which the tray is operable is slowly decreased until the fall-off point and the flooding and/or entrainment point coincide. Thus the model satisfies all the limitations of perforated tray operation. Conclusions
The experimental data reaffirm conclusions of previous investigators with respect to the effect of gas mass velocity, liquid mass velocity, weir height, and hole diameter upon Murphree tray efficiency and pressure drop. In addition, this work probably presents the first systematic study of the effect of tray length upon efficiency. All experiments were performed in the operating range of a perforated tray as defined by Figure 9, and the system methanol-air-water is strongly gas-phase-controlled. T h e Murphree tray efficiency decreases and the total tray pressure drop increases with increased gas rate. T h e Murphree tray efficiency and the total tray pressure drop are both slightly increased with increased liquid rate. T h e Murphree tray efficiency is not affected by change in hole diameter between 1/16 and 5 / 1 6 inch, while the total tray pressure drop is greatly increased with decreasing hole diameter. The dry tray pressure drop, which is a large portion of the total tray pressure drop, is a function of reciprocal hole diameter. VOL. 3
NO. 2
APRIL
1964
115
T h e Murphree tray efficiency and the total tray pressure drop are increased by increased weir height. The Murphree tray efficiency is increased by tray length. The total tray pressure drop is slightly increased by tray length; however, the significance of the effect is marginal compared to experimental accuracy. The effect is of the right magnitude and sign to be accounted for by the small hydraulic gradient across perforated plates which increases with tray length. This experimental work has served as a basis for a model for the scale-up of the Murphree tray efficiency of small perforated tray columns. The equation which is the basis for this model is E\fV
f
=
ALoL
(12)
The gas and liquid contact times are functions of the operating variables, gas mass velocity and liquid mass velocity, and the design variables, weir height and tray length. The coefficients ale functions of the syPtem properties alone (this means also at the same concentration). The use of the model for scale-up is limited mainly by the accuracy in determination of the total bed height, B,. It is not recommended that this equation be used for a priori prediction of Murphree tray efficiencies for the following reasons : A systematic investigation of the effect of system properties upon A , and A , is not yet available, and the bed height of a system is a strong function of the type of surface tension gradients in the system (36). The recommended procedure for calculation of pressure drop data is: Calculate dry plate pressure drop by the method of Kolodzie, Smith, and Van Winkle (20, 30), calculate the tray liquid pressure drop by Equation 6, correct the tray liquid pressure drop by clear liquid density ratio, and add the dry tray pressure drop and the tray liquid preswre drop to obtain the total tray pressure drop. Nomenclature
Ac. AL
= efficiency coefficients of gas and liquid
contact times, respectively, in Equation 12. efficiency % per second coefficients of coded regression equation height of total bed,. gas bed, and clear liquid bed, respectively, upon operating tray, feet diameter of holes, inches efficiency, Murphree tray, Murphree point and over-all column, respectively,
At, Bt, Bo, B L
=
dh
=
E.,fr., E.uv*, EoA
=
F
= F fictor based on tray bubbling area,
G
= gas mass velocity, pounds per hour per
=
%
UdP R hw
K
Q t u o , UL
116
P
square foot of tray bubbling area = height of weir, inches of clear liquid = vaporization equilibrium ratio = liquid mass velocity, pounds per hour per foot of average tray width = number of stages, actual and theoretical, respectively = pressure drop of total tray and dry tray, respectively, inches of water = volume liquid rate, gallons per minute per foot of average tray width = temperature, degrees Fahrenheit = velocity of gas and liquid, respectively, feet per second, based on tray bubbling area
I & E C PROCESS D E S I G N A N D DEVELOPMENT
= average tray width, feet
= independent variable in regression equa-
tion = mole fraction in liquid
X
r
= dependent variable in regression equation
Y Z
= mole fraction in vapor
ZL
= tray length between inlet and outlet weir,
ec, or.
of gas and liquid, respectively, upon an operating tray, seconds = density of gas and liquid, respectively, pounds per cubic foot
= coded independent variable in regression
equation
PC> PL
inches, feet = residence times
References
(1) Am. Inst. Chem. Enprs., “Tray Efficiencies in Distillation Columns,” Final Report from UiiiversityofDelaware toA.1.Ch.E. Research Committee, 1958. (2) Bolles, W.L., Petrol. Processing 11, No. 2. 64; No. 3, 82; No. 4, 72; No. 5, 109 (1956). (3) Box, G. E. P., Biometrics 10, 16 (1954). (4) Box, G. E. P., Wilson, K . B., J . Roy. Siati.rticn/ Soc. 13, 1 (1951). (5) Butler, J. A. V., Thompson, D. W.,MacClennan, TV. H., J . Chem. Soc. (London) 1933, 674. (6) Chaiyavech, Pramote, Van LVinkle, M., I d . Eng. C/7em. 53, 187 (1961). (7) Dechman, D. A.. Van LVinkle, M., Ibid., 51, 1015 (1959). (8) Drichamer, H. G., Bradford, J. R., Tran.7. A.1.Ch.E. 39, 319 (19 43) . (9) Foss, A. S., Gerster, ,J. A . , Pigford, R. L., A.1.Ch.E. J . 4, 231-9 (1958).
(10) Gautreaux, M. F., O’Connell, H. E., Chem. Eng. PrOg7. 51, 232 (1955). (11) Geddes. R. L.. Tram. A.1.Ch.E. 42. 79 (1946) (12) Gersteri J. A.,’Bonnett, W.E., Hein. Irhin, Chem. Eng. Progr. 47, 523, 621 (1951). (13) Gerster, J . A . , Colburn, A . P., Bonnett, I V . E., Carmody. T. IV., Ibid,, 45, 716 (1949). (14) Gerster, .J. A . , Miznna, T., Marks, T. N., Catanach, A. i V . , A.1.Ch.E. J . 1, 536 (1955). (15) Gunnesq, R. C., Raker. J. C., Trans. A.1.Ch.E. 34, 707 (1938). (16) Hrllums, J. D.: Braulick, C. J . , Lyda, C. D., Van IVinkle, M., A.I.CIi.E. J . 4 , 4 6 5 (1958). (17) Jonrs, J. B., Pyle, C., C k m . Eng. Progr. 51, 424 (1955). (18) Kirschbaum, E., “Distillation and Rectification,” pp, 227, 276. Chemical Publishincz Co.. New York. 1948. (19) Kocatas, B. M., Ph.D: dissertation, University of Texas, 1962. (20) Kolodzie, P. A , , Jr., Van IVinkle, M., A.1.Ch.E. J . 3, 305 (19571. (2i) Lek, D. C., Jr., Chem. En?. 61,179 (May 1954). (22) Lewis, LV K., J . Ind. En?. . Chern. 14, 492 (1922). (23) Ibid., 28, 399 (1936). (24) Mayfield, F. D., Church. W.L., Green, A. C., Lee, D. C., Jr., Rasmussen, R. W., 44, 2238-49 (1952). (25) Muruhree. E. V.. Ibid.. 17. 747 11925).
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RECEIVED for review December 26, 1962 ACCEPTED November 12. 1963
